Measurement Quantization:
A Count-Based Resolution of the Quantum Measurement Problem

Jody A. Geiger

Informativity Institute

Chicago, IL USA 60605

jodygeiger@informativity.org

ORCID: 0000–0001–5389–0447

ABSTRACT

Measurement Quantization (MQ) resolves the quantum measurement problem through conserved, pre-probabilistic relational counts on a non-spatiotemporal Internal Frame mapped through an admissible Frames mapping to a System Frame measure, not through discreteness alone. Normalization gives outcome weights; completed contextual realization plus an exclusive outcome partition induce a single-valued record map whose pushforward distribution equals those weights, identifying empirical probabilities, with restriction and renormalization afterward. The wave function is constructed as an amplitude-phase representation; arbitrary-observable Born rule follows after Hilbert completion through spectral/POVM representation. Schrödinger-form evolution is conditionally selected with independently fixed phase-action scale. Translation covariance yields Weyl relations and

rather than postulated canonical kinematics. A Lorentz-covariant System Frame completion on emergent metric yields Klein–Gordon and Dirac sectors. Combining L=ħ/2, rotational covariance, and Clifford factorization fixes S²=3ħ²/4 and j=1/2, yielding Pauli generators, 4πperiodicity, Stern–Gerlach probabilities, conserved positive hypersurface current, and the Pauli–Schrödinger limit without primitive spin. Local composition extends to observable algebras: GNS fixes representations, Fock space is recovered in free/stationary regimes, U(1) charge labels superselection, and MQ Maxwell sector supplies QED; non-Abelian and Standard-Model completion are not claimed. Composite availability is non-factorizable, reproducing Bell–CHSH violations with measurement independence and operational no-signaling. Novelty is the closed count-to-record architecture across probability, realization, dynamics, Bell correlations, and parameter-free finite-count metrology. MQ predicts upper-count gravitational transition frequencies 11.6426596 ppm above standard quantum mechanics, untested at required precision. Finally, finite distinguishability, additive composition, relational relabeling invariance, closed conservation, and no primitive weighting field uniquely select nonnegative integer multiplicity as minimal finite-domain pre-probabilistic carrier.

Keywords: Measurement Quantization, quantum foundations, measurement problem, Bell correlations, measurement, wave function

1. INTRODUCTION

Quantum mechanics is among the most empirically successful frameworks in the history of physics, yet the ontological status of its foundational object, the wave function, remains unresolved [1–4]. Whether the wave function represents a physical entity, a state of knowledge, or a representational structure remains an open question despite nearly a century of experimental confirmation. This ambiguity underlies persistent tensions surrounding the measurement problem and the emergence of classical behavior from quantum systems [5–8].

Standard quantum theory encodes physical dynamics in a continuous wave function defined over configuration space and governed by linear evolution together with probabilistic measurement rules. While operationally successful, this structure leaves unresolved foundational questions regarding the origin of quantization, the role of discreteness, and the emergence of single stable measurement records from unitary quantum evolution. In particular, quantum mechanics presupposes probabilistic outcomes and the Born rule [1] without deriving a deeper kinematic origin for either.

A central unresolved issue in standard implementations of quantum mechanics is that unitary evolution can produce entangled superpositions of system–apparatus record states, whereas completed measurements yield single stable records unless the formalism is supplemented by collapse, branching, hidden-variable structure, or another account of outcome realization. This tension constitutes the measurement consistency problem developed formally in Sec. 2.5 and Appx. B [22].

The issue is well established in the measurement-problem literature. Schlosshauer emphasizes that decoherence suppresses interference without, by itself, settling the outcome problem [7]; Adler argues explicitly that decoherence does not solve the measurement problem [9]; and Gomes likewise identifies the incompatibility between unitary evolution and definite outcomes as requiring a physical account of outcome selection [10]. Bell-related results further require any viable framework reproducing quantum predictions to accommodate nonclassical correlations while preserving operational no-signaling [3,16].

Measurement Quantization (MQ) [11] addresses this problem by treating the wave function as derived rather than primitive. Its primitive structure is conserved count structure in the Internal Frame, while wave-function encoding, probability density, amplitudes, phase structure, operators, and empirical records arise only after realization in the System Frame. The two domains are related by the Frames mapping (Appx. A.2–A.4). For a completed measurement, contextual realization together with the mutually exclusive outcome partition establishes one record class; restriction and renormalization then condition subsequent predictions on that realized record rather than introducing fundamental collapse dynamics (Secs. 3.1–3.2.1; Appx. H.3–H.9). The same MQ publication develops multiple quantitative constructions from this framework [11], but those constructions are not presumed to be statistically independent empirical tests.

Many programs already reconstruct quantum structure from nonstandard primitives, including deterministic discrete models, exact-uncertainty and Fisher-information schemes, stochastic mechanics, entropic dynamics, scale-relativity constructions, hydrodynamic reformulations, causal-set approaches, and operational or informational reconstructions [15,53–59,72,91]. Accordingly, the novelty claimed here is not discreteness, an emergent wave-function representation, recovery of the Schrödinger equation, or outcome actualization by itself, but the conjunction of the complete MQ derivational chain.

That claim is specific. MQ begins with conserved measurement-availability counts carrying neither probability nor spacetime geometry. An admissible Frames mapping produces a realized measure whose normalization yields relative outcome weights; their empirical probability meaning follows only when completed contextual realization and the measurable outcome partition induce one record label and the pushforward of the same complete-interaction count measure yields the identical label distribution. Post-outcome restriction and renormalization then update predictions. Composite mapped availability yields non-factorizable Bell statistics while preparation remains measurement independent and the realized marginals remain operationally no-signaling; the same architecture also fixes the coefficient-resolved non-relativistic quantum correspondence and finite-count metrological predictions without target-specific adjustable parameters. Secs. 4.4–4.6 and Appx. AU compare these elements with representative reconstruction programs. No priority is claimed for any individual mathematical device.

The count premise is likewise bounded. MQ asks for the minimal pre-metric, pre-probabilistic carrier representing finitely distinguishable stable-record precursors whose multiplicities compose additively, are invariant under relational relabeling, and are conserved under closed redistribution. Sec. 2.4.1 and Appx. W show that, under those conditions, Theorem 2.1 selects the free commutative multiplicity structure of finite relational units, canonically represented by nonnegative integer counts. The remaining assumption is that this minimal operational carrier is ontologically complete rather than merely an effective record description.

Fig. 1. Count of l_f describing realized separation in the System Frame, with discrete count evaluated in the Internal Frame. Modified from [11].

, corresponding to a realized separation

, and is not integer-valued for any nonzero n_L. The mapping-induced residual count contribution

therefore gives the equivalent realized count n_L+Q_L. Realized geometric composition is consequently not closed over the primitive integer-count structure.

Arithmetic non-closure alone does not uniquely select MQ: one could enlarge the primitive number field, abandon integer closure, or adopt another discrete-to-continuum construction. Within the realization class considered here, MQ excludes those alternatives by requiring conserved integer counts, a fixed unit reference, no intrinsic metric structure in the count domain, and observable geometric separation only after an admissible Frames mapping. Appx. M proves the resulting conditional three-role necessity: the fixed Reference Frame normalization (side a), the discrete Internal Frame count (side b), and the non-discrete realized System Frame separation (side c). The Reference Frame is an operational normalization/comparison role, not a third primitive ontological domain.

The same construction produces the Informativity differential 2Q_Ln_L, the exact count-dependent finite-count contraction factor of the local mapping-normalization rule (Eqs. (M.6)–(M.8); Sec. 2.9.1). It depends on count separation and mapping normalization rather than relative observer velocity and is therefore distinct from Lorentz contraction. The associated metrological comparisons are treated in Appx. J as residual, provenance, and dependence assessments; reused or algebraically related inputs are not counted as independent empirical evidence, and prospective discrimination is developed in Sec. 2.9.2 and Appx. J.8–J.9.

MQ therefore formalizes a discrete Internal Frame carrying conserved count structure over irreducible fundamental measures and a realized System Frame in which observables arise relationally through the Frames mapping (Appx. A.1–A.4; Appx. C.1–C.22). The Frames Principle requires physically meaningful observables to arise through this mapping. Within the admissibility class established in Sec. 2.8.1 and Appx. C.4.1–C.4.3, realization preserves count conservation, relational adjacency, finite normalization, and the required consistency invariants. Distinct admissible count configurations may share a coarse-grained realized encoding, while every realized structure within the proved class is supported by conserved count structure. The resulting normalization, continuum, operator, and dynamical constructions are developed in Secs. 2.5–2.8 and 3.2–3.7 and Appx. C–G.

Importantly, MQ discreteness is a quantitatively testable structural hypothesis because its finite-count realization produces metrological consequences distinct from the corresponding upper-count limit. In particular, MQ predicts a small realization-dependent correction to the gravitational coupling G associated with the Informativity differential. This is an MQ prediction, not a community-established classification of separate gravitational constants. Conventional metrology reports a single Newtonian gravitational constant G; the 2022 CODATA recommended value is 6.67430(15)×10^-11 m³ kg⁻¹ s⁻² [62]. Within MQ, the finite-count and upper-count values are alternative predicted realizations of the same coupling. With c denoting the speed of light and t_f and m_f the fundamental time and mass measures, respectively, Eq. (1) gives the count-resolved relation. Experimental discrimination requires the applicable interaction realization to be specified independently of the measured result.

The numerical proximity of the historical CODATA 2010 and 2014 recommendations to two MQ realization values in Appx. J is retained only as a historical residual comparison. CODATA adjustments combine heterogeneous measurements into one recommended value and do not classify experiments by an MQ interaction realization; successive adjustments therefore cannot establish the predicted realization dependence. A more informative differential target became available when Schlamminger et al. independently repeated the BIPM torsion-balance experiment at NIST using the original apparatus and gravitational geometry [80]. Their combined result, G=(6.67387±0.00038)×10^-11 m³ kg⁻¹ s⁻², differs from the MQ finite-count value 6.67384483×10^-11 m³ kg⁻¹ s⁻² by about 0.07 standard deviations using the quoted combined uncertainty. The same source geometry was measured in free-deflection and electrostatic-servo modes, with the servo result lower for both copper and sapphire source masses. Although the authors investigated systematic effects extensively, residual mode- and configuration-dependent uncertainty remained. The result therefore does not demonstrate an MQ effect; it provides a post-publication differential consistency target whose prospective interpretation is developed in Sec. 2.9.2 and Appx. J.8–J.9.

A second finite-count correspondence concerns the classical inverse fine-structure constant

. With θ_si denoting the dimensionless MQ coefficient, Eq. (2) gives its MQ realization.

The resulting value differs from the tabulated reference by a diagonal standardized residual of approximately 0.0077 using the quoted uncertainties (Appx. J.2 and J.5–J.7). Because the comparison shares electromagnetic input structure with the CODATA quantity, it is classified as reused-input concordance rather than independent validation. Appendix K distinguishes the dimensional representations of θ_si: the untagged θ_si is dimensionless, the corresponding angular quantity is Θ_si=θ_si rad, and the momentum realization is

, where ℏ is the reduced Planck constant. These quantities may share the same numerical coefficient in the adopted realization but are not the same physical quantity and are not equated dimensionally. This convention is consistent with the SI treatment of plane angle as a quantity with the unit radian, whose symbol remains useful in identifying the quantity [81]. MQ derives additional first-principles relations among physical constants through the Frames mapping, with examples given in the summary publication ([11], Appx. L, M, P–R, AA, AD, AM, AP–AR, BA, BL, BM). Here “first-principles” describes derivational provenance within MQ and does not imply statistical independence from empirical inputs; Appx. J.6–J.8 audits that distinct evidentiary question.

The corresponding control construction, obtained by removing the mapping-induced Informativity differential and using continuum Planck-scale normalization, is retained only as a model comparison. Inferential comparison of the finite-count and control constructions requires the residual, uncertainty, and covariance treatment of Appx. J.1–J.3; the earlier conversion of a right-tail p-value into a positive “agreement sigma” is therefore not used.

Finite-count corrections are called fixed only when the universal MQ inputs and interaction realization are determined independently of the target measurement; this does not imply that every numerical input is free of empirical anchoring. Appendix AG records that provenance and separates universal inputs, independently measured subsystem and apparatus quantities, and predicted quantities with no target-specific adjustable parameter. Non-injectivity of coarse-grained realization explains why realized observables need not retain all Internal Frame information, but it does not by itself generate Bell correlations; those additionally require the composite non-factorizable availability structure and contextual outcome partitions of Sec. 3.7 and Appx. G–H.

This distinction is essential for Bell–CHSH constraints [3,16]. Bell-inequality violations exclude Bell-local factorizable descriptions satisfying the corresponding assumptions; they do not imply controllable faster-than-light signaling. MQ therefore does not claim Bell local causality for realized System Frame outcome statistics. Its composite construction instead yields non-factorizable joint probabilities while retaining preparation measurement independence and operational no-signaling, together with adjacency-local redistribution of conserved count structure in the Internal Frame (Sec. 3.7; Appx. G.1–G.12 and H.7). The Bell–CHSH result further requires the dichotomic analyzer-response relation that resolves the composite availability into angle-dependent System Frame outcome partitions; non-injectivity of the Frames mapping alone is insufficient to determine either the Bell violation or the quantum angular dependence (Appx. G.5–G.12).

MQ likewise does not begin from Hilbert space [17], operators, probabilistic axioms, or a Bell-local hidden-variable completion. The primitive object is conserved count structure defined through the discrete measurement count field (Sec. 2.6; Appx. A–C). Mapped availability subsequently supplies amplitudes and phase-sensitive interference, while Internal Frame redistribution remains deterministic (Secs. 2.11 and 3.1–3.6; Appx. C–E). The realized System Frame representation reconstructs the linearity, superposition structure, and non-factorizable correlations required for quantum correspondence without imposing Bell factorization (Secs. 3.3–3.7; Appx. D, G, and H).

The framework is applied explicitly to two-slit interference, quantum measurement, entanglement correlations, and the conditional recovery of Schrödinger dynamics in the continuum realization limit. Successive discrete differences supply the local Laplacian structure, while the remaining non-relativistic admissibility conditions and independently fixed MQ kinetic coefficient are treated separately in Sec. 3.5 and Appx. O. The resulting Schrödinger equation [2] is therefore a coefficient-resolved recovery within the stated admissibility class rather than a consequence of discreteness alone. Quantum, electromagnetic, and gravitational observables are subsequently treated as distinct System Frame realizations developed from the common count-based kinematic architecture (Appx. A.3–A.10 and Appx. C–H).

2. METHODS

2.1 Nomenclature and Terminology

Because MQ relates discrete count structure to continuum quantum dynamics, the paper uses terminology drawn from several established research domains. The full vocabulary and symbol list used throughout the work is collected in Appx. A, including the definitions of the Internal Frame, System Frame, Frames mapping, conserved count structure, encoded availability, adjacency flux, relational regions, normalization, and realized observables. The terminology is therefore not introduced ad hoc in the body of the paper. Rather, each technical term is either a standard term from an established discipline or an MQ-specific compound term defined from such structures in Appx. A.

For orientation, terms associated with discrete mathematics and conservation methods include relational cells, adjacency links, adjacency boundary, adjacency divergence, and adjacency count-flux. Terms associated with quantum information theory include encoded availability, conditioning, no-signaling, Bell–CHSH constraints, and contextual restriction. Terms associated with quantum mechanics include wave function, phase field, Hilbert space state vector, Schrödinger evolution, and Born-rule probability. Terms associated with measure theory and mathematical physics include normalization, probability density, marginal density, inner product, self-adjoint operator, and spectral structure. Terms associated with kinematics and continuum physics include configuration domain, realized separation, continuum encoding, emergent metric, and continuum dynamics.

This nomenclature is used to keep distinct the primitive count-based structures of the Internal Frame from the realized relational structures of the System Frame. In particular, the Internal Frame carries no intrinsic metric, continuum coordinate, or differential structure. Such structures arise only after application of the Frames mapping. Readers encountering unfamiliar terms should therefore consult Appx. A before interpreting them by analogy with standard spacetime, Hilbert-space, or continuum-mechanical usage.

2.2.1 Verification and Dependency Architecture

The breadth of the present construction is intentional because the central measurement claim depends on a linked chain extending from primitive count structure to realized probability, dynamics, composite states, and experimentally distinguishable consequences. It does not follow, however, that every claim must be verified as one indivisible argument. Appendix AE decomposes the manuscript into independently reviewable modules and records the explicit dependencies between them. This separates the core count-to-record result from later geometric, relativistic, field-theoretic, gravitational, and metrological extensions, while preserving every logical link needed to determine which downstream claims depend on a disputed upstream step.

The verification rule is deliberately asymmetric. A specialist need only validate the assumptions, derivations, and external inputs assigned to a module together with the upstream results on which that module explicitly depends; no module may rely silently on a later result. Table AE.1 gives the resulting claim-to-proof crosswalk, and Table AE.2 gives the corresponding dependency and evidence gates. The map also distinguishes MQ-derived results, standard mathematical representation theorems used after MQ structure has been established, external experimental correspondence, and prospective MQ-specific tests. Thus, a referee can audit the manuscript by domain without treating breadth itself as evidence for, or against, any individual derivation. Section 2.2.2 provides the front-of-paper entry point and identifies the minimum core audit before optional extensions are considered.

Independent formal verification is a separate evidentiary status from author-side derivational completeness. Appendix AV therefore defines an external specialist sign-off protocol for the major proof gates identified in Appendix AE. Until a gate has been checked by a reviewer who did not participate in the derivation and has returned a reproducible acceptance or defect report, that gate is described here as author-verified but not independently verified. This distinction prevents the manuscript architecture itself from being mistaken for external validation.

2.2 Verification and Review Architecture

2.2.2 Referee Roadmap and Minimal Proof Paths

For editorial triage, the manuscript can be audited through four bounded paths. Path I is the minimum argument for the central count-to-record measurement claim; Paths II-IV test dynamical correspondence, optional universality extensions, and MQ-specific empirical discrimination. Appendix AE contains the complete claim-to-proof and failure-propagation map.

The roadmap changes no proof burden. A referee may issue a bounded judgment after completing the path relevant to that claim, and any rejection propagates only through the explicit dependency gates of Table AE.2.

Appendix AV converts these four reading paths into five specialist verification packets with explicit independence, reproducibility, acceptance, and failure-propagation criteria. The packets are intended for separate review so that agreement in one discipline cannot substitute for verification in another.

2.3 MQ Resolution Strategy

MQ addresses the measurement consistency constraint of Sec. 2.5 by separating conserved encoded availability from realized measurement structure. Rather than requiring the same wave-function formalism to account simultaneously for unitary evolution and realized measurement records, MQ distinguishes between discrete conserved count structure and its continuum realization under the Frames mapping.

In MQ, the conserved count structure is defined in the Internal Frame. The Internal Frame carries no primitive continuum coordinate, metric, or differential structure. Instead, it consists of conserved count relations defined on a discrete relational adjacency configuration domain. Structure at this level is represented through count availability, adjacency relations, and admissible count redistribution under allowed updates.

Realized observables arise only after realization in the System Frame. The System Frame is the continuum encoding generated through an admissible Frames mapping constrained by the Frames Principle. The mapping first fixes the admissible realization structure. For a completed measurement, referenceability fixes a single contextual realization; the measurable outcome partition then induces exactly one outcome label, while the pushforward of the underlying complete-interaction count measure through the same realization reproduces the normalized outcome distribution derived from encoded availability (Sec. 3.2.1; Appx. H.9). Contextual restriction does not perform this selection but updates the predictive encoding after the realized outcome is established. Wave-function encoding, probability density, amplitudes, phase structure, operators, and differential evolution arise only after realization of conserved count structure in the System Frame encoding.

This separation resolves the measurement consistency constraint by assigning different roles to normalized outcome weighting, empirical probability identification, outcome realization, ensemble frequency, and post-outcome conditioning. Conserved availability evolves through deterministic redistribution in the Internal Frame; normalization of its mapped System Frame encoding assigns relative weights to admissible outcome classes; the complete admissible interaction and its outcome partition induce one realized label; the pushforward of the same complete-interaction count measure proves that those normalized weights are exactly the distribution of realized labels; and only then is the predictive encoding restricted and renormalized to the corresponding record class (Sec. 3.2.1; Appx. H.9). No stochastic collapse of the underlying count structure or simultaneous physical realization of all alternatives is required.

Within this framework, normalization of encoded availability first produces a normalized outcome-weight measure in the System Frame. Its interpretation as empirical probability is not inferred from normalization alone: Sec. 3.2.1 and Appx. H.9 identify the same measure as the pushforward distribution of the completed-interaction outcome map. Superposition correspondingly represents linear structure within the System Frame encoding rather than simultaneous realization of multiple macroscopic measurement records. Interference, conditioning, and entanglement therefore arise from realization structure induced through the Frames mapping.

MQ consequently preserves Schrödinger dynamics, Bell–CHSH violations, Hilbert-space structure, and operational no-signaling while avoiding the requirement that globally unitary evolution correspond to simultaneous realization of multiple stable measurement records. The framework thereby provides a non-branching conserved-availability resolution of the quantum measurement problem consistent with the operational constraints established in Secs. 2.4–2.5.

2.3.1 Primitive Assumptions and Derived Structures

MQ separates its primitive kinematic assumptions from the quantum structures claimed as consequences. Primitive content is limited to the discrete measurement-availability count structure on the Internal Frame relational adjacency domain, admissible count-conserving redistribution, and the requirement that observable realization proceed through a Frames mapping constrained by the Frames Principle. The mapping is therefore primitive as a structural relation, but its existence is not left as an unchecked assumption. Section 2.8.1 and Appx. C.4.2 give an explicit constructive existence result for finite connected domains, while Sec. 2.10.3 and Appx. AO extend the measure, mapping, probability, and topology construction to connected locally finite domains by compatible finite exhaustion. Appx. C.4.1 states the admissibility conditions and equivalence relation. No primitive continuum coordinate, metric, probability measure, wave function, operator algebra, or physical time is assigned to the Internal Frame.

The logical status of the subsequent constructions is deliberately separated. First, the Frames mapping induces a realized System Frame measure from conserved Internal Frame count structure; when that measure is density-representable, normalization produces a finite additive unit-normalized outcome-weight measure. Second, single-valued completed realization is a referenceability property of the admissible mapping, and the outcome label map is induced by composition of the contextual realization with the disjoint, exhaustive System Frame outcome partition. The pushforward of the normalized complete-interaction count measure under this induced outcome map equals the normalized System Frame outcome-weight measure, thereby establishing its empirical probability interpretation and realization-frequency compatibility (Sec. 3.2.1; Appx. H.9). Third, post-outcome conditioning is a separate predictive update. Fourth, the MQ wave function is a complex amplitude-phase representation constructed from the already normalized density together with the realized redistribution phase; Proposition C.3 fixes the minimal local complex U(1) representation. Fifth, Appx. F reconstructs the complex pre-Hilbert geometry from the MQ availability norm and coherent amplitude composition: the availability norm obeys the parallelogram identity, the Jordan-von Neumann polarization theorem therefore fixes the inner product uniquely, and separable composite realization induces the Hilbert tensor product through the universal property of bilinear composition. Hilbert completion then adds only norm-limit points, after which the spectral/POVM representation of arbitrary measurements is available. Sixth, within the admissibility class of Sec. 3.5, mapped successive-difference structure conditionally selects Schrödinger-form evolution. Measure construction, probability identification, individual outcome realization, ensemble frequency, post-outcome conditioning, amplitude representation, spectral measurement completion, and dynamical selection are therefore distinct but explicitly linked claims.

Pointwise Frames mapping representatives are not required to be unique. Section 2.8.1 and Appx. C.4.3 instead identify the physical mapping with an admissible equivalence class: representatives related by invariant-preserving Frames automorphisms are the same realization law, while inequivalent global classes are distinguished by observable content and defect holonomy. Completed-interaction outcome uniqueness remains a separate statement and does not depend on a preferred coordinate representative.

A further operator-kinematic step is now derived rather than merely represented. Appendix S starts from the realized System Frame translation action induced by adjacency-local count redistribution under the Frames mapping. Strong continuity and norm preservation make that translation a unitary one-parameter group; the independently derived MQ phase-action scale fixes its self-adjoint generator as p̂=-iℏ∇. Because the mapped position coordinate acts by multiplication, translation covariance gives T(a)†x̂_jT(a)=x̂_j+a_jI. Exponentiating the position action then yields the Weyl relations, whose infinitesimal form on a common dense invariant core is [x̂_i,p̂_j]=iℏδ_ij. Thus the canonical commutator is downstream of mapped translation and the MQ action scale, not an additional Internal Frame postulate.

For finite composites, this hierarchy is now closed beyond the binary product case. Appendix F.14-F.14.1 first reconstructs the bipartite tensor product from separable MQ composition and then extends it uniquely to any finite number of subsystems by associativity. Within that derived composite space, Schmidt decomposition, reduced-state spectra, purification, and the standard monogamy consequences are theorems of the representation rather than additional quantum postulates. The non-factorizable joint availability developed in Sec. 3.7 therefore occupies the complete multipartite state space fixed by the same count-to-amplitude construction.

The primitive measure entering this chain is itself fixed on finite MQ domains: Proposition 2.1 shows that the integer unit-count ontology, finite additivity, and relabeling invariance leave only the counting measure, with any overall scale removed by normalization. The later probability law therefore does not inherit an arbitrary family of pre-probabilistic weights. Appendix T combines this primitive uniqueness with the completed-context pushforward and coarse-graining consistency into a single pre-Hilbert theorem: for every finite MQ measurement context, the empirical record-probability measure is the unique admissible probability assignment inherited from the count ontology. No alternative context-dependent weighting survives the same relabeling, additivity, normalization, and realization-consistency requirements.

The assumption count can therefore be audited globally rather than inferred from the number of later mathematical conditions. Appendix AF collects the independent roots of the present construction and distinguishes five statuses: primitive ontological commitments, operational premises defining the measurement class, realized-domain regularity conditions, extension-specific physical conditions, and standard representation theorems used only after the corresponding MQ structure has been obtained. It also lists familiar quantum structures that are not independent assumptions in this manuscript because they are downstream results. This ledger is intended to answer the reconstruction objection directly: a later condition is counted as a new assumption only if it contributes physical content not already fixed by an upstream MQ result or by the empirical domain being modeled.

Parameter economy is audited separately from assumption economy. Appendix AG partitions every numerical quantity used by the flagship derivations into universal externally anchored quantities, exact structural relations, quantities derived from those inputs, measured subsystem inputs, apparatus calibration variables, and predicted outputs. A quantity is called “target-parameter-free” only when no parameter is estimated from the dataset used to test that prediction. This stricter usage prevents a derived universal quantity that ultimately inherits a CODATA input from being mistaken for an ab initio numerical prediction.

The integer character of the primitive carrier is separately addressed in Sec. 2.4.1 and Appx. W. It is not inferred from the later Born rule, Hilbert geometry, or the quantitative MQ mapping corrections. Rather, it follows conditionally from finite distinguishability, compositional additivity, relational relabeling invariance, nonnegative conserved multiplicity, and the requirement that no further primitive scalar label or weighting field be introduced. This keeps the ontology argument logically upstream of the quantum structures that MQ subsequently derives.

2.4 Motivation and Scope of Discreteness

No physical measurement can access arbitrarily fine operational resolution, nor can it distinguish an unbounded number of stable outcomes within a finite physical region [14,18]. This reflects a fundamental constraint associated with finite information capacity and entropy bounds in physical systems [14,18–21] and is distinct from ordinary instrumental resolution limits. Here “finite information capacity” is used in the conservative operational sense relevant to bounded regions and stable physical records, not as a claim that the universe possesses a finite-dimensional state space.

A century of foundational analysis has documented a persistent tension between (i) globally linear norm-preserving evolution, (ii) empirically observed definite measurement records, and (iii) Born-rule probability assignment. In modern terms, the measurement problem [1,7,8,22,23] concerns the apparent incompatibility between universally linear quantum evolution and the existence of single stable records with well-defined probabilities, including the preferred-basis, outcome-selection, and Born-rule problems. Early quantum theory formalized measurement through projection and state-collapse postulates [6,8,23], while later interpretations such as Many Worlds [24,25] attempted to avoid collapse by treating branching structure as emergent within a universal wave function. Decoherence mitigates part of this tension by explaining suppression of interference through environment-induced entanglement, but decoherence alone does not derive either unique records or the Born rule from linear evolution [5,7]. Various approaches attempt to supplement this gap through symmetry, decision-theoretic, or typicality arguments, yet the status of such derivations remains unsettled [7,26,27].

Recent work at the intersection of quantum information, cosmology, and foundations has further questioned the universality of globally unitary Schrödinger evolution in contexts where spacetime itself evolves. In expanding universe settings, Cotler and Strominger have argued that time evolution may be represented more naturally through isometries rather than strictly unitary operators, preserving inner products while allowing effective embedding into enlarged state structures during cosmological evolution [28,29]. In quantum information language, isometries describe norm-preserving embeddings that need not be invertible on the realized state space. A quantum channel may therefore be realized through an isometric embedding or equivalently through unitary evolution on a larger space followed by restriction. We use “isometric” only in this structural sense, without commitment to any particular underlying extension mechanism. Importantly, isometric evolution alone does not resolve the problem of definite records or probability assignment.

The present work takes definite records, finite record capacity, repeatability, conditional probability updating, and operational no-signaling as explicit premises for the class of measurement descriptions examined here. Accordingly, before introducing the MQ count structure, we establish a conditional record-consistency result. The result is not a blanket no-go theorem against Everettian, relative-state, or other branching interpretations; it applies only to frameworks that accept Operational Constraints 1–5 in the sense stated below, including the single-realized-record requirement.

We therefore adopt the following operational constraints:

Operational Constraint 1 (Definite Records)
Every completed laboratory measurement terminates in a single definite macroscopic record, such as a pointer position, detector-click pattern, or registered bit string that remains stable under immediate rereading. This operational definiteness constitutes the minimal empirical content of a measurement outcome and defines the object to which probabilities are experimentally assigned [8,23].

Operational Constraint 2 (Finite Record Capacity)
Any finite physical system — finite energy, finite spatial extent, finite entropy resources — can encode only a finite number of mutually distinguishable and stably retrievable macroscopic records. Infinite information density or unbounded stable outcome multiplicity within a bounded region is physically inadmissible under standard thermodynamic and information-theoretic constraints. This assumption concerns stable records and bounded resources only and does not assert that the global universe possesses finite dimension [18–21].

Operational Constraint 3 (Linear Norm-Preserving Micro-Evolution)
Between record-generating interactions, evolution of an isolated system is represented by a linear norm-preserving one-parameter family of transformations on the relevant state space. This corresponds to the standard dynamical postulate of quantum theory, typically represented through unitary Schrödinger evolution on Hilbert space [2,30].

Operational Constraint 4 (Repeatability and Consistent Probability Updating)
If a record is interrogated twice in immediate succession using the same apparatus, with no intervening dynamics capable of altering the record, the second interrogation returns the same outcome with unit probability. Probabilities assigned to records are additive over mutually exclusive outcomes and are updated through conditioning on the realized record. These operational requirements render records stable empirical objects and ensure consistency of sequential probability assignment [6,8,23,31,32].

Operational Constraint 5 (Operational Relativistic Causality)
Statistics of local measurement records cannot be controllably altered through space-like separated measurement choices. Equivalently, local marginal outcome statistics remain independent of distant measurement settings. This operational no-signaling requirement is strictly weaker than Bell local causality (factorizability) and constitutes the relevant relativistic compatibility condition [33,34].

Taken together, these operational constraints define the measurement class analyzed in this paper: stable observer-accessible records, finite record capacity, repeatability, conditional probability assignment, and operational relativistic causality. They are not claimed to be interpretation-neutral at the level of global ontology. In particular, Operational Constraint 1 is a single-record premise. A relative-state or Everettian framework may instead regard each observer record as definite only relative to a branch while retaining multiple branches globally [25]. Such a framework therefore does not satisfy the global single-record reading used in the conditional theorem below, even though it may reproduce branch-relative definite records.

The resulting consistency constraint is developed formally in Sec. 2.5 and rigorously established in Appx. B. MQ addresses this constraint by separating conserved encoded availability in the Internal Frame from realized observational structure in the System Frame, with realized observables arising only after contextual restriction and normalization under the Frames mapping and the Frames Principle.

MQ is then introduced as a minimal count-based kinematic structure in which discreteness is built directly into the measurement description. MQ distinguishes the discrete Internal Frame, in which conserved count structure is defined over irreducible count configurations, from the realized System Frame, in which realized observables arise through mapping-normalized realization induced by the Frames mapping under the Frames Principle (Secs. 2.6–2.12; Appx. A.2–A.5 and Appx. C.1–C.22). In MQ, discreteness pertains to encoded measurement availability and stable record structure, while probability appears only after normalization of encoded availability in the realized System Frame (Secs. 3.1–3.2; Appx. C.7–C.8, Appx. C.9–C.21, Appx. D.1–D.2, and Appx. G.11–G.12). Determinism applies to the evolution of encoded availability in the Internal Frame, not to realized System Frame outcomes (Sec. 2.11; Appx. C.5, Appx. G.4, and Appx. H.4).

This positioning is essential for the later foundational analysis developed in Secs. 3.3–3.11 and Appx. G–I. MQ reproduces Bell–CHSH violations while preserving operational no-signaling in the realized System Frame together with adjacency-local redistribution of conserved count structure in the Internal Frame (Sec. 3.7; Appx. G.7, Appx. H.1–H.7, and Appx. G.8–G.12). MQ therefore does not satisfy Bell local causality at the level of realized System Frame outcome statistics. Instead, non-factorizable joint probabilities arise through contextual restriction and renormalization induced by the Frames mapping (Secs. 3.6–3.7; Appx. G.5–G.6, Appx. H.2, and Appx. G.8–G.12).

2.4.1 Operational Minimality of the Primitive Integer-Count Carrier

Operational Constraint 2 establishes only that a bounded finite-resource apparatus cannot support an unbounded number of mutually distinguishable stable records; by itself, that statement does not prove that microscopic reality is integer-valued. The stronger MQ question is whether a more general primitive carrier is required once the measurement description is stripped of continuum coordinates, metric, probability, and arbitrary scalar weights. Operational reconstructions of quantum theory likewise treat distinguishability and composition as physically meaningful primitives, although they do not infer the MQ count ontology from them [143]. The mathematical step needed here is therefore a minimality statement about multiplicity rather than an appeal to discreteness by assertion.

Theorem 2.1 (Operational minimality of the finite integer-count carrier). Consider the finite measurement class of Operational Constraints 1–2 and require a primitive pre-probabilistic carrier with the following properties: elementary record precursors are finitely distinguishable; independent juxtaposition and disjoint recombination are associative and additive; the order and arbitrary labels of otherwise identical primitive units have no physical significance; multiplicity is nonnegative and is conserved under closed redistribution; and no additional primitive metric, probability, continuous amplitude, or scalar weighting field is admitted. Then the configuration carrier is, up to relabeling of its generators, the free commutative monoid of finite multisets of primitive relational units. For a single primitive unit type its additive multiplicity is canonically represented by the nonnegative integers; over the relational adjacency domain a configuration is equivalently a finitely supported nonnegative-integer count assignment. Thus the integer count used by MQ is the unique minimal carrier of conserved distinguishable multiplicity under the stated assumptions, not an independently adjustable numerical choice.

The proof is given in Appx. W. The key point is that finite distinguishability supplies identifiable elementary occurrences, composition supplies repeated juxtaposition, relabeling invariance removes ordering and naming information, and conservation preserves multiplicity. The universal mathematical object with exactly those operations and no additional relations is the free commutative monoid, equivalently the finite-multiset construction [144]. A real-valued primitive weight would add a scalar structure not supplied by distinguishability; signed multiplicity would contradict nonnegative availability; and an idempotent carrier would erase experimentally distinguishable multiplicity by making repeated occurrences equivalent to one occurrence.

This result strengthens but does not overstate the physical claim. Entropy and information bounds support finite stable record capacity in bounded systems [14,18–21], while Theorem 2.1 shows why integer multiplicity is the minimal carrier once that finite record structure is required to compose, relabel, and conserve without importing later quantum or geometric structure. Neither those bounds nor Theorem 2.1 can prove the metaphysical nonexistence of every deeper variable. The physically relevant question is narrower: whether any additional primitive structure can distinguish two admissible realizations or records once the complete MQ count configuration and measurement context are fixed. Theorem 2.2 and Appx. AN close that question for the finite measurement domain by separating empirically equivalent extensions from physically distinct ones.

Theorem 2.2 (Observable completeness of the finite MQ count carrier). Fix the finite measurement class of Theorem 2.1, the conserved count configuration and adjacency data, the admissible Frames mapping equivalence class, and a complete measurement context. Consider any proposed extension that appends additional primitive data while leaving the MQ structures intact. Exactly one of two cases obtains. If varying the additional data while holding the MQ data fixed leaves every admissible System Frame observable, completed-interaction record, normalization, current, holonomy, and other realization invariant unchanged for every context in the claimed domain, then the enlarged descriptions are physically equivalent under the same quotient principle already used for admissible Frames mapping representatives; the additional data do not label distinct physical states in this domain. If instead some variation of the additional data changes any such invariant or record, the extension is physically inequivalent and supplies an empirical discriminator against the count-only MQ ontology. Therefore, the finite MQ carrier is physically complete modulo empirically equivalent extensions; any physically consequential completion beyond it is necessarily testable in principle rather than an undetectable hidden freedom. The proof and scope are given in Appx. AN.

The theorem does not assert that logic can exclude an empirically inert metaphysical substrate. No physical theory can distinguish two descriptions that differ only in structure that has no possible effect on any observable or admissible interaction in the theory's domain. The result instead removes such structure from the scientific completeness question by quotienting it as representational surplus. This is consistent with the role of physical equivalence in gauge and generally covariant theories [178] and with the use of operational equivalence as a constraint on ontological descriptions of laboratory procedures [179]. MQ therefore retains a sharp Level-I falsification condition: reproducible evidence that stable records, their conserved precursors, or any other admissible System Frame observable depend on irreducible primitive information not determined by the MQ count configuration would select the second branch of Theorem 2.2 and reject count-only completeness.

2.5 The Measurement Consistency Problem

The operational constraints introduced in Sec. 2.4 jointly impose a conditional consistency requirement on measurement frameworks that retain a globally single realized record. Linear norm-preserving evolution can produce a superposition of mutually orthogonal system-apparatus record components, whereas Operational Constraint 1 requires one realized macroscopic record for the completed measurement. The issue addressed here is therefore not whether unitary quantum mechanics mathematically contains several record components, but whether those components may all be interpreted as simultaneously realized records within the same observer-accessible record ontology while the single-record premise is retained.

Under linear evolution, an initial superposition evolves into an entangled system-apparatus state containing multiple correlated record components. Standard single-outcome formulations require one of those alternatives to be the realized record. Everett's relative-state formulation instead retains the full unitary state and treats records as relative to correlated components rather than asserting one globally selected outcome [25]. Modern decoherence-based Everettian accounts further use decoherence to identify dynamically robust branch structure [7]. These are different responses to the measurement problem and must not be conflated.

Projection-based collapse theories alter the state-update dynamics; hidden-variable theories supplement the quantum state with additional outcome-relevant structure; MQ uses contextual restriction and renormalization of encoded availability; and Everett-type theories retain global unitary evolution while rejecting a globally unique realized outcome. Consequently, the argument below does not establish that branching theories are internally inconsistent. It establishes only that simultaneous global multiplicity and a globally single realized record cannot both be imposed on the same observer-accessible record description.

Finite record capacity and repeatability sharpen the operational setting but do not, by themselves, refute Everettian branching. A finite apparatus can support only finitely many mutually distinguishable stable records at a given time, while an Everettian ontology may distribute distinct records across branch-relative observer states rather than place them all in one observer-accessible record space. The relevant contradiction therefore arises only when a framework simultaneously demands global single-record realization and interprets more than one orthogonal record component as realized within that same record ontology.

The resulting conditional consistency requirement may therefore be stated as follows.

Theorem 2.3 (Conditional Record-Consistency Constraint).

Consider a framework satisfying Operational Constraints 1–5 with Operational Constraint 1 interpreted as requiring exactly one globally realized observer-accessible macroscopic record for each completed measurement. If linear norm-preserving evolution produces more than one mutually orthogonal record component, those components cannot all be interpreted as simultaneously realized records within that same observer-accessible record ontology while the single-record requirement is retained.

globally linear norm-preserving evolution,

simultaneous physical realization of more than one orthogonal macroscopic record component within one observer-accessible record ontology,

finite stable record capacity,

repeatability with conditioning on the recorded result, and

exactly one globally realized observational record per completed measurement,

cannot all hold together. In particular, conditions (ii) and (v) are mutually incompatible under the stated meaning of a realized record. A framework can evade the conclusion by rejecting the global single-record premise, by modifying the dynamics, by adding outcome-relevant structure, or by denying that every nonzero record component is itself a realized record. The theorem therefore classifies the assumption that must change; it does not select among those alternatives.

The derivation, scope conditions, and MQ application of Theorem 2.3 are developed in Appx. B.

MQ resolves this inconsistency by separating conserved encoded availability from realized observational structure through the distinction between the Internal Frame and the System Frame. Conserved count structure evolves in the Internal Frame independently of realized observational outcomes. The Internal Frame therefore does not represent simultaneously realized branching structure. Instead, it defines the conserved availability structure from which realized observables are obtained through the Frames mapping under the Frames Principle. Realized observables, normalized encoded availability densities, amplitudes, and operator representations arise only after normalization and realization in the System Frame.

MQ adopts the non-branching alternative developed in Secs. 3.1–3.2: realized records are represented at the System Frame level without altering the conserved count evolution in the Internal Frame.

This separation permits MQ to preserve linear Schrödinger evolution, Hilbert-space structure, Bell–CHSH violations, and operational no-signaling without requiring simultaneous realization of multiple stable observational records. Probability consequently arises from normalization of encoded availability after realization rather than from primitive stochastic postulates or ontological branching assumptions.

The following sections formalize this structure beginning with the definition of the conserved measurement count field and the distinction between the Internal Frame and the System Frame.

2.6 Definition of the Measurement Count Field

The primitive object of MQ is the discrete measurement-availability count field defined in the Internal Frame. The field is defined on a relational adjacency structure rather than on continuum spatial coordinates and possesses no primitive metric, geometric vector, or continuum differential structure. We denote the discrete measurement-availability count assignment by n_I(ξ,A), where ξ labels an Internal Frame relational cell and A is the discrete Internal Frame update index. The value n_I(ξ,A)∈N₀ gives the non-negative count associated with relational cell ξ at update step A. This count represents encoded measurement availability prior to normalization and probabilistic interpretation and defines the conserved count structure from which realized System Frame structure is subsequently derived through the Frames mapping.

Theorem 2.1 supplies the upstream status of this definition. On the finite measurement domains treated here, the nonnegative-integer codomain is the canonical representation of the minimal conserved multiplicity carrier selected by the operational and structural conditions of Sec. 2.4.1. Proposition 2.1 then answers a different question: once that count carrier is fixed, relabeling invariance and finite additivity uniquely determine the primitive measure used to aggregate its unit counts. Theorem 2.1 therefore concerns the ontology of multiplicity; Proposition 2.1 concerns the measure on that ontology.

Let ω^'⊆ω denote an admissible relational region consisting of a finite adjacency-connected subset of Internal Frame relational cells, where adjacency is defined solely through admissible relational links of the Internal Frame count structure. The associated Internal Frame regional count measure is defined by

which gives the total encoded availability count over relational region ω^' at update index A. The quantity N_I(ω^',A) denotes the Internal Frame regional count measure and is distinct from the mapping normalization functional N[ω^'] introduced later through realized System Frame normalization (Appx. A.3 and Appx. A.5).

The discrete measurement availability count field satisfies basic structural constraints implied by conserved count structure. Non-negativity requires

and finite additivity holds for disjoint admissible relational regions,

Finite additivity ensures consistency under admissible coarse-graining and relational domain decomposition and guarantees that the conserved count structure admits a well-defined realized System Frame encoding after application of the Frames mapping (Sec. 3.4; Appx. C.6–C.8 and Appx. C.9–C.10).

Proposition 2.1 (Primitive count-measure uniqueness on a finite MQ domain). Fix an Internal Frame update and a finite admissible count domain. Decompose each integer availability count n_I(ξ,A) only as a bookkeeping device into that many unit-count tokens. Let U_A denote the resulting finite set of primitive count units. A nonnegative finitely additive measure on subsets of U_A that uses the MQ unit count as its reference and is invariant under relabeling of physically indistinguishable unit-count tokens is uniquely the counting measure. The physical count configuration n_I may vary with preparation and deterministic evolution; the measure functional that aggregates those counts does not contain an additional weighting freedom.

Proof. Relabeling invariance makes every primitive unit count carry the same measure. The fixed MQ unit reference sets that common value to one. Finite additivity then fixes the measure of every finite subset D of U_A by cardinality,

and the regional count measure of Eq. (3) is precisely the cardinality of the corresponding token subset. If the unit convention is temporarily left unfixed, every admissible alternative differs only by one common positive scale c; normalization removes even that freedom:

Thus, a cell- or outcome-dependent primitive weight is not another admissible MQ count measure: it would introduce an additional scalar weighting field beyond the integer count ontology. Proposition 2.1 establishes uniqueness of the primitive measure conditional on the MQ count-only substrate; it does not assert that every preparation has the same count configuration. Nonuniform quantum probabilities can therefore arise only from different count multiplicities of the dynamically generated event preimages under the Frames mapping, not from unequal primitive weights assigned by hand.

Under the Frames mapping, the Internal Frame count measure induces a realized System Frame measure on realized continuum regions. When the induced measure is absolutely continuous with respect to the realized System Frame continuum volume element d³r, it admits a density representation written ρ_S(r,t) (Sec. 3.4; Appx. C.6 and Appx. A.4). All coordinate-dependent densities, gradients, and continuum differential structures therefore belong exclusively to the realized System Frame and do not appear in the primitive Internal Frame definition (Secs. 3.4–3.5; Appx. C.6–C.8, Appx. C.9–C.21, and Appx. E.1–E.3).

Count evolution in the Internal Frame is defined through deterministic redistribution over the relational adjacency structure. Let J_I denote the Internal Frame adjacency count-flux defined on oriented admissible adjacency links connecting neighboring relational cells ξ and ξ^' (Appx. A.2). The orientation is purely combinatorial and introduces no geometric or continuum directional structure. The associated adjacency-divergence operator div_adj acts on the adjacency count-flux and returns the net signed count flow out of a relational cell. It is a purely combinatorial operator defined on the adjacency structure of the Internal Frame and does not represent a realized continuum differential divergence (Appx. A.2).

The local conservation law for the discrete measurement-availability count field is therefore written at the relational-cell level as

where Δ_A is the discrete update operator acting on the Internal Frame update index A. Equation (9) states that any update variation of local encoded availability count is exactly balanced by the net adjacency count-flux through the relational cell. Conserved count structure is therefore redistributed only through adjacency relations and not through any background geometric evolution (Appx. C.4–C.5 and Sec. 2.11).

For any admissible relational region ω^'⊆ω, define the adjacency boundary ∂ω^' as the set of oriented adjacency links connecting cells in ω^' to cells outside ω^' (Appx. A.2). The net boundary count-flux is then defined by

which gives the signed outward adjacency flux across the relational boundary. The boundary definition is entirely relational and combinatorial and does not involve geometric surfaces, normal vectors, or continuum metric structure.

Summing Eq. (9) over all relational cells satisfying ξ∈ω^' yields the regional conservation law

Using Eq. (3), the summation may be expressed directly in terms of the regional count measure N_I(ω^',A), giving

where Eq. (12) is the compact regional conservation law written in terms of the regional count measure N_I(ω^',A).

Equation (11) states that the update variation of total encoded availability within an admissible relational region equals minus the net outward adjacency count-flux across its relational boundary. Equation (12) is the equivalent measure-level representation obtained through the definition of the regional count measure. Together, Eqs. (11) and (12) establish conservation of encoded availability at the regional level and provide the discrete basis for continuum mapping and realized encoded-availability conservation developed later in Secs. 3.4–3.5 and Appx. C.5–C.8.

For closed admissible domains satisfying ϕ_I(∂ω^',A)=0, total Internal Frame count is conserved:

This closed-domain conservation condition provides the count-conservation basis for normalization preservation in the realized System Frame encoding (Appx. C.5, C.7–C.8, C.11, and Appx. E.1–E.2).

After application of the Frames mapping to the realized System Frame, this discrete conservation structure becomes the basis for normalization preservation in the encoded-availability representation (Sec. 3.4; Appx. C.7–C.8, Appx. C.11, Appx. E.1–E.2, and Appx. G.11–G.12), with norm-preserving realized dynamics subsequently developed within the realized System Frame formalism (Secs. 3.5 and 3.11; Appx. F.4–F.6, Appx. F.7–F.11, and Appx. F.15–F.17).

We denote the realized System Frame phase field by ϕ(r,t) (Appx. A.5 and Appx. E.6), while the Internal Frame boundary count-flux is denoted by ϕ_I(∂ω^',A) (Appx. A.2). These symbols represent distinct structures and are not interchangeable.

At this stage the discrete measurement-availability count field is the only primitive object. Probability density, phase structure, interference behavior, operator structure, and realized continuum dynamics arise only after normalization and realized System Frame encoding through the Frames mapping under the Frames Principle (Secs. 3.1–3.11; Appx. C–F and Appx. H). The realized quantum state representation is therefore derived from conserved count structure rather than introduced axiomatically.

2.7 Internal and System Frames

MQ separates primitive count kinematics from realized continuum encoding through two ontologically distinct but mapping-related domains together with a third operational reference role. The Reference Frame fixes the unit normalization used to compare a variable count with its realization; it does not carry an independent primitive state space. The Internal Frame is the discrete count domain, and the System Frame is the realized non-discrete relational encoding. The Internal Frame is defined by the discrete measurement count field together with its admissible redistribution dynamics. This structure is specified on a relational adjacency domain and carries no continuum coordinate, metric, geometric vector, or continuum differential structure. Physical structure at this level is determined entirely by count distribution, adjacency relations, and adjacency count-flux. The primitive count objects obey non-negativity, finite additivity, and conservation under admissible updates (Sec. 2.6; Appx. C.2–C.5). This kinematic layer supplies the conserved count structure from which realized continuum observables are derived.

The System Frame is the realized continuum encoding of this conserved count structure in terms of spatial and temporal coordinates (r,t). It is obtained through the deterministic, many-to-one Frames mapping under the Frames Principle (Secs. 2.7–2.8; Appx. A.3 and Appx. C.1–C.22). Under this mapping, the Internal Frame count measure induces a realized System Frame measure which, when absolutely continuous with respect to the realized continuum volume element, admits an encoded availability density representation ρ_S(r,t). Coordinate-dependent densities, gradients, differential operators, and metric relations therefore belong exclusively to the realized System Frame, not to the primitive Internal Frame. Wave function encoding and operator representations arise only after this realization step (Secs. 3.4–3.11; Appx. C.6–C.22 and Appx. F). The realized geometric structure induced by the Frames mapping is developed explicitly in Sec. 2.8, Appx. A.3 and Appx. M, where realized metric structure is derived from count-based relational composition. The local metric sufficiency conditions for this realization are proved in Appx. N.

The conserved count structure is preserved across the Frames mapping, while the mathematical representation differs between the two descriptions. Primitive count redistribution and adjacency conservation are defined in the Internal Frame, whereas amplitudes, normalized probability densities, differential generators, and continuum conservation laws arise only after realization in the System Frame. MQ therefore constructs amplitudes, probabilities, and dynamical operators from normalized encoded availability rather than introducing them as primitive ontological objects (Secs. 3.4–3.11; Appx. C–F).

The frame distinction established above is assumed in the later derivations; Secs. 3.1 and 3.6–3.7 apply it directly to conditioning, interference, and entanglement.

With respect to Bell constraints, the discrete measurement count field is not used as a Bell-local hidden-variable response space. MQ introduces no setting-independent local response variable λ carrying response functions P(A∣a,λ) and P(B∣b,λ). Bell and CHSH assumptions, including factorizability, P(A,B∣a,b,λ)=P(A∣a,λ)P(B∣b,λ), apply only to realized System Frame outcome statistics. In MQ those statistics are computed from normalized encoded availability under contextual restriction, so Bell factorizability is not imposed. What is established directly is operational no-signaling: the observed local marginals are invariant under space-like separated setting choices. This operational statement is weaker than λ-conditioned parameter independence in a Bell model. Bell–CHSH violations [3] therefore arise from contextual many-to-one encoding under the Frames mapping, not from nonlocal update dynamics (Sec. 3.7; Appx. G.1–G.7, Appx. H.1–H.7, and Appx. G.8–G.12).

2.8 The Frames Mapping: Admissibility and Physical Uniqueness

The connection between the discrete Internal Frame description and the continuum System Frame representation is established through the Frames mapping, which realizes continuum geometric structure from conserved count relations. The primitive measurement count field n_I(ξ,A) is defined only on a relational adjacency domain and possesses no primitive coordinate, metric, geometric vector, or continuum differential structure. Under the deterministic, many-to-one Frames mapping and the Frames Principle, admissible System Frame realizations correspond to the image of the Internal Frame realization mapping operator M_I, which maps deterministic count redistribution to constraint manifolds in the realized continuum encoding (Sec. 2.11; Appx. A.3 and Appx. C.1).

The connection between the discrete Internal Frame description and the continuum System Frame representation is established through the Frames mapping. In the present paper the symbol M denotes the structural realization map from an admissible Internal Frame count configuration to its admissible System Frame realization class. This class may contain more than one coordinate representative before a measurement context is imposed, while distinct Internal Frame configurations may also possess the same coarse-grained System Frame encoding. The mapping is therefore deterministic at the level of the admissible realization class but is not asserted to be a one-to-one correspondence between primitive configurations and individual continuum points. Once the full measurement context and complete admissible interaction are fixed, the completed contextual realization M_c is single-valued by referenceability; the disjoint outcome partition then induces the realized label before post-outcome restriction and renormalization (Sec. 3.2.1; Appx. H.9). This distinction removes the earlier conflation between a deterministic structural mapping rule, non-injective coarse-graining, completed-context realization, and predictive conditioning.

2.8.1 Admissibility, Equivalence, and Uniqueness Status

The Frames mapping is admissible only when it preserves the structures independently required by the MQ primitive and operational assumptions. At the count-measure level these are non-negativity, finite additivity on disjoint relational regions, local finiteness of the induced System Frame measure, and closed-region count conservation. At the realization level the mapping must preserve adjacency-local redistribution, admit finite normalization, remain invariant under admissible relabeling of purely relational Internal Frame labels, assign exactly one realized outcome to each complete admissible interaction configuration, encode distinct realized outcomes injectively as stable macroscopic records, and remain path-consistent on every contractible reference cycle. The last requirement is the MQ holonomy-free closure condition. These conditions are collected formally in Appx. A.3 and Appx. C.4.1, while their measurement application is made explicit in Secs. 3.1-3.2 and Appx. H.8.

For every realized chart on which metric reconstruction is invoked, Frames mapping admissibility additionally requires a local C³ realization, a full-rank realization Jacobian, and C³-diffeomorphic chart transitions. In a pseudo-Riemannian extension the realized tangent image must also be nondegenerate with respect to the chosen constant-signature reference form. These are explicit differential-geometric regularity conditions rather than consequences of count conservation. Appx. N proves that they are sufficient to make the induced System Frame metric symmetric, nondegenerate, of constant prescribed signature, sufficiently smooth for curvature, coordinate covariant, and independent of admissible reparametrization.

The motivations for these conditions are logically distinct. Positivity, additivity, and conservation follow from treating the primitive substrate as conserved count rather than probability. Adjacency locality follows from the definition of admissible Internal Frame redistribution. Relabeling invariance follows because relational cell labels carry no physical coordinate meaning. Finite normalization is required before encoded availability can define a probability measure. Stable record distinguishability follows from Operational Constraints 1 and 4. Holonomy-free closure is required so that transporting reference data around a contractible loop does not assign path-dependent realized observables; this parallels the standard role of holonomy as a loop-level consistency diagnostic in gauge and geometric descriptions [74]. None of these motivations presupposes a primitive Hilbert space or wave function.

These requirements define an admissibility class of representatives, not an unconstrained family of physical laws. The physically relevant object is the quotient of that representative space by admissibility-preserving Frames automorphisms. Two representatives are realization-equivalent when they produce the same realized observables for every admissible Internal Frame region and preserve normalization, stable-record encoding, and contractible-cycle closure. Local representatives related in this way differ only by a change of realization description; they cannot be used to tune an observable prediction. On overlapping realization charts the associated transition maps must be invertible and satisfy the identity cocycle, so locally equivalent descriptions glue consistently. If the domain contains a genuine global obstruction, non-contractible defect holonomies are additional invariant data. Accordingly, the complete physical class is specified by the local automorphism orbit together with the regular domain and its invariant defect-holonomy set. This is the standard mathematical distinction between gauge-like representative freedom and physical inequivalence, with holonomy acting as a global invariant [74]; the MQ specialization and proof are given in Appx. C.4.3.

Proposition 2.2 (Restricted physical uniqueness of the Frames mapping). Fix an admissible Internal Frame domain U, its conserved count data and boundary/interaction context, and the MQ realization invariants used in the present construction. Then (i) all Frames mapping representatives connected by an admissibility-preserving automorphism belong to one physical mapping class [M]U; (ii) local representatives define a consistent global realization only when their overlap maps satisfy the cocycle condition; and (iii) on a non-contractible domain, two global realizations are physically equivalent only if their defect sets and all defect holonomies agree. For the bounded contractible finite measurement domains of Proposition 2.3, the defect data are trivial, so the physical mapping is unique up to the automorphism orbit [M]U. The explicit construction and classification are given in Appx. C.4.3.

Proposition 2.2 closes the ambiguity relevant to the quantum derivations. The arbitrary positive basis functions used in the existence construction of Appx. C.4.2 are auxiliary representatives, not adjustable physics: once two choices preserve the same realized measure/current, record map, normalization, and closure invariants, they lie in the same physical class. A choice that changes those observables is not an alternative representative of the same MQ law; it is an inequivalent mapping class and must independently satisfy the count-normalization relations of Sec. 2.8.2, the continuity/current constraints, the composite-system constraints, and the restricted dynamical conditions of Sec. 3.5. Thus, admissibility cannot be used as a free fitting function to insert a desired continuum law.

Proposition 2.3 (Constructive existence of an admissible MQ realization). For a finite connected Internal Frame relational region with nonzero finite count measure, count-conserving adjacency redistribution, and a finite System Frame outcome partition represented on a bounded contractible realization chart, there exists at least one Frames mapping representative satisfying the measure, conservation, adjacency, finite-normalization, relational-relabeling, stable-record, and contractible-cycle closure requirements of Sec. 2.8.1. The construction is explicit in Appx. C.4.2. Where metric reconstruction is invoked, the same realization chart can simultaneously satisfy the smoothness, full-rank, and nondegeneracy conditions of Appx. N.

The proof uses a positive normalized basis of System Frame densities to carry each finite Internal Frame count exactly into a finite realized measure, a divergence lift of each adjacency transfer to preserve the discrete continuity law, and an exact System Frame availability connection so that every contractible reference cycle has zero holonomy. Relabeling only permutes the basis functions, while the finite outcome partition and the record construction of Sec. 3.2.1 provide single-valued realized labels and injective stable records. Thus the advertised conditions are jointly satisfiable on the physical class used here rather than merely imposed separately.

The consequences of the mapping must likewise be separated by which admissibility properties they use. Positivity, additivity, local finiteness, and finite normalization support the induced System Frame measure and its normalized probability representation [42]. Count conservation together with adjacency-local redistribution supplies the mapped continuity/current structure. The absence of primitive metric structure, combined with count-composition closure and the specific MQ normalization rule, yields the emergent geometric relations of Sec. 2.8.2. Holonomy-free closure guarantees path consistency on contractible cycles but does not by itself derive interference or the Schrödinger equation. Likewise, non-injectivity of the coarse-grained encoding explains loss of recoverable Internal Frame information but does not by itself imply Bell violation; the latter additionally requires the composite non-factorizable availability and contextual partition structure of Sec. 3.7 and Appx. G–H. Schrödinger-form evolution additionally requires the restricted dynamical admissibility class stated in Sec. 3.5. Finally, the quantitative Informativity differential and associated metrological predictions require the specific MQ count-normalization construction summarized in Eqs. (14–17) and Appx. J rather than merely the abstract existence of a Frames mapping.

This hierarchy is important because the Frames Principle is stronger than the statement that some map exists, but weaker than a claim that every later quantum result follows from mapping alone. The principle fixes where observables reside and requires admissible realization; the additional measure, closure, composite-system, and dynamical conditions determine which specific continuum and quantum consequences follow.

The remaining question is whether the specific finite-count normalization used by MQ is freely selectable within that physical class. Appendix AI now separates two issues that were previously conflated. The primitive Internal Frame supplies no angular metric and therefore cannot simply assume Euclidean geometry. However, on a regular local free-realization neighborhood whose count and boundary data contain no direction-selecting invariant, relational relabeling invariance and the physical-equivalence rule of Proposition 2.2 require all unit System Frame spatial directions to lie in one orbit of admissibility-preserving linear isometries. For a finite-dimensional normed tangent realization, transitivity of the linear isometry group on the unit sphere forces the unit ball to be an ellipsoid and hence the norm to be induced by an inner product [160]. The Jordan-von Neumann characterization then gives the equivalent parallelogram law [110]. With the MQ unit reference fixed and no directional scale available, that inner product is isotropic up to an overall scale already fixed by l_f. Thus, the positive quadratic spatial norm used in Appendix M is downstream of the direction-free local-realization condition rather than an independently selectable metric ansatz. Appendix AI states the scope and the residual anisotropic-sector boundary precisely.

Proposition 2.4 (Conditional uniqueness of the MQ local mapping-normalization rule). Fix the finite, bounded, contractible measurement class of Propositions 2.2-2.3 and a regular direction-free local spatial realization satisfying Appendix AI. Then Appendix AI selects the positive isotropic inner-product norm from relational direction equivalence; with (i) a fixed unit reference of norm l_f, (ii) collinear additivity of n_L identical primitive count units, (iii) orientation-preserving realization, and (iv) asymptotic identity of the realized separation with n_L l_f as n_L grows, the realized right-triangle separation is uniquely l=(n_L+Q_L)l_f=sqrt(1+n_L^2)l_f, the positive residual is uniquely Q_L=sqrt(1+n_L^2)-n_L, and the associated finite-count normalization factor is uniquely 2Q_L n_L=1-Q_L^2. Any different local normalization either is only a representative transformation that leaves these invariants unchanged, introduces a physical direction-selecting invariant excluded from the local free sector, or defines an inequivalent theory. The metric-selection proof is given in Appx. AI and the residual algebra in Appx. M, Eqs. (M.4-M.8).

Proposition 2.4 remains conditional to the direction-free sector, but the complementary anisotropic sector is now constructive rather than unspecified. Section 2.8.3 and Appx. AQ show that once primitive adjacency, matter, boundary, or defect data break direction equivalence, the same MQ Action fixes the local quadratic response on the admissible edge-transfer space. Mapping that action Hessian to the realized tangent space produces a unique positive anisotropic principal tensor, and the mapped count measure together with the fixed unit scale reconstructs the corresponding System Frame spatial metric. Thus neither an arbitrary anisotropic tensor nor a freely chosen Finsler function is admitted: changing the anisotropy while holding the physical source invariants and MQ Action fixed changes the action response and defines an inequivalent theory. The direction-free limit reduces to Appx. AI and the Appendix M normalization rule.

2.8.2 Geometric Consequences of the Admissible Mapping

To understand the geometric origin of the Frames mapping, consider the right-triangle realization construction shown in Fig. 1. Orthogonal realized separations are constructed from whole-count relations defined relative to a fundamental reference measure l_f, with one leg fixed at count one and the second assigned an integer count n_L. The realized hypotenuse separation is then determined by

where 2Q_Ln_L is the Informativity differential (i.e., length contraction) induced by mapping normalization under realized continuum encoding. Since

is not integer-valued for nonzero integer n_L, realized geometric composition is not closed over discrete count structure. By itself this establishes arithmetic non-closure only. Under the independent MQ admissibility conditions proved in Appx. M, preserving conserved integer count structure while admitting realized continuum geometry requires a distinct realization layer in which non-integer geometric relations become admissible under the Frames mapping.

Under the stated MQ admissibility conditions the construction is stronger than a heuristic. Appx. M gives the elementary closure proof: for every positive integer count nₗ, the Euclidean norm √(1+nₗ²) lies strictly between nₗ and nₗ+1 and therefore cannot be an Internal Frame integer count. Side a is simultaneously noninterchangeable with side b because it is fixed as the normalization reference, whereas side b is the variable conserved count. Accordingly, the construction requires three distinct roles—Reference, Internal, and System—while only the Internal and System roles define distinct ontological domains. In the terminology of metric geometry, this is a source/target distinction for a realization map rather than a claim that the triangle alone proves a unique microscopic ontology [82]. The Internal Frame preserves exact count conservation and adjacency redistribution, while realized continuum geometry emerges only after mapping normalization and continuum realization.

The realized System Frame interval is written

where the displayed tensor is the realized System Frame metric induced by mapped count structure. Equation (15) is a local interval statement. For differential-geometric sufficiency, the reconstruction is understood through the two-point realized separation and coincidence limit stated precisely in Appx. N; Eq. (16) is retained as the corresponding local single-chart shorthand. This distinction is essential because an unconstrained Hessian of a one-point scalar is not, by itself, a global definition of a metric tensor.

Using the count-to-coordinate realization relation,

the realized metric may be written directly in terms of count structure as

Accordingly, Eqs. (16–18) are evaluated locally at coincidence for neighboring realized points. The manifestly covariant mixed-endpoint reconstruction is Eq. (N.5), and its count-resolved form is Eq. (N.6). The fundamental-length factors cancel under the count-to-coordinate relation of Eq. (17), but the tensorial definition remains the two-point coincidence-limit construction.

Under the regularity and rank conditions of Appx. N, Eqs. (15–18) therefore represent a genuine emergent System Frame spatial metric rather than merely a matrix of count derivatives. The right-triangle construction fixes the positive-definite spatial reference norm used in the non-relativistic sector; it does not by itself determine a spacetime signature. Section 2.9.1 and Appx. X now close that distinct step for the regular four-dimensional realization class used here. Adjacency-local redistribution together with the single primitive update ordering induces a locally finite reachability order on count-update events; its monotone Frames mapping gives one realized evolution direction and a finite nonzero propagation boundary. Because Appx. M–N already fix a three-dimensional positive spatial tangent subspace, any nondegenerate quadratic metric representing that nontrivial causal boundary must have exactly one eigenvalue of opposite sign. By Sylvester inertia this selects the Lorentzian class, conventionally (-,+,+,+), with future orientation inherited from increasing update order (Proposition 2.6; Appx. X) [145–147]. Symmetry, nondegeneracy, smoothness, covariance, and reparametrization independence then follow from the standard pullback/induced-metric construction proved in Appx. N [83], while the equivalence with local squared-separation reconstruction follows from the standard world-function coincidence identities [84]. Continuum derivatives, geometric intervals, and differential operators consequently belong exclusively to the realized System Frame and do not exist in the primitive Internal Frame description.

When the induced System Frame measure is absolutely continuous with respect to the realized continuum volume element, it admits an encoded availability density representation written ρ_S(r,t) (Sec. 3.4; Appx. C.6 and Appx. A.4). The normalized count-gradient field is then defined on the realized encoded availability density by

where ∇ is taken with respect to realized System Frame coordinates. Equation (19) represents the continuum realization of admissible discrete count differences under the Frames mapping and serves as a local fractional variation measure of mapped encoded availability. Uniform encoded availability density yields vanishing fractional variation and therefore locally uniform continuum realization structure.

The scalar θ_si characterizes fractional encoded availability scaling with realized separation in the System Frame representation and functions as a kinematic expansion parameter derived from count ratios. MQ nomenclature retains the single symbol θ_si across regimes to reflect scalar invariance of the underlying count ratios. In MQ fundamental-measure form, θ_si is fixed by irreducible count measures as ([11], Appx. V)

and is dimensionless in the System Frame ([11], Appx. AX). The dimensional rule is stated explicitly in Appx. K: the scalar coefficient θsi is dimensionless; angular and momentum realizations are separate quantities carrying their own units. A published Shwartz–Harris calculation for polarization-entangled X-ray down-conversion gives a degenerate-frequency Bell-state solution with signal and idler angles of 3.26239 rad [35]. That number is a result of their numerical phase-matching/current-density model, not a reported experimental measurement of θ_si. The later X-ray down-conversion and X-ray/optical wave-mixing experiments [36,37] establish the relevant nonlinear X-ray processes but do not report an experimental determination of the 3.26239-rad Bell-state angle. The evidentiary status and MQ correspondence are separated explicitly in Appx. L.

The Frames mapping therefore acts locally on discrete count differences and adjacency relations when producing realized continuum encoding. Spatial structure in the System Frame arises from mapped count gradients and count-composition geometry, while symmetry and conservation of count redistribution constrain the associated realized continuum metric and operator structures. Metric structure, directional structure, amplitudes, and differential generators consequently enter only through realization encoding and are not primitive components of the Internal Frame.

Because realization encoding is performed on a shared underlying availability structure, contextual restriction and renormalization produce single-measurement marginal probabilities determined by local measurement partitions, while joint probabilities retain the geometric structure of the mapped count distribution. Consequently, MQ does not introduce a setting-independent Bell-local response variable with factorizable local outcome functions. Standard Bell and CHSH inequalities [3,38,39], derived from factorizability P(A,B∣a,b,λ)=P(A∣a,λ)P(B∣b,λ) together with measurement independence, apply only to realized System Frame outcome probabilities, not to the primitive count level. In MQ those probabilities are computed from normalized mapped encoded availability under measurement context, so Bell factorizability is not imposed, while the realized marginals satisfy operational no-signaling under space-like separated setting choices. The latter should not be identified automatically with λ-conditioned parameter independence. The resulting Bell–CHSH violations therefore arise from contextual many-to-one realization encoding under the Frames mapping, rather than from nonlocal update dynamics, consistent with the Bell-constraint analysis developed later (Appx. G–H).

The Internal Frame therefore supplies only discrete ordering, adjacency, and conserved count redistribution, whereas continuum gradients, realized directions, metric structure, and differential operators arise exclusively in the realized System Frame encoding derived from mapped count-composition structure.

2.8.3 Invariant-Selected Anisotropic Realization

Appendix AI proves that the regular local free-realization sector is isotropic when no physical datum distinguishes spatial directions. The converse problem is not solved by merely allowing a general tensor: anisotropy must be generated by the same primitive data and action that define the MQ realization. For a regular anisotropic neighborhood, Appx. AQ therefore forms the second variation of the single MQ Action on the finite adjacency-transfer space after the actual count state, matter content, boundary conditions, and defect/holonomy class are fixed. This finite response matrix contains the complete local direction-selecting information supplied by the physical state; no additional metric coefficients are introduced.

Under the Frames mapping, each admissible primitive edge variation has a realized tangent increment. Contracting those increments with the action response matrix gives a symmetric positive principal tensor A^{ij} governing the continuum quadratic response. This is the same graph-to-elliptic structure used in weighted diffusion and Dirichlet-form constructions [188]. In three realized spatial dimensions the standard conductivity-metric correspondence then reconstructs the spatial metric uniquely from A^{ij}, with the remaining overall unit fixed by the mapped count measure and l_f [187]. The isotropic case is recovered when the action response is proportional to the identity on the realized tangent space.

Proposition 2.5 (Invariant selection of the regular anisotropic MQ metric). Fix a regular three-dimensional local System Frame realization, a finite-support Internal Frame neighborhood, its conserved count and adjacency data, the realized matter state, boundary conditions, defect/holonomy class, and the single MQ Action. Assume the action is twice differentiable on the admissible local transfer variables and that its stationary second variation is positive definite on the realized spatial transfer subspace. Then the action Hessian and the mapped edge increments determine a unique symmetric positive principal tensor A^{ij}. The locally finite mapped measure and fixed unit scale determine from A^{ij} one positive-definite System Frame spatial metric h_{ij}; representatives related by relational relabeling or Frames automorphism give the same tensor geometrically. Any different anisotropic tensor changes the quadratic action response and is physically inequivalent. A nonquadratic Finsler norm is not an additional representative freedom: it can arise only if the MQ Action itself contains independently derived direction-dependent higher-order response data. The construction and uniqueness proof are given in Appx. AQ.

Proposition 2.5 closes the anisotropy gap at the level required by the present continuum construction: regular local anisotropy is determined by physical invariants through the fixed MQ Action rather than selected by ansatz. It does not claim that every anisotropic medium has a nondegenerate local quadratic response. Degenerate action Hessians, genuinely nonlocal constitutive response, or higher-order direction-dependent sectors require their corresponding MQ action terms to be derived before a metric can be assigned; those cases remain explicit extension boundaries rather than free metric choices.

2.9 Relativistic and Metrological Consequences of the Mapping

2.9.1 Discreteness, Lorentz Symmetry, and the Informativity Differential

The discreteness postulated by MQ is not a discrete spacetime lattice. The Internal Frame carries count labels, adjacency, and update ordering but no spacetime coordinates, metric, inertial-frame velocity, or boost parameter. Lorentz transformations therefore have no primitive domain of action on those count labels. Lorentz symmetry is instead a property of the realized System Frame spacetime description, where coordinates, metric structure, propagation velocities, and observer transformations are defined. This distinction is essential because discreteness of a non-spatiotemporal relational substrate does not by itself imply a preferred inertial frame or observable Lorentz violation.

The Informativity differential is likewise not a Lorentz-violating contraction law. It is the count-dependent normalization residual fixed by the right-triangle realization and the Frames mapping (Eq. (14); Appx. J). Its value depends on count separation and mapping normalization, not on relative observer velocity, boost direction, photon energy, polarization, or inertial-frame orientation. Consequently, the present MQ construction generates none of the characteristic signatures by which Lorentz violation is experimentally sought: preferred-frame anisotropy, boost- or orientation-dependent couplings, vacuum birefringence, or energy-dependent vacuum propagation speed. The term “mapping-induced finite-count contraction” is used throughout this paper which states the intended distinction without implying violation of Lorentz symmetry.

This distinction is experimentally consequential. Modern Lorentz tests are commonly organized within the Standard-Model Extension (SME), where departures from Lorentz symmetry are parameterized by explicit coefficients in matter, photon, neutrino, and gravity sectors; the accumulated bounds are extremely restrictive and show no established Lorentz-violating signal [85,86]. Astrophysical propagation tests provide complementary constraints: the GW170817/GRB 170817A association limits the fractional difference between gravitational-wave and light speeds to approximately the 10⁻¹⁵ level [87], while gamma-ray burst timing constrains linear energy-dependent photon dispersion at or beyond the Planck scale for the tested models [88]. Pulsar timing and spin-precession analyses likewise place stringent contemporary bounds on gravity-sector SME coefficients [89].

The status of realized inertial symmetry can now be stated more strongly. MQ does not claim that count conservation by itself generates a Lorentz group. Proposition 2.6 and Appx. X first select the local Lorentzian metric class and future orientation from the single update ordering, adjacency-local finite reachability, the invariant fundamental length/time ratio, and the positive three-dimensional spatial metric. Proposition 2.7 and Appx. Y then use the physical-equivalence rule of Proposition 2.2: two local inertial representatives of the same Frames mapping class must preserve the realized interval, the future-directed causal cone, spatial orientation and isotropy, and the invariant conversion c = l_f/t_f. The connected homogeneous transformation group preserving those already-selected invariants is the proper orthochronous Lorentz group. Appendix S independently supplies realized translations; on a locally flat homogeneous realization the combined connected inertial kinematics is therefore the proper orthochronous Poincaré group. Its low-velocity c → ∞ contraction is the Galilei group, so the Galilean boost law used in Appx. O is no longer an independent non-relativistic symmetry postulate. Existing preferred-frame, birefringence, dispersion, and gravity-sector bounds [85–89] remain direct empirical constraints. What remains open is global symmetry on curved or topologically nontrivial realizations, not the local boost group within the finite regular class proved here.

Proposition 2.6 (Count-order selection of the local Lorentzian signature). Fix a regular four-dimensional System Frame realization satisfying Appx. N. Let H_p be the three-dimensional tangent subspace generated by realized spatial count-separation variations. The Appx. M right-triangle norm and Theorem N.1 make the restriction of the realized metric to H_p positive definite. Let the fourth independent realized tangent direction be the monotone image of the single Internal Frame update-order degree of freedom. Require the realized quadratic metric to represent the accessibility relation induced by adjacency-local redistribution, whose finite mapped propagation boundary contains nonzero null directions. Then every nondegenerate metric representing these structures has inertia (1,3) or, under overall sign reversal, (3,1). With the positive spatial convention used throughout this paper, its local signature is (-,+,+,+), and increasing update order selects the future orientation. Thus Lorentzian causal signature is no longer an independently selectable System Frame admissibility datum within this realization class.

Proof. H_p is a three-dimensional positive subspace of the four-dimensional tangent space. A nondegenerate symmetric bilinear form containing such a subspace can have at most one negative eigenvalue. The finite nontrivial accessibility boundary requires nonzero null directions, which excludes a positive-definite four-metric; nondegeneracy excludes zero eigenvalues. Exactly one eigenvalue must therefore have sign opposite to the spatial three-subspace. Sylvester’s law of inertia makes this index invariant under admissible coordinate changes. A second timelike direction would require a second negative direction and is incompatible with the already-fixed three-dimensional positive spatial subspace. The overall sign is conventional, giving (-,+,+,+); the monotone image of increasing Internal Frame update order fixes time orientation. Appx. X constructs the reachability relation and gives the causal-reconstruction comparison [145–147].

Proposition 2.7 (Mapped-invariant selection of local Lorentz and Galilean symmetry). Fix the finite regular four-dimensional realization class of Propositions 2.3–2.6 and consider two physically equivalent local inertial representatives of one admissible Frames mapping class. Require the representative change to preserve all realized invariants already fixed upstream: the nondegenerate System Frame metric and its (-,+,+,+) inertia, the future orientation induced by increasing Internal Frame update order, isotropic spatial orientation, and the invariant conversion ratio c = l_f/t_f. Then the connected homogeneous group of admissible tangent-frame transformations is uniquely SO⁺(1,3), the proper orthochronous Lorentz group. Where the realized metric is locally flat and translations act as derived in Appx. S, the connected affine inertial group is the proper orthochronous Poincaré group. In the non-relativistic limit obtained by the standard İnönü-Wigner contraction as c → ∞, this realized kinematics reduces to the Galilei group. Thus Lorentz and Galilean boost covariance are downstream consequences of the MQ realization invariants within this class rather than separately selectable continuum symmetries.

Proof. Proposition 2.2 identifies admissibility-preserving representatives with the same physical realization only when they preserve the realized observables and closure invariants. Proposition 2.6/Appx. X fixes a time-oriented Lorentzian metric, while the MQKB fixes c = l_f/t_f as a path-independent mapping ratio and Appx. M fixes isotropic positive spatial geometry. Consequently the differential of any continuously connected local inertial representative change must be a metric-preserving, orientation-preserving, future-preserving linear automorphism of a (1,3) tangent space. The complete connected group of such automorphisms is SO⁺(1,3); any alternative continuous boost law changes the interval, cone, orientation, or invariant speed and therefore leaves the physical mapping class. Appx. S supplies the translation subgroup. Standard kinematical classification independently reaches the Lorentz/Galilei alternatives from homogeneity, isotropy, group structure, and causality [148,149], and the Galilei group is the group-contraction limit of Poincaré kinematics [150]. Appx. Y gives the MQ-specific provenance and scope of each step.

2.9.2 Status of the Realization-Dependent Gravitational-Coupling Prediction

The gravitational discussion in this paper uses two notational realizations of the same MQ coupling. The finite-count value is obtained when the mapping-induced normalization is evaluated at the fixed interaction count associated with the electromagnetic demarcation; the upper-count value is the asymptotic realization in which the Informativity differential tends to zero. These quantities are predictions internal to MQ. They must not be read as a community-standard 'electromagnetic G' and 'macroscopic G,' because gravitational metrology defines and reports a single Newtonian constant G. An experimental test of the MQ proposal must therefore predict in advance which measurement feature changes the applicable realization rather than relabeling published G values after measurement.

The 2026 NIST redetermination of the BIPM torsion balance is unusually informative in this respect because it holds the source-mass geometry essentially fixed while changing the torque-readout and calibration method [80]. In free deflection, the gravitational torque is inferred from the mechanical response of the torsion balance, whereas in electrostatic servo an applied electrostatic torque supplies the balancing calibration. Schlamminger et al. report four determinations: copper-servo 6.673642 × 10⁻¹¹, copper-free 6.674021 × 10⁻¹¹, sapphire-servo 6.672637 × 10⁻¹¹, and sapphire-free 6.673636 × 10⁻¹¹ m³ kg⁻¹ s⁻². Thus the servo value is lower than the free-deflection value for both source-mass materials.

That ordering is qualitatively consistent with the MQ expectation that a measurement channel containing an explicit electromagnetic balancing interaction can retain the finite-count correction more strongly than a curvature-resolved mechanical determination. This interpretation remains a hypothesis, however. The NIST/BIPM experiment was designed as a replication and systematic study of G, not as a prospective test of the Frames mapping; both modes measure the same underlying gravitational torque, and the published analysis allows residual unmodeled instrumental effects. The experiment therefore opens a discriminating test channel but does not by itself establish more than one physical gravitational constant.

A decisive test would use one apparatus and source geometry, pre-register the MQ assignment of the two measurement channels, alternate those channels under otherwise unchanged conditions, and require the inferred difference to follow the MQ finite-count correction with the predicted sign and magnitude. Repetition with different source materials, separations, electrostatic drive amplitudes, and independent laboratories would then distinguish a mapping-dependent effect from apparatus-specific systematics. Under this criterion, failure of the pre-specified differential relation would falsify the realization-dependent G component of the MQ mapping without requiring rejection of the broader count-based quantum construction.

The existing NIST data also permit a covariance-resolved magnitude check, although not an independent second test. The two MQ realizations differ by 2.3310 × 10⁻¹⁵ m³ kg⁻¹ s⁻², or 34.9274 ppm relative to the finite-count value. From the four published NIST determinations [80], the free-minus-servo differences are 56.7889 ppm for copper and 149.6888 ppm for sapphire. Using the combined standard uncertainties and the full four-result correlation matrix reported by Schlamminger et al., the corresponding difference uncertainties are 29.4252 ppm and 91.9937 ppm with correlation 0.1708. Testing the two observed differences against one pre-existing MQ separation of 34.9274 ppm gives χ² = 1.85 for two residual components (right-tail p = 0.40). The result is therefore statistically compatible with the MQ separation at the published uncertainty level, but it is only a post-publication differential consistency diagnostic: the apparatus-specific servo/free regime assignment was not preregistered before the NIST data were analyzed. Appendix J.8 records this distinction explicitly.

The apparatus-specific regime assignment can now be fixed independently of the measured value by a channel-selection rule inherited from the MQKB interaction-regime distinction. The MQKB defines electromagnetic interactions as fixed-count realizations at the electromagnetic demarcation and gravitational interactions as count-resolved realizations over variable separation. For a mixed measurement chain, the applicable realization is therefore assigned by the physical interaction that supplies the calibrated force or torque used to close the estimator for the inferred observable, not by a passive sensing channel that merely reports the state of the apparatus. This criterion is fixed by interaction provenance before examining the resulting value of G.

Applied to the BIPM/NIST torsion balance, the classification is unambiguous at the level of the two published torque equations. In Cavendish/free-deflection mode the gravitational torque difference is inferred from the freely displaced equilibrium angle and the torsional stiffness, with the stiffness obtained from the torsional frequency and calculated moment of inertia. In electrostatic-servo mode the pendulum is held at a fixed angle and the gravitational torque is inferred from the equal-and-opposite electrostatic counter-torque generated by the servo. Schlamminger et al. documented this distinction, including the relations between equilibrium angle, torsional stiffness, electrostatic counter-torque, voltages, and capacitance gradients, in 2022, before the 2026 redetermination [142]. The optical angle readout is present in both modes and therefore does not select the MQ regime; what differs is the interaction that supplies the calibrated torque entering the G estimator.

The resulting prospective MQ assignment is therefore fixed before any future measurement: electrostatic-servo torque closure selects the finite-count electromagnetic-demarcation realization, whereas free mechanical-deflection torque closure selects the count-resolved gravitational realization approaching the upper-count value. This converts the earlier qualitative interpretation into a theory-side intervention rule and removes freedom to reverse the assignment after a result is known. It does not retroactively make the 2026 NIST/BIPM mode difference a causal confirmation because this MQ selector was not preregistered for those data. Appendix AD therefore separates prediction from demonstration by specifying a controlled intervention in which the torque-closing interaction is changed while source geometry, pendulum, sensing, environmental state, and analysis are held fixed. The causal claim is supported only if the inferred G difference follows the active torque-closing interaction with the pre-specified +34.9274-ppm free-minus-servo ordering, while negative controls that alter sensing or energize electrostatic hardware without closing the torque estimator do not reproduce the shift.

2.10 Global Relational Topology and Domain Extension

2.10.1 Global Topological Classification from Relational Cycle Structure

The local constructions above do not require the realized domain to be simply connected. For a finite connected Internal Frame relational adjacency graph, global one-dimensional topology is already finitely enumerable: choosing any spanning tree leaves one independent fundamental cycle for each non-tree adjacency link. Appendix Z proves that this cycle basis is independent of labels and therefore survives the relational-relabeling requirement. The Frames mapping may make some of those cycles contractible in the realized System Frame or may exclude them behind a defect set; only the surviving non-contractible classes carry global physical data.

Proposition 2.8 (Global topological classification of finite MQ realizations). Fix a finite connected Internal Frame adjacency domain and an admissible regular System Frame realization. Let the primitive adjacency graph have V relational cells and E admissible undirected adjacency links. Then its independent cycle rank is β₁=E−V+1. After quotienting cycles that become contractible under the realized mapping, every remaining global loop-sensitive structure used by this manuscript is classified on the surviving cycle classes as follows: (i) mapping-equivalent representatives are distinguished globally only by their defect/connection holonomy representation, consistent with Proposition 2.2; (ii) U(1) phase sectors are the integral winding classes H¹(Ω;Z), equivalently the homotopy classes of maps Ω→S¹; (iii) the count-order future direction extends globally exactly when no surviving loop reverses the monotone update-order image; and (iv) the local Dirac spinor sector extends globally exactly when w₂(TΩ)=0, after which inequivalent spin structures form an affine space over H¹(Ω;Z₂). Thus non-contractible phase, holonomy, time-orientation, and spin alternatives are discrete global classes rather than unclassified continuum freedom.

Proposition 2.8 classifies the global loop data once a regular realized complex is available. Section 2.10.2 and Appx. AJ now close the distinct higher-dimensional incidence question for the finite adjacency domains used by this paper. Because the Internal Frame ontology contains adjacency but no independent higher-order incidence field, the no-extra-incidence completion is the flag (clique) complex of the primitive adjacency graph: every pairwise-adjacent finite set spans one simplex and no set containing a nonadjacent pair can do so. The resulting cell attachments, homology, and cohomology are therefore computable from primitive adjacency rather than freely specified. The regular System Frame manifold locus is then selected by the existing mapping-admissibility, defect, and local link conditions. Appendix Z continues to classify holonomy, winding, time orientation, and spin structure on that selected topology.

2.10.2 Adjacency-Determined Higher-Dimensional Realization

The primitive topological input available to MQ is relational adjacency, not a pre-existing manifold or an independently specified list of two- and higher-dimensional cells. A global realization rule must therefore avoid inserting higher-order incidence information that is absent from the Internal Frame. Appendix AJ makes this requirement constructive by using the standard flag-complex completion of a graph [161]. In a flag complex, a finite set of vertices spans a simplex exactly when every pair in that set is joined in the one-skeleton. Consequently, once the primitive adjacency graph is fixed, all higher-dimensional simplices are fixed as well.

Proposition 2.9 (Adjacency-completion selection of finite realized topology). Fix a finite Internal Frame relational adjacency graph at an admissible update and require, consistently with the count-and-adjacency ontology of Secs. 2.4.1 and 2.6, that no independent primitive higher-order incidence variable be introduced. Require realized incidence to be downward closed and to contain no empty simplex: whenever all pairwise adjacencies among a finite set of relational cells are present, the corresponding higher-order incidence is admitted, while any set containing a nonadjacent pair is excluded. Then the unique simplicial completion is the clique, or flag, complex of the primitive adjacency graph. Its simplicial homology, cohomology, fundamental-group presentation, and candidate higher-dimensional cell attachments are therefore functions of the primitive adjacency data rather than independently selectable System Frame inputs. The regular realized manifold domain is the maximal connected referenceable subcomplex containing the physical support of the preparation and satisfying the local manifold-link criterion together with the Frames mapping regularity, holonomy, and defect conditions already imposed by Proposition 2.2, Appx. N, and the MQKB universe-origin construction. For a specified finite adjacency history, topology can change only when the adjacency relation itself or the admissible defect/referenceability data change.

Proof. Downward closure forbids a higher-dimensional simplex whenever one of its edges is absent. Conversely, if every edge among a finite vertex set is present but the corresponding simplex is withheld, the distinction between that clique and an otherwise identical filled clique requires an additional higher-order exclusion datum not contained in primitive adjacency. Under the stated no-extra-incidence requirement, every clique must therefore be filled. The family of all cliques is exactly the clique complex and is uniquely determined by the graph [161]. Standard PL topology then supplies an intrinsic regularity test: a simplicial complex is a combinatorial manifold precisely when the relevant face links have the required PL-sphere or PL-ball structure [162]. Applying that test together with MQ mapping admissibility removes non-manifold or obstructed cells from the regular realized domain without adding a second topology variable. The resulting topology is consequently selected from adjacency plus the already-existing defect/referenceability invariants.

Proposition 2.9 is proved first on finite domains because the flag-completion argument is elementary there, but the topology it selects is local. Section 2.10.3 and Appx. AO now show that for any connected locally finite adjacency domain, an increasing exhaustion by finite connected induced subgraphs carries the finite flag complexes, mapping classes, locally finite measures, and finite-context record probabilities into a unique compatible limit whenever the finite restrictions agree on overlaps. Thus finite-domain consistency survives the locally finite thermodynamic/global limit without generating a new freely selectable realization class. What remains outside that theorem is a non-locally-finite domain for which bounded regions can contain infinite primitive incidence or availability. The distinct dynamical question of changes in the adjacency relation is addressed in Sec. 2.10.4 and Appx. AP.

2.10.3 Locally Finite and Thermodynamic-Limit Extension

The finite constructions above can be extended without postulating a second continuum law when the primitive adjacency domain is connected and locally finite. A connected locally finite graph is countable and admits an increasing exhaustion by finite connected induced subgraphs [180]. MQKB independently requires local finiteness of mapped availability—every finite realized System Frame region carries finite encoded availability—and defines a referenceable universe as a maximal subdomain on which a valid Frames mapping exists. These conditions supply the natural domain on which the finite realization results can be assembled rather than replaced.

Proposition 2.10 (Compatible-exhaustion extension of an MQ realization). Let U be a connected locally finite Internal Frame adjacency domain and let U1⊂U2⊂… be finite connected induced subdomains whose union is U. Suppose each Um carries the finite-domain MQ structures of Propositions 2.1–2.9 and an admissible physical mapping class [Mm]. Require restriction compatibility: whenever m<n, the restriction of [Mn] to Um agrees with [Mm] on count measure, adjacency current, realized observables, completed record maps supported in Um, normalization on finite contexts, and all defect/holonomy invariants visible in Um. Require also that every bounded realized region and every finite measurement context is supported in some Um. Then there is a unique physical limit realization class [MU] on the algebra of local observables and finite-context records whose restriction to each Um is [Mm]. The induced System Frame measure is locally finite; finite-context record probabilities are unique and projectively consistent; and the flag topology of U is the directed union of the finite flag complexes. Any two limit representatives agreeing on every finite restriction are physically equivalent under Proposition 2.2 and Theorem 2.2. Appx. AO gives the construction and scope.

The probability statement does not normalize an infinite total count. For an infinite domain, MQ probabilities remain attached to finite completed contexts. Their finite event laws must agree under restriction and marginalization. The Kolmogorov extension theorem then supplies a unique probability measure on the cylinder σ-algebra generated by those finite record events when the standard consistency conditions hold [181]. This is exactly the limiting operation needed by the count-to-record construction: every laboratory record remains finite, while the ambient domain may be unbounded.

Topology behaves compatibly for the same reason. Every simplex of the flag complex has finitely many vertices and therefore lies in some finite exhaustion element. The infinite flag complex is consequently the directed union of the finite flag complexes, and ordinary homology commutes with such directed unions under the standard compact-support condition [182]. Global winding, defect holonomy, time orientation, and spin-structure data therefore arise as stabilized global invariants of the compatible limit rather than as new cell-attachment freedom.

Proposition 2.10 closes issue #74 for the physically relevant connected locally finite extension class, including thermodynamic limits of finite measurement regions and maximal referenceable MQKB domains that admit a compatible exhaustion. It does not assert existence on arbitrary non-locally-finite graphs, nor does it manufacture a global realization across a genuine MQKB obstruction: if finite restrictions fail overlap compatibility or a persistent defect prevents extension, the construction terminates at the corresponding maximal referenceable subdomain. Local edge-adjacency evolution is treated separately in Sec. 2.10.4 and Appx. AP so that the locally finite limit theorem is not asked to supply a dynamical graph law.

2.10.4 Count-Action Selection of Primitive Adjacency Evolution

Propositions 2.9-2.10 determine realized topology once an Internal Frame adjacency graph or compatible fixed-adjacency exhaustion is specified. They do not by themselves determine how the primitive edge relation changes. The MQKB supplies the missing upstream structure: the MQ Action is defined over Internal Frame count structure and its relational configuration, and the MQ variational principle requires physical realizations to be stationary under admissible variations consistent with count conservation and the Frames mapping. Adjacency is part of that relational configuration rather than a System Frame geometric observable. The natural MQ extension is therefore to vary adjacency within the same action rather than append an independent graph Hamiltonian or stochastic edge process.

Write the primitive state at update A as the pair (n_I(·,A), G_A), with G_A=(V,E_A). Over one elementary adjacency update the relational-cell set V is held fixed and only the edge set may change. Count redistribution is evaluated first on G_A by Eq. (9), producing n_I(·,A+1); the subsequent edge rewrite changes E_A but does not itself transfer, create, or destroy count. Hence every admissible rewrite preserves the exact regional and closed-domain conservation statements of Eqs. (11-13). This ordering is important: topology can change through incidence while the conserved carrier remains unchanged.

An elementary rewrite is admissible only when it has finite relational support K, leaves all edges outside K unchanged, is equivariant under arbitrary relabeling of Internal Frame cells, introduces no dangling or half-defined incidence, preserves the update-order orientation used by Proposition 2.6, and yields a locally finite graph whose flag completion satisfies the mapping, defect, record, and local-link admissibility conditions already imposed by Propositions 2.2 and 2.9. Algebraic graph-transformation theory supplies a standard local replacement formalism for this kind of constrained edge creation/deletion [183]. Within a regular PL-manifold sector, Pachner moves provide the corresponding complete local retriangulation diagnostics for topology-preserving changes [184]; they are used here as a consistency test, not as an independent MQ dynamics. Causal-set sequential growth and quantum-graphity models likewise show that relabeling-covariant discrete structure can be dynamical, but their stochastic coupling families or model-specific Hamiltonians are not imported into MQ [185,186].

The physical selector is instead the already-existing MQ variational principle. For the post-redistribution count state, form the finite set of locally supported adjacency replacements that pass the preceding admissibility gates and quotient them by the same relational automorphism equivalence used in Proposition 2.2. Evaluate the single MQ Action on the corresponding admissible realization classes. A nontrivial adjacency change occurs only when the action selects one stationary physical equivalence class under those admissible local replacements. If several stationary representatives differ only by relational relabeling or invariant-preserving automorphism, they are one physical update. If no nonidentity stationary class exists, the identity edge relation is retained. If two genuinely inequivalent stationary classes survive with different realization invariants and the action does not distinguish them, unique referenceability fails at that update and the locus is treated as an MQ defect/obstruction rather than resolved by an added label-dependent tie breaker.

Proposition 2.11 (Local count-conserving adjacency-rewrite law). Fix an admissible locally finite Internal Frame state (n_I(·,A),G_A) and let Eq. (9) determine the next count state on G_A. Among finite-support edge replacements of G_A, retain exactly those that preserve count, adjacency locality, relational relabeling covariance, update-order consistency, local finiteness, flag-completion admissibility, and all Frames mapping/referenceability conditions visible on their support and boundary. Quotient that candidate set by admissibility-preserving relational automorphisms. The next primitive adjacency relation is the unique stationary equivalence class of the existing MQ Action on this candidate quotient; when no nonidentity stationary class exists the identity rewrite is retained. Any unresolved degeneracy between inequivalent stationary classes is a failure of referenceability and defines a defect/obstruction boundary rather than additional physical freedom. Consequently, within the fixed-cell locally finite sector, adjacency creation/deletion is local, count conserving, relabeling invariant, and selected by the same first-principles MQ Action that governs the realized continuum dynamics. The construction and conservation proof are given in Appx. AP.

Proposition 2.11 closes the specific universality gap identified in item #75 without asserting more than the MQKB supports. It supplies a first-principles edge-rewiring rule for a fixed set of primitive relational cells and shows why a topology change does not create count or introduce a second causal ordering. It does not yet derive birth or annihilation of primitive relational cells themselves, nor does it guarantee that every conceivable count state admits a unique nonidentity stationary rewrite. Those stronger questions are separated in Sec. 4.7.2; they are not required by the fixed-cell adjacency evolution used in the present quantum construction.

2.11 Determinism and Information Availability

Evolution of the measurement-availability count field in the Internal Frame is deterministic. The primitive count assignment n_I(ξ,A) and the associated regional count measure N_I(ω^',A) evolve through count-conserving update rules that redistribute encoded availability across the relational adjacency configuration domain (Sec. 2.6; Appx. C.2–C.5). These updates preserve total count in closed domains and uniquely determine subsequent count configurations from admissible initial conditions.

Probabilistic behavior enters only after realization and normalization in the System Frame. Observers access conserved count structure through the Frames mapping under the Frames Principle, which produces a continuum relational encoding. When the realized count measure admits density representation (Sec. 3.4; Appx. C.6–C.8), the resulting System Frame encoding defines the encoded-availability density ρ_S(r,t). Statistical predictions are obtained only after normalization and contextual restriction of this encoded-availability structure (Sec. 3.4; Appx. C.9–C.11). Probability in MQ is therefore associated with normalized realized availability rather than with the underlying conserved count structure itself.

Deterministic count evolution and probabilistic realized outcomes therefore belong to distinct descriptive levels. Conserved count structure evolves uniquely in the Internal Frame, while multiple realized System Frame configurations may be associated with the same underlying count redistribution through the mapping-normalized encoding induced by the Frames mapping. Determinism therefore applies to Internal Frame count evolution rather than to realized observational outcomes.

The origin of this multiplicity is informational rather than dynamical. Internal Frame evolution specifies count relations and adjacency structure only. Under System Frame realization, a fixed count-level separation may admit multiple realized configurations consistent with the same conserved count structure and the consistency conditions imposed by the Frames mapping. Multiple realized System Frame configurations may therefore correspond to the same underlying count redistribution.

Continuum conservation laws and the consistency conditions imposed by the Frames mapping restrict the set of realizable System Frame configurations but do not uniquely determine a single realized configuration prior to contextual restriction (Sec. 3.5; Appx. C.11 and Appx. E.1–E.2). Normalized encoded availability consequently assigns nonzero weight across compatible realizable outcomes until physical interaction produces a stable realized measurement record through contextual restriction and subsequent normalization (Secs. 3.1–3.2 and Appx. H).

The logical sequence – conserved Internal Frame count evolution, Frames mapping, realization constraints, and contextual probabilistic realization – is summarized below. The “Outcome Realization” stage is physically completed by the first-stable-record criterion of Appx. H.8; the completed interaction is therefore generated by deterministic count evolution rather than supplied as an unspecified terminal configuration.

For a finite end-to-end realization of that sequence, Appx. AH fixes an explicit relational adjacency graph and preparation class and evaluates every admissible initial microconfiguration through count redistribution, completed contextual realization, first-stable-record closure, outcome labeling, and record-frequency counting. Section AH.4 now adds a dependency-free, machine-readable reference implementation and frozen expected output, making the construction directly reproducible from the manuscript without manual intervention. This executable example is the computational counterpart of the abstract map used in Sec. 3.2.1 and Appx. H.8-H.9.

2.12 What We Do Not Assume

MQ does not take the usual quantum formalism as primitive. In the standard textbook architecture (e.g., Sakurai, Griffiths, Dirac; axiomatizations following von Neumann), one typically begins by positing: (i) a complex Hilbert space of states, (ii) an operator/observable correspondence, (iii) a Born-rule probability map, and (iv) a measurement update rule (projection or its generalizations), together with unitary time evolution generated by a Hamiltonian.

In MQ, none of the following are assumed as primitive objects or postulates:

Hilbert space kinematics (state vectors ∣Ψ⟩, inner products, spectral postulates) as a starting axiom.

Operator algebra as ontology (observables as self-adjoint operators) at the primitive level.

Born-rule probability as a primitive axiom. System Frame outcome-weight measure from encoded availability and then identifies that measure with empirical record probability through the completed-interaction pushforward relation of Eq. (32) and Appx. H.9. The position-space identity with the squared wave-function modulus follows from the subsequent amplitude representation, while arbitrary-observable Born probabilities enter only through the later spectral-measure completion (Appx. F.18–F.19). MQ first obtains a normalized

A dynamical collapse postulate or any additional stochastic law at the primitive level; “conditioning” is implemented only after encoding.

Setting-independent Bell-local response variables λ with Bell factorizable response functions P(A∣a,λ),P(B∣b,λ); consequently, Bell/CHSH premises are not imposed at the primitive level (they are assessed only for System Frame outcome statistics).

Continuum coordinates, metric structure, or geometric vectors in the primitive description; these appear only in the System Frame encoding (e.g., r∈R³, ∇, ∇⋅).

Instead, MQ takes as primitive only the discrete Internal Frame availability count structure and its admissible redistribution/conservation on the adjacency domain, together with the deterministic Frames mapping mandated by the Frames Principle. The System Frame density is introduced only when the mapped count measure admits a density representation. Normalization then supplies outcome weights; their empirical probability meaning is fixed by the completed-interaction pushforward identity of Sec. 3.2.1 and Appx. H.9; and the MQ wave function is constructed only after this measure-level structure is established.

The canonical position-momentum algebra is likewise not assumed at the primitive level. The Internal Frame has neither a continuum position variable nor a momentum operator, so a commutator cannot be a primitive count relation. Appendix S derives the corresponding System Frame kinematics from the realized translation group: mapped count redistribution supplies translations, the MQ phase-action scale fixes the generator normalization, and covariance of the mapped coordinate under those translations yields the Weyl form and hence the canonical commutator on the regular common core.

What remains aligned with standard quantum mechanics once the encoding is established is representational rather than primitive. Appendix F now distinguishes Hilbert geometry from Hilbert completion. The MQ availability norm, coherent complex amplitude composition, and Proposition C.3 determine a complex normed linear state representation; its parallelogram identity uniquely reconstructs the sesquilinear inner product by polarization, and separable composite realization uniquely linearizes as a tensor product. Completing that already determined pre-Hilbert geometry adds norm-limit points but no new physical weighting, angle, or composition law. Realized observables may then be represented by self-adjoint operators and their projection-valued spectral measures, with generalized measurements represented by POVMs. The full operational Born rule is recovered at this stage as the standard spectral probability representation of the already normalized MQ state, not as a consequence of normalization alone. Closed evolution may likewise be represented by the unitary Schrödinger form, and Bell/CHSH predictions are evaluated at the level of realized outcome statistics. These mathematical representations preserve the empirical content of the MQ probability and redistribution structure but do not constitute additional Internal Frame postulates.

3. RESULTS

3.1 Measurement, Superposition, and the Absence of Collapse

Measurement in MQ is represented as information restriction applied to the normalized System Frame encoding. The primitive structure remains the measurement count field n_I defined on a relational adjacency domain. No continuum coordinates, metric variables, or probabilities exist at this level. Coordinate-dependent quantities – including the encoded availability density ρ_S(r,t) and amplitude Ψ_MQ(r,t) – arise only after the deterministic encoding of the Frames mapping under the Frames Principle.

To formulate measurement probabilistically, we introduce a partition of the encoded outcome domain in the System Frame. Let {A_k} denote a measurable, pairwise disjoint partition of the relevant System Frame outcome region O_out⊂R³. Each region A_k⊂O_out corresponds to a distinct macroscopic record class (e.g., pointer position or coarse-grained alternatives). Unlike an Internal Frame relational region Ω, a partition {A_k} is defined within the encoded coordinate space and is mutually exclusive and exhaustive:

These sets define outcome classes rather than geometric subdivisions.

kUsing the normalized MQ encoding, define the normalized outcome weight assigned to outcome k by

i.e., the normalized availability contained in region A_k. Prior to restriction, multiple regions A_k may carry nonzero availability. Multi-region support of ψ_MQ represents shared encoded availability across outcome partitions and corresponds to superposition in the System Frame representation only, not to multiple realized Internal Frame outcomes.

Equation (22) assigns a pre-realization normalized availability weight to each admissible record class; normalization alone does not yet establish that this weight is an empirical probability. Section 3.2.1 supplies that missing physical identification: the individual outcome label is induced by the completed contextual realization and the mutually exclusive outcome partition, and Eq. (32) proves that the pushforward distribution of the same complete-interaction count measure equals the weights of Eq. (22). Thus the same quantity is both normalized availability over record regions and the distribution of realized records. Multiple regions may carry nonzero encoded availability before realization without representing simultaneous realized records.

Conditioning on outcome k corresponds to restriction to A_k and renormalization,

where

is the indicator function of the outcome region. The denominator equals

from Eq. (22). Conditioning is defined for P_k>0. This is standard conditionalization of a normalized representation under restricted information and is mathematically equivalent to normalized projection onto a partition element in Hilbert space (Appx. F.18–F.19). The update rule is used only for subsequent predictions (Appx. H.3–H.6).

Macroscopic alternatives therefore correspond to disjoint partitions of encoded availability. Their normalized weights specify the distribution of possible records before the complete interaction is resolved. The actual admissible interaction has one realized outcome; restriction then acts on the already realized region in the predictive representation and renormalizes it without modifying the underlying Internal Frame adjacency counts.

There is thus no primitive dynamical collapse in MQ. The update is representational: it modifies the normalized System Frame encoding used for inference while leaving the underlying count evolution unchanged. Superposition and measurement update are properties of encoded availability and its conditioning rules, not of a fundamental wave ontology (Appx. D and H).

3.2 Definite Outcomes and Record Realization

Let Ω_out denote the realized outcome domain and let ρ_S(r,t) denote the encoded availability density induced through the Frames mapping. The total encoded availability measure on the realized outcome domain is

and the encoded availability measure associated with outcome region A_k⊆Ω_out is

N_S(A_k,t)N_S(Ω_out,t)A_kThe quantities N_S(Ω_out,t) and N_S(A_k,t) quantify realized encoded availability and do not themselves constitute probabilities. Their ratio in Eq. (27) is first a normalized outcome weight. Its empirical probability status is established only by the completed-interaction realization-frequency identity of Eq. (32). The normalized weight associated with outcome region A_k is therefore

Equation (27) defines unit-normalized relative availability weights for the admissible outcome regions. By itself it is a measure-normalization statement and does not prove that those weights equal observed outcome frequencies. That identification is the content of Proposition 3.1 below: the completed-interaction outcome map has a pushforward distribution exactly equal to Eq. (27). Once that equality is established, the weights may be interpreted as the probabilities with which the corresponding record classes occur across repeated preparations.

3.2.1 Outcome Realization and Frequency Compatibility

The current MQ construction contains an explicit completed-interaction realization structure. Let E denote the admissible interaction set and O the outcome-label space associated with a referenceable Frames mapping. An element x in E is the complete admissible interaction configuration for the measurement context. We denote the resulting outcome map by R:E→O. Equation (28) records its single-valued completed-interaction action; the construction of R from the contextual realization and the outcome partition, and its relation to the ensemble probabilities of Eq. (27), are derived below.

Equation (28) supplies the outcome-realization step that probability normalization alone cannot supply. The normalized availability of Eq. (27) describes the relative weights of possible record regions before the complete interaction is resolved. The interaction x is not a context-independent pre-existing eigenvalue assignment; it is the complete admissible interaction configuration on which the contextual Frames mapping acts. The construction therefore does not impose the noncontextual value assignment excluded by Kochen-Specker [38]. Nor does it equip the primitive count configuration with Bell-local factorized response functions. Deterministic Internal Frame redistribution fixes the count configuration entering the interaction, while the realized outcome exists only after the complete interaction is evaluated under the Frames mapping.

The physical content of “complete interaction” is fixed rather than left as an unspecified terminal condition. For a fixed measurement context c, let C_j denote the actual joint Internal Frame configuration of the measured system, apparatus, and physically coupled environment after update j. The count-conserving adjacency law determines the next configuration from the present one. A measurement interaction is complete at the first finite update for which the contextual Frames mapping places the realized configuration in exactly one outcome region and the corresponding macroscopic record satisfies the repeatability/stability condition of Operational Constraint 4 in the absence of a new record-altering interaction. Thus x is the finite interaction history generated from the actual initial count configuration and the imposed measurement context up to that first stable-record closure; it is not an arbitrarily chosen member of E and no additional stochastic selector is introduced. If no such finite stable-record closure occurs, the attempted interaction is not a completed measurement and does not define an element of E. Appendix H.8 gives the explicit construction in Eqs. (H.35–H.38).

Appendix AH supplies a constructive finite computation of this rule rather than another existence statement. In that example the primitive initial count configuration, the apparatus adjacency graph, the permitted count transfer, the contextual Frames mapping, the macroscopic outcome partition, and the record-stability interval are fixed before evaluation. The Internal Frame update law then produces a unique count trajectory; the mapped detector region fixes the corresponding label; and the same primitive counting measure over the allowed preparation microconfigurations gives the ensemble record frequencies. Section AH.4 supplies a dependency-free reference implementation that performs this complete chain automatically for all eight admissible micro-configurations and emits the frozen record map, outcome counts, and exact frequencies. The example is deliberately elementary: it demonstrates that the record map can be executed and independently reimplemented from specified count data without using the observed label as an input. It is not offered as a Bell-local hidden-variable model or as a replacement for the interference and entanglement constructions of Secs. 3.6-3.7.

A realized outcome must also be encoded as a stable, distinguishable macroscopic record. Let R_i denote the observer-specific macroscopic record-state space and rho_i the record-encoding map. MQ requires rho_i to be injective on the realized outcome subset O_R,

Equations (28-29) establish single-run outcome and record distinguishability. For each completed admissible interaction there is one realized label o_k; that label corresponds to one record region A_k in the System Frame output partition; and the record encoding assigns it a distinguishable stable macroscopic state. What remains is to show that this same realization structure reproduces the normalized ensemble weights of Eq. (27), rather than introducing an independent outcome-selection probability law.

Accordingly, the measure space used in Proposition 3.1 is not an unconstrained sample space. E is the image of the deterministic completion construction of Appx. H.8 over admissible actual initial joint configurations, and μ_E is the count measure inherited from conserved Internal Frame availability under that construction.

Nor is μ_E an independently selectable weighting on that sample space. By Proposition 2.1, conserved integer availability admits a unique primitive counting measure on the finite interaction domain, up to an irrelevant common scale. The deterministic completion map of Appx. H.8 carries that measure to E; outcome probabilities are then fixed by the cardinalities, with multiplicity, of the preimages R⁻¹({o_k}). A different nonuniform weighting would constitute an additional primitive field and lies outside the MQ assumptions.

Proposition 3.1 (Realization-frequency compatibility). For a fixed preparation and measurement context, let (E, Σ_E, μ_E) be the finite measure space of complete admissible interaction configurations, where μ_E is the pre-probabilistic count measure induced from conserved Internal Frame availability on E. Let M_c:E→Ω_out denote the completed contextual System Frame realization induced by an admissible Frames mapping, and let {A_k} be the measurable, pairwise-disjoint and exhaustive outcome partition of Ω_out. Define the label map κ:Ω_out→O by κ(y)=o_k exactly when y∈A_k. Then the realized-outcome map is the composition

Because M_c is single-valued for a complete referenceable interaction and the regions A_k are pairwise disjoint and exhaustive, exactly one k exists for every x∈E. Thus the particular outcome of an individual completed interaction is fixed by the contextual realization M_c(x) and its unique partition membership; no additional stochastic selector is introduced.

The realized System Frame encoded-availability measure is the standard image (pushforward) measure induced by the completed realization. For every measurable B⊆Ω_out [42],

Normalize the complete-interaction measure by P_E(D)=μ_E(D)/μ_E(E). Since Eq. (30) gives R⁻¹({o_k})=M_c⁻¹(A_k), the distribution of realized labels is the pushforward R_*P_E, and

Equation (32) is the realization-frequency compatibility relation and the physical probability-identification step. It states that the normalized availability of A_k is not merely an arbitrary unit-normalized measure: it is exactly the measure of the subset of complete admissible interactions whose realized record is o_k. The same conserved count measure that generates mapped encoded availability therefore determines the distribution of single realized records. Under the usual repeated-identically-prepared sampling conditions, standard law-of-large-numbers reasoning then makes the observed relative frequency of o_k converge to this induced probability. Equation (28) supplies single-run definiteness; Eqs. (30–32) supply the event structure and its empirical probability measure; no independent frequency or outcome-selection postulate is inserted. The discrete proof and its relation to the realization manifold are given in Appx. H.9.

Together, Proposition 2.1 and Eq. (32) close two different uniqueness questions. Proposition 2.1 fixes which pre-probabilistic measure is permitted by the MQ primitive count structure; Eq. (32) then fixes how that unique measure is transported into empirical record probabilities. The first prevents arbitrary primitive reweighting, while the second prevents an independent probabilistic assignment after realization.

Appendix T now makes the combined result explicit as a uniqueness theorem rather than leaving it distributed across these two constructions. Once primitive unit counts are exchangeable under relational relabeling, finite additivity fixes equal atomic weight; normalization fixes that weight to the reciprocal total count; and the completed contextual realization fixes every record event by its preimage. Coarse-graining consistency then forces all coarser event probabilities to be the corresponding pushforwards. Hence no second count-compatible record-probability measure exists for the same preparation and context.

A realized measurement record associated with outcome region A_k restricts admissible encoded availability to that realized region. The resulting conditioned normalized probability density is

The outcome region appearing in Eqs. (33-34) is therefore not selected by the normalization in those equations. It is the region associated with the already realized label o_k from Eq. (28). Eqs. (33-34) implement the post-outcome predictive update.

where

is the restriction function defined in Eq. (24). Using the normalized System Frame density and the outcome probability of Eq. (27), Eq. (33) may equivalently be written

Equations (33–34) give equivalent forms of the conditioned realization associated with measurement outcome A_k, respectively in terms of encoded availability and its normalized probability density. Because the conditioned normalized probability density is nonzero only on the realized outcome region A_k, immediate repetition of the same idealized measurement returns the same outcome with unit probability. Repeatability therefore follows from conditioning and normalization of encoded availability rather than from any modification of the underlying conserved count structure.

Accordingly, repeatability follows from the combination of three structures: completed contextual realization and unique partition membership fix one outcome, injective record encoding makes that outcome a distinguishable stable record, and restriction with renormalization conditions subsequent predictions on that record.

The existence of a realized measurement record does not alter the conserved count structure of the Internal Frame. Encoded availability continues to evolve according to the deterministic count-conservation relations developed in Secs. 2.6–2.11 and Appx. C, while Eqs. (33–34) describe the realized System Frame encoding associated with the established measurement record.

This distinction between conserved count structure and realized outcome encoding provides the measurement-update mechanism required by the measurement-consistency constraint established in Sec. 2.5. Prior to conditioning, encoded availability may possess support over multiple admissible outcome regions A_k. Following establishment of a realized measurement record, the System Frame encoding is conditioned onto the realized outcome region while remaining fully normalized. The resulting update is representational rather than dynamical and leaves the underlying conserved count structure unchanged.

The Hilbert space representation of the conditioned realization defined by Eqs. (33–34), developed formally in Appx. F.18–F.19, Appx. H.3–H.6, and Appx. H, is obtained through projection onto the realized outcome subspace,

where Π_k denotes the projection operator associated with outcome region A_k. The relations

express idempotence and self-adjointness of the projection operator. Within MQ, Eq. (35) is the Hilbert space representation of the conditioning operation defined by Eqs. (33–34). It is not interpreted as a primitive dynamical process.

Equations (28-35) now close the complete measurement chain: Eq. (27) supplies the normalized outcome weights; Eq. (28) records the single-valued completed-interaction outcome map; Eq. (29) supplies stable record distinguishability; Eqs. (30-32) derive the outcome map from contextual realization and prove that its pushforward distribution equals those same normalized weights; and Eqs. (33-35) supply post-outcome conditioning and repeatability. No additional stochastic collapse or independent outcome-selection probability law is required.

3.2.2 Device-Level Realization of a Physical Measurement Channel

The outcome regions A_k and record map of Secs. 3.1-3.2 specify the kinematic event structure, but a laboratory apparatus adds a calibrated response between an ideal realized outcome and the macroscopic electrical or digital record actually registered. This response belongs to the System Frame because detector geometry, gain, thresholds, timing windows, and noise are realized experimental quantities. It does not alter the unique Internal Frame counting measure of Proposition 2.1 or the completed-interaction probability identity of Eq. (32). Appendix V makes this layer explicit for a representative on/off photodetector.

The general device bridge is a calibrated conditional response matrix. If o_k is the already defined ideal outcome label and r_j is the macroscopic detector record, the matrix element C_jk is the conditional probability that the physical device reports r_j when the realized input belongs to outcome class o_k. The observed record distribution is therefore the classical post-processing of the MQ outcome distribution by C. After Hilbert-space completion the same channel is represented by an effective POVM obtained by applying that response matrix to the ideal effects. This is the operational instrument structure used in standard measurement theory [137], while experimental detector tomography can determine the POVM of an actual apparatus without relying on a microscopic detector model [138].

Appendix V then specializes the construction to a gated avalanche photodiode. A finite detection efficiency η_d and a per-gate dark-click probability p_dc give explicit click and no-click probabilities for an n-photon input; the corresponding noisy POVM, the observed MQ click probability, and the stable threshold-record criterion are written in Eqs. (V.4-V.8). Finite efficiency and noise therefore change the apparatus response matrix, not the underlying MQ record-probability measure. This separation is standard in realistic photocounting models, where finite efficiency and noise counts are treated explicitly [139], and it is consistent with Glauber photodetection theory [140].

The construction closes the gap between an abstract partition and one concrete laboratory detector without claiming that every apparatus has the same response function. For a different device, C must be supplied by a calibrated physical model or detector tomography and then composed with the same MQ event measure. The first-stable-record criterion of Appx. H.8 consequently acquires an operational apparatus meaning: the completion update is the first update at which the calibrated raw detector output lies in one record region and remains there for the specified stability interval. Device imperfections can move or blur that closure, but they do not supply an additional outcome-selection law.

3.3 Foundational Consistency Criteria

Sections 3.1–3.2 establish measurement conditioning, definite record realization, and repeatability. The purpose of the present subsection is narrower: to state the foundational Bell–CHSH and operational no-signaling criteria that the composite-system construction must satisfy. The entanglement mechanism itself is derived in Sec. 3.7 and Appx. G.

At the System Frame level, the required empirical structure consists of normalized contextual outcome probabilities, operationally no-signaling local marginals, and joint correlations capable of violating Bell–CHSH bounds. At the Internal Frame level, conserved count redistribution must remain deterministic and adjacency-local. These conditions apply at different descriptive levels and provide the criteria against which the later joint-availability construction is evaluated.

MQ does not introduce a setting-independent Bell-local response variable λ carrying response functions P(A∣a,λ), P(B∣b,λ), nor does it assume Bell factorization of the form P(A,B∣a,b,λ)=P(A∣a,λ)P(B∣b,λ).

Accordingly, Bell, CHSH, and Kochen–Specker constraints [3,38,39] are applied to realized System Frame outcome statistics, not to the primitive count labels themselves. The question for MQ is whether the System Frame statistics generated by the Frames mapping satisfy the experimentally required correlation and no-signaling conditions without converting Internal Frame count indices into setting-independent outcome variables.

For binary outcomes A,B∈{±1}, define the correlator

The Clauser–Horne–Shimony–Holt (CHSH) combination is

Under Bell factorizability together with measurement independence, the standard Bell–CHSH derivation implies S≤2, whereas quantum-consistent correlations satisfy the Tsirelson bound

.

Equations (36–37) are used here as evaluation benchmarks. Their role is to define the correlator and CHSH combination against which the explicit composite-system probabilities of Sec. 3.7 are tested; the present subsection does not independently derive the entanglement probability structure.

Relative to the Jarrett–Shimony decomposition [40,41], parameter independence (PI) and outcome independence (OI) are conditional statements defined at fixed complete state λ. MQ directly establishes the weaker operational requirement needed here: local observed marginals are independent of the distant setting. The manuscript therefore does not infer λ-conditioned PI merely from no-signaling. If the complete normalized System Frame state is used as the conditioning state, the resulting quantum probability representation has the familiar state-conditioned pattern of remote-setting-independent marginals together with outcome dependence of the joint statistics. No claim is made here about PI or OI for an additional primitive λ-completion that MQ does not introduce.

Thus, Sec. 3.3 functions as a diagnostic statement of the foundational constraints rather than a second derivation of entanglement. The physical origin of non-factorizable joint statistics, the exceptional separability condition, and the explicit no-signaling marginal relations are developed once, in Sec. 3.7.

3.4 Continuum Mapping and Encoded Availability

With the discrete count structure in the Internal Frame and the Frames mapping to the System Frame established (Sec. 2), we introduce a continuum encoding suitable for comparison with standard quantum formalism. The primitive MQ variable is the measurement count field n_I, defined on a relational adjacency domain and therefore not a function of continuum coordinates. Under the deterministic encoding of the Frames mapping M_I and the Frames Principle, this count structure induces a System Frame count measure N_S(Ω,t) on regions Ω⊂R³. When N_S(⋅,t) is representable by a density with respect to d³r, its Radon–Nikodym derivative [42] defines the encoded availability density ρ_S(r,t).

3.4.1 Construction of the MQ Wave-Function Representation

The MQ wave function is then defined by

where ϕ(r,t) is a real phase field on the System Frame encoding. For a single subsystem with configuration space R³, the total mapped count is obtained from the System Frame encoded availability measure introduced in Eq. (39). The corresponding total encoded availability is

Where N₁(t)≡N_S(t) denotes the normalization measure associated with the single-subsystem encoding. The notation N₁ is introduced to distinguish this normalization from the higher-dimensional normalization measures that arise for composite systems. No new physical quantity is introduced; N₁ is simply the single-subsystem specialization of the total System Frame encoded availability.

With 0<N₁(t)<∞, the normalization appearing in Eq. (38) is well defined. For composite systems, distinct normalization measures, such as N₂(t), are defined on the corresponding higher-dimensional configuration spaces (Appx. F.14 and Appx. G.8–G.9).

The phase field in Eq. (38) is not an additional primitive variable appended to the count structure. It exists only in the realized System Frame and is reconstructed as a scalar-potential representation of the mapped redistribution flow. The primitive directional information is the oriented Internal Frame adjacency count-flux; after the Frames mapping and normalization, that conserved redistribution is represented by the System Frame probability current and velocity field (Appx. C.11). Proposition C.3 then establishes why the minimal reversible norm-preserving linear representation of this derived continuous phase coordinate is U(1), so the complex amplitude is not selected merely for mathematical convenience.

Accordingly, the phase depends on count evolution rather than supplying an independent source of it. Combining the flux factorization and phase-flow relations of Eqs. (C.35) and (C.36), the phase gradient is fixed by the mapped current wherever the normalized availability density is nonzero. On a connected, simply connected region this reconstructs the phase locally up to a spatially uniform additive constant, which changes neither probability density, current, momentum expectation, nor interference observables. Only phase differences and phase gradients enter the measurable System Frame structure.

For supports containing nodes or non-contractible loops, the global phase sectors are no longer left as an unspecified representability condition. Section 2.10.1 and Appx. Z classify them from the surviving loop classes of the mapped relational domain. The local phase gradient remains fixed by mapped count flow as above, while homotopically distinct U(1) phase maps are indexed by the first integral cohomology class of the realized support, equivalently by their integer winding on an independent cycle basis. Contractible cycles retain zero mapping holonomy; only non-contractible cycles can carry topological phase or defect data. Thus topology changes the global representative class without introducing a new Internal Frame field [72–74,153].

3.4.2 Normalized Availability, Empirical Probability, and the Born-Form Identity

The normalization of the MQ wave-function representation is

Equation (40) is a normalization statement for the wave-function representation, not a derivation of the full Born rule and not, by itself, the physical identification of probability. Positivity and finite additivity of the mapped count measure together with normalization on the realized System Frame domain determine a unit-normalized position-space availability measure (Appx. C.6–C.7). The additional event structure supplied by Sec. 3.2.1 is what identifies that measure empirically: Eq. (32) proves that the normalized measure of each record region equals the pushforward measure of all complete interactions that realize that record. Only after this realization-frequency equality is established is the normalized availability measure used as the position-space probability measure. The equality of its density with the squared modulus of the MQ wave function is then representational, following from the amplitude definition in Eq. (38). Neither normalization nor the modulus-squared identity alone establishes the probability rule for an arbitrary observable.

The logical bridge is therefore measure → event → empirical probability, not normalization → probability by fiat. The measurable outcome partition defines the events, the completed contextual realization maps every complete interaction to exactly one event, and Eq. (32) identifies the normalized availability of that event with the pushforward distribution of realized labels. For repeated identically prepared trials, ordinary statistical sampling then connects that probability distribution to long-run observed frequencies. The probability interpretation is thus carried by the realization map and its event preimages, not supplied by wave-function notation.

The word “measure” in this chain is consequently not shorthand for an arbitrary normalized prior. On each finite MQ domain its primitive form is the unique normalized counting measure established by Proposition 2.1. Preparation dependence enters through the physically different count configuration and through the completed-context event preimages, not through a freely chosen weighting functional.

This statement is strengthened in Appx. T. The admissible probability law on a finite completed-interaction domain is unique before any Hilbert-space completion: if U_E is the finite multiset of primitive interaction-count units and R is the completed contextual record map, then every record event B has probability |R⁻¹(B)|/|U_E|. The square-modulus form introduced by Eq. (38) is therefore a representation of an already-fixed event measure, not a separate choice among powers of an amplitude.

The phase ϕ(r,t) parameterizes the encoded count redistribution structure and associated conserved current representation under count conserving evolution (Appx. C.11–C.21 and Appx. E.3–E.6). Interference and superposition arise from phase relations in the System Frame representation, while the underlying Internal Frame count field remains single-valued and coordinate-free. The wave function Ψ_MQ is therefore a compact continuum encoding of discrete availability structure under the Frames Principle, not a primitive ontological field. More precisely, its gradient represents the mapped conserved redistribution current; the phase is therefore downstream of the Internal Frame count-flux and is not an independently evolving ontic field (Appx. C.8.1 and C.11).

3.4.3 Full Born Rule as Spectral-Measure Completion

For a general observable, the operational Born rule requires more structure than position-space normalization. Once the normalized System Frame encoding has generated the pre-Hilbert geometry reconstructed in Appx. F.8-F.14.1 and that space is completed in its derived norm, a realized observable represented by a self-adjoint operator admits the spectral theorem and its projection-valued measure over the observable spectrum. The probability assigned to a measurable spectral set is then represented by the state functional of Eq. (F.66). Generalized measurements are represented by the POVM form of Eq. (F.69). Accordingly, MQ recovers the arbitrary-observable Born rule only at the Hilbert-space completion stage; it does not claim that the spectral theorem, operator algebra, or POVM structure follows from count normalization alone.

This separation is consistent with the standard foundations literature. Gleason's theorem shows that, for Hilbert spaces within its stated dimensional and additivity assumptions, a noncontextual probability measure on projections has density-operator form [70]. Busch's generalized-observable formulation establishes the analogous result for normalized probability assignments on quantum effects [71]. These theorems constrain the form of the probability representation once Hilbert-space measurement structure is assumed; they do not supply the earlier MQ derivation of normalized availability. The MQ claim is therefore deliberately layered: normalized probability is derived from mapped availability, the modulus-squared relation follows from amplitude representation, and the full projective/POVM Born rule is recovered as the unique standard functional-analytic representation under the corresponding measurement assumptions.

Appendix T closes the distinct pre-Hilbert question for the finite MQ record algebra. Its theorem does not use projections, effects, density operators, or a Hilbert-space dimension assumption. It uses the MQ-specific count ontology, permutation invariance of primitive unit counts, finite additivity, normalization, and functorial consistency of contextual realization under event coarse-graining. Gleason and Busch remain necessary at the later stage because they establish the unique operator-theoretic representation of probabilities once projective or POVM measurement structure is introduced [70,71].

3.4.4 Canonical Translation Kinematics from the Frames Mapping

The operator construction of Appx. F already identifies realized position with multiplication by the mapped System Frame coordinate and momentum with the differential generator associated with the phase-gradient field. Appendix S now closes the missing kinematic bridge between these operators. Adjacency-local Internal Frame redistribution is realized as continuous System Frame displacement; on the completed MQ state geometry, the resulting norm-preserving strongly continuous displacement family is unitary. Stone’s theorem therefore supplies its self-adjoint generator, while the independently derived MQ phase-action scale fixes the physical normalization of that generator to p̂=-iℏ∇ rather than an arbitrary multiple of ∇.

The key relation is translation covariance of the mapped coordinate. A realized displacement a changes x̂_j by a_jI. Combining this covariance with the translation generator gives the exponentiated Weyl relation T(a)V(b)=exp[-i(a·b)/ℏ]V(b)T(a), where V(b)=exp[i b·x̂/ℏ]. Differentiation on the shared invariant core gives [x̂_i,p̂_j]=iℏδ_ij. This order of derivation is important: MQ does not quantize a pre-existing classical phase space by replacing Poisson brackets with commutators. It derives a regular System Frame translation representation from count redistribution and then obtains the canonical algebra as the compatibility condition between that translation and the mapped coordinate. The detailed domain statement, finite-degree-of-freedom uniqueness, and scope conditions are given in Appx. S [30,134,135].

3.5 Conditional Selection of Schrödinger Dynamics from Discrete Count Evolution

The continuum dynamical law is obtained by encoding discrete count redistribution from the Internal Frame into the System Frame through the deterministic encoding of the Frames mapping under the Frames Principle. Primitive evolution acts on the measurement count field n_I (or region count measure N_I) by local, count-conserving adjacency updates (defined on the adjacency graph and not on continuum coordinates). After encoding, this redistribution induces the differential generator governing the MQ wave function introduced in Sec. 3.4.

Let C_I(ξ,A) denote the discrete Internal Frame count assignment indexed by relational adjacency label ξ (adjacency index) and update index A. Under the Frames mapping, the induced System Frame encoding yields the complex continuum amplitude Ψ_MQ(r,t) together with the encoded availability density ρ_S(r,t). Discrete first-difference operators on the adjacency graph map to continuum gradient terms, and discrete second-difference operators map to curvature terms whose continuum limit is the Laplacian (Appx. E). For additive, locally conservative difference schemes on partitioned domains, the induced continuum operators are self-adjoint with respect to the System Frame inner product, ensuring unitary (norm-preserving) evolution (Appx. F.15–F.17) [30] on the encoded Hilbert space domain.

Derivations of the Schrödinger equation from more primitive structure require an explicit separation between quantities derived by MQ and conditions that delimit the target non-relativistic realization. Standard Galilean-invariance derivations obtain the Schrödinger form only after fixing symmetry and representation data [68,69]. In MQ, that symmetry datum is now downstream: Proposition 2.7/Appx. Y derives the Galilei group as the non-relativistic contraction of the count-selected local Lorentz/Poincaré realization. Closed-domain count conservation and adjacency-local redistribution remain primitive structural requirements; first-order update structure and normalization preservation descend directly from them. Regular-lattice representability, sufficient continuum smoothness, the isotropic nearest-neighbor stencil, a self-adjoint realized domain, admissible boundary behavior, and a real multiplicative potential remain continuum-representation conditions. Appendix O.4 then derives the inertial dispersion from the realized phase-action/Hamilton-Jacobi structure together with the now-derived Galilean boost action rather than assuming either the quadratic law or boost covariance independently.

Within this restricted class, the operator form and its dimensional coefficient can be separated. Adjacency-local redistribution supplies the local first-difference structure; composing opposite first differences produces centered second differences, and isotropic summation gives the discrete Laplacian developed in Appx. E.1–E.2. The controlled Frames mapping limit then gives the System Frame Laplacian. The phase-action scale is not introduced at this continuum step: Appx. K and the MQ relations collected in Appx. O give p_si = l_f m_f/(2t_f) = ħ/(2l_f), hence ħ = 2l_f p_si = l_f²m_f/t_f. The subsystem mass is likewise represented as a realized mass count m = n_m m_f. Appendix Y first fixes the Galilean boost law as the c → ∞ contraction of the local Lorentz/Poincaré symmetry selected by the same Frames mapping invariants. Appendix O.4 then combines that derived boost action with the MQ phase-action relation and realized Hamilton-Jacobi characteristic relation to fix the free inertial Hessian and integrate uniquely to T = p²/(2m). Substitution of the already-derived momentum generator yields the Schrödinger kinetic operator without inserting either the factor 1/2, the quadratic dispersion, or an independent Galilean symmetry axiom.

Here ∇²is the continuum limit of the discrete adjacency Laplacian acting on encoded availability, and V(r,t) represents System Frame constraint structure induced by external, boundary, or interaction restrictions on availability (Appx. E and Appx. F.4–F.5). Spatial derivatives act only on encoded continuum fields, never on primitive Internal Frame variables.

The reduced Planck constant in Eq. (41) therefore has a prior MQ derivation rather than the status of a coefficient chosen to reproduce Schrödinger dynamics. In the dimensionally explicit notation of Appx. K, the momentum realization p_si obeys p_si = ħ/(2l_f), so ħ = 2l_f p_si; using p_si = l_f m_f/(2t_f) gives ħ = l_f^2 m_f/t_f. The mass m is the System Frame realization of the subsystem mass count, m = n_m m_f. Its numerical value is system-specific input, not a new universal MQ constant or a fit parameter. Appendix O traces these identities to the MQKB and separates them from the non-relativistic kinetic-sector relation that supplies the inertial response 1/(2m).

The MQKB further resolves this phase-action scale by count separation. In the canonical MQ relation, the realized reduced Planck scale is hbar(n_L) = theta_si l_f/[Q_L(n_L)n_L], with Q_L(n_L) = sqrt(1+n_L^2) - n_L. At the electromagnetic demarcation n_L = 276/theta_si this relation reproduces the ordinary SI-scale value, whereas its upper-count limit is 2 theta_si l_f. Equation (41) therefore uses the realization-appropriate phase-action scale rather than postulating one count-independent value across all MQ regimes. Ordinary electromagnetic quantum tests remain in the demarcation realization; the distinct upper-count branch becomes experimentally discriminating for gravitationally bound quantum states whose level spacing is set by a macroscopic gravitational realization. Appendix L.2 derives the resulting parameter-free transition-frequency ratio and states the scope conditions required for a clean test.

Because the generator in Eq. (41) is self-adjoint on its dense domain, normalization of encoded availability is preserved:

For time-independent V, the associated Hamiltonian is the self-adjoint generator of System Frame time translations of the encoded state (Appx. F.4-F.6 and Appx. F.15-F.17). Self-adjointness is not inferred from count ontology alone; it is the functional-analytic realization condition corresponding to reversible norm-preserving closed evolution on the chosen continuum domain. What MQ supplies independently is the count-conservation basis for norm preservation and the discrete local operator structure from which the Laplacian limit is constructed. The detailed provenance split is summarized in Appx. O.

Stone's theorem then supplies uniqueness of the self-adjoint generator for the strongly continuous unitary group generated by this realized evolution [30]. It does not determine the kinetic coefficient or select the Schrödinger Hamiltonian from arbitrary laws. Those tasks are separated here: MQ fixes the phase-action scale ħ from its fundamental measures, the subsystem fixes m through its realized mass count, the non-relativistic sector fixes T = p^2/(2m), and the discrete-to-continuum construction fixes the Laplacian operator. Their combination yields Eq. (41) without treating ħ^2/(2m) as an adjustable coefficient (Appx. O).

Accordingly, the manuscript claims a conditional but coefficient-resolved recovery whose kinetic dispersion is derived within the stated non-relativistic realization. The count ontology alone still does not uniquely imply every element of non-relativistic quantum dynamics: regular continuum representability, self-adjoint domain conditions, admissible boundary behavior, and the conservative real-potential representation delimit the realization being recovered. The inertial symmetry is no longer on that assumption list: Proposition 2.7/Appx. Y derives the local Lorentz group from the mapped metric and causal invariants and obtains Galilean boosts by its non-relativistic contraction. Within those conditions, the dispersion T = p²/(2m), differential operator, norm-preserving structure, phase-action scale, subsystem mass realization, and therefore the coefficient ħ²/(2m) are fixed as stated in Appx. O. The relativistic continuation is treated separately in Sec. 3.9.1 and Appx. P.

3.6 Interference and the Two-Slit Experiment

Interference provides a central test of whether the MQ framework reproduces standard quantum behavior while maintaining single realized measurement records. In conventional quantum theory, interference arises from linear superposition of wave-function amplitudes defined over configuration space. Within MQ, the same observable interference structure emerges from coherent encoded availability realization under the Frames mapping.

In the MQ framework, conserved count redistribution is defined in the Internal Frame through adjacency-local evolution of encoded availability. Observable interference arises only after realization in the System Frame, where admissible realization configurations combine through the normalized realization structure developed in Secs. 3.4–3.5 and Appx. C–E. Interference therefore appears at the realization level of encoded availability rather than from simultaneous realized macroscopic outcomes.

Let

denote the System Frame encoded availability amplitudes associated with two admissible realization configurations. Here ρ_P,j(r,t) denotes the normalized encoded availability density associated with realization configuration j, and ϕ_j(r,t) is the corresponding System Frame redistribution phase field induced through coherent encoded availability redistribution under the Frames mapping.

The total encoded availability amplitude is then

with corresponding normalized encoded availability density

Substituting Eq. (44) into Eq. (45) yields

Equation (46) shows that the interference contribution arises from relative redistribution phase within the normalized System Frame encoded availability structure. Observable interference therefore emerges from encoded availability realization algebra under the Frames mapping.

Consistent with the measurement consistency structure established in Sec. 2.5 and Appx. B, interference in MQ does not require simultaneous realized macroscopic records. Before completion of the interaction, encoded availability may remain distributed across admissible realization structure. For the completed interaction, the contextual realization and outcome partition induce exactly one realized label by Eqs. (28) and (30), while Eq. (32) fixes the ensemble distribution of such labels. Post-outcome restriction and renormalization then update only the predictive System Frame encoding.

When which-path measurement restriction is introduced, contextual restriction modifies the admissible encoded availability structure and suppresses the interference contribution in Eq. (46). MQ therefore reproduces the standard transition between coherent interference and which-path measurement statistics through contextual restriction and normalization of encoded availability rather than through physical wave-function collapse or ontological branching (Appx. D.1–D.3 and Appx. H.1–H.6).

Detailed derivations, numerical realizations, and explicit slit configurations are developed in Appx. D.

3.7 Entanglement and Bell Correlations under the Frames Principle

Entanglement in MQ is the composite-system realization used to test the foundational criteria stated in Sec. 3.3. The primitive object is a joint count configuration in the Internal Frame, while correlation statistics arise only from its realized System Frame encoding under the Frames mapping and the Frames Principle. Spacetime separation and analyzer orientation therefore enter only at the realized System Frame level.

An entangled preparation corresponds to a single joint measurement count configuration in the Internal Frame, defined on the composite adjacency domain. This relational domain carries no intrinsic metric or geometric structure. The joint availability structure is generally non-factorable and is not expressible as a Cartesian product of subsystem count measures. It constrains which joint outcomes are accessible under measurement partitions without assigning realized outcomes in advance and without introducing stochastic hidden variable response functions.

For correlation tests, outcomes are represented using binary ±1 coding defined by a two-channel partition of System Frame availability for each analyzer setting. This is the standard correlator normalization convention (Appx. G.6, Appx. G.11–G.12) and does not assume pre-existing ontic ±1 values.

For subsystems A and B, the primitive joint count object is a composite Internal Frame measure

, not a coordinate function. Under the Frames mapping, when density-representable, it induces a joint encoded availability density ρ_S(r_A,r_B,t) on composite System Frame configuration space

Analyzer settings and spatial orientations enter only through this encoding. Settings define rotated availability partitions used to construct binary outcome regions in the encoded coordinate space.

The total mapped joint count is

The corresponding joint encoded MQ state takes the continuum form

The normalized joint probability density is therefore

Correlations arise from non-factorability of this joint encoded availability structure:

A measurement at subsystem Aimplements restriction to a selected partition of the joint encoded domain and conditions the normalized distribution accordingly (Secs. 3.1–3.2 and Appx. H). This produces global renormalization in the System Frame encoding, while the underlying Internal Frame count configuration remains unchanged. Correlations are therefore computed from conditioned access to shared availability geometry under the Frames mapping.

Factorization can arise only in special separable or symmetry-restricted cases. These occur when both the encoded joint availability density separates, ρ_AB(r_A,r_B)=ρ_A(r_A)ρ_B(r_B), and the measurement-induced restriction partitions factorize, A_ab=A_a×A_b, yielding P(A,B∣a,b)=P(A∣a)P(B∣b). Such separability conditions are exceptional. Outside these restricted cases, MQ joint probabilities remain non-factorizable.

The ambient composite architecture is not chosen independently to accommodate these correlations. Proposition F.2 derives the bipartite tensor product from separable count composition, and Proposition F.3 extends that result to finite multipartite systems. Every normalized finite-dimensional coefficient array then lies in the completion of the count-generated amplitude space, while bipartite Schmidt structure, purification of mixed reduced states, and monogamy of pure entanglement follow from that same tensor-product geometry [113-116]. Thus non-factorizable MQ availability does not merely resemble an entangled state after the fact; it is represented in the unique finite composite space generated by the already-derived subsystem composition law.

To place MQ in relation to Bell analysis, we now state the standard Bell-local factorizability assumption for hidden variable models. Bell–CHSH constraints apply to models satisfying factorizability at fixed hidden state λ,

together with measurement independence and standard probability axioms (Appx. G–H). Here λ denotes a Bell hidden variable specifying outcome-relevant properties which may be deterministic or stochastic. Equation (51) defines the Bell-local model class and is not assumed by MQ.

MQ does, however, satisfy the separate measurement-independence premise for the preparation ensemble used here. Appendix G.1 expresses that condition directly on the complete pre-setting Internal Frame count configuration Cᵢ and derives P(Cᵢ|a,b)=P(Cᵢ) from the preparation-setting separation of the MQ Bell construction (Eqs. (G.3–G.5)). The later setting-dependent partitions therefore do not constitute a measurement-dependence loophole.

The benchmark can now be evaluated on the MQ construction. For analyzer settings (a,b), the realized joint probability is assigned directly from the normalized encoded-availability counts associated with the setting-dependent System Frame partition:

where N_ab(A,B) is the encoded joint availability count for outcome pair (A,B) under the specified settings, and

is the corresponding total availability for those settings. The probabilities therefore arise from normalization of encoded availability rather than from an assumed factorized response structure.

The preceding separability condition shows precisely when product structure can be recovered. In the generic case, the measurement partitions remain non-Cartesian in joint availability space and the joint distribution is non-factorizable. The local marginals nevertheless satisfy operational no-signaling:

Equation (53) establishes operational no-signaling: each observed local marginal depends only on its local setting. It does not, by itself, establish λ-conditioned parameter independence, because no Bell hidden state λ has been introduced in Eqs. (52–53). Likewise, generic non-factorizability of the unconditioned joint distribution is not by itself a proof of Jarrett outcome-independence failure. At the complete System Frame state-conditioned level, however, the joint quantum probability representation exhibits outcome dependence while retaining remote-setting-independent marginals, matching the standard quantum pattern. More generally, because MQ reproduces Bell–CHSH violations while retaining measurement independence, no Bell-local factorized hidden-variable completion can reproduce the same statistics.

The division of labor is therefore explicit. Section 3.3 states the Bell–CHSH and no-signaling tests; Sec. 3.7 derives the composite non-factorizable availability structure and applies those tests. Appendix G.11 now closes the remaining analyzer-response step from the same MQ hierarchy. Count conservation fixes normalization, the derived System Frame amplitude supplies a two-channel linear representation, and continuous rotation of an ideal lossless analyzer acts as a norm-preserving relabeling of that realized channel plane. Those conditions force the SO(2) rotation matrix; the already-derived MQ probability identification then gives the cos²/sin² channel weights and hence the Bell angular correlation. Equations (G.48–G.54) therefore derive the angular law without inserting Malus' law, a Hilbert-space inner product, or a Born-rule postulate as an independent input. The CHSH evaluation remains the standard algebraic benchmark (Appx. G.12).

3.8 Classical Limit and Stable Record Formation

The classical limit in MQ is not a transition to a second ontology. It is a realized System Frame regime in which normalized encoded availability is coarse-grained over macroscopic record classes, while phase-sensitive structure below the resolution of those records becomes operationally unresolved. The underlying Internal Frame count structure continues to evolve through the same deterministic, adjacency-local conservation relations established in Secs. 2.6–2.11. Classical and quantum behavior therefore remain two realization regimes of the same conserved count substrate, consistent with the MQKB classical-correspondence construction.

The coarse-graining relevant to a macroscopic record is already supplied by the mutually exclusive and exhaustive System Frame outcome partition {A_k} introduced in Secs. 3.1–3.2. Aggregating normalized encoded availability over those record regions preserves total availability because the regions partition the realized output domain. Using Eq. (27), the coarse-grained normalization condition is

Equation (54) makes the correspondence operation explicit: coarse-graining changes the resolution of the System Frame description but does not alter normalized total encoded availability or the conserved Internal Frame count structure. The classical regime is therefore obtained by aggregation of realized alternatives, not by replacing the underlying count kinematics with a primitive continuum state.

Macroscopic definiteness additionally requires that distinct realized outcome labels remain distinguishable as persistent records. Let ρ_i:O→R_i denote the record-encoding mapping from the realized outcome label space to the observer-specific macroscopic record-state space defined in Appx. A.3. On the realized outcome subset, stable distinguishability requires the injective relation

Equation (55) identifies effective macroscopic exclusivity with record distinguishability: different realized alternatives correspond to different observer-accessible record states. This is a System Frame statement about realized records and must not be projected backward as a primitive partition or classical geometry in the Internal Frame.

Persistence requires more than distinguishability. Under Operational Constraint 4, an established record must survive immediate rereading in the absence of intervening record-altering dynamics. For a realized outcome o_k and rereading interval Δt, the ideal stability condition is

Equation (56) is the operational stability condition already implicit in Eqs. (33–35): after contextual restriction to A_k, the conditioned normalized density has support only on that realized record region, so immediate repetition returns the same outcome with unit probability. A persistent classical record is therefore a restricted, normalized System Frame realization that remains invariant under admissible evolution over the interval in which no record-changing interaction occurs.

Environmental decoherence is compatible with this construction but is not identified with realized-outcome selection. Within MQ, decoherence operates on the realized System Frame quantum encoding, where environmental entanglement can suppress phase coherence between macroscopic alternatives and stabilize robust record classes [5,7]. Outcome realization is instead fixed by the completed contextual realization and unique outcome-partition membership of Eqs. (28) and (30), with Eq. (32) giving the corresponding ensemble distribution; Eqs. (33-35) then condition the predictive encoding on the already realized record. The relation between these mechanisms is developed explicitly in Sec. 4.2.

The effective exclusivity of macroscopic alternatives follows from these two structures together. Before record realization, normalized encoded availability may have support across several macroscopic regions A_k. After realization, conditioning restricts predictive support to the realized region, while Eq. (55) keeps distinct realized outcomes mapped to distinct records and Eq. (56) makes the selected record repeatable. Fine interference structure may remain part of the underlying System Frame encoding before contextual restriction, but once it is below macroscopic resolution it no longer changes the stable record classification.

For position-like macroscopic observations, the correspondence limit is consequently approached when coarse-grained normalized encoded availability remains concentrated within a sequence of stable realized record regions over the relevant observational timescale. The resulting record sequence may be represented by classical trajectories at the empirical level, but neither those trajectories nor their continuum geometry are primitive in the Internal Frame. Classical behavior is therefore an operationally stable, coarse-grained System Frame realization of the same conserved count structure that produces quantum interference and entanglement at finer realized resolution. No additional stochastic collapse law is required for this correspondence.

Proposition 3.2 (Quantitative classical-limit scaling). The qualitative correspondence criteria above admit a quantitative sufficient-condition theorem without assigning classicality to the Internal Frame. Appendix U proves three System Frame scaling results for the same normalized MQ state. First, if two macroscopic record alternatives become correlated with m conditionally independent environment fragments whose branch-overlap factors satisfy |γₐ|≤exp(-ηₐ), then the magnitude of their off-diagonal coherence, and hence every interference contribution between the two alternatives, is bounded by exp[-Σ_a ηₐ] and by exp(-mηmin) when ηₐ≥ηmin>0. Second, coarse record resolution suppresses unresolved phase structure independently of environmental tracing: for a locally linear phase difference sampled by a top-hat record bin of width Δx, the normalized interference factor is |sinc(ΔkΔx/2)|. Third, if a macroscopic record variable is the average of Nc mapped count-resolved contributions with uniformly bounded variance and only summably weak correlations, its realization variance scales as O(1/Nc), so its standard deviation scales as O(Nc^{-1/2}). These statements are derived with respect to the already-established MQ record-probability measure; they introduce no primitive stochastic count law.

The three mechanisms are cumulative rather than interchangeable. Environmental coupling suppresses branch coherence, record coarse-graining averages phase structure that lies below the chosen resolution, and large count aggregation concentrates macroscopic record variables. A sufficient realized classical regime is therefore obtained when the relevant decoherence exponent is large, the record bin resolves many phase oscillations, and the effective count number Nc is large compared with the residual correlation scale. The result strengthens Eqs. (54-56): those equations establish normalization, distinguishability, and persistence, while Proposition 3.2/Appx. U supplies an explicit approach-to-classicality rate. The theorem is conditional at the correct level: count conservation alone does not determine an environmental scattering rate, guarantee fragment independence, or force weak correlations among every possible macroscopic observable. Those are realized System Frame dynamical conditions and are experimentally testable [5,7,97,98,136].

3.9 Relativistic and Field-Theoretic Completion

3.9.1 Lorentz-Covariant System Frame Free-Field Completion

The MQKB contains a distinct general-relativistic construction in which the Frames mapping induces a System Frame metric, its Levi-Civita connection and curvature tensors, a covariantly conserved stress-energy tensor, the Einstein tensor, and the geodesic law. The quantum manuscript can therefore test relativistic closure without assigning spacetime geometry to the Internal Frame. Appendix P imports only those already-derived geometric structures and combines them with the phase-action scale ħ and subsystem mass realization m = n_m m_f established above.

The relativistic completion is formulated on the locally Lorentzian, time-oriented System Frame realization selected by Proposition 2.6 and Appx. X. The signature is not inferred from integer count multiplicity alone: it follows, within the regular four-dimensional quadratic realization class, from the conjunction of the single Internal Frame update ordering, adjacency-local finite propagation, the mapped fundamental length/time scale, and the positive-definite three-dimensional spatial metric of Appx. M–N. For a complex scalar MQ amplitude, locality, linearity, second-order covariance, and minimal free-field coupling give the covariant Klein–Gordon equation on the emergent metric. Its WKB phase satisfies the relativistic Hamilton–Jacobi mass shell g_sys^{μν}∂_μS∂_νS + m²c² = 0, and the flat-space limit gives E² = p²c² + m²c⁴. Removing the rest-energy phase and taking |p|/(mc) ≪ 1 reproduces the free Schrödinger generator and the same coefficient ħ²/(2m) obtained independently in Sec. 3.5 and Appx. O.

For the spinor realization, Appx. P introduces no preferred frame. Section 3.9.3 and Appx. R fix the local j=1/2 Clifford representation, while Sec. 2.10.1 and Appx. Z now classify the global extension. On an orientable realized System Frame manifold a spin lift exists exactly when the second Stiefel-Whitney obstruction vanishes; when it exists, inequivalent spin structures form an affine family over H^1(Ω;Z₂) [129,152]. The finite contractible charts used for the local quantum derivations have trivial relevant cohomology and therefore one local spin class. On a non-contractible realization the available spin boundary conditions are consequently discrete topological data fixed only after the global realized topology is specified. This separates the already-closed local spin value from the distinct global spin-structure question.

This construction closes the absence of a Lorentz-covariant one-particle formulation at the level supported by the present derivation: MQ contains explicit scalar and spin-1/2 covariant sectors, conserved relativistic currents, the relativistic mass shell, and a controlled non-relativistic reduction to Eq. (41). Section 3.9.4 and Appx. Q continue this structure into quantum field theory, while Sec. 3.10/Appx. AB close the additional mean-backreaction bridge between those quantum matter fields and the MQKB-derived emergent geometry. The remaining distinction is therefore not “particle versus no field theory” or “fixed geometry versus no feedback,” but which deeper field-theoretic and quantum-geometric structures are actually derived from MQ.

3.9.2 Selection of the Minimal Massive Charged Matter Representation

The preceding spin result still leaves a logically prior question: why should the elementary massive charged carrier realized from primitive MQ count structure belong to the Dirac/Clifford class at all? The answer uses constraints already established upstream rather than an economy choice among known particle representations. The primitive measurement-availability count field has no independent spinor, tensor, chirality, or species index; its local law, Eq. (9), is a one-update adjacency redistribution law. Proposition 2.7 supplies the local proper-orthochronous Poincaré symmetry after realization, while Appx. P supplies the relativistic mass shell, the nonzero subsystem mass realization, and the positive conserved hypersurface current required of the one-particle matter carrier. The MQKB electromagnetic construction separately supplies the nonzero U(1) charge scale used in Appx. Q. Appendix AK combines these ingredients into a representation-selection theorem.

Proposition 3.3 (Selection of the minimal massive charged MQ matter carrier). Fix the finite regular System Frame realization class used in Secs. 3.9.1–3.9.3. Consider an elementary one-particle matter carrier obtained from the scalar Internal Frame count ontology with no independent primitive internal multiplet. Require that its realized free propagation preserve the inherited one-update locality without introducing an additional state-memory variable, transform covariantly under the derived proper-orthochronous Poincaré group, carry nonzero realized mass and nonzero MQ U(1) charge, reproduce the already-derived relativistic mass shell, and admit a positive conserved hypersurface availability norm. Then the minimal faithful complex local realization is, up to similarity and admissible representative change, the four-component Dirac Clifford module. The first-order realized operator must factor the mass shell and therefore obey the Clifford relation already written in Eqs. (P.8) and (R.2); the complex Clifford algebra in four spacetime dimensions has a unique simple four-dimensional module up to equivalence [129,165]. Its Lorentz content is the paired chiral representation needed for a massive charged fermion [163,164]. Consequently the Dirac/Clifford class is downstream of the MQ count-update, causal, mass, and charge structure for this elementary carrier rather than an independently selected matter ansatz.

The alternatives are separated by which premise they fail, not by numerical economy. A single Weyl representation is a chiral massless carrier; a nonzero Lorentz-invariant mass couples opposite chiralities. A Majorana condition can describe a neutral massive fermion but is incompatible with the nonzero unbroken U(1) charge carried by the present electromagnetic matter channel [164]. The scalar Klein–Gordon sector of Appx. P remains an admissible bosonic System Frame field and is retained in Appx. Q, but its second-order one-particle equation and indefinite Klein–Gordon current do not satisfy the present elementary positive-availability-carrier criteria. Higher-spin Rarita–Schwinger or more general tensor-spinor representations require additional realized tensor indices and subsidiary constraints [151,163]; those structures are not generated by the scalar primitive count carrier and must be derived from additional MQ invariants before they can represent a distinct elementary matter class.

This result is deliberately narrower than a derivation of the complete particle spectrum. It selects the representation class of the elementary massive charged carrier actually used by the quantum-measurement and Abelian-QED continuation. It does not assert that Nature contains no neutral Majorana field, no massless chiral field, no scalar excitation, or no higher-spin field. Nor does it determine species multiplicity, flavor, non-Abelian internal multiplets, chirality assignments, Higgs/Yukawa structure, or anomaly cancellation. Those are separate matter-content questions and remain part of the universality program described in Sec. 4.7.2 and Appx. AA.

3.9.3 MQ Rotational Completion and the Spin-1/2 Sector

The remaining one-particle structural issue is whether spin-1/2 is merely appended to the realized System Frame description or is selected by structures already present in MQ. Appendix R gives a conditional derivation at the same level of rigor used elsewhere in the paper. Its inputs are not a Pauli postulate: they are the MQKB half-Planck angular-momentum geometry, the complex state geometry already reconstructed in Appx. F, the locally isotropic realized rotation group of the System Frame metric, and the first-order Clifford factorization already implicit in the Dirac continuation of Appx. P.

The MQKB relation used in this step is L=rΘ with r=l_f and Θ represented by the same invariant coefficient that gives θ_si=ℏ/(2l_f), so that the corresponding fundamental rotational action is L=ℏ/2. This equality does not by itself create a spin representation; the representation is fixed by the already-required local Clifford factorization. On the finite bounded contractible System Frame charts of Proposition 2.3, the oriented orthonormal frame bundle is trivial and admits a spin lift. Quantum states are rays in the derived complex Hilbert geometry, so continuous physical rotations lift from SO(3) to Spin(3)≅SU(2) [128,129]. The first-order Clifford factorization then fixes the spatial spinor module: its rotation generators have Casimir S²=3ℏ²/4, and the SU(2) identity S²=j(j+1)ℏ² has the unique nonnegative solution j=1/2. The half-Planck MQ relation supplies the same component scale ±ℏ/2. Thus j=3/2,5/2,... are not competing representations of the same Dirac Clifford sector and no economy assumption about minimal internal multiplicity is needed.

The resulting generators are S_i=ℏσ_i/2. They satisfy the angular-momentum algebra, yield S²=3ℏ²/4, and generate U(n,ϑ)=exp[-iϑ n·σ/2]. Therefore U(2π)=-I while U(4π)=I: the double-valued spinor rotation law is a consequence of the selected SU(2) realization rather than an independent spin postulate. Werner et al. directly observed the 2π sign reversal with neutron interferometry, and the companion 1975 neutron-interferometer work observed the expected 4π periodicity [132,133].

Because MQ probability has already been identified with realized record frequency before Hilbert completion, the Stern–Gerlach law follows without introducing a separate spin probability rule. For a state polarized along a and an analyzer axis n, the two spectral projectors of S·n give P_MQ(±|a,n)=[1±a·n]/2=cos²(ϑ/2), sin²(ϑ/2), where ϑ is the angle between a and n. The original Stern–Gerlach experiment established discrete directional splitting [131]; Eq. (R.10) gives the complete two-channel probability law in the MQ representation. In the electromagnetic sector, the non-relativistic limit of the Appx. P/Q Dirac equation is the Pauli equation, with the leading magnetic coupling corresponding to g=2 [130]. QED corrections to g belong to the field-theoretic representation of Appx. Q and are constrained at approximately the part-per-trillion level by the modern electron magnetic-moment measurement already cited in Ref. [126].

The status is therefore stronger and its global qualification is now classified. Spin-1/2 is not inferred from integer count conservation in isolation; it is fixed within the elementary massive charged System Frame matter carrier by the MQ-derived Lorentzian geometry, local spin lift, first-order Clifford factorization, and independently derived half-Planck rotational scale. Section 3.9.2 and Appx. AK now close the preceding representation-selection gap: the first-order character is inherited from the primitive one-update count law when realized locally without additional evolution memory; Lorentz covariance promotes that first-order structure to the full tangent four-vector; consistency with the already-derived relativistic mass shell forces the Clifford relation; and the complex four-dimensional Clifford algebra has one simple module up to equivalence. Nonzero MQ U(1) charge and nonzero mass then require the paired chiral Dirac realization rather than an isolated Weyl or Majorana carrier. The Clifford-Casimir relation consequently forces j=1/2 on every simple block. Appendix Z states the global spin criterion, and Appx. AJ fixes the finite higher-dimensional carrier on which that criterion is evaluated. Higher-spin tensor-spinor, neutral Majorana, and massless chiral sectors remain possible only as distinct matter classes requiring additional physical data; they are not alternative representations of the same minimal massive charged carrier.

3.9.4 Quantum Field Theoretic System Frame Completion

A relativistic quantum theory is incomplete if its state space, local observables, charge classes, vacuum status, and interactions remain globally selectable after the one-particle construction. Appendix F already reconstructs the complex Hilbert geometry and finite composition, Appx. P supplies the covariant scalar and Dirac one-particle structures, and the primitive MQ update law supplies relational locality before spacetime realization. Appendix Q retains Fock space as the explicit free/stationary occupation representation. Appendix AR now takes the stronger route: local System Frame observable algebras are generated from the mapped local MQ observables, their inclusions follow relational-region inclusion, spacelike commutativity follows the already-derived causal realization and microcausality, and the global algebra is the inductive completion of that local net [122,190,191].

The admissible state is then not chosen by selecting an unrelated Hilbert representation. The finite-context probabilities already fixed by Proposition 3.1 and Appx. T define positive normalized state functionals on the local observable net; the GNS theorem supplies a representation unique up to unitary equivalence for each such state. In a stationary positive-energy realization, an invariant ground-state functional gives the vacuum representation and its free limit is the Fock construction of Appx. Q. On a generic nonstationary curved System Frame no preferred global vacuum is required; the local algebra and its admissible state functional remain primary, with Bogoliubov particle language attached only when additional mode structure exists [120,124,193].

The gauge interaction is not inserted as an unrelated external field. The MQKB Unification sector already contains a single realized action, an emergent Abelian gauge transformation, gauge-invariant field strength, Maxwell-type sourced equations, and a gauge stress-energy tensor. Appendix Q combines that MQKB connection with the derived Dirac field through the unique first-order U(1)-covariant derivative D_μ = ∇_μ + iqA_μ/(ħc), where q is the System Frame charge realization and the universal charge scale is supplied by the MQ charge construction rather than fitted here. The resulting curved-System-Frame QED action contains the Maxwell term and minimally coupled Dirac term, is invariant under the local U(1) transformation, and reduces to ordinary QED in a locally Minkowski realization [124,126].

This extension closes the interacting relativistic state-space question to the level supported by the present MQ matter content. Appendix AR derives the local observable-net architecture, count-compatible state class, state-selected representation, and the present Abelian charge superselection structure; Appendix Q supplies the corresponding free/stationary Fock realization and perturbative QED fields. The existing MQKB action fixes the Abelian interaction rather than a freely chosen interaction Hamiltonian. Appendix AA proves that the current gauge algebra is U(1), and Appendix AL proves that primitive adjacency alone does not select a larger continuous non-Abelian group. Appendix AL now also gives the non-Abelian reconstruction route forced by the local-algebra result: if MQ derives a complete localized superselection category from count-local physical sectors, Doplicher-Roberts duality reconstructs the corresponding compact gauge group and field algebra rather than permitting the group to be chosen independently. Consequently, the remaining universality problem is narrowed to derivation of the missing non-Abelian charge sectors, their fusion/statistics data and chiral matter assignments, not an unresolved freedom to append an arbitrary continuous gauge group to the scalar/Dirac/U(1) state-space architecture used by this paper.

Proposition 3.4 (Count-local selection of the scalar/Dirac/U(1) relativistic state space). Fix the regular realized domain and the MQ-derived scalar/Dirac/U(1) matter content of Secs. 3.9.1–3.9.4. For every bounded referenceable System Frame region, generate the local observable algebra only from bounded gauge-invariant observables obtained from mapped MQ fields supported in that region. Require inclusion covariance under region inclusion, spacelike commutativity, compatibility with the MQ local equations of motion, and state restrictions whose finite measurement probabilities equal the count-derived record probabilities of Proposition 3.1/Appx. T. Then the local net and its quasi-local completion are fixed by the MQ realization data up to the representative equivalence already quotiented by Proposition 2.2; each admissible state determines its Hilbert-space representation uniquely up to unitary equivalence by GNS. In the presently derived U(1) matter class, inequivalent localized charge representations are labeled by the conserved realized charge, while the neutral vacuum representation is selected by invariance and positive energy whenever the realized geometry admits such a state. Appx. AR gives the proof, the DHR comparison, and the curved-background qualification [190–194].

3.10 Matter–Geometry Backreaction and the Quantum-Gravity Boundary

The gravitational and quantum constructions developed above can now be coupled at a stronger, but still carefully delimited, level. The MQKB General Relativity and Unification constructions already derive the System Frame metric, Einstein tensor, gravitational coupling, matter action, variational stress-energy tensor, and covariant conservation from one MQ action. Appendix Q independently supplies quantum scalar, Dirac, and Abelian gauge fields on that realized geometry. Appendix AB first recovers the semiclassical mean-source equation and then carries the same count-derived quantum state one step further: the renormalized stress-energy fluctuation kernel is fixed by that state and drives a linearized stochastic System Frame metric response. Appendix AS now closes the additional coherence step without promoting a metric to an Internal Frame primitive. Because the realized geometry is itself a function of count, adjacency, matter, boundary, defect, and MQ-action data, inequivalent geometry classes partition the same MQ configuration state already represented in the derived Hilbert space. The corresponding sector amplitudes inherit the existing MQ phase-action structure and can therefore remain coherent until a geometry-distinguishing record is realized.

At the classical realized level, mutual matter–geometry response is already present: the same mapping-induced action that generates the metric also generates its stress–energy source by metric variation, and the resulting MQKB field equation closes geometry against matter. For quantum matter, Appx. AB defines the corresponding semiclassical System Frame correspondence by replacing the classical source with the renormalized expectation value of the QFT stress–energy operator in the realized state. Because the source and metric then determine one another self-consistently, this supplies mean quantum-matter backreaction rather than a fixed-background test-field approximation [124,158,159].

The stochastic construction alone does not amount to quantum geometry because a probability distribution over c-number metrics contains no relative phase. Appendix AS supplies the missing structural distinction. It does not append an independent gravitational Hilbert space or postulate canonical metric commutators. Instead, the physical geometry map already derived from MQ count and adjacency data is promoted to a derived observable on the same count-generated System Frame state space. A state whose support spans more than one inequivalent geometry class then has coherent geometry components with relative phase inherited from the existing phase-action field; invariant geometric functionals act as operators on those classes, and evolution that mixes classes generates noncommuting Heisenberg geometric observables and measurable off-diagonal interference terms. The metric remains non-primitive: its quantum character is inherited from the quantum state of the count configurations from which it is realized.

Accordingly, item #60 remains closed at the mean-backreaction level and item #71 remains the stochastic-fluctuation correspondence. Item #79 is now closed at the structural quantum-coherence level for the finite and connected locally finite regular realization classes proved in this paper: Appendix AS constructs coherent superpositions of inequivalent physical geometry classes inside the already-derived MQ Hilbert state, defines invariant geometric observables without adding a primitive metric field, and derives stochastic and semiclassical geometry as successive decohered/weak-fluctuation limits. What remains outside this result is a stronger universality program: nonperturbative control of all topology-changing classes, a complete interacting gravitational observable algebra on arbitrary domains, and experimental confirmation that gravity preserves the predicted coherence.

Proposition 3.5 (Count-induced coherent quantum geometry). Fix a finite or connected locally finite regular MQ realization in which the physical geometry class is uniquely determined from the count, adjacency, matter, boundary, defect/holonomy, and MQ-action data up to the representative equivalence of Proposition 2.2, and let the complete realized state belong to the MQ Hilbert space already reconstructed in Appx. F/AR. Partition that state space by the inequivalent physical geometry classes and let the corresponding orthogonal projectors resolve the identity on the supported geometry-class subspaces. Then every MQ state decomposes uniquely into geometry-class components. If at least two components have nonzero amplitude and no geometry-distinguishing record has been realized, their relative phase is the phase already carried by the common MQ state and is therefore physically available to any admissible observable with off-diagonal matrix elements between the classes. Every real relationally invariant geometric functional defines a self-adjoint operator by its values on these sectors; if the realized evolution mixes geometry classes, its Heisenberg geometric observables need not commute at distinct update intervals. Thus coherent quantum geometry is a derived property of the count-generated state rather than a separately postulated metric operator. Appendix AS gives the construction, the gauge/representative quotient, and the stochastic and semiclassical limits.

3.11 Emergence of Quantum Structure

The preceding sections establish a hierarchy whose physical derivations and mathematical representations must be kept distinct. Sections 3.1–3.3 derive the outcome partition, completed-interaction record map, realization-frequency compatibility, conditioning, and record structure used for realized measurements. Section 3.4 first constructs the normalized System Frame availability measure, uses the pushforward identity of Sec. 3.2.1 to justify its empirical probability interpretation, then constructs the MQ wave function as an amplitude-phase representation of that measure, and finally identifies the full arbitrary-observable Born rule with the later spectral/POVM completion of Appx. F.18–F.19. Section 3.5 conditionally selects Schrödinger-form evolution within the stated MQ admissibility class, while Secs. 3.6–3.8 apply the resulting representation to interference, entanglement, and the stable coarse-grained classical-record regime. The complex linear norm, inner-product geometry, and tensor-product composition are reconstructed in Appx. F from the mapped MQ amplitude structure; Hilbert completion then enters only as the canonical addition of norm-limit points, while operator and spectral correspondence remain functional-analytic representations rather than additional primitive physical structure.

For composite systems, Appx. F.14-F.14.1 additionally shows that the binary product reconstructed from separable MQ realization extends canonically to finite multipartite composition. The usual bipartite Schmidt decomposition, reduced-state spectrum, purification representation, and pure-state monogamy structure then follow without changing the underlying MQ probability or realization rules. Standard multipartite entanglement classes are consequently classes within the derived composite state space rather than extra kinematic assumptions.

At the foundational level, MQ begins with conserved count structure defined on the Internal Frame relational adjacency configuration domain. The primitive count field n_I(ξ,A)∈N₀ assigns a nonnegative discrete availability count to relational cell ξ at Internal Frame update index A. Here ξ labels an adjacency-connected relational configuration element of the Internal Frame, and A denotes the discrete update ordering associated with admissible count redistribution. The primitive count field therefore defines the conserved encoded availability structure from which later System Frame realization is derived.

Conservation of encoded availability is expressed locally through the adjacency continuity relation

where J_I(ξ→ξ^',A) denotes the Internal Frame adjacency count-flux object and div⁡_adjJ_I(ξ,A) is the adjacency-divergence operator defined by the net count imbalance over admissible adjacency links incident on relational cell ξ. Both quantities are purely combinatorial objects defined on the adjacency structure of the Internal Frame and carry no metric, geometric, vector, directional, or differential structure. Equation (57) therefore states that local encoded availability evolves through count-conserving adjacency-local redistribution across neighboring relational cells. No continuum coordinates, metric structure, or differential geometry are assumed at this level.

The Frames mapping induces an encoded-availability measure in the System Frame. Let μ_S denote the induced System Frame encoded availability measure generated by admissible count realization. When μ_S is absolutely continuous with respect to the System Frame realization measure dΩ, the corresponding encoded availability density is defined through the Radon–Nikodym relation

which defines the local encoded-availability density associated with the induced System Frame realization measure. The quantity ρ_S(r,t) therefore represents a System Frame realization of conserved encoded availability under the Frames mapping. Normalized realized probabilities do not arise at this stage. Realized probabilities appear only after normalization and contextual restriction.

Let Ω_out denote the realized System Frame realization domain associated with an admissible measurement context. Thus,

defines the normalized encoded-availability density associated with the realized measurement context.

The System Frame wave-function encoding is constructed from the normalized encoded availability density together with the redistribution phase field fixed locally by mapped count flow. Proposition C.3 shows that the minimal continuous reversible norm-preserving linear representation of this density-plus-phase data is uniquely complex, up to global phase and complex conjugation:

where ϕ(r,t) denotes the realized System Frame redistribution phase induced through coherent count redistribution. Equation (60) is therefore not postulated as a primitive continuum field. Rather, it is a derived encoding generated from normalized encoded-availability structure. Its ontological status is representational: the phase is the scalar-potential encoding of mapped count redistribution, fixed locally by the normalized current up to an unobservable additive constant, rather than an additional primitive variable (Appx. C.8.1 and C.11).

Successive difference operators defined on the Internal Frame adjacency structure induce continuum differential operators in the System Frame. Under the continuum limit developed in Sec. 3.5 and Appx. E.1–E.6, admissible count redistribution yields the standard Schrödinger evolution relation

Equation (61) is therefore the System Frame continuum generator selected by the admissibility conditions of Sec. 3.5 and Appx. E.

Once the normalized System Frame encoding satisfies the Hilbert-space regularity conditions of Appx. F, observable quantities may be represented by the standard self-adjoint operator machinery. The expectation-value functional is fixed first by normalized encoded availability; the operator language supplies its equivalent representation on the completed state space.

where

denotes the realized System Frame operator associated with observable quantity O. As developed in Appxs. G–I, inner products, self-adjoint operators, and realized evolution generators arise from realized representations induced through admissible count redistribution.

Superposition, interference, and entanglement are consequences of the respective System Frame constructions in Secs. 3.6–3.7; their Bell behavior is assessed at the realized probability layer.

The resulting emergence structure is hierarchical. Conserved encoded availability evolves deterministically in the Internal Frame through count-conserving adjacency-local redistribution governed by Eq. (57). The Frames mapping induces encoded availability in the System Frame, yielding the mapped density, normalized density, and wave-function encoding. Under the controlled continuum limit of Sec. 3.5 and Appx. E.1–E.6, the difference construction supplies the gradient and Laplacian operators; Appx. O separately fixes the phase-action scale from MQ fundamental measures, represents subsystem mass by its mass count, and derives the free-particle dispersion from the realized phase-action/Hamilton-Jacobi structure using the Galilean boost law that Proposition 2.7/Appx. Y obtains as the non-relativistic contraction of the count-selected local Lorentz/Poincaré symmetry. Within those conditions, the construction recovers Schrödinger evolution together with the operator, interference, and entanglement structures developed above.

Probability uniqueness is now part of that hierarchy rather than an implicit consequence. Proposition 2.1 fixes the primitive finite count measure, Eq. (32) identifies record probabilities as its contextual pushforward, and Theorem T.1 proves that relabeling invariance plus additivity, normalization, and coarse-graining consistency leave no alternative count-based empirical measure on the same finite record algebra. Hilbert completion subsequently represents that unique probability law; it does not select it.

The measurement hierarchy now also reaches a laboratory record channel. Section 3.2.2/Appx. V composes the already-fixed MQ outcome distribution with a calibrated System Frame detector response, so finite efficiency, dark counts, and thresholding appear as apparatus-specific post-realization effects rather than changes to the primitive count measure. A noisy detector therefore changes the effective POVM and observed record frequencies in the standard way while leaving Eq. (32) as the underlying ideal-event probability law.

This completes the emergence construction summarized in Sec. 3.11; the Discussion now turns to its conceptual and comparative implications.

The canonical kinematic algebra is now fixed within this same hierarchy. Appendix S derives the realized translation group from mapped adjacency redistribution, identifies its generator using the independently fixed MQ action scale, proves the Weyl position-translation relation, and recovers the canonical commutator on a common dense invariant core. Consequently, Eq. (F.62) is no longer merely the standard position-representation identity adopted after Hilbert completion; it is the infinitesimal System Frame consequence of the MQ translation realization.

The same hierarchy now has an explicit relativistic continuation and global topological classification. Section 3.9.1 and Appx. P place the derived amplitude and phase-action scale on the MQKB-derived Lorentzian System Frame geometry and obtain the covariant scalar and Dirac equations. Section 3.9.3/Appx. R fixes the local j=1/2 spinor representation. Proposition 2.8/Appx. Z then classifies the global extension: the primitive adjacency cycle basis identifies all candidate non-contractible loop generators; mapped U(1) phase sectors are indexed by integral winding data; mapping defect classes are fixed by their holonomy representation on surviving loops; and global spin structures exist exactly when w₂ vanishes and then differ by H^1(Ω;Z₂). These structures remain realized System Frame data and do not promote topology, spinors, or a primitive phase field to Internal Frame ontology.

The field-theoretic hierarchy is now explicit at both algebraic and occupation-representation levels. Section 3.9.4/Appx. AR construct the local System Frame observable net from mapped MQ-local observables, transport the already-derived finite-context probability law into positive state functionals, and use the state itself to select the GNS representation. Appendix Q then appears as the free/stationary Fock realization of that algebraic structure, with creation/annihilation operators and Bogoliubov particle creation where a mode split is available. The MQKB Abelian connection fixes the U(1) interaction, and the conserved realized charge labels the presently derived localized superselection classes. Section 3.10/Appx. AB returns the renormalized quantum stress-energy mean and fluctuation kernel to the MQKB gravitational sector. Thus variable particle number, representation selection, Abelian interaction, and matter–geometry feedback all remain downstream System Frame structures rather than additional Internal Frame primitives.

4. DISCUSSION

Taken together, the results establish MQ as a two-domain, three-role kinematic reconstruction in which conserved count structure, continuum quantum representation, measurement conditioning, and the stable-record classical regime occupy distinct but formally linked levels. The Discussion therefore begins with the principal conceptual consequences of that construction before turning to Bell locality and comparison with alternative quantum interpretations.

4.1 Conceptual Position and Principal Gains

Given the two frame kinematic structure, conserved discrete availability in the Internal Frame and contextual encoding in the System Frame under the Frames mapping and the Frames Principle, quantum formal structure appears as a derived representation rather than a primitive postulate. This section summarizes the principal structural gains that follow from that construction.

The principal gains are:

Removal of primitive wave–particle duality. Physical ontology is assigned to the conserved discrete count availability field in the Internal Frame, whose encoded System Frame representations are generically non-factorizable under contextual restriction. Wave-like behavior appears only in the context-dependent System Frame encoding. Dual aspect language is replaced by a single conserved substrate with frame-dependent representations.

Measurement update without a primitive collapse postulate. Measurement conditioning is implemented as contextual restriction and renormalization of the System Frame encoding used for subsequent prediction. Definite outcomes are fixed earlier by the completed contextual realization and the disjoint outcome partition, while their ensemble weights are the pushforward of the same pre-probabilistic complete-interaction count measure (Eqs. 30-32; Appx. H.9). The framework therefore requires neither a separate stochastic collapse law nor an independent probability rule for choosing among already weighted alternatives.

Explicit separation of conservation and probability layers.
Count conservation and continuity relations are defined in the Internal Frame, while probabilities arise only after normalization in the System Frame. Constraint-compatible correlations and probabilistic outcomes are therefore produced by distinct but formally linked layers, without Bell-type hidden outcome variables.

Coefficient-resolved conditional reconstruction of non-relativistic Schrödinger dynamics.
Within the non-relativistic admissibility class of Sec. 3.5, Schrödinger-form evolution is conditionally reconstructed by combining the controlled continuum limit of successive discrete difference operators with the independently fixed MQ phase-action scale and the free inertial dispersion derived in Appx. O.4. The needed Galilean boost action is no longer an independent realization postulate: Proposition 2.7/Appx. Y obtains it as the non-relativistic contraction of the mapped-invariant local Lorentz/Poincaré realization. Linearity and norm preservation follow from discrete conservation and admissible operator structure rather than being imposed as primitive axioms.

A further closure concerns canonical kinematics. The Frames mapping carries adjacency-local redistribution into continuous realized translations, and the derived phase-action scale fixes their generator. Covariance of the realized coordinate under that translation produces the Weyl relations and hence the canonical commutator. The usual position-momentum algebra is therefore recovered as a System Frame consequence of the count-to-translation map rather than assumed as primitive operator ontology (Appx. S).

Taken together, these results place the standard Hilbert-space formalism downstream of the MQ kinematic construction. Normalized probability is derived from encoded availability; the MQ wave function is constructed as an amplitude-phase encoding of that probability and redistribution phase; Proposition C.3 fixes its minimal local complex U(1) representation; and Appx. F shows that the resulting availability norm and coherent linear composition satisfy the parallelogram law, thereby fixing the complex inner product uniquely by polarization. Separable composite realization similarly fixes tensor-product composition up to canonical isomorphism. Hilbert completion therefore completes an already reconstructed pre-Hilbert geometry rather than supplying that geometry by assumption. The full Born rule for arbitrary observables is recovered only at the subsequent spectral/POVM representation stage. Predictive equivalence therefore does not imply that every element of the standard formalism has the same derivational status.

The probability step is correspondingly stronger than normalization alone. Appendix T proves pre-Hilbert uniqueness of the finite MQ record-probability measure: equal primitive count units cannot receive unequal probabilistic weight without adding a new scalar field, and contextual outcome maps cannot introduce an independent reweighting without violating pushforward/coarse-graining consistency. The later modulus-squared and spectral formulas are representations of this already-fixed event measure.

The composite result is correspondingly stronger than representability alone. The tensor product is fixed by the universal linearization of separable MQ composition; repeated composition fixes the finite multipartite product up to canonical unitary isomorphism; Schmidt and purification structure are then consequences of that product; and standard entanglement-sharing restrictions, including the three-qubit Coffman-Kundu-Wootters monogamy inequality, apply to the resulting state geometry [113-116]. These results do not assert that MQ has solved the classification of all multipartite entanglement or derived every entanglement monotone from primitive counts. They establish that no alternative finite composite-state architecture can be inserted while preserving the separable composition law already derived by MQ.

A second closure is now available for relativistic covariance, causal signature, elementary charged-matter representation, spin, and the associated global topological data. The MQKB general-relativistic construction supplies the emergent System Frame metric, connection, covariant conservation structure, and geodesic dynamics. Proposition 2.6/Appx. X selects the local Lorentzian signature class, Proposition 2.7/Appx. Y fixes the connected local Lorentz/Poincaré symmetry, and Proposition 2.8/Appx. Z classifies the non-contractible cycle data carried by a global realization. Proposition 2.9/Appx. AJ fixes the finite higher-dimensional simplicial carrier from primitive adjacency. Section 3.9.2/Appx. AK then selects the Dirac Clifford module for the minimal massive charged elementary carrier by combining the primitive one-update law, derived Lorentz covariance, nonzero MQ mass and U(1) charge, mass-shell factorization, positive hypersurface availability, and the absence of an independent primitive internal multiplet. Appendix R fixes j=1/2 inside that selected module. The remaining matter-content problem is therefore no longer why the present charged elementary carrier is Dirac-type; it is whether additional primitive or realized invariants generate neutral, massless chiral, scalar, higher-spin, flavor, or non-Abelian matter classes beyond it.

A third closure now extends the same realized hierarchy to quantum fields without making global Fock space the defining state-space postulate. Appendix AR reconstructs a net of local gauge-invariant System Frame observables from mapped MQ-local fields and relational support, carries the count-derived event probabilities into positive normalized state functionals, and lets each admissible state select its representation through GNS. Superselection is correspondingly a statement about inequivalent localized representations of the same quasi-local observable algebra; in the presently derived Abelian matter class, conserved U(1) charge supplies the explicit charge-class label, while DHR/Doplicher-Roberts theory provides the standard mathematical classification benchmark [190–192]. Appendix Q then supplies the occupation-number/Fock representation in free or stationary regimes and the Bogoliubov comparison between admissible mode decompositions. The remaining field-theory problem is therefore additional matter/gauge content and genuinely nonperturbative dynamics, not arbitrary global state-space choice.

4.2 Relation to Environmental Decoherence and Record Selection

Environmental decoherence and MQ conditioning address different stages of the measurement description and should not be conflated. In standard open-system quantum mechanics, interaction between a system, measuring apparatus, and environment entangles the apparatus with environmental degrees of freedom. When the environment is not observed, the reduced state of the measured subsystem loses experimentally accessible coherence between environmentally distinguished alternatives. This process suppresses interference between macroscopically distinct components and dynamically favors robust pointer states through environment-induced superselection [5,7]. It therefore explains why certain macroscopic record structures are stable and why interference between them becomes operationally inaccessible.

That dynamical achievement does not by itself establish which one of the decohered alternatives is the realized record in an individual measurement. The total system-environment state remains governed by the same linear quantum dynamics, and suppression of off-diagonal coherence in a reduced description is not equivalent to selecting one member of the resulting mixture as the unique observed outcome [7,9]. This distinction is already required by the measurement-consistency analysis of Sec. 2.5 and is preserved explicitly in MQ.

In MQ, decoherence belongs entirely to the realized System Frame encoding. The wave function, phase field, Hilbert-space tensor-product structure, reduced subsystem description, environmental coupling, and pointer-state representation are all continuum quantum structures that arise only after the Frames mapping has produced the System Frame representation. None of these structures acts directly on the primitive Internal Frame count field. The conserved Internal Frame availability counts continue to evolve through the adjacency-local redistribution laws of Secs. 2.6-2.11 regardless of whether the realized System Frame representation is treated as isolated or environmentally open.

The effect of decoherence can therefore be stated directly in MQ terms. Before record realization, normalized encoded availability may have support across several macroscopic outcome regions. Environmental interaction can suppress observable phase coherence between those regions, including the interference cross terms represented by Eq. (46), and can make particular coarse-grained outcome partitions dynamically robust. This modifies the coherence structure of the System Frame representation without changing the underlying Internal Frame count conservation and without performing the contextual restriction defined by Eqs. (33-35).

Realized-record selection is supplied by completed-context realization, not by conditioning. For a completed measurement interaction x, Eqs. (28) and (30) give R(x)=κ(M_c(x))=o_k, so the realized outcome is fixed by the unique System Frame partition containing M_c(x); Eq. (29) encodes that outcome as a stable distinguishable macroscopic record. Equation (32) simultaneously shows that the pushforward distribution of these individual realized labels equals the normalized availability weights. The associated System Frame record region A_k is therefore fixed before the predictive state update. Contextual restriction then confines the System Frame encoding to A_k and renormalizes the remaining encoded availability for subsequent predictions. Immediate repetition returns the same record with unit probability under the ideal stability condition of Eq. (56). Decoherence may strengthen persistence and practical distinguishability, but it neither chooses o_k nor performs the restriction.

This separation also clarifies the preferred-basis issue. Environmental coupling can identify a dynamically stable pointer basis or approximately robust set of macroscopic alternatives [5]. MQ does not require those pointer states to exist as primitive Internal Frame objects. Instead, they define particularly stable System Frame partitions within the realized measurement context. The outcome partition and its environmental stability therefore belong to the same realized representational level, while the conserved count substrate remains basis-free and coordinate-free.

MQ is consequently compatible with the established decoherence program without assigning decoherence the outcome-selection role. Standard decoherence dynamics may be used wherever the System Frame encoding satisfies the Hilbert-space and open-system regularity conditions required for that description. MQ keeps three mechanisms separate: decoherence suppresses coherence and stabilizes record alternatives; the unique-realization map assigns one outcome to the complete admissible interaction; and contextual restriction with renormalization performs the post-outcome predictive update.

The quantitative status of this separation is given by Proposition 3.2 and Appx. U. For a branching System Frame representation, environmental overlap multiplies rather than replaces the MQ interference term, giving exponential suppression in the number or cumulative strength of decohering fragments. Independent record coarse-graining supplies a resolution factor, while count aggregation gives a 1/Nc variance law under explicitly stated weak-correlation conditions. Thus MQ no longer identifies the classical limit only by qualitative coarse-graining plus decoherence compatibility: it provides sufficient scaling conditions for vanishing observable interference and concentration of macroscopic records, while retaining completed-context realization as the distinct outcome-selection step.

4.3 Relation to Bell Locality and Quantum Interpretations

From a structural perspective, the interpretational tension in quantum theory is not due to predictive failure but to formal compression: a single Hilbert-space layer is used both to propagate constraint structure across space-like separated regions and to represent single-outcome probabilistic realization. Standard quantum theory assigns linear evolution and the Born rule within one representational layer, leaving conservation-constrained correlations and irreducible outcome statistics formally coupled but structurally unresolved.

Bell’s theorem isolates this constraint for factorizable hidden-variable models. Models satisfying Bell local causality, defined by the conditional relation P(A,B∣a,b,λ)=P(A∣a,λ)P(B∣b,λ) together with measurement independence, admit the representation

where a,b are settings, A,B are outcomes, and λ is a setting-independent underlying state parameter. From this structure follow Bell-type inequalities, including the CHSH bound S≤2, which are violated by quantum predictions and experiment (Appx. G.1–G.7 and G.11–G.12). Modern loophole-free Bell tests preserve operational no-signaling while excluding factorizable hidden variable models[44]. The excluded premise is therefore factorizable local causality, not operational locality.

The Bell premises must be kept distinct from operational no-signaling (Appx. G.1–G.7). PI and OI are defined only after a complete conditioning state λ is specified; observed marginal invariance without such conditioning is the separate no-signaling condition:

Measurement independence (MI): measurement settings are statistically independent of underlying state parameters, P(λ∣a,b)=P(λ). For MQ, this premise is implemented at the primitive preparation level rather than merely asserted for an unspecified λ: Appx. G.1 defines the complete pre-setting count configuration Cᵢ and establishes P(Cᵢ|a,b)=P(Cᵢ) together with its equivalent joint-factorization form in Eqs. (G.3–G.5).

Parameter independence (PI): at fixed complete state λ, remote-setting independence – the probability of a local outcome does not depend on the distant setting, P(A∣a,b,λ)=P(A∣a,λ).

Outcome independence (OI): at fixed complete state λ, conditional independence of one outcome from the other, given the underlying state description, P(A∣B,a,b,λ)=P(A∣a,b,λ).

Bell factorization (BF): joint conditional factorization at fixed λ, equivalent to PI + OI. Measurement independence is a distinct additional assumption, not part of the PI + OI equivalence.

Hidden variable outcome model (HV): existence of setting independent underlying variable that parameterizes or determines outcome probabilities in Bell’s sense.

Single-outcome realism (SO): each experimental run yields one realized outcome rather than ontological branching.

Empirically adequate frameworks can be classified only after the conditioning state and ontology are specified. Bohmian mechanics provides a hidden-variable completion and violates parameter independence at the hidden-variable level. Standard quantum mechanics, when the quantum state is used as the complete conditioning state, preserves PI but violates OI. Branching, collapse, operational, superdeterministic, and retrocausal approaches require separate model-specific qualifications. The PI/OI labels therefore should not be treated as synonyms for observable signaling behavior.{MI,PI,OI,SO,HV}

Two-level constructions are common in foundations, such as ontological models separating an ontic state λfrom an epistemic quantum state, or pilot-wave theories adding additional beables. These remain within the Bell architecture when λparameterizes outcome probabilities.

MQ introduces a two-domain, three-role kinematic structure. The Reference Frame is the fixed operational comparison/normalization role; it is not an independent primitive hidden-state space. MQ distinguishes the Internal Frame, a discrete relational adjacency domain carrying the measurement-availability count field, from the System Frame, a continuum relational encoding induced through the deterministic Frames mapping under the Frames Principle. Conserved measurement-availability counts and adjacency count redistribution are defined only in the Internal Frame. Observable quantum behavior appears as normalized System Frame probability density and amplitude structure. Constraint propagation and probabilistic realization are therefore structurally separated without introducing Bell-type hidden-variable outcome variables.

At the foundational level, MQ begins with a conserved discrete measurement-availability count field defined on the Internal Frame adjacency structure. Local conservation is expressed by the adjacency continuity relation

where n_I(ξ,A)is the Internal Frame measurement-availability count associated with relational cell ξ, J_Iis the associated adjacency count-flux defined on admissible adjacency links, Ais the Internal Frame update index, and div⁡_adjdenotes the adjacency-divergence operator acting on the adjacency count-flux. Here the update index Ais distinct from the Bell outcome symbol Aused above. No continuum coordinates, metric structure, geometric vectors, or differential operators are assumed at this level. These structures arise only through realization under the Frames mapping into the System Frame.

Under the Frames mapping and the Frames Principle, conserved count structure admits a System Frame encoding in which successive discrete difference relations induce the gradient and Laplacian operators of the realized representation. The Schrödinger generator is then recovered only within the additional non-relativistic representation conditions stated in Sec. 3.5. Appendix O makes this separation explicit and resolves the kinetic coefficient independently of the continuum operator choice: the MQ fundamental measures fix ħ, the subsystem mass count fixes m, and the Galilean boost law derived by Proposition 2.7/Appx. Y combines with realized phase-action dynamics to give T = p²/(2m), including the factor 1/(2m).

The normalized encoded wave-function representation of Eq. (60) is therefore a derived System Frame representation of normalized probability density and mapped redistribution flow, not a primitive ontic field. Appendix C.8.1 now closes the additional local representation question: once the flow-derived phase coordinate is continuous and its encoding is required to be reversible, linear, norm-preserving, and minimal, the representation is uniquely SO(2) ≅ U(1), yielding the complex phase form up to global phase and complex conjugation. Measurement updates remain contextual restriction and renormalization under the Frames mapping (Secs. 3.1–3.2 and Appx. H.1–H.7).

For MQ, the strongest directly derived locality statement is operational no-signaling of realized System Frame marginals (Secs. 3.3 and 3.7; Appx. G.7 and G.10). The primitive Internal Frame count configuration is underlying ontic structure, but MQ does not equip it with setting-independent local response functions of the Bell form. Accordingly, it is more precise to say that MQ does not posit a Bell-local hidden-variable response model than to claim the absence of hidden structure in every broader ontological sense. If the complete System Frame encoded state is used as the conditioning state, the realized probability representation has the standard quantum pattern of remote-setting-independent marginals and outcome-dependent joint probabilities.

This two-domain, three-role construction therefore reproduces Bell-test outcome structure without a Bell-local factorized hidden-variable response model, ontological branching, collapse dynamics, or rejection of measurement independence. Deterministic availability redistribution is assigned to the Internal Frame, while contextual non-factorizable probabilities arise in the System Frame encoding. The manuscript makes no claim that Bell violation proves the absence of all possible hidden variables; rather, it excludes Bell-local factorization for any measurement-independent completion reproducing the derived CHSH statistics.

Table 1 provides only a schematic comparison across representative frameworks. PI and OI entries are meaningful only relative to the complete conditioning state adopted by each framework, whereas operational no-signaling is an observable marginal statement. The MQ row is therefore annotated separately so that its directly derived no-signaling result is not conflated with an unintroduced λ-conditioned hidden-variable model.

Table 1. Schematic Bell-premise classification across representative quantum frameworks. MI = measurement independence; PI = λ-conditioned parameter independence; OI = λ-conditioned outcome independence; BF = Bell factorization (PI ∧ OI); HV = hidden-variable outcome model; SO = single-outcome realism. Operational no-signaling is a distinct empirical condition and must not be identified automatically with PI. Entries are model- and conditioning-state dependent; the MQ row is qualified by notes b and c.

^a Bell factorization is equivalent to PI and OI at fixed complete state λ; measurement independence is a separate auxiliary assumption in the standard Bell–CHSH derivation.
^b MQ directly proves operational no-signaling of realized marginals. Its primitive count structure is not assigned Bell-local response functions P(A|a,λ) and P(B|b,λ). Therefore PI and OI are not asserted for an additional primitive λ-completion. At the complete System Frame state-conditioned level, the probability representation has remote-setting-independent marginals and outcome dependence, as in standard quantum mechanics.

Accordingly, the Bell analysis in this paper is explanatory only at the level explicitly derived. MQ derives the pre-setting preparation independence, non-factorizable joint availability, contextual normalization, operational no-signaling, and—in the ideal lossless two-channel polarization specialization—the analyzer angular response. Appendix G.11 obtains that response from the continuous norm-preserving rotation of the derived System Frame channel amplitude, so the cos²/sin² law is no longer an external measurement-model input. The resulting correlation and Tsirelson value follow without instantiating a Bell-local factorized response model. This does not establish a general theorem that every hidden-variable ontology or every alternative interpretation is excluded.

Accordingly, the Bell analysis in this paper is explanatory only at the level explicitly derived: MQ reproduces Bell-violating, operationally no-signaling statistics and does not instantiate a Bell-local factorized response model. It does not establish a general theorem that every hidden-variable ontology or every alternative interpretation is excluded.

4.4 Historical Context: Deriving Quantum Dynamics from More Fundamental Principles

A substantial amount of literature derives the Schrödinger equation, and in some cases broader quantum dynamics, from assumptions more primitive than the standard Hilbert space postulates. These approaches are useful points of comparison because they make explicit which additional structures are required to recover quantum evolution rather than taking the formalism as primitive.

Most derivation programs introduce one of three classes of additional structure: information- or inference-based constraints, modified classical or stochastic dynamics, or discrete or geometric constructions whose continuum limits reproduce quantum equations. In each case, quantum behavior emerges after extending an underlying classical, probabilistic, or geometric framework with further axioms or constraints.

MQ adopts a different primitive layer: conserved discrete measurement availability in the Internal Frame. MQ conditionally reconstructs Schrödinger-form dynamics in the continuum System Frame limit under the deterministic encoding of the Frames mapping specified by the Frames Principle, subject to the explicit non-relativistic sector conditions of Sec. 3.5 and Appx. O. The following subsections compare MQ with representative derivation and reconstruction programs at the level of primitive assumptions and construction methods.

4.5 Relation to Existing Reconstruction Programs

4.5.1 Exact-Uncertainty and Fisher-Information Approaches

Exact-uncertainty and Fisher-information programs derive Schrödinger dynamics by supplementing classical ensemble mechanics with information-theoretic or uncertainty constraints [53-55]. In these approaches, quantum evolution emerges after introducing fluctuation structure at the ensemble level through Fisher-information functionals or uncertainty-constrained momentum fluctuations. MQ differs structurally in that its count ontology supplies conserved encoded availability, adjacency-local redistribution, and the discrete operator structure, while the final Schrödinger form is recovered conditionally within an explicitly stated non-relativistic sector. Appendix O distinguishes these sector conditions from the independently MQ-derived phase-action scale and mass-count realization, so the kinetic coefficient is not treated as a fitted continuum input.

4.5.2 Stochastic Mechanics

Nelson-type stochastic mechanics reproduces Schrödinger dynamics through an assumed underlying diffusion process together with additional constraints needed to recover quantum phase structure [56]. These approaches therefore introduce an ontic stochastic micro-dynamics beneath the quantum formalism. MQ instead derives its conserved availability and mapped redistribution structure from the Internal Frame without introducing stochastic trajectory-level dynamics. Outcome statistics arise from many-to-one realization and contextual normalization under the Frames mapping rather than from microscopic diffusion processes (Appx. D–E).

4.5.3 Entropic and Inference-Based Dynamics

Entropic and inference-based approaches derive quantum evolution from constrained Bayesian or entropic updating rules defined on configuration space [57]. In such frameworks, probabilities and dynamics are both grounded in inference principles. MQ differs in assigning probability only to normalized realized System Frame encoding, while the underlying Internal Frame count structure evolves deterministically through conserved encoded availability redistribution. Quantum probabilities therefore arise from realization structure under the Frames mapping rather than from primitive inferential axioms (Appx. C–E).

4.5.4 Scale-Relativity Constructions

Scale-relativity and fractal-space programs derive Schrödinger-type dynamics by modifying continuum geometric structure through non-differentiable or fractal spacetime assumptions [58]. Quantum behavior is consequently attributed to altered continuum geometry. MQ instead retains ordinary continuum spacetime structure in the realized System Frame while introducing discreteness only through the Internal Frame count configuration domain and its realization under the Frames mapping. Quantum structure therefore emerges from conserved count kinematics rather than modified continuum geometry.

4.5.5 Causal-Set and Discrete-Substrate Constructions

Causal-set theory makes microscopic spacetime itself a locally finite partially ordered set whose order encodes causal precedence [91]. Cellular-automaton and related discrete-substrate approaches likewise place deterministic discrete structure beneath quantum phenomenology [15]. These programs establish that discreteness and deterministic substrate dynamics are not unique to MQ. The MQ primitive differs in kind: its Internal Frame is explicitly non-spatiotemporal and carries conserved measurement-availability counts rather than a discrete spacetime manifold, causal order, or preassigned particle-state trajectory. Geometry, continuum coordinates, and probability all enter only after an admissible Frames mapping. The novelty claim therefore cannot rest on discreteness; it rests on the subsequent count-to-measure, measure-to-probability, and interaction-to-record closure built on that substrate.

4.5.6 Hydrodynamic and Phase-Flow Reformulations

Hydrodynamic formulations rewrite an already available quantum state in terms of density and phase or probability current [72,73,111]. They are important analogues for the System Frame representation used here, and MQ does not claim novelty for amplitude-phase decomposition, continuity equations, or phase-flow variables themselves. The distinction is that MQ derives the normalized measure and mapped current first and then proves, in Proposition C.3, the local minimal U(1) representation of the resulting continuous flow coordinate.

4.5.7 Operational and Informational Reconstructions

Operational and informational reconstruction programs derive Hilbert space quantum theory from axioms governing probabilities, transformations, composition rules, and operational consistency [59]. These approaches characterize the structure of admissible quantum state spaces at the operational level. MQ addresses a complementary structural layer by deriving the probability and redistribution structures from which the encoded System Frame wave function Ψ is constructed, while Schrödinger-form evolution is conditionally reconstructed within the stated non-relativistic admissibility class; interference and contextual probability realization then follow at the realized representation level under the Frames mapping. The resulting System Frame formalism remains compatible with standard Hilbert space operational structure while supplying an underlying count-based kinematic realization (Appx. F).

For MQ, the comparison requires a stricter vocabulary. The normalized System Frame probability measure is derived from mapped encoded availability; the phase gradient is reconstructed from mapped conserved redistribution; and Proposition C.3 fixes the complex U(1) amplitude representation as the unique minimal continuous reversible norm-preserving linear realization of that density-plus-flow data. The position-space squared-modulus identity is then representational, while the arbitrary-observable Born rule remains the later spectral/POVM completion.

4.5.8 Classical-Statistical Reconstructions

Wetterich's classical-statistical reconstruction is a particularly close comparator because it derives wave-function and density-matrix structure, unitary evolution, conditional measurement correlations, entanglement, Bell-inequality violations, and Born-rule probabilities from an underlying classical statistical ensemble [201]. This materially overlaps several downstream MQ claims and therefore must be included in any novelty assessment. The distinction is upstream: Wetterich begins with a classical probability distribution and probabilistic observables, with a purity constraint and suitable time evolution selecting the quantum subsystem. MQ instead begins with a pre-probabilistic conserved integer-count carrier whose counting measure is fixed before normalization, and it identifies empirical record probability only after the completed-context pushforward construction. Wetterich also does not derive the MQ count-normalization residual or the associated fixed finite-count metrological discriminators. The overlap therefore narrows the MQ novelty claim but does not reproduce its full count-to-record-to-differential-prediction chain.

4.5.9 Indivisible Stochastic Reconstructions

The stochastic-quantum correspondence developed by Barandes is an even more recent and broad comparator. It reconstructs quantum theory from generalized or indivisible stochastic processes on ordinary configuration spaces, demotes the wave function to a secondary mathematical representation, incorporates measuring devices as subsystems, and accounts for interference, decoherence, entanglement, noncommutative observables, and collapse-like updating [202]. Subsequent work derives the Tsirelson bound from the causal-locality structure of indivisible stochastic processes [205]. This program therefore overlaps MQ in treating the wave function as non-primitive and in supplying a lower-level account of quantum statistics and Bell correlations. Its primitive starting point, however, is explicitly stochastic and probabilistic, not a conserved pre-probabilistic count measure; its probability law is therefore not obtained by the MQ sequence of counting-measure uniqueness, Frames mapping, completed-context record realization, and pushforward identification. Nor does the stochastic-quantum correspondence presently tie its reconstruction to the MQ finite-count metrological residual or the fixed QG-PPM/G-DIFF discriminators. It is thus a major near-neighbor, but not an equivalent complete architecture under the criteria used here.

4.5.10 Hidden-Measurement Reconstructions

Aerts and Sassoli de Bianchi provide a measurement-centered reconstruction in which the Born rule arises as a universal average over hidden measurement interactions and an individual measurement is represented as a tripartite deterministic-indeterministic-deterministic process [203]. This is directly relevant to MQ because it addresses outcome actualization rather than only Schrödinger dynamics. The construction nevertheless starts from the generalized Bloch representation of quantum states and from a distribution over measurement interactions; it does not derive a pre-probabilistic conserved count carrier or identify probability as the pushforward of a completed-context count measure. Its explicitly indeterministic middle stage also differs from MQ's deterministic count redistribution plus single-valued completed contextual realization. The hidden-measurement program therefore supplies an important analogue of probability-and-record closure without reproducing the MQ upstream ontology or cross-domain finite-count predictions.

4.5.11 Physical-Requirement and Informational Reconstructions

Operational reconstructions are broader than the Hardy example alone. Chiribella, D'Ariano, and Perinotti derive quantum theory from informational principles, with purification selecting quantum theory within a wider operational class [143], while Masanes and Müller reconstruct the full formalism from physical requirements on preparations, transformations, and measurements [204]. These works show that Hilbert-space structure, composition, and measurement probability need not be primitive mathematical postulates. Their primitives are nevertheless operational probabilities, transformations, and distinguishability relations. They do not posit or derive a pre-probabilistic conserved count substrate, a completed-context count-to-record pushforward law, or a common finite-count realization rule generating independent metrological corrections. They therefore overlap strongly with MQ at the level of formal reconstruction but not at the proposed upstream physical carrier and differential-prediction link.

4.6 Precise Novelty Claim: The Referenceable Count-to-Record Construction

The comparison above, supplemented by the systematic architecture audit in Appx. AU, narrows the novelty claim. No single result in this paper is asserted to be without precedent merely because it uses discreteness, normalization, a continuity equation, a phase field, a Hilbert-space completion, a discrete-to-continuum limit, a lower-level stochastic or statistical description, or a measurement-specific actualization mechanism. Theorem 2.3 is a conditional record-consistency statement rather than a claim to have discovered the measurement problem; the arithmetic closure result of Appx. M is likewise not presented as a new theorem of number theory. What is proposed as distinctive is the conjunction of the following requirements within one upstream construction.

Pre-probabilistic conserved substrate. The primitive Internal Frame object is a nonnegative, finitely additive, adjacency-conserved availability count. Probability, metric structure, continuum coordinates, and wave-function variables are absent at this level (Secs. 2.6–2.8; Appx. A–C).

Mapping-derived probability rather than primitive probability. Frames mapping induces the System Frame availability measure; normalization supplies dimensionless outcome weights, and the completed-interaction pushforward identity then identifies those same weights as the empirical record-probability measure before a wave function is constructed (Secs. 3.1–3.4; Appx. C and H). An admissible

Primitive-measure closure. The pre-probabilistic measure is not selected from a family of normalized distributions. Integer unit counts, finite additivity, and admissible relabeling invariance uniquely fix the counting measure on each finite MQ domain (Proposition 2.1); only the physical multiplicities generated by preparation, deterministic count evolution, completion, and realization vary. The pushforward probability law therefore has no independent primitive weighting parameter.

Outcome realization linked to probability without collapse. The completed contextual realization M_c and the mutually exclusive System Frame outcome partition induce R=κ∘M_c, so each complete admissible interaction has one realized label. The pushforward theorem of Sec. 3.2.1 and Appx. H.9 then proves that the distribution of those single realized labels is exactly the normalized availability measure, while injective record encoding makes the realized label distinguishable and only afterward do restriction and renormalization update the predictive state. This closes the individual-outcome/ensemble-frequency link without adding a stochastic selection postulate. The admissible interaction x is itself generated by the deterministic completion map X_c from the actual initial joint count configuration and context to the first stable record, so R does not act on an unspecified “whatever occurred” terminal state.

Bell closure without a measurement-dependence loophole. The preparation configuration is explicitly independent of analyzer settings, P(C_i|a,b)=P(C_i), while MQ-derived contextual System Frame partitions are non-factorizable and operationally no-signaling (Sec. 3.7; Appx. G.1–G.10). Appendix G.11 further derives the ideal polarization analyzer's rotation law from the normalized two-channel MQ amplitude: count-derived norm preservation, linear channel representation, continuity, and composition under physical rotation force the SO(2) map, whose squared channel weights give the cos²/sin² dependence. The Bell angular response is therefore included in the first-principles MQ chain for this ideal benchmark rather than inherited as an external projection law.

Rotational spin closure. The MQKB independently fixes a half-Planck fundamental angular-momentum relation. On the finite contractible System Frame realization class, Appx. R combines that scale with the spin lift of realized rotations and first-order Clifford factorization. The spatial Clifford generators satisfy S²=3ℏ²/4, so the SU(2) Casimir relation fixes j=1/2 without a separate elementary-minimality assumption. Pauli generators, 4π spinor periodicity, the two-channel Stern–Gerlach law, and the Pauli magnetic coupling then follow as representations of the already-derived MQ probability and relativistic structures rather than as primitive spin axioms.

Coefficient-resolved dynamics and cross-domain falsifiability. The discrete-to-continuum construction supplies the Laplacian within the stated non-relativistic sector, while the phase-action scale is fixed independently from MQ fundamental measures rather than fitted to the Schrödinger equation (Sec. 3.5; Appx. O). The same specific mapping-normalization rule also generates the Informativity differential and its parameter-free finite-count metrological predictions (Sec. 2.9.2–2.9.1; Appx. J). The same count-resolved phase-action relation also yields the quantum-specific gravitational-resonance prediction of Appx. L.2: a fixed +11.6426596 ppm transition-frequency shift relative to standard quantum mechanics under the stated upper-count realization conditions.

The systematic comparison in Appx. AU includes the principal reconstruction families most capable of challenging this conjunction, including exact-uncertainty/Fisher-information methods, stochastic mechanics, entropic dynamics, scale relativity, causal-set and cellular-automaton approaches, hydrodynamic reformulations, operational/informational reconstructions, Wetterich's classical-statistical reconstruction, Barandes's indivisible stochastic reconstruction, and the hidden-measurement program [15,53-59,72,91,143,200-205]. Several of these programs reproduce substantial subsets of the MQ chain. Wetterich, in particular, derives Born probabilities, unitary dynamics, entanglement, and Bell-inequality violations from classical statistics [201]; Barandes reconstructs quantum theory from indivisible stochastic processes and now supplies a causally local route to the Tsirelson bound [202,205]; and Aerts and Sassoli de Bianchi derive Born probabilities from a measurement-interaction actualization model [203]. Accordingly, the defensible novelty claim is not that MQ uniquely derives probability, records, Bell correlations, or quantum dynamics in isolation. It is the more restricted conjunction that one non-spatiotemporal pre-probabilistic conserved count carrier is used to fix the primitive measure, generate a System Frame measure through the Frames mapping, identify empirical record probability by the completed-context pushforward before conditioning, recover the stated quantum dynamical and Bell structures, and generate fixed finite-count differential predictions from the same realization rule. Within the surveyed peer-reviewed and foundational reconstruction literature, we did not identify another program establishing this entire conjunction. This is a bounded literature claim, not a theorem of historical uniqueness: discovery of a prior construction satisfying all of these criteria would require the priority language to be narrowed while leaving the internal MQ derivations unaffected.

This distinction also makes the claim falsifiable. If another reconstruction is shown to realize the same complete chain, the priority claim should be narrowed without affecting the internal MQ derivations. More importantly, failure of any required MQ link—count conservation, mapping admissibility, unique record realization, preparation independence, non-factorizable availability, operational no-signaling, analyzer-rotation closure for the ideal polarization benchmark, coefficient closure, or the pre-specified mapping-dependent metrological correction—would invalidate the corresponding part of the construction. The contribution is therefore a concrete linked architecture with identifiable failure points rather than a generic assertion that quantum mechanics can emerge from discreteness.

4.7 Limitations and Scope of the Present Framework

The MQ construction developed here provides a first-principles kinematic foundation for System Frame availability, probability, redistribution, and pre-Hilbert state geometry, with coefficient-resolved Schrödinger recovery in the non-relativistic realization, a locally count-order-selected Lorentzian causal signature (Proposition 2.6; Appx. X), a mapped-invariant derivation of the connected local Lorentz group and its Galilean contraction (Proposition 2.7; Appx. Y), a finite adjacency-selected higher-dimensional topology (Proposition 2.9; Appx. AJ), and a global cycle/holonomy classification (Proposition 2.8; Appx. Z). For the elementary massive charged matter channel actually used by the Abelian quantum continuation, Proposition 3.3/Appx. AK now selects the Dirac Clifford module rather than taking that representation class as an extension-specific input; Appx. R then fixes its local j=1/2 rotational content. The explicit scalar field sector is retained as a distinct bosonic System Frame realization, while the Lorentz-covariant one-particle and scalar/Dirac/Abelian QFT continuations remain as developed in Secs. 3.9.1–3.9.2 and Appxs. P–R, AK. Appendix AL additionally proves that the current adjacency-only and scalar-gauge data do not uniquely select a continuous non-Abelian gauge group, converting that question from an implicit missing derivation into an explicit universality boundary with stated necessary inputs.

Measurement theory is formulated at the operator and POVM level within the Hilbert space encoding, while Sec. 3.2.2 and Appx. V now supply an explicit device-level realization for a representative physical detector. The general bridge is a calibrated System Frame response matrix from ideal MQ outcome labels to stable apparatus records; composing that response with the MQ event measure gives the observed record distribution, and after Hilbert completion the same response gives the corresponding noisy POVM. Appendix V instantiates this construction for a gated on/off avalanche photodetector with finite quantum efficiency, dark counts, threshold readout, and detector-tomography calibration [137-140]. The completed-interaction criterion remains the kinematic first-stable-record closure of Appx. H.8; the device model determines when a particular apparatus enters a stable record region, rather than supplying a second outcome-selection law. Other detector architectures require their own calibrated response functions, but the abstract-to-device composition rule is no longer left unspecified.

The Frames mapping is specified here by an explicit admissibility hierarchy: measure-level count preservation and local finiteness; adjacency and boundary-flux compatibility; finite normalization; relational relabeling invariance; single-valued completed-context realization; injective stable record encoding; holonomy-free closure on contractible cycles; and, wherever a continuum metric is reconstructed, the local C³, full-rank, nondegenerate-image and smooth-chart conditions of Sec. 2.8.1 and Appx. N. Proposition 2.3 and Appx. C.4.2 construct at least one representative on every finite connected measurement domain used here. Proposition 2.2 and Appx. C.4.3 then remove representative ambiguity by quotienting admissible maps by invariant-preserving Frames automorphisms and by classifying global inequivalence through the regular domain and defect holonomies. On the bounded contractible domains required by the present quantum derivations, no nontrivial defect class is present and the mapping is physically unique up to this automorphism freedom. This is the relevant uniqueness result for the paper: it is uniqueness of physical realization content, not uniqueness of coordinates, auxiliary basis functions, or other gauge-like representatives.

The primitive-measure question is correspondingly closed at two nested scales. Proposition 2.1 proves uniqueness of the count measure up to a common scale on every finite domain, and normalization removes that scale. Proposition 2.2 closes the finite mapping-representative question by identifying all invariant-preserving representatives as one physical equivalence class. Proposition 2.10/Appx. AO then shows that a compatible family of those finite classes has a unique physical extension on any connected locally finite domain, including a maximal referenceable domain when such an exhaustion exists. What remains outside the present result is non-locally-finite structure or failure of overlap compatibility, not a freedom to choose different continuum laws in the thermodynamic limit.

Appendix T closes the record-probability question on every finite measurement context. Theorem T.1 proves uniqueness of the normalized record-event measure for each completed context, not merely uniqueness of the underlying pre-probabilistic count measure. Proposition 2.10/Appx. AO extends these laws to a connected locally finite ambient domain by requiring projective consistency of the finite-context marginals; the Kolmogorov extension theorem then fixes the cylinder-event probability measure [181]. No normalization of an infinite total count is required. Arbitrary projective and generalized quantum measurements continue to use the Hilbert-space spectral/POVM completion and the Gleason/Busch uniqueness results of Sec. 3.4.3 and Appx. F.18–F.19.

The local metric-selection question is now closed for both regular direction-free and regular quadratic anisotropic sectors. Proposition 2.4/Appx. AI show that absence of a direction-selecting invariant forces the isotropic inner-product norm and Appendix M then fixes Q_L(n_L)=sqrt(1+n_L^2)-n_L and 2Q_Ln_L=1-Q_L^2. Proposition 2.5/Appx. AQ treats the complementary case: inequivalent primitive adjacency channels, anisotropic matter response, boundary constraints, or defect/holonomy data enter the Hessian of the same MQ Action, whose mapped principal tensor uniquely fixes the local anisotropic metric. Thus anisotropy is not a free mapping representative or adjustable Finsler function. Remaining metric boundaries are limited to degenerate, nonlocal, or genuinely higher-order direction-dependent action sectors for which the required MQ response has not yet been derived.

These scope conditions bound the domain of the present derivations while leaving intact the internal mathematical results obtained for the MQ System Frame encoding, its non-relativistic quantum limit, the Lorentz-covariant one-particle extension of Sec. 3.9.1/Appx. P, the scalar/Dirac/Abelian quantum-field-theoretic extension of Sec. 3.9.4/Appx. Q, and the semiclassical and fluctuation-sensitive stochastic backreaction closure of Sec. 3.10/Appx. AB. The QFT result supplies variable particle number, local field operators, vacuum/particle sectors, particle creation, and a relativistic electromagnetic interaction; Appx. AB then feeds the renormalized quantum matter mean and derived stress–energy covariance back into the emergent gravitational response. Neither result is yet a derivation of the full Standard Model or a coherently quantized gravitational field.

Finite-partite composite kinematics are not an additional limitation within this domain: Appx. F.14-F.14.1 derives the N-fold tensor product and establishes the standard Schmidt, purification, and pure-state monogamy consequences. Remaining composite-system questions concern dynamical reach and classification on unbounded, relativistic, field-theoretic, or superselected state spaces, not freedom to replace the finite tensor-product law used in the present quantum construction.

Because these results span several specialist domains, scope is also controlled structurally rather than only narratively. Appendix AE provides the verification map for the complete manuscript. The core measurement reconstruction is isolated from its later extensions, and every extension lists the upstream MQ results it consumes. A failure in a downstream relativistic, field-theoretic, topological, gravitational, or empirical extension therefore does not automatically invalidate an upstream measurement result unless the dependency map identifies that path; conversely, failure of a listed upstream prerequisite propagates only to the dependent claims. This makes the breadth auditable as a directed derivational architecture rather than as one undifferentiated omnibus claim. For editorial triage, Sec. 2.2.2 compresses this architecture into four bounded reading paths before the full dependency tables are encountered.

4.7.1 Empirical Consequences and Falsifiability

Within the domain derived here, MQ is intended to reproduce standard non-relativistic quantum statistics, including the position-space probability measure, the spectral/POVM Born rule after Hilbert-space completion, interference, Schrödinger evolution, operator expectations, and Bell–CHSH correlations. Agreement with these phenomena is therefore a consistency requirement rather than, by itself, a discriminating test. A useful falsification analysis must instead identify which layer of the MQ construction a failed prediction would invalidate. This distinction is standard in quantum-foundations testing: experimentally distinguishable alternatives require a quantitative prediction that differs from standard quantum mechanics within a stated domain and uncertainty model [76], while assumption-specific no-go results test narrower ontology classes rather than every interpretation at once [77].

The following audit separates evidence for the physical phenomena that MQ must reproduce from evidence that discriminates MQ from standard quantum mechanics. The former can strongly support the physical premises and correspondence structure without independently confirming the MQ ontology; only a result fixed by MQ before an independent target measurement can provide strong theory-specific evidence.

Single-event interference accumulation. MQ requires individual realized records to accumulate into an interference distribution rather than requiring each detection event itself to be spatially extended. Tonomura et al. directly recorded the buildup of an electron interference pattern from successive single-electron detections, and Bach et al. later realized a controlled two-slit electron experiment in which individual detection events accumulated into the expected diffraction pattern [94,95]. This is strong empirical support for the coexistence of discrete realized records and ensemble interference, but it is a Level-III correspondence test rather than an MQ-specific discriminator.

Context-dependent suppression of interference. MQ assigns which-path restriction to the realized System Frame measurement context. Dürr, Nonn, and Rempe experimentally showed in an atom interferometer that acquiring which-way information reduces or removes fringe visibility through correlations with the path marker, with the measured visibility-distinguishability relation agreeing with quantum complementarity [96,64]. This strongly supports the required context-sensitive interference behavior.

Coherent phase structure and environmental decoherence. MQ treats relative phase as a System Frame representation of mapped redistribution and allows environmental coupling to suppress observable coherence without changing primitive count conservation. Brune et al. observed the progressive decoherence of a mesoscopic cavity-field superposition, and Myatt et al. measured controlled decoherence of trapped-ion motional superpositions coupled to engineered reservoirs [97,98]. These experiments strongly support the physical decoherence mechanism used by MQ at the realized representation level.

Persistence of superposition beyond microscopic particles. MQ does not introduce a separate classical ontology at larger mass; instead it predicts a coarse-grained classical regime of the same realized quantum structure. Matter-wave interference has been observed for C60 molecules and, more stringently, for molecular systems above 25 kDa containing up to about 2,000 atoms [99,100]. These results strongly support the continuity of superposition behavior into increasingly complex systems and constrain any MQ classical-limit construction to recover that persistence.

Stable, repeatable measurement records. MQ requires completed measurements to yield distinguishable records that can be reread with unit probability in the ideal limit. Quantum-nondemolition cavity experiments have repeatedly interrogated the same photon-number state: Gleyzes et al. resolved individual photon birth and death events, while Guerlin et al. demonstrated progressive photon-number determination and correlations between repeated measurements [101,102]. This is strong empirical support for the repeatability and stable-record premises used in Secs. 2.4, 3.2, and 3.8. These experiments operationally instantiate the stable-rereading criterion used in the completed-interaction construction of Appx. H.8; they support the physical availability of such record closure without selecting MQ over standard quantum mechanics.

Born-rule probability structure. MQ must reproduce the standard probability calculus after its realization-frequency identification. Sinha et al. performed a dedicated three-slit test of Born's rule and bounded genuine third-order interference to below 10^-2 of the ordinary second-order interference scale in their experiment [103]. The result does not validate the MQ derivation of probability, but it strongly supports the empirical target that MQ's normalized record probabilities must match.

Contextuality of realized measurement outcomes. The MQ outcome map is explicitly contextual and does not assign setting-independent pre-existing eigenvalues. Kirchmair et al. experimentally demonstrated state-independent quantum contextuality with trapped ions, ruling out the tested noncontextual description even after accounting for experimental imperfections [104]. This provides strong external support for the necessity of contextual realized-outcome structure of the type respected by MQ.

Non-factorizable Bell correlations. MQ's composite availability construction must reproduce Bell-inequality violation while not imposing Bell-local factorization. Hensen et al. closed the principal locality and detection loopholes using electron spins separated by 1.3 km [44], while independent photonic experiments by Giustina et al. and Shalm et al. also obtained statistically significant loophole-free violations [105,106]. These results are among the strongest empirical constraints on any MQ composite-system realization.

Measurement independence in viable Bell tests. MQ separately requires preparation independence from later analyzer settings. The loophole-free Bell experiments used space-like separation and independently generated, rapidly selected measurement settings to prevent ordinary subluminal coordination of setting choice and outcome formation [44,105,106]. They do not prove the MQ preparation model, but they strongly support the relevance of preserving measurement independence rather than explaining Bell violation by a setting-dependence loophole.

Operational no-signaling. MQ directly requires local observed marginals to remain invariant under the remote measurement setting. The same space-like-separated Bell experiments produced quantum correlations without an operational faster-than-light signaling channel and are consistent with the standard no-signaling marginal structure [44,105,106]. This is strong compatibility evidence for the MQ Bell layer, although it does not uniquely select MQ over other no-signaling quantum descriptions.

Decoherence stabilizes alternatives but does not experimentally identify an ontic selector. The cavity and trapped-ion decoherence experiments directly establish loss of coherence and record stabilization effects [97,98], but they do not experimentally identify a unique microscopic mechanism that chooses one outcome from all mathematically represented alternatives. The evidence therefore strongly supports the decoherence component of Sec. 4.2 while leaving MQ's completed-context realization map as a theory-specific explanatory proposal rather than an independently observed process.

Non-relativistic Schrödinger correspondence. Atomic, ionic, electron, molecular, and cavity experiments cited above are quantitatively described by standard linear quantum evolution over their coherent intervals [94-103]. MQ is required to recover that same domain, and Sec. 3.5/Appx. O deliberately limits the claim to coefficient-resolved conditional reconstruction in the Galilean non-relativistic sector. The experimental support for the target dynamics is overwhelming, but because standard quantum mechanics already predicts it, successful recovery remains compatibility evidence rather than distinctive confirmation.

Hilbert-space and generalized-measurement representability. MQ postpones spectral and POVM structure until Hilbert-space completion. Generalized measurements are not merely formal devices: Martínez et al. implemented and certified a seven-outcome non-projective measurement in a four-dimensional photonic Hilbert space with 99.7% fidelity [107]. This strongly supports the empirical adequacy of the POVM-level representation that MQ adopts after, rather than before, its count-to-probability construction.

Absence of observed Lorentz-violating signatures in the tested sectors. MQ's Internal Frame discreteness is non-spatiotemporal and the Informativity differential is not a boost-dependent contraction. Existing SME compilations, gravitational-wave/light-speed comparison from GW170817/GRB 170817A, gamma-ray timing bounds, and pulsar tests place stringent limits on preferred-frame, birefringent, dispersive, and gravity-sector Lorentz violation [85-89]. The absence of such signals is therefore compatible with MQ's present claim that its finite-count mapping correction is not itself a Lorentz-violation operator.

Experimental accessibility of the X-ray nonlinear platform. The Shwartz-Harris 3.26239-rad Bell-state angle remains a model-derived value rather than an independent MQ measurement [35], but X-ray parametric down-conversion and X-ray/optical wave mixing have been experimentally observed in the relevant class of nonlinear processes [36,37]. This is strong evidence that the proposed platform is physically accessible, while Appx. L correctly withholds any claim of Level-IV confirmation until a genuinely differential MQ prediction is measured.

Prospective realization-dependent gravitational-coupling test. The strongest presently available theory-specific empirical comparison remains the 2026 NIST/BIPM torsion-balance redetermination [80]. The MQ finite-count value predates that measurement and differs from the combined NIST result by only 0.066 diagonal standard deviations using the quoted combined uncertainty. More importantly, the experiment provides two readout modes at fixed source geometry. The pre-existing MQ finite-to-upper separation is 34.9274 ppm; the published free-minus-servo differences are 56.7889 ppm for copper and 149.6888 ppm for sapphire. Propagating the published four-result covariance matrix gives a two-component differential χ² = 1.85 (ν = 2, p = 0.40) against the common 34.9274-ppm separation. This is a stronger magnitude-and-covariance consistency check than the sign comparison alone, but it is not counted as a second independent confirmation because the exact apparatus-specific regime assignment was not preregistered and the NIST analysis retains configuration-dependent unexplained uncertainty. Appendix J.8 therefore classifies the combined out-of-sample numerical comparison as the sole completed independent MQ-specific target and the mode comparison as a covariance-resolved post-publication diagnostic. A decisive follow-up remains a blinded, preregistered same-apparatus differential protocol with fixed geometry and alternated electrostatic-servo versus free-deflection readout. The remaining causal ambiguity is now removed at the theory level by the channel-selection rule of Sec. 2.9.2 and Appx. J.9. The MQKB fixed-count/count-resolved distinction assigns the electrostatic-servo mode to the finite-count realization because an electrostatic counter-torque closes the torque estimator, while free-deflection mode is assigned to the count-resolved gravitational realization because its torque is obtained from mechanical deflection and torsional stiffness. The two-mode apparatus architecture and torque equations were published independently in 2022 [142]. A future blinded repetition therefore has a frozen sign and magnitude prediction rather than a post-hoc mode label.

Mechanically driven gravitational quantum-state spectroscopy. The gravity-resonance platform required by Appx. L.2 has already been demonstrated with ultracold neutrons [108,109], and the later Ramsey implementation measured the |1⟩→|6⟩ transition as ν₁₆ = 972.81(12) Hz while identifying frequency shifts of order 100 mHz from instrumental and dynamical effects [141]. The MQ Level-IV prediction is not a refit of those data: at independently measured local g it is the pre-specified common +11.6426596 ppm displacement of every mechanically driven gravitational transition assigned to the upper-count realization, corresponding to +0.011326 Hz for ν₁₆. The published 0.12-Hz uncertainty is about 123 ppm, roughly 10.6 times the MQ displacement at 1σ, and a nominal 5σ discrimination requires a combined relative uncertainty below 2.33 ppm together with systematic residuals below the 11.3-mHz signal. The published experiments therefore establish accessibility of the observable but do not test the MQ prediction at the required precision. No existing result used in this manuscript is promoted to Level-IV confirmation; the decisive test remains prospective and its locked analysis conditions are collected in Appx. AC.

Interacting relativistic QED target. The field-theoretic extension of Sec. 3.9.4/Appx. Q must recover the extraordinary precision of ordinary QED in its flat-background electromagnetic domain. The 2023 single-electron magnetic-moment measurement tests the Standard Model/QED prediction at approximately one part in 10^12 [126]. This does not confirm MQ-specific field ontology, but it provides an exceptionally stringent correspondence target: any MQ QED realization that altered the ordinary local electromagnetic sector above this level would already be excluded. This does not test the non-Abelian universality program separated in Appx. AA; it tests the interacting U(1) correspondence actually derived here.

Spin-1/2 rotational structure and double-valued rotations. The MQ rotational completion of Sec. 3.9.3/Appx. R has three independent correspondence targets. First, the Stern–Gerlach experiment established discrete directional splitting of atomic magnetic moments [131]. Second, neutron interferometry directly observed the fermion wave-function sign reversal after a 2π spin rotation [132], while an independent perfect-crystal neutron experiment found interference oscillations with the predicted 4π periodicity [133]. Third, the Pauli/Dirac magnetic coupling gives g=2 at leading order [130], with the modern electron magnetic-moment measurement constraining the complete QED realization at the approximately 10^-12 level [126]. These experiments do not independently prove the MQ half-Planck-to-SU(2) derivation, but they sharply constrain every observable consequence of the resulting spinor sector.

Quantum-matter backreaction and the full-quantum boundary. Quantum field theory in curved spacetime supplies the renormalized stress–energy operator and semiclassical mean source [124,158], while stochastic gravity shows that the connected stress–energy noise kernel drives metric fluctuations omitted by a mean-field equation [159,169]. Appendix AB now imports that distinction into MQ without adding a primitive random source: the Appx. Q state is itself downstream of the unique count measure, so its renormalized stress–energy covariance fixes the stochastic System Frame source and the resulting linear-response metric covariance. The mean-source equation is recovered when connected fluctuations are negligible. Page and Geilker still exclude promotion of the simplest universal expectation-value rule to a fundamental microscopic law [157]. The remaining full-quantum gap is coherent superposition/interference and operator algebra for inequivalent realized geometries, not the absence of a fluctuation-sensitive backreaction law.

Taken together, this evidence materially strengthens the empirical footing of the construction while also defining its present maturity without ambiguity. Items (1)-(15) show that the principal phenomena and operational constraints MQ is built to reproduce are independently established across electron, atom, ion, molecular, cavity-QED, photonic, Bell, nonlinear-X-ray, and relativistic tests. They therefore support compatibility of the MQ architecture with the observed quantum domain, not a unique causal inference to MQ. Item (16) remains the only completed out-of-sample MQ-specific metrological target; its internal servo/free comparison adds a covariance-resolved magnitude diagnostic but not another independent empirical degree of freedom. Item (17) is the strongest quantum-specific discriminator and remains unexecuted at the required precision. Items (18-23) constrain the interacting relativistic, spinor, and matter-geometry extensions but remain correspondence or boundary tests. Appendix AC freezes the two principal confirmatory experiments—QG-PPM and G-DIFF—while Appx. AM now audits their experimental and theoretical dependence and identifies the strongest non-gravitational diversification route already present in the MQKB. That third route is not promoted to independent confirmation because its presently used cosmological age input is itself CMB-derived; the paper therefore does not manufacture empirical independence by relabeling a correlated comparison.

Level I — Count-based ontology. The deepest falsification target is the claim that physically realized structure can be grounded in conserved discrete availability counts on the Internal Frame relational adjacency domain. Within the present paper, this layer is constrained by non-negativity, finite additivity, adjacency-local redistribution, and closed-domain count conservation (Eqs. (3–13) and (57)). Theorem 2.2/Appx. AN now makes the completeness criterion operational: additional primitive data that never alter any admissible System Frame observable or completed record are physically redundant within the claimed domain, whereas any additional data that do alter an observable define a distinct theory and a Level-I discriminator. Evidence that the claimed domain requires primitive continuum coordinates, a non-count additive state label, non-conserved creation or destruction of availability count, or any other independently observable primitive degree of freedom would therefore falsify this ontological layer. Failure here would invalidate the subsequent realization construction rather than merely one particular representation.

Level II — Specific Frames mapping and mapping-normalization realization. Conditional on the count ontology, the next target is the specific admissible realization rule connecting conserved count structure to observable System Frame quantities. The fixed, non-fitted Informativity differential, the associated mapping-induced corrections to realized measures, and the cross-domain residual comparisons summarized in Appx. J are predictions of this stronger structure, not consequences of discreteness alone. A statistically interpretable failure requires a pre-specified independent comparator or a covariance-resolved joint residual analysis. Persistent disagreement under that standard would falsify the specific MQ realization rule even if a more general count-based ontology remained logically possible.

Level III — Derived quantum correspondence. The third target is the claim that the admissible MQ realization reproduces the established non-relativistic quantum domain. Failure of the realization-frequency identity in Eq. (32)—that is, a failure of the pushforward distribution of completed interactions to equal the normalized outcome measure—would directly falsify the MQ measurement correspondence before any Hilbert-space completion is invoked. Failure to recover the normalized position probability measure, the spectral/POVM Born rule under the stated Hilbert-space completion, interference, the conditionally selected Schrödinger generator, or the observed Bell–CHSH/no-signaling structure would likewise falsify the quantum correspondence developed in Secs. 3.1–3.11 and Appx. C–H. Because standard quantum mechanics already predicts these phenomena, successful recovery at this level establishes consistency but does not by itself discriminate MQ from standard quantum theory.

Level IV — Quantum-specific differential prediction. Appendix L.2 now supplies a target-parameter-free observable that differs from standard quantum mechanics. The MQKB canonical count-resolved phase-action relation gives hbar(n_L) = theta_si l_f/[Q_L(n_L)n_L], with Q_L(n_L) = sqrt(1+n_L^2) - n_L. At the electromagnetic demarcation n_L = 276/theta_si this yields the ordinary SI-scale reduced Planck value, while the upper-count limit is 2 theta_si l_f. For the gravitational quantum bouncer, the standard Airy spectrum scales as nu_jk proportional to hbar^(-1/3). Under the pre-specified MQ assignment that the gravitational level spacing is an upper-count realization, the resulting prediction is nu_jk^MQ/nu_jk^QM = 1.0000116426596: an upward fractional shift of 11.6426596 ppm for every transition at fixed independently measured local g and neutron mass. Standard quantum mechanics predicts unity because it uses one universal hbar for the same Hamiltonian. The mechanically driven ultracold-neutron gravity-resonance experiments of Jenke et al. and Cronenberg et al. provide a demonstrated platform [108,109], but their published precision does not constitute a test of this ppm-scale prediction. The earlier X-ray theta_si correspondence in Appx. L.1 remains a non-discriminating cross-correspondence because the same angle follows from the standard nonlinear-optics calculation [35-37].

The hierarchy prevents a negative result from being overinterpreted. A sufficiently precise null result for the pre-registered gravitational transition-frequency ratio of Appx. L.2 would reject the count-dependent phase-action realization used for that Level-IV prediction while leaving the already recovered standard-quantum correspondence intact; a Level-III failure would reject the MQ quantum realization; a Level-II failure would reject the specific Frames mapping realization used across MQ; and a Level-I failure would challenge the count-based ontology itself. Conversely, agreement at one level does not prove the stronger levels above it.

This layered structure also clarifies the status of the existing metrological evidence. The comparisons in Appx. J probe Level-II consequences only to the extent that their reference data are independent of the MQ source quantities. Rows with shared CODATA inputs or algebraic dependence are traceability checks, not independent Level-II tests; successive CODATA releases are historical adjustments and are not separate degrees of freedom. The quantum correspondence results of Sec. 3 remain Level-III tests because MQ is constructed to recover standard quantum statistics in the stated regime. In contrast, Appx. L.2 states a Level-IV observable for which MQ and standard quantum mechanics make different numerical predictions before measurement. Its evidential status remains prospective until the MQ regime assignment, independent g calibration, transition frequencies, systematic corrections, and uncertainty threshold are fixed before data analysis.

4.7.2 Open Problems and Required Future Work

Open problems no longer include existence or uniqueness of the realization limit on connected locally finite fixed-adjacency domains, local edge rewiring on a fixed primitive cell set, regular anisotropic metric selection, the scalar/Dirac/U(1) relativistic state-space architecture, or the existence of a coherent geometry-class representation on the same count-derived MQ state. Proposition 3.5/Appx. AS now show that inequivalent physical geometry classes define geometry-class subspaces of the already-derived Hilbert state; relative phase is inherited from the MQ phase-action field, invariant geometry functionals become derived observables, and geometry-class-mixing evolution supplies the noncommuting/ interference structure absent from stochastic c-number geometry. Remaining structural problems include genuinely non-locally-finite primitive domains, creation or annihilation of primitive relational cells, action-degenerate graph-rewrite loci, degenerate/nonlocal/nonquadratic anisotropic response, and derivation of additional matter classes not covered by Proposition 3.3. The non-Abelian problem is more sharply delimited by Appx. AL than before: MQ need not guess SU(2), SU(3), or another compact group directly. Once a count-derived family of localized, transportable charge classes with composition, conjugation, and statistics has been obtained, the Doplicher-Roberts reconstruction fixes the associated compact gauge group up to the standard physical equivalence. What remains open is therefore the genuinely physical antecedent: derivation from MQ invariants of the required non-Abelian sector category and its chiral matter content, followed by anomaly cancellation, symmetry breaking, Yukawa structure, confinement, and associated renormalization flow. For gravity, the remaining universality question is no longer whether coherent geometry can be represented, but whether the geometry-class-mixing dynamics can be extended nonperturbatively to all admissible topology-changing domains and confirmed experimentally.

The finite composite-state architecture and the interacting state-space selection for the presently derived Abelian relativistic class are no longer included among these open items. Proposition F.3 fixes finite multipartite tensor composition. Proposition 3.4/Appx. AR then promotes locality and the MQ probability functional to the quasi-local observable algebra and its admissible state representations, so Fock space is a derived free/stationary realization rather than a globally imported choice. What remains open is the physical derivation of additional superselection content associated with matter or gauge structures that MQ has not yet generated, together with nonperturbative existence/phase questions for specific interacting models. Those residual questions do not leave the scalar/Dirac/U(1) state space used here arbitrary.

The distinction is important: Appx. AI removes quadratic/isotropic metricity from the independent assumption list for the direction-free local sector, while Proposition 2.5/Appx. AQ closes the constructive regular anisotropic sector. Once direction-selecting adjacency, matter, boundary, or defect invariants are fixed, the local Hessian of the single MQ Action determines the anisotropic principal tensor and hence the spatial metric uniquely up to the already-quotiented representative freedom. What remains open is not the choice of an anisotropic tensor, but only sectors in which the action response is degenerate, nonlocal, or genuinely nonquadratic and therefore requires additional MQ-derived physical response data.

Primitive probability weighting is not included among those open problems. On the finite measurement domains used here, Proposition 2.1 excludes nonuniform primitive weights without adding new physical structure; Appx. H.9 then shows that the unique count measure is pushed forward to the observed outcome distribution. What remains preparation-dependent is the dynamically realized count configuration, not the rule by which its unit counts are measured.

The bare choice of nonnegative integers is likewise no longer left as an unexplained numerical primitive for the finite measurement class. Theorem 2.1 and Appx. W derive it as the unique minimal multiplicity carrier compatible with finite distinguishability, additive composition, relational relabeling invariance, and closed conservation when no additional primitive scalar structure is admitted. Theorem 2.2/Appx. AN then closes the distinct physical-completeness question to the strongest scientifically decidable level. A proposed deeper structure that is invisible to every admissible realization and completed record is empirically equivalent and does not enlarge the physical state space of the theory; a proposed deeper structure that changes any admissible observable is an inequivalent, falsifiable completion. Absolute denial of empirically inert metaphysical structure is neither required nor claimed. A future experiment requiring an irreducible primitive continuum label, a non-count additive conserved quantity at the same ontological level, or a stable-record precursor not representable by finite multiplicity would therefore falsify the count-only completion rather than merely alter its normalization.

The Level-IV derivational gap is closed at the present theory level by Appx. L.2, but its experimental realization remains open. A decisive protocol must pre-register the MQ count assignment for the gravitational-state dynamics and downstream record channel; determine local g independently of neutron transition frequencies; measure at least two mechanically driven gravitational transitions; fit one common fractional displacement rather than transition-specific offsets; and propagate mirror-motion, vibration, surface, magnetic, neutron-velocity, detector, and local-g systematics into the full frequency covariance. The MQ prediction is the fixed common +11.6426596 ppm shift of Appx. L.2, whereas the standard-QM null is zero fractional shift after conventional systematic corrections. For the published Ramsey transition ν₁₆ = 972.81(12) Hz [141], the predicted displacement is +0.011326 Hz; the quoted uncertainty is about 10.6 times larger than that signal, while a nominal 5σ test requires combined uncertainty below 2.33 ppm and systematic control below 11.3 mHz. A result consistent with zero at that sensitivity would falsify this count-dependent phase-action realization. Appendix AC.2 freezes the corresponding confirmatory analysis rule before any new dataset is considered.

For the gravitational-coupling channel test, the theoretical assignment is frozen but causal demonstration remains open. Appendix J.9 derives the selector from MQKB interaction provenance: the interaction that supplies the calibrated force or torque at estimator closure determines the realization, while passive sensing does not. Appendix AD converts that statement into a prospective causal experiment. The decisive comparison uses randomized or blinded free-deflection and electrostatic-servo blocks with the same gravitational source geometry and common analysis, supplemented by a negative-control block in which the electrostatic hardware remains energized and sensed but is prevented from supplying the balancing torque. MQ predicts that this control stays on the free/count-resolved branch, whereas active electrostatic torque closure moves to the finite-count branch. The pre-specified free-minus-servo separation is +34.9274 ppm. A null or reversed active-closure contrast, or any shift that instead tracks passive sensing or merely energized electrostatic hardware, would falsify the proposed realization-regime selector. The existing NIST/BIPM data cannot satisfy this causal criterion retrospectively because the selector and controls were not preregistered before unblinding.

The empirical limitation identified here cannot be removed by further algebraic reconstruction. The strongest repair available within a theoretical manuscript is therefore to freeze the theory-side decision rules before new data are acquired, eliminate post-result regime reassignment, and expose rather than hide cross-test dependence. Appendices AC and AD already record the fixed numerical targets, null hypotheses, minimum precision, required controls, covariance treatment, and evidentiary classification for QG-PPM and G-DIFF. Appendix AM shows that those tests are experimentally distinct but share upstream MQ calibration and realization structure. Appendix AT now converts that audit into a single prospective adjudication protocol: the QG-PPM and G-DIFF predictions are treated as immutable targets; any joint inference carries their shared theory-input covariance; and a third-arm result receives independent-confirmation status only if both its target observable and its source-input vector are independent of the data used in the two gravitational tests. The currently available MQKB CMB-temperature chain is retained only as a cross-domain candidate, not as completed independent evidence. Until these conditions and new data exist, MQ should be assessed as a theoretically developed and experimentally falsifiable framework with narrow independent theory-specific evidence, not as an experimentally established replacement for standard quantum mechanics.

The third-arm gate is experimentally concrete but not yet met. Direct CMB thermometry is already much more precise than the present MQ temperature uncertainty [175], while the MQKB temperature chain inherits its dominant source dependence from the adopted cosmic age [176]. Late-Universe cosmic-chronometer measurements provide a genuinely different observational route to the expansion history: the DESI DR1 chronometer analysis obtains direct H(z) measurements from differential ageing of passive galaxies without assuming a specific cosmological model [199]. Its present systematic uncertainties are, however, orders of magnitude too large to turn the small MQ-versus-monopole temperature difference into a decisive independent test. Appendix AT therefore requires the age/geometry source likelihood to exclude CMB information, to be frozen before evaluation of the MQ temperature chain, and to reach sufficient propagated precision before CMB-AGE may be promoted from cross-domain consistency to a confirmatory discriminator.

A reconstruction can always be made formally successful by narrowing its admissible class until the desired theory is the only survivor. The relevant test is therefore not whether MQ uses conditions, but whether those conditions are independent physical inputs, ordinary regularity assumptions needed to state a continuum problem, or consequences already fixed upstream. Appendix AF makes that distinction globally. In the core count-to-record chain, the independent physical roots are substantially fewer than the number of propositions used to propagate them: nonnegative conserved relational multiplicity, adjacency-local closed redistribution, the requirement that observable structure be realized through the Frames mapping, and the operational record constraints define the starting class; counting measure, empirical record probability, complex amplitude, pre-Hilbert geometry, tensor composition, canonical translation algebra, local Lorentz symmetry, and the j=1/2 Dirac realization are then downstream at the levels stated in the manuscript. Conditions such as smooth charts, self-adjoint domains, or detector calibration delimit the mathematical or experimental realization being represented and are not reinterpreted as new microscopic ontology. The residual conditional assumptions are stated separately rather than hidden inside the word 'admissible.' This is the same methodological discipline used in modern operational reconstruction programs, which make explicit which physical principles are postulated and which formal structures follow from them [59,143].

5. CONCLUSION

Measurement Quantization develops a two-domain, three-role count-based kinematic reconstruction of quantum behavior whose detailed dynamical derivation begins in the non-relativistic sector and now has an explicit Lorentz-covariant one-particle continuation, a count-constrained selection of the minimal massive charged Dirac matter carrier, a local MQ-selected spin-1/2 rotational sector, and an explicit scalar/Dirac/Abelian quantum-field-theoretic continuation in which conserved measurement availability is primitive and the standard quantum state formalism is downstream of realization. Its specific novelty claim is the referenceable count-to-record chain established across Secs. 2.6–3.11 and distinguished from representative reconstruction programs in Secs. 4.4–4.6 and the systematic audit of Appx. AU: pre-probabilistic conserved count → admissible Frames mapping → normalized availability → completed-context realization and outcome partition → single realized record with pushforward-compatible ensemble frequency → post-outcome conditioning → quantum representation, with Bell/no-signaling closure and independently resolved dynamical scale. The Internal Frame contains discrete count assignments, relational adjacency, update ordering, and count-conserving redistribution; observable continuum structure arises only in the System Frame. The present matter-selection theorem does not derive the complete particle spectrum: it fixes the Dirac/Clifford representation for the elementary massive charged carrier under the already-established MQ conditions, while neutral, chiral, scalar, higher-spin, flavor, and non-Abelian matter content remain separately testable extensions. Appendix AL now reduces the non-Abelian group-choice problem to the derivation of a count-selected superselection category: once such sectors are physically derived, standard Doplicher-Roberts reconstruction fixes the corresponding compact gauge group rather than allowing it to be appended freely.

Within the stated assumptions, normalization of mapped encoded availability constructs a System Frame outcome-weight measure, but empirical probability is not identified by normalization alone. For each complete admissible measurement interaction, the referenceable completed-context realization M_c is single-valued and the measurable, pairwise-disjoint outcome partition induces R = κ∘M_c, fixing exactly one realized label. The pushforward of the normalized complete-interaction count measure through R is identical to the normalized System Frame outcome-weight measure (Eq. 32; Appx. H.9). This equality is the probability-identification step: the weight assigned to a record region is exactly the measure of complete interactions that realize that record, and under standard repeated-trial sampling it is the limiting outcome frequency. Individual definiteness and ensemble probabilities are therefore two consequences of the same realization structure. Injective record encoding maps the realized label to a stable macroscopic record, after which contextual restriction and renormalization update the predictive encoding. The MQ wave function is constructed as an amplitude-phase representation of the empirically identified normalized measure and realized redistribution phase, while the arbitrary-observable Born rule is recovered after Hilbert-space completion through spectral measures and POVMs. The admissible nearest-neighbor successive-difference structure conditionally selects Schrödinger-form evolution in the non-relativistic System Frame. Proposition 2.6/Appx. X selects the local Lorentzian signature class from the single Internal Frame update ordering, adjacency-local finite accessibility, the mapped fundamental propagation scale, and the positive three-dimensional spatial metric. Proposition 2.7/Appx. Y then fixes the connected local Lorentz group from preservation of those mapped invariants and obtains the Galilei group by non-relativistic contraction; the boost covariance used in Appx. O is therefore derived rather than imposed. Section 3.9.1/Appx. P consequently places the same phase-action and mass realization on a locally Lorentz-covariant geometry supporting Klein-Gordon and Dirac free-field realizations with conserved currents and a controlled return to the Schrödinger limit. Global isometry and causal/spin topology remain open. MQ also derives a measurement-independent, non-factorizable and operationally no-signaling Bell probability architecture. For the ideal polarization benchmark, Appx. G.11 derives the analyzer-axis response from the continuous norm-preserving rotation of the two-channel MQ amplitude, producing the cos²/sin² joint weights and the cosine double-angle correlation before the standard CHSH evaluation. Environmental decoherence suppresses coherence and stabilizes record classes without choosing the realized outcome. Finally, the canonical MQ count-resolved phase-action relation yields a prospective Level-IV discriminator in mechanically driven gravitational quantum-state spectroscopy: under the pre-specified upper-count gravitational realization, MQ predicts a common +11.6426596 ppm shift of transition frequencies relative to standard quantum mechanics at the same independently measured local gravitational acceleration. Existing gravity-resonance experiments establish the required platform but have not yet tested this prediction at sufficient precision [108,109].

The primitive measure in that statement is itself unique within the finite count class used in the paper. Proposition 2.1 shows that the MQ unit-count definition and finite additivity force ordinary counting measure, while admissible relabeling forbids unequal weights on otherwise identical count units; an unfixed overall scale cancels on normalization. Accordingly, the observed nonuniform probabilities arise from the multiplicities of completed-interaction event preimages rather than from a hidden choice of prior measure.

The underlying integer multiplicity now has a correspondingly sharper provenance. Theorem 2.1/Appx. W show that, for the finite stable-record class analyzed here, a pre-metric and pre-probabilistic ontology satisfying finite distinguishability, additive composition, relational relabeling invariance, nonnegative closed conservation, and no-extra-scalar minimality has the free commutative finite-multiset structure and hence a canonical nonnegative-integer count representation [143,144]. Theorem 2.2/Appx. AN then removes the stronger completeness objection at the level accessible to physics: any additional primitive datum that leaves every admissible realization, record, and observable invariant belongs to the same physical equivalence class and adds no physical state distinction, whereas any datum that changes one of those invariants defines a different, empirically discriminable theory. MQ therefore does not claim to prove the metaphysical absence of empirically inert hidden structure. It claims physical completeness of the finite count carrier modulo such empirically equivalent extensions, with every physically active enlargement exposed as a Level-I falsifier.

The finite-to-global scope is likewise sharper. Proposition 2.10/Appx. AO shows that connected locally finite fixed-adjacency domains inherit the finite MQ measure, mapping, record-probability, and flag-topology constructions whenever the finite restrictions are compatible. Finite-context probabilities extend by standard projective consistency, while no infinite total count is normalized. Accordingly, the finite-domain realization theorems are not isolated approximations whose physical class can drift in the thermodynamic limit; their compatible restrictions determine one limit class. The residual domain question is confined to genuinely non-locally-finite structure or adjacency-changing dynamics.

The quantum-field-theoretic gap is therefore no longer an absence of field architecture, local algebra, representation selection, or an interacting relativistic example. Appx. AR fixes the local observable net and state-selected representation from MQ locality and probability provenance; Appx. Q supplies its free/stationary Fock realization, Bogoliubov particle creation, and the Abelian gauge-coupled interaction. The remaining field-theory boundary is the derivation of additional non-Abelian/chiral matter content and model-specific nonperturbative phase structure, together with coherent quantization of the emergent geometry.

The global numerical economy is summarized in Appx. AG. The manuscript does not claim an ab initio universe with no empirically anchored universal quantity: in the current MQKB, θ_si is fixed from the 2022 CODATA inverse fine-structure constant and several fundamental measures inherit that calibration through the canonical MQ relations. The stronger claim supported here is absence of hidden target-specific fitting in the flagship tests. Once the universal MQ calibration vector, the physical subsystem, and independently measured apparatus inputs are frozen, the 11.6426596-ppm QG-PPM displacement and the 34.9274-ppm G-DIFF separation have no adjustable coefficient that can be retuned to the test dataset.

The outcome-realization step is also constructive on finite domains rather than purely extensional. Appendix AH gives a complete finite instance in which a specified primitive count configuration and apparatus context determine the realized record by direct execution of the Internal Frame update and Frames mapping, while the counting measure over the preparation class reproduces the corresponding ensemble frequencies. Section AH.4 provides the same calculation as executable, dependency-free reference code with frozen machine-readable output for independent reproduction.

The principal remaining empirical weakness is therefore explicit rather than structural. The 2026 NIST/BIPM result supplies one temporally out-of-sample MQ-specific Level-II comparison, but the causal G-DIFF realization assignment was not preregistered for that dataset; the QG-PPM +11.6426596-ppm gravitational-state prediction likewise has not been measured at sufficient precision. Appendices AC and AD convert those weaknesses into locked prospective tests, while Appendix AM establishes that the two experiments are experimentally independent but not theoretically orthogonal because both remain downstream of common MQ calibration and realization structure. Appendix AT now supplies the manuscript-wide empirical adjudication gate: QG-PPM and G-DIFF must be executed prospectively with frozen prediction and nuisance models, shared theory uncertainty must enter their covariance rather than be counted twice, and no third observable is promoted to independent confirmation unless its target and source data are demonstrably outside both gravitational test chains. The MQKB CMB-temperature construction is the strongest presently identified non-gravitational candidate, and contemporary cosmic-chronometer work shows that CMB-independent expansion-history inputs are experimentally obtainable [199]; current precision is nevertheless insufficient for decisive CMB-AGE discrimination. Issue #80 is therefore closed only at the protocol and falsifiability level. Decisive prospective experimental confirmation remains open and cannot be supplied by further manuscript algebra.

A final limitation concerns external proof verification rather than missing manuscript structure. The author-side derivations have been decomposed into explicit dependency gates, but complete independent sign-off of every major gate has not yet occurred. Appendix AV therefore specifies the minimum evidence package, specialist domain, independence condition, decision record, and failure propagation required for external verification. Until those certificates exist, the manuscript does not describe its major proof chain as independently verified. This remaining step is intrinsically external to the theory and is properly resolved by peer review or equivalent independent specialist audit rather than by adding another internal derivation.

APPENDIX A: DEFINITIONS OF KEYWORDS, TERMS AND SYMBOLS

Notes

In earlier publications (prior to 2026), the roles of the Internal Frame and System Frame were described using a dimension-based distinction. With the formal identification of the Frames mapping, this characterization has been refined to align with the kinematic structure defined in the present work. The Internal Frame is now understood strictly as a discrete, count-based configuration domain without intrinsic geometric or continuous measure, while the System Frame represents the relational encoding in which observables are realized following application of the Frames mapping.

This refinement does not constitute a reversal of physical interpretation, but rather a clarification: geometry, continuity, and measurement arise only through the mapping from discrete count structure to relational observables. The realization of observables corresponds to the mapping-normalized encoding of conserved count, as defined by the mapping normalization functional. In this sense, normalization specifies the conditions under which Internal Frame configurations admit consistent, repeatable representation in the System Frame, and thereby defines the operational meaning of measurement.

Keywords

Adjacency divergence — A combinatorial divergence operator on the Internal Frame adjacency graph that computes net count change at a relational cell as incoming minus outgoing adjacency flux, without using metric geometry or spatial direction.

Adjacency flux — A relational count-transfer quantity defined on edges of the Internal Frame adjacency graph that describes redistribution of availability counts between neighboring relational cells and is not a geometric vector field.

Availability count — A nonnegative integer measure of accessible measurement realizations associated with an Internal Frame relational cell or domain. It is the primitive MQ quantity and is mapped under the Frames mapping to a System Frame encoded availability density before normalization into probability.

Primitive counting measure — The unique finitely additive measure on a finite MQ unit-count domain obtained by assigning one measure unit to each primitive availability count. Relabeling of physically indistinguishable unit counts leaves it invariant; after normalization, any common scale cancels. It is pre-probabilistic and is distinct from the later System Frame probability measure.

Availability geometry — The relational structure formed by discrete counts and adjacency relations in the Internal Frame, which constrains accessible outcomes and induces interference and correlation structure after System Frame encoding.

Availability partition — A setting-dependent decomposition of System Frame configuration space into disjoint regions used to define measurement outcome channels and restriction operations through indicator functions; acts on encoded availability density and amplitudes after Frames mapping.

Bell factorizability — The Bell-local model condition that joint probabilities factor into local response probabilities at fixed complete hidden state, P(A,B∣a,b,λ)=P(A∣a,λ)P(B∣b,λ). Used to define the Bell-local model class and explicitly not assumed in MQ.

Bell–CHSH inequality — A correlation bound derived under Bell-local factorizability and measurement independence that limits the CHSH combination to magnitude ≤ 2; violated by quantum theory and recovered in the MQ Bell benchmark from the MQ-derived non-factorable availability structure together with the count-normalized two-channel analyzer rotation derived in Appx. G.11.

Blackbody Demarcation — System Frame boundary condition and partition rule that defines an idealized perfectly absorbing and re-emitting surface used to delimit an availability domain for thermal radiation. Under the Frames mapping, the blackbody demarcation specifies a maximal contextual restriction in which all incident radiative availability modes are admitted and re-emitted (the interaction distance, n_L=84.60055l_f) with spectrum determined solely by temperature, independent of material composition. In MQ analysis it functions as a thermodynamic boundary that fixes the encoded radiative availability density and associated spectral distribution.

Composite availability structure — The joint Internal Frame count configuration defined over a composite adjacency domain for multiple subsystems, forming the ontological basis of entanglement and joint measurement statistics in MQ.

Contextual availability — Dependence of encoded availability partitions and resulting probabilities on analyzer settings and measurement context through the Frames mapping, without introducing setting-independent hidden response variables.

Count separation

The number of fundamental length units l_f between relational configurations in the Internal Frame. This quantity determines the scale at which mapping–induced deformation becomes significant.

Realization manifold

A realization manifold is the set of all admissible System Frame realizations compatible with a specified Internal Frame configuration under the Frames mapping R(Ω_I). The realization manifold is not a primitive geometric object of the Internal Frame. Rather, it is the image of conserved Internal Frame count structure after application of the Frames mapping and any applicable realization constraints. Probabilities are assigned only after a normalized measure is defined on the admissible realization manifold.

Encoded availability density — The System Frame field ρ_S(r,t) obtained by mapping Internal Frame availability counts into continuum configuration space under the Frames mapping. It is nonnegative and integrable and is the density-level object used under spatial integrals and differential operators prior to normalization into probability density.

Frames mapping

The Frames mapping is the admissible kinematic realization relation that carries conserved Internal Frame count structure into a System Frame realization class. It preserves the mapping invariants required by the framework, including count conservation, normalization consistency, relational-label invariance, stable record realizability, and holonomy-free closure on contractible reference cycles. It need not be injective under coarse-graining. Pointwise representatives are not physical observables: admissibility-preserving Frames automorphisms define the equivalence orbit [M]U, and the global physical class is completed by the regular domain and invariant defect holonomies as proved in Appx. C.4.3. Thus uniqueness is asserted at the level of physical realization class for the finite contractible measurement domains used here, not at the level of a preferred coordinate or auxiliary-basis representative.

Frames Principle

The Frames Principle asserts that all physically meaningful observables arise from the Frames mapping between the Internal Frame and the System Frame. No observable quantity is intrinsic to the Internal Frame alone. All measurement is relational and defined through the mapping. This principle replaces the assumption of a pre–existing spacetime manifold with a kinematic origin of geometry. The right-triangle construction resolves this relation into a fixed Reference Frame normalization, a variable Internal Frame count, and a realized System Frame separation; the first is an operational role rather than an additional primitive ontology (Appx. M).

Hilbert space representability — The condition that normalized encoded availability densities and phases admit representation as square-integrable wave functions in Hilbert space with standard operator expectations and inner-product structure.

Indicator function — A binary function χ_A(r) equal to 1 inside a specified System Frame region and 0 outside, used to implement restriction and conditioning operations in MQ measurement updates.

Internal Frame

The Internal Frame is the discrete, count–based configuration domain in which all fundamental measures are defined relative to the fundamental units l_f, m_f and t_f. All relational structure is encoded as integer count configurations over this domain. The Internal Frame does not possess a continuous geometric structure; rather, geometry arises only through its mapping into the System Frame.

Mapping–induced deformation

The modification of realized separation in the System Frame arising from the discrete–continuous Frames mapping between the Internal Frame and the System Frame. This deformation is fully determined by the mapping structure and does not constitute an independent physical field. It has two coupled manifestations depending on the scale of the interaction. At fixed count separation, it appears as a local, scale–invariant modification governed by the Informativity differential, corresponding to the mapping-induced finite-count contraction of realized lengths. Over extended count separation, the same mapping–induced deformation accumulates through the mapping structure and is realized as geometric curvature in the System Frame. These two manifestations are not distinct mechanisms but represent different scale–dependent realizations of a single underlying mapping effect.