Measurement Quantization:

A First Principles Origin of the Universe

Jody A. Geiger

Informativity Institute

Chicago, IL USA 60605

Editorial Board Member for Scientific Reports/Nature

jodygeiger@informativity.org

ORCID: 0000–0001–5389–0447

ABSTRACT

We present a first principles deterministic theory for the origin, interaction structure, and long-term behavior of universes within Measurement Quantization (MQ). Physical observables arise from a discrete count structure on an Internal Frame configuration domain ω, while classical spacetime and dynamics emerge through a constrained Frames mapping M to an encoded System Frame. We show that universe initiation occurs when a persistent mapping defect class D⊂ω permits local realization of M but obstructs its extension to a globally consistent domain satisfying holonomy–free closure on all contractible reference cycles. A universe corresponds to a maximal subdomain U⊂ω on which M satisfies normalization, unique realization, and holonomy–free closure. Initiation is therefore a deterministic consequence of global mapping obstruction within ω, without stochastic vacuum processes or an external temporal parameter. From this structure, we derive closed-form population and interaction relations, including a strict upper bound of three simultaneously stable universes as a consequence of global mapping constraints, a constant mass accretion rate M_acr, and finite lifetimes set by count separation limits under the Frames mapping. The framework yields falsifiable signatures: a scale-dependent offset between gravitational and electromagnetic length measures arising from the Informativity differential, with fractional deviations on the order of 10^-5-10^-6, and corresponding systematic variation in inferred values of G. MQ reproduces observed CMB energy density and temperature and provides a kinematic resolution of the Hubble tension at 3.35σ. More broadly, concordance between MQ predictions and independent measurements yields a combined 8.74σ agreement supporting the Frames Principle. Universe initiation is thus mathematically grounded, structurally deterministic, and fundamentally atemporal, yielding a mapping–defined theory of cosmogenesis with testable consequences.

Keywords: Measurement Quantization, origin of universe, big bang, early universe, quantum epoch, foundations of physics

1. INTRODUCTION

The origin of the Universe remains one of the most fundamental open problems in physics. The standard Big Bang model provides a successful empirical description of cosmic evolution from a fraction of a second after the initial singularity to the present epoch, yet it does not by itself explain the physical origin of spacetime or the mechanism by which an initial referenceable universe is established. In the absence of a first principles account, prevailing cosmological theories typically invoke stochastic or externally imposed structures, including quantum vacuum fluctuations, inflationary scalar fields, or multiverse ensembles, to account for cosmic inception [1–3]. While mathematically rich, these approaches rely on additional postulates whose physical necessity and empirical accessibility remain unresolved. As Ellis and Silk argue [4], attempts to exempt such cosmological models from direct testability can undermine the scientific method itself.

A complementary strategy is to seek a deterministic origin for cosmic structure grounded in minimal and operationally meaningful assumptions about physical measurability. Measurement Quantization (MQ) provides such a framework by positing that physical phenomena arise from a discrete count-based structure rather than from a primitive continuous spacetime manifold [5]. In MQ, the Internal Frame is the discrete count-based configuration domain in which all fundamental measures are defined relative to l_f, m_f, and t_f. The System Frame is the non-discrete relational encoding induced by the Frames mapping, in which observable quantities are realized as continuous relational separations derived from discrete count structure [6]. The Frames Principle asserts that all physically meaningful observables arise from the Frames mapping between the Internal Frame and the System Frame. Empirical support for this principle is reported across eleven physically independent implementations of MQ, yielding a combined 8.74σ concordance (Appx. K).

This discrete–continuous structure motivates a central interpretation of quantization: quantization is not introduced as an independent dynamical postulate but follows from finite measurement availability under mapping-normalized count structure. In MQ, the primitive object is not the wave function, field, or spacetime metric, but conserved measurement availability defined over the Internal Frame and encoded into the System Frame through the deterministic Frames mapping ([6], Appx. A–C). Probability enters only after normalization of encoded availability, while phase information arises in the System Frame representation of coherent count structure. This construction has been developed to recover Hilbert-space quantum dynamics [7] and contextual Bell/CHSH statistics [8] without introducing primitive stochastic collapse or setting-independent hidden variables ([6], Appx. H–J). The empirical success of quantum mechanics therefore supports the possibility that probabilistic behavior reflects normalized access to encoded availability rather than fundamental indeterminism.

A second, independent line of physical evidence arises from precision measurements of fundamental constants. Within MQ, physical constants ([5], Appx. C, M, AA, AD, AM, AP–AR, BM) are not independent inputs but arise from mapping-consistent relations between Internal Frame count structure and System Frame observables. In particular, MQ predicts systematic differences between electromagnetic and macroscopic determinations of the gravitational constant G, arising from the Informativity differential, the non-Lorentz length contraction associated with discrete count structure ([5], Appx. C, D, M). Using the CODATA 2010 and 2014 recommended values as independent benchmarks, this comparison yields a 3.94σ concordance; when combined with the corresponding fine-structure-constant relation, the concordance yields 5.08σ. The observed spread in measured values of G [9–12] is therefore treated in MQ as a possible structural signature of discrete measurement geometry rather than as a purely instrumental effect.

A third line of physical evidence is provided by cosmological expansion. MQ predicts that expansion evaluated relative to the System Frame and Internal Frame yields distinct Hubble parameters because the two frames encode inequivalent operational notions of distance and time. This prediction is tested by a first-principles MQ resolution of the Hubble tension ([13], Eqs. 84–85), yielding H_sys=68.259(19) and H_int=73.508(21) km s^-1 Mpc^-1, with a right-tail 3.35σ concordance relative to the SPT-3G CMB-based [14] and PANTHEON+SH0ES local distance-ladder determinations [15], respectively. In this framework, the observed discrepancy is interpreted as a geometric consequence of the Frames Principle, rather than as an independent new dynamical component or an observational inconsistency. Cosmic expansion is therefore treated as a frame-dependent structural relation rather than a single absolute scalar.

A fourth empirical anchor arises from the cosmic microwave background (CMB). MQ predicts a finite quantum epoch ([5], Appx. BF–BH) during which quantum referencing permits mass–energy allocation before classical thermalization. The mass–energy allocated during this epoch is shown to correspond to the present-day CMB energy density, allowing the observed CMB temperature T_CMB=2.725 K to be calculated within the MQ formalism (Appx. E). This calculation reproduces the measured CMB temperature [16] without invoking inflationary reheating or stochastic initial conditions. Moreover, the inverse calculation, in which cosmic age is inferred from the measured CMB temperature, agrees with independent observational age estimates [17–19] ([5], Appx. BI). These results suggest that early-universe thermodynamics may encode discrete reference structure rather than purely classical field dynamics.

Together, these independent lines of evidence – quantum statistics, gravitational-constant systematics, cosmological expansion measurements, and the CMB temperature – support the structural possibility that observable physics arises through a constrained Frames mapping between discrete count structure and continuous relational encoding. MQ is therefore not introduced as an interpretation layered upon existing cosmology or quantum theory, but as a kinematic framework in which quantum behavior and cosmic structure are both treated as consequences of finite measurement availability. Within this framework, the origin of a universe is not a stochastic dynamical event in external time but a structural transition in referenceability within the Internal Frame configuration domain ω.

The purpose of the present work is to formalize this transition. We derive the necessary and sufficient geometric conditions under which a referenceable universe can originate from a discrete count medium and classify the resulting interaction, persistence, and long-term outcome structures. The subsequent sections establish the deterministic count medium, the conditions for referenceability, the role of closure and mapping defects, and the minimal orthogonal structure required for universe initiation. Later sections then derive the Universe Initiation Theorem, the classification of inequivalent referenceable domains, and the conditions under which a universe may lose referenceability. In this way, the paper develops a deterministic, mapping-defined theory of cosmogenesis whose primitive assumptions are constrained by existing observational data.

Before proceeding, a brief methodological note is warranted regarding the role of geometric and mathematical structure in physical theory [20]. Historically, this debate has centered on whether mathematical form merely describes empirical regularities or reflects physically real structure. Over the last century, substantial empirical support has accumulated for frameworks such as special and general relativity [21], in which length, mass, and time are relational quantities determined by geometric relations between observers and systems. Modern physics therefore already treats geometry not as a descriptive convenience but as a structural component of physical law.

MQ adopts this perspective in a minimal operational form. Physical descriptions are expressed as discrete counts n_L, n_M, and n_T of irreducible fundamental measures l_f, m_f, and t_f ([5], Appx. W, BM). This representation separates the discrete geometric relations underlying physical structure from their continuous measurement representation. As a change of nomenclature, it introduces no additional physical assumptions. In practice, however, it provides an analytical tool: expressions in which the fundamental measures cancel correspond to purely geometric relations independent of scale or unit choice. For example, relations such as escape velocity or other invariant ratios may be written entirely in terms of count variables, revealing the underlying geometric structure of the physical law (e.g., escape velocity v_e=(2GM/r)^1/2 ⇒ n_L=2n_M) ([5], Appx. I). In this sense, MQ encodes geometric relations directly into count variables. The discrete count representation therefore isolates those aspects of physical law that arise from reference geometry itself, separating invariant structural relations from their dimensional realization in measurement units.

2. METHODS

2. Assumptions, Constraints, and Design Philosophy

This section establishes the minimal first principles constraints required to derive universe initiation and the conditions for referenceable relational structure within MQ. Because the framework introduces a nonstandard set of primitives, domains, and mappings, precise definitions are essential. A complete and formally ordered set of terms is provided in Appx. A and should be consulted as needed for exact meanings and dependencies. To assist the reader in navigating the nested domain structure, mapping hierarchy, realization structure, and equivalence relations used throughout the paper, a corresponding set of relational overview diagrams is provided in Appx. L. These diagrams summarize the containment, dependency, and mapping relations among the principal MQ structures and are intended as a global organizational reference for the formal development that follows.

The introduction of an expanded mathematical vocabulary reflects the broader geometric scope of the MQ framework rather than an arbitrary extension of terminology. Modern physics already recognizes that geometric relations possess direct physical significance, as established most prominently through the development of special and general relativity. MQ extends this principle by treating referenceability, realization structure, closure conditions, and conserved count relations as physically meaningful geometric constraints within the Frames mapping. Consequently, the framework necessarily incorporates additional relational, topological, combinatorial, and mapping-theoretic structures that are more commonly encountered in advanced mathematical disciplines than in conventional physical theory. The physical relevance of these structures is not assumed a priori but is established operationally through explicit derivations of measured quantities appearing throughout the research literature.

The guiding principle throughout is geometric sufficiency, understood here as the restriction that no postulates beyond mapping consistency, normalization, holonomy–free closure, and conservation of Internal Frame count are introduced [22]. All observable structure arises exclusively through the Frames mapping, consistent with the Frames Principle. Whenever multiple modeling choices are available, the construction is fixed by the weakest admissible constraints imposed by conserved count structure and the existence of a valid mapping realization.

In particular, assumptions are restricted to kinematic relations derived from the conservation of measurement availability count in the Internal Frame and the existence of conditions for a valid Frames mapping, namely normalization, repeatable realization, and closure on all contractible reference cycles. No geometric, stochastic, or ensemble assumptions are introduced at the primitive level. Probabilistic structure arises only through normalization of encoded availability in the System Frame.

Any residual parameters are made explicit, symbolically parameterized, and propagated through the derivations within the MQ action functional. This ensures that qualitative conclusions do not depend on arbitrary modeling choices and remain invariant under admissible reparameterizations of the conserved count structure. Within this framework, the goal of the present section is to identify the minimal mapping-consistent constraints required for the subsequent derivation of the geometric condition for universe initiation.

2.1 Deterministic Count Medium

MQ posits a deterministic discrete substrate described by count redistribution within the Internal Frame. Let ξ denote a relational cell label in the configuration domain ω, and let μ denote the base measure defined over the Internal Frame relational adjacency domain. Let ρ_I(ξ,A) denote the Internal Frame count density, meaning the density representation of the regional count measure induced by the underlying discrete measurement availability count at relational cell ξ and update index A. For any bounded Internal Frame relational region ω^'⊂ω, the total count is

N_{I} (ω^{'}, A) = \sum_{ξ \in ω^{'}} n_{I} (ξ, A) .

(1)

Here ρ_I represents the Internal Frame count-density representation of measurement availability within the relational domain, not a System Frame encoded availability density. Deterministic evolution of the count medium is governed by a conservation law defined entirely in terms of relational adjacency. Count may only be redistributed between adjacent relational cells, and no creation or annihilation of count is permitted. This conservation is expressed as the adjacency balance relation

Δ_{A} n_{I} (ξ, A) + (di v_{adj} J_{I}) (ξ, A) = 0 .

(2)

Therein,

\partial / \partial A

(S1)

denotes the formal update derivative with respect to the Internal Frame update index, not differentiation with respect to physical time. The summation over ∂ω^'(ξ) denotes the discrete aggregation of signed count-transfer contributions across all admissible adjacency links connecting ξ to neighboring relational cells, and J(ξ→ξ^',A) is the Internal Frame adjacency count-transfer function describing directed redistribution of count between relational cells.

Importantly, J does not represent a physical current or flow in spacetime. It is a purely combinatorial object defined on the adjacency structure of the Internal Frame and carries no geometric, metric, or spatial direction. Likewise, the summation over ∂ω^'(ξ) is not a geometric divergence. It is the discrete aggregation of signed count transfer over admissible relational transitions.

The parameter A denotes the Internal Frame update index and is not a physical time coordinate. Evolution in A reflects ordered updates of relational count assignments rather than temporal progression in a System Frame. No stochastic source terms are introduced. The evolution is strictly deterministic at the level of count redistribution.

Summing Eq. (2) over a bounded relational region ω^' cancels all internal transfers pairwise and leaves only adjacency links connecting ω^' to its complement. For regions closed under the adjacency relation, the boundary contribution vanishes, giving

Δ_{A} \sum_{ξ \in ω^{'}} n_{I} (ξ, A) + ϕ_{I} (\partial ω^{'}, A) = 0 .

(3)

Thus, closed relational regions conserve total count as a kinematic constraint of the Internal Frame. Open relational regions need not conserve regional count, because their total count can change through boundary count-transfer. This distinction is essential for the later treatment of open-boundary defects and universe initiation.

All probabilistic statements in MQ arise only after application of the Frames mapping, where conserved Internal Frame count structure is represented as encoded availability in the System Frame. Quantum probabilities therefore emerge from normalization of encoded availability, rather than from intrinsic randomness in the Internal Frame count dynamics.

Consequently, Eq. (2) is the primitive combinatorial conservation relation for relational updates within the Internal Frame. Only after application of the Frames mapping does this relation admit a continuous System Frame representation in which geometric notions such as vector fields, divergence operators, and physical currents become well defined.

2.2 Internal Frame, System Frame, and Referenceability

A universe in MQ is not identified with a count configuration alone. Rather, a universe corresponds to a referenceable realization of a valid Frames mapping from the Internal Frame to the System Frame. The Frames Principle requires that this mapping satisfy three conditions: normalization of measurement availability, unique realization of interaction outcomes, and stability of macroscopic records [23,24].

In MQ, the Internal Frame encodes the discrete count structure of measurement availability, whereas the System Frame represents the continuum description that emerges through the mapping. Physical observables therefore correspond to System Frame encodings of conserved Internal Frame count relations. A Frames mapping is admissible only if it preserves the total Internal Frame count under mapping while producing a well-defined encoded availability distribution in the System Frame.

A Frames mapping is said to be uniquely realized if multiple admissible Internal Frame configurations correspond to the same interaction context, yet exactly one System Frame outcome is instantiated per interaction event. The mapping is therefore many-to-one at the level of configuration equivalence classes, but single-valued at the level of realized outcomes.

The probabilistic structure observed in experiments reflects incomplete accessibility to the underlying count configurations rather than the existence of multiple realized outcomes.

Let M denote the Frames mapping. Referenceability requires that the mapping satisfy

ϕ_{I} (\partial ω^{'}, A) = \sum_{\begin{array}{l} ξ \in ω^{'} \\ ξ^{'} \notin ω^{'} \end{array}} J_{I} (ξ \to ξ, A),

(4)

where 𝒞_s denotes the equivalence class of admissible Internal Frame configurations that encode the same System Frame state s, and P_i(s) are the normalized encoded availability weights induced by the conserved Internal Frame count over 𝒞_s.

Normalization expresses conservation of Internal Frame count under the Frames mapping expressed as encoded availability in the System Frame, while single-valued realization ensures that each interaction event produces exactly one stable record in the System Frame.

Failure of either condition results in loss of System Frame validity. If normalization fails, encoded availability is not conserved under the Frames mapping, violating count conservation inherited from the Internal Frame. If unique realization fails, stable macroscopic records cannot be maintained. This distinction becomes important in later sections, where loss of referenceability is used to classify disappearance as de-referenceability within the MQ framework, corresponding to breakdown of admissible Frames mapping structure rather than stochastic loss of state.

2.3 Closure, Open Boundaries, and Defects

A strictly closed region of the count medium, for which the boundary admits zero net count transfer, does not admit conditions sufficient to initiate a universe. In such regions global closure enforces conservation of total count and ensures that a globally consistent Frames mapping can be defined without obstruction across the domain. Universe initiation therefore requires a relational domain whose boundary conditions are structurally non-removable while preserving referenceability.

Within MQ this requirement is expressed as the existence of a persistent, non-removable boundary condition admitting sustained net count transfer (Appx. B). Persistence is not a dynamical assumption but a structural non-removability condition. A region capable of supporting a referenceable System Frame must retain this condition over a finite sequence of Internal Frame updates. Boundary conditions that can be removed without altering global mapping consistency do not satisfy this requirement [25,26].

The mathematical consequence of such boundary constraints is structural obstruction. When a global realization constraint cannot be satisfied consistently over the relational domain, any attempt to impose it produces localized incompatibility loci ([27], pp. 417–424). Such incompatibility cannot be resolved by local modification and therefore localizes as persistent obstruction loci within the mapping structure. These phenomena are well known in multiple mathematical and physical contexts including obstruction theory, constraint systems, and discrete gauge structures ([27], Sec. 4.2).

Within MQ, a mapping obstruction locus is defined as a localized failure of closure or unique realization induced by global constraint incompatibility. It represents the minimal relational structure required to resolve the incompatibility between global referenceability and persistent non-removable boundary conditions. Under these conditions, a universe originates when such a persistent obstruction locus permits sustained count transfer and supports the emergence of a maximal referenceable relational structure corresponding to an independent System Frame (Appx. B).

2.4 Geometric Minimality and Orthogonality

MQ geometry is characterized by orthogonality in the sense of algebraic independence of count dimensions. Length, mass-equivalent, and time counts correspond to algebraically independent axes in the Internal Frame ([5], Appx. W; [6], Appx. A) [25,27,28]. A referenceable domain therefore requires closure across mutually independent count dimensions in order to maintain unique realization of the Frames mapping. Lower-dimensional or non-orthogonal constructs do not, in general, admit consistent realization across all relational loops and therefore fail to support a globally referenceable mapping structure.

This condition motivates the use of minimal orthogonal three-axis constructs as the lowest-dimensional domains capable of supporting closure under independent count dimensions. Here, orthogonality refers strictly to algebraic independence and not to angular relations in a manifold geometry. Consequently, the analysis does not assume preferred lattices, background metrics, or additional geometric structure beyond that required by deterministic count geometry.

2.5 Structural Principles

The MQ framework developed here is defined by a minimal set of structural conditions. Deterministic count evolution with local conservation is specified by Eq. (2). Referenceability is defined through a uniquely realized Frames mapping between the Internal Frame and the System Frame. Universe initiation arises from a persistent, structurally non-removable boundary condition together with mapping obstruction. Orthogonal closure provides a necessary condition for minimal referenceable geometry. Structural persistence is defined by non-removability under admissible mapping deformations. Residual parameters are treated as explicitly derived and non-fundamental.

These conditions provide a kinematic characterization of referenceable physical structure without introducing additional geometric or dynamical assumptions. They form the minimal structural basis required for subsequent derivations of the geometric conditions governing universe initiation and for the classification of interaction structures between independent systems.

3. RESULTS

3. Mathematical and Geometric Foundations

Original preprint figure

Fig. 1. Domain structure in MQ. The Internal Frame configuration domain ω contains a maximal referenceable subdomain U, within which bounded regions ω^' are used for local analysis. The System Frame region Ω arises as the image of the Frames mapping M:U→Ω and is not defined independently of the mapping.

This section develops the mathematical infrastructure required to establish universe initiation and to characterize the class of referenceable interaction structures within MQ. The emphasis is on constraints arising from deterministic count evolution in the Internal Frame and the structural requirements of a valid Frames mapping. All results are derived from kinematic, topological, and combinatorial considerations without invoking stochastic dynamics, ensemble assumptions, or phenomenological rate parameters. Where appropriate, key statements are formulated as lemmas and propositions, with full proofs provided in the appendices.

All constructions in this section are defined over the Internal Frame configuration domain unless explicitly mapped into the System Frame. The domain structure used throughout is ω^'⊆U⊆ω, where ω denotes the Internal Frame configuration domain consisting of all admissible relational assignment structures, U denotes a maximal referenceable subdomain admitting a globally consistent Frames mapping, and ω^' denotes bounded Internal Frame relational regions used in local conservation and structural analysis (Fig. 1).

The referenceable domain U is defined in the Internal Frame as the maximal subdomain of ω that admits a valid Frames mapping. However, referenceability in the physical sense arises only after application of the Frames mapping, through which U is realized as a System Frame domain Ω=M(U).

The System Frame does not introduce an independent domain at this level; rather, it arises through the Frames mapping as a realized representation of structures within U. Any System Frame region Ω therefore corresponds to the mapped image of an admissible subdomain of U, and is not defined independently of the mapping.

3.1 Count Density and Boundary Flux

Let ρ_I(ξ,A) denote the local Internal Frame count density, defined as the number of fundamental measurement counts per relational cell labeled by ξ at update index A. For any bounded Internal Frame relational region ω^'⊂ω, the total count is defined using Eq. (1),

N_{I} (ω^{'}, A) = \int_{ω^{'}} ρ_{I} (ξ, A) d μ (ξ)

(S2)

, where dμ denotes the base measure on the Internal Frame relational adjacency domain.

Deterministic evolution of the count medium imposes a local conservation law defined purely through relational adjacency. Count redistribution occurs only between adjacent relational cells, yielding the local balance relation given in Eq. (2). Notably, the object J represents the local redistribution of discrete measurement counts along adjacency links. It is not a physical current or flow in spacetime, but a combinatorial object defined on the adjacency structure of the Internal Frame.

To relate local conservation to global balance, we consider the net count transfer across the adjacency boundary of a bounded Internal Frame relational region ω^'. Let ∂ω^' denote the set of oriented adjacency links (ξ,ξ^') such that ξ∈ω^', ξ^'∉ω^', and ξ^' is adjacent to ξ. The net outward boundary count-flux is defined as

ϕ_{I} (\partial ω^{'}, A) = \sum_{(ξ, ξ^{'}) \in \partial ω^{'}} J_{ξ ξ^{'}} (A) .

(5)

This expression represents the total directed transfer of count across all adjacency links comprising the boundary. No geometric notion of surface, normal vector, or metric orientation is assumed. The sign of each contribution is determined solely by the directionality of the adjacency transfer. For a closed relational region, the boundary count-flux vanishes, and regional count conservation follows as

ϕ_{I} (\partial ω^{'}, A) = 0 \Rightarrow Δ_{A} N_{I} (ω^{'}, A) = 0 .

(6)

A bounded relational region is admissible for realization only if its conserved count content and boundary count-flux admit finite normalization under the Frames mapping. Accordingly, ω^' satisfies the first referenceability condition when

N [N_{I} (ω^{'}, A), ϕ_{I} (\partial ω^{'}, A)] < \infty .

(7)

Here 𝒩[⋅] denotes the mapping normalization functional introduced in Appx. A. The functional determines whether conserved Internal Frame count structure can be normalized into a finite realized configuration. Equation (7) therefore does not define unique realization or closure. It supplies only the finite normalization condition required before a relational region can support a valid Frames mapping.

Summing Eq. (2) over all ξ∈ω^', all internal transfers cancel pairwise due to antisymmetry of the adjacency count-transfer function, leaving only boundary contributions. This yields the global balance law

Δ_{A} N_{I} (ω^{'}, A) = - ϕ_{I} (\partial ω^{'}, A) = 0 .

(8)

Equation (2) expresses local conservation of measurement counts, while Eq. (6) shows that the total count in a relational region can change only through boundary count-transfer across adjacency links. Equations (6) and (7) separate the two requirements needed at this stage: closed regions conserve regional count, while candidate referenceable regions must also admit finite mapping normalization.

A strictly closed relational region for which ϕ_I(∂ω^',A)=0 conserves total count and cannot support growth of Internal Frame count measure. Universe initiation therefore requires nonzero boundary count-transfer at the level of the count substrate, implying that any realizable universe must arise from a relational region whose boundary condition is not globally closed.

This formulation constitutes a purely combinatorial conservation structure defined on the Internal Frame adjacency domain, with boundary behavior encoded by adjacency-link count-transfer. Only after application of the Frames mapping does this relation admit a continuous representation, in which geometric constructs such as vector fields, divergence operators, and Gauss-type relations emerge as derived System Frame descriptions.

3.2 Frames Mapping and Unique Realization Mappings

A universe in MQ is not identified solely by the existence of conserved count structure within the Internal Frame. Rather, a physically referenceable universe requires the existence of a valid Frames mapping M relating Internal Frame count configurations to realizable observables in the System Frame. The Frames Principle asserts that all physically meaningful observables arise only after application of this mapping. Consequently, geometry and measurement are not primitive structures. Both arise through realization of conserved count structure under the Frames mapping. The Internal Frame therefore serves as the count-based configuration domain, while the System Frame denotes the realized relational structure induced by the mapping.

Let U⊂ω denote a referenceable universe domain, where ω is the full Internal Frame configuration domain and U is a maximal subdomain on which a valid Frames mapping exists. The corresponding realized relational structure is denoted by Ω. A valid Frames mapping assigns admissible Internal Frame configurations to elements of the realized System Frame structure,

M : U \to Ω .

(9)

Unique realization requires that repeated realization of the same admissible count structure produce a single realized configuration. Let R_M(ω^')⊆R_M denote the realization manifold associated with the bounded relational region ω^'⊂U under the mapping M. This manifold contains admissible Internal Frame configurations that are indistinguishable under realization because they map to the same System Frame state. Thus, for relational cell ξ and update indices A and A^',

M (ξ, A) = M (ξ, A^{'}) \Rightarrow ((ξ, A), (ξ, A^{'})) \in R_{M} (ω^{'}) .

(10)

Equation (10) does not introduce an additional physical degree of freedom. It states the equivalence condition required for repeatable realization: configurations that realize identically under M belong to the same realization manifold rather than to distinct physical outcomes. This prevents multiple inequivalent System Frame realizations from being assigned to the same admissible mapping structure.

A valid Frames mapping must also satisfy holonomy-free closure on contractible reference cycles. For any contractible reference cycle γ⊂ω^', ℋ(γ)=0. Here, ℋ denotes the holonomy functional associated with the Frames mapping. The condition states that transport of reference information around a contractible cycle returns the same realized configuration, thereby preventing closure inconsistency within the realized relational structure. Unique realization and holonomy-free closure therefore act together: Eq. (10) enforces single-valued realization of admissible configurations, while ℋ(γ)=0 enforces path-independent realization on contractible cycles.

The mapping does not introduce new physical content. Rather, it provides the realization rule by which conserved count relations become observable quantities. The realized structure therefore inherits its physical information from the underlying count configuration while expressing that information in a form compatible with observation.

The primitive quantity preserved under realization is the conserved count measure. For a bounded Internal Frame relational region ω^'⊂U,

N_{I} (ω^{'}, A) = \sum_{ξ \in ω^{'}} n_{I} (ξ, A) .

(11)

Here, n_I(ξ,A) denotes the discrete measurement-availability count associated with relational cell ξ at update index A, while N_I(ω^',A) is the total conserved count contained within ω^'. The quantity N_I is defined entirely within the Internal Frame and exists prior to realization. Under application of the Frames mapping, conserved count structure is normalized and encoded into Ω, from which measurable System Frame observables subsequently arise.

Equations (9–10) separate the requirements necessary for physical referenceability at this stage. Equation (9) states the existence of a mapping from conserved count structure to realized relational structure. Equation (10) states the unique-realization condition on admissible configurations through the realization manifold R_M(ω^'). ℋ(γ)=0 states the corresponding holonomy-free closure condition on contractible reference cycles. Equation (11) identifies the conserved count measure preserved under realization. Together with finite mapping normalization, these conditions define the referenceable universe domains on which physically meaningful observables can exist.

Observable quantities arise only after this conserved count structure is acted upon by the mapping normalization functional. Let 𝒞(ω^') denote the count functional associated with ω^'. The corresponding realized quantity is

R_{M} (ω^{'}) = N_{M} [C (ω^{'})],

(12)

where ℛ_M denotes the mapping-normalized realization functional induced by the Frames mapping. Equation (12) summarizes the role of the Frames mapping in MQ. Conserved count structure is primary, while observable quantities correspond to mapping-normalized realizations of that structure within the System Frame.

The geometric implications of the Frames Principle are illustrated in Fig. 2. The construction represents the relation between discrete count structure in the Internal Frame and realized separation in the System Frame. Side a is assigned a unit count and serves as the reference count. Side b corresponds to the Internal Frame count separation n_L, which when multiplied by l_f defines the associated discrete

Original preprint figure

Fig. 2. Count of l_f describing the separation between two phenomenon in the System Frame, with discrete length evaluated in the Internal Frame. Modified from [5].

interaction distance. The hypotenuse represents the realized separation and therefore encodes the corresponding System Frame distance.

Within this construction, n_L+Q_L characterizes the mapping-resolved separation. The quantity Q_L is the mapping-induced residual count contribution that arises from realization of discrete count structure as a continuous observable. The resulting geometry provides a count-based analogue of the Pythagorean relation and establishes the operational basis from which realized separations are constructed.

The geometric interpretation is not merely illustrative. It provides the foundational realization structure used throughout MQ. The orthogonality of the construction reflects the distinction between discrete count assignment and realized separation, while the hypotenuse represents the observable quantity obtained through the Frames mapping. The construction therefore establishes the operational basis from which realized separations are defined.

The most direct consequence of the mapping is the emergence of realized separation. MQ expresses this relation as

l^{2} = {(n_{L} + Q_{L})}^{2} l_{f}^{2},

(13)

where n_L is the Internal Frame count separation, Q_L is the mapping-induced residual count contribution, and l_f is the fundamental length measure. The quantity l therefore arises from realization of conserved count structure under the Frames mapping and is not assumed as a primitive geometric distance.

The continuum metric of the System Frame is subsequently obtained from the realized separation structure. The corresponding line element is

d l^{2} = g_{μ ν}^{sys} (x) d x^{μ} d x^{ν},

(14)

with metric tensor

g_{μ ν}^{sys} (x) = {⟨ \frac{1}{2} \frac{\partial^{2} l^{2} (x)}{\partial x^{μ} \partial x^{ν}} ⟩}_{cg} .

(15)

Equations (13–15) establish that the metric of the System Frame is obtained from realized separation under the Frames mapping. The mapping therefore provides a direct connection between conserved Internal Frame count structure and observable geometric quantities.

This connection extends beyond the metric itself. The same count-resolved construction generates the curvature relations associated with gravitational phenomena. Using the right-triangle realization structure of Fig. 2, curvature may be expressed as ([5], Eqs. C1–C3)

\frac{Q_{L} l_{f}}{n_{Lr} t_{f}^{2}} .

(16)

\frac{Q_{L} l_{f}}{n_{Lr} t_{f}^{2}} (\frac{c}{θ_{si}}) = \frac{Q_{L} c^{2}}{n_{Lr} t_{f} θ_{si}} = \frac{Q_{L} l_{f} c^{2}}{n_{Lr} l_{f} t_{f} θ_{si}} = \frac{Q_{L} c^{3}}{r θ_{si}} .

(17)

\frac{Q_{L} c^{3}}{r θ_{si}} \approx \frac{G}{r^{2}} .

(18)

Such that θ_si = l_fm_f/2t_f, these relations show how geometric curvature emerges from the same count-resolved realization structure that generates measurable separation. In MQ, the mapping-induced residual count contribution Q_Ltherefore appears in both the realized metric and its associated curvature description.

The resulting expression for the gravitational measure is

G = 2 Q_{L} n_{Lr} {(\frac{l_{f}}{t_{f}})}^{3} (\frac{t_{f}}{m_{f}}) .

(19)

Evaluation of Eq. (19) against the 2010, 2014, and 2018 CODATA determinations yields a 3.94σ concordance (Table 3). This agreement provides an empirical assessment of the count-resolved realization structure underlying the Frames mapping and demonstrates that the mapping is associated with quantitatively testable geometric observables.

A valid Frames mapping must therefore satisfy three conditions. First, realization must preserve consistency of the underlying conserved count structure through the mapping normalization functional. Second, each admissible interaction must admit exactly one realized outcome within the System Frame. Third, realized macroscopic records must remain stable under repeated interrogation. These requirements ensure that observables are uniquely realized and remain globally referenceable.

Failure of any of these conditions destroys referenceability even though deterministic count evolution within the Internal Frame may continue. The existence of a referenceable universe therefore depends not only on conservation of count structure but also on the existence of a mapping capable of producing consistent realizations. The global consistency conditions associated with closed reference cycles and the resulting constraints on admissible mappings are developed in the following section.

3.3 Holonomy, Loops, and Mapping Defects

Holonomy is the net transformation acquired when a reference is transported around a closed relational loop. Within MQ, holonomy measures the consistency of the Frames mapping under closed transport. It measures the path dependence of the Frames mapping and therefore determines whether identity is preserved after loop transport. Holonomy quantifies identity preservation rather than conservation of a physical quantity and does not represent local dynamical change.

Non-vanishing holonomy indicates the presence of intrinsic topological obstruction: after traversing a loop, the reference fails to return to its original identity. Vanishing holonomy corresponds to path independence of all contractible loops and therefore defines the condition under which references are globally well defined. Unique realization under the Frames mapping therefore requires vanishing holonomy for all contractible loops, consistent with global referenceability.

Let γ ⊂ U be a contractible closed reference cycle in the Internal Frame adjacency domain. The holonomy functional ℋ measures the closure mismatch obtained by transporting reference information around the cycle under the Frames mapping. A valid Frames mapping requires vanishing holonomy on all contractible reference cycles.

Δ (γ) : = \sum_{e \in γ} σ (e) .

(20)

A mapping defect is present within a relational region ω′ ⊂ ω when the boundary count structure fails to admit holonomy-free closure. This condition is expressed as

D (ω^{'}) \neq 0 \Leftrightarrow ϕ_{I} (\partial ω^{'}, A) \neq 0 and \exists γ_{i} \subset ω^{'} : H (γ_{i}) \neq 0 .

(21)

Here D(ω′) denotes the mapping defect class associated with the relational region ω′, ϕ_I(∂ω^',A) is the Internal Frame boundary count-flux evaluated on the adjacency boundary of the region at update index A, and ℋ(γ_i) is the holonomy functional evaluated on a reference cycle contained within ω′. A non-vanishing value of ℋ(γ_i) indicates failure of holonomy-free closure on that cycle and therefore signals obstruction to extension of a globally consistent realization.

For all contractible loops γ ⊂ U, the Frames mapping validity condition requires ℋ(γ)=0. This condition enforces path independence of realized observables and guarantees that the realized reference structure remains globally consistent under transport around all contractible reference cycles.

A mapping defect is a global obstruction in the Frames mapping that prevents extension of a holonomy-free realization across the full referenceable domain. Operationally, such defects are detected through the existence of at least one reference cycle for which the holonomy functional does not vanish. Such defects arise when boundary conditions prevent global closure and are well known in topology and lattice gauge systems [25,27,29]. Within MQ, such defects correspond to localized violations of Frames mapping consistency and therefore obstruct global referenceability rather than local count evolution.

3.4 Orthogonality and Minimal Referenceable Geometry

MQ geometry is fundamentally orthogonal: length, mass, and time counts define independent algebraic axes in the Internal Frame ([5], Appx. W; [6], Appx. A). Any referenceable domain must therefore admit three independent, commuting, holonomy-free loop generators under the Frames mapping, consistent with the algebraic independence of count measures.

Lemma 3.1 (Orthogonal Closure Requirement)

Any bounded Internal Frame region ω^' that supports a valid Frames mapping must admit a three-dimensional orthogonal closure. Lower-dimensional or algebraically dependent constructs cannot sustain global unique realization ([27], Sect. 1.1; Sect. 4.2).

This follows because the cohomological dimension of the relational mapping space induced by the count algebra is three. By the Eilenberg–Ganea theorem, its minimal geometric realization is therefore also three-dimensional [30,31]. Equivalently, the three-torus T³ provides the minimal algebraic model admitting three independent commuting holonomy-free loop generators since π₁(T³)=ℤ³[32] (Appx. B). Thus, T³ constitutes the minimal closure structure capable of supporting a globally consistent Frames mapping without redundancy in generator structure.

3.5 Defect Persistence and Structural Coherence

Not all mapping defects initiate universes. A removable violation of holonomy that admits a globally consistent re-extension of the Frames mapping does not produce a sustained referenceable domain ([25], pp. 603–606). We therefore introduce the notion of defect persistence.

A defect in region ω^'⊆U is persistent if no admissible deformation of the Frames mapping exists that restores global unique realization and holonomy-free closure on ω^'. Persistence therefore corresponds to topological non-removability within the configuration domain ω.

This condition is geometric rather than temporal. It expresses that the global realization constraints imposed by the Frames Principle generate an irreducible obstruction to referenceability within admissible mapping space. Under these conditions a locally consistent System Frame may emerge as a consequence of constraint satisfaction, without invoking dynamical evolution or causal generation.

3.6 Ontological Status of the Frames Mapping and Defects

The Frames mapping is not a physical field defined over spacetime and does not reside in a background manifold possessing metric or vector structure. Outside a referenceable universe there exists no notion of length, mass, time, or spatial separation and therefore no embedding geometry in which the mapping could possess magnitude, direction, or location. Instead, the Frames mapping exists as a pre-physical relational consistency structure defined over the configuration domain ω, consisting of admissible relational assignment rules connecting Internal Frame count relations with potential System Frame observables.

Its domain is therefore not a set of spatial points but the abstract space of admissible relational encodings subject to the MQ variational principle and the Frames Principle. This structure may be illustrated with a minimal example. Consider three abstract measurement labels A,B,C, together with the following relational constraints: A=B,B=C,A≠C. Each constraint is locally consistent, yet no global assignment of values satisfies all three simultaneously. The inconsistency arises purely from the relational constraint structure and therefore constitutes a mapping defect: a configuration for which no globally consistent Frames mapping exists.

In MQ a defect is therefore not a physical object nor a disturbance in space. It is a pattern of relational constraints lacking a globally consistent realization. The configuration domain ω is the space of all relational rule systems describing possible assignments between Internal Frame counts and System Frame observables. Most such rule systems are globally inconsistent. A universe corresponds to a maximal referenceable domain U⊆ω, within which bounded local regions ω^'⊆U may be used for local conservation, defect, and closure analysis. On U, the relational constraints admit a globally self-consistent realization under the Frames mapping, thereby supporting the realized System Frame image Ω=M(U).

Geometry in MQ therefore refers to the structure of constraint satisfaction within relational assignment space rather than spatial relations between material objects. Mapping defects correspond to unsatisfiable global constraint sets, while universe initiation corresponds to the emergence of a maximal satisfiable subdomain supporting a valid Frames mapping. In this way physical structure arises from the internal logical conditions required for consistent realization, measurement, and repeatability.

4. DISCUSSION

4.0 Universe Initiation: Necessary and Sufficient Geometric Conditions

This section establishes the central theoretical result of this work: a precise necessary and sufficient structural condition for universe initiation within MQ. The result is purely relational and is formulated entirely within the configuration domain of the Internal Frame count medium. It does not invoke stochastic processes, dynamical nucleation mechanisms, or external temporal evolution.

Theorem 4.0 (Universe Initiation Theorem).

A referenceable universe exists if and only if there exists a bounded Internal Frame relational region ω^'⊆ω for which the Frames mapping M admits a persistent open-boundary mapping defect satisfying orthogonal closure constraints.

In such a region, global holonomy-free closure of M on ω^' is impossible, yet a locally consistent restriction M∣_U exists on a maximal referenceable subdomain U⊆ω containing ω^'. This restriction defines an emergent System Frame region Ω=M(U), rendering U a referenceable universe in the physical sense.

4.0.1 Geometric Necessity of Open-Boundary Defects

Suppose a referenceable universe exists on a maximal subdomain U⊆ω contained within a bounded Internal Frame relational region ω^'. By definition the Frames mapping M must be uniquely realized, normalizable, and repeatable on that domain.

If ω^' were globally closed, then by Lemma D.1 (Appx. D) the mapping M would admit a holonomy-free extension to the boundary of ω^' and therefore would not be obstructed by the regional boundary condition. In that case U would not constitute an independent maximal referenceable domain, since no obstruction would prevent the same mapping from extending to larger admissible relational domains. This contradicts the requirement that a universe corresponds to a maximal referenceable subdomain of the configuration domain. Consequently, the initiating region ω^' cannot be globally closed and must instead admit a nonzero adjacency-boundary condition.

Furthermore, if the open boundary were structurally trivial, a continuous admissible deformation would exist that restores global holonomy-free closure of M, again contradicting the independence of U as a referenceable domain. The boundary must therefore be structurally non-removable.

By the general obstruction principle for relational systems ([27], pp. 417–423; [25], Sect. II.B; [29], Sect. II & IV; Appx. B), the incompatibility between global closure and the imposed boundary condition implies the necessary existence of at least one mapping defect. The presence of a persistent open-boundary defect is therefore a necessary structural condition for universe initiation.

4.0.2 Geometric Sufficiency of Persistent Mapping Defects

Conversely, suppose a bounded Internal Frame relational region ω^'⊆ω contains a persistent open-boundary mapping defect. By definition of persistence, no admissible deformation of the Frames mapping M can restore global holonomy-free closure on all of ω^'. Consequently, a globally consistent mapping cannot exist on the entire region.

However, a restriction M∣_U exists on at least one maximal subdomain U⊆ω^' for which the mapping remains normalizable, holonomy-free on contractible reference cycles, uniquely realizable, and repeatable. This restricted domain therefore supports a valid System Frame image Ω=M(U). Because the mapping cannot be extended beyond U without encountering the defect, U is maximal with respect to referenceability and therefore constitutes a universe.

No dynamic process is required for this emergence. The universe is not created in time. It is identified as a maximal satisfiable substructure U of the Internal Frame configuration domain, with physical realization only through the mapped System Frame image Ω=M(U).

4.0.3 Minimal Geometric Construct for Initiation and the √3 Referencing Criterion

The Universe Initiation Theorem (Appx. C) establishes the necessary and sufficient structural condition for the existence of a referenceable universe. However, the theorem alone does not determine the minimal algebraic realization required for global referenceability. The present section identifies that minimal realization and its corresponding geometric interpretation under the Frames mapping.

Let R_min denote the minimal realized separation associated with the first admissible referenceable domain. Let U_min⊂ω denote the corresponding minimal referenceable universe domain and let Ω_min=M(U_min) denote its realized System Frame image under the Frames mapping M. A valid realization requires normalization, unique realization, and holonomy-free closure on all contractible reference cycles. Consequently, every irreducible generator required for referenceability must admit simultaneous closure. Failure of closure along any required generator produces path dependence in the realization and prevents extension of a globally referenceable domain.

The relevant dimensionality at initiation is therefore not pre-existing spatial dimensionality but the number of algebraically independent closure generators that must be simultaneously satisfied. Within MQ, this requirement is associated with the three irreducible count generators corresponding to the fundamental measures l_f, m_f, and t_f. These generators are algebraically independent in the Internal Frame and become physically interpretable only after realization under the Frames mapping.

The minimal construct satisfying this condition is the smallest algebraic closure supporting three mutually independent commuting generators. Topologically, the corresponding loop algebra has the closure structure of the 3-torus T³, with fundamental group π₁(T³)=ℤ³. The Eilenberg–Ganea argument is used here only to identify the minimal realization dimension of this three-generator closure algebra. Within MQ this does not assume a pre-existing geometric manifold. It states only that a two-generator closure structure is insufficient for three independent closure requirements, whereas a three-generator structure is minimal.

Accordingly, define

U_{\min} = {γ_{1}, γ_{2}, γ_{3}}, γ_{i} ⊥ γ_{j} .

(22)

Here γ_i denotes a contractible reference cycle and γ_i⊥γ_j denotes algebraic independence of the corresponding closure generators. Equation (22) defines the minimal referenceable generator set required for simultaneous holonomy-free closure.

The realized geometric interpretation arises only after application of the Frames mapping. At first admissible closure, the three independent generators contribute one normalized count unit each. The diagonal of the resulting orthogonal 3-cell gives the minimal realized separation,

R_{\min} = \sqrt{1^{2} + 1^{2} + 1^{2}} l_{f} = \sqrt{3} l_{f} .

(23)

Equation (23) defines the minimal mapping-consistent closure distance and establishes the √3 criterion. The quantity R_min is not a primitive geometric radius of the Internal Frame. Rather, it is the first realized separation appearing in the System Frame once simultaneous closure of all three irreducible generators is achieved.

The √3 relation therefore represents the minimal closure condition required to support three independent holonomy-free generators. Any realized separation R satisfying R<R_min necessarily fails to admit closure along at least one required generator. In such regions the Frames mapping becomes multi-valued, unique realization is lost, and no globally consistent System Frame realization can exist. Only after simultaneous closure of all three generators is achieved does a globally referenceable realization become possible.

Once a valid System Frame realization exists, the same closure and holonomy constraints apply to all subsequent realized loops. Spatial dimensionality is therefore determined operationally by the collection of loops capable of holonomy-free closure within the realized reference structure. The present argument addresses the prerequisite condition: the minimal algebraic closure structure that must exist before any realized spatial universe can emerge.

Crossing the threshold defined by Eq. (23) establishes the first globally valid System Frame realization and therefore the attainment of global referenceability under the Frames mapping. Prior to satisfaction of this condition, no globally meaningful observables can be defined because unique realization does not exist. Satisfaction of the condition permits a self-consistent System Frame realization together with the onset of classical referencing. The threshold follows directly from the minimal orthogonal closure condition implied by the three irreducible generators of the Internal Frame and their realization through the Frames mapping (Appx. D.2; Appx. D.4; [5]). The associated physical implications and empirical consequences are developed in Sec. 4.0.5.

The √3 criterion therefore defines the minimal algebraic condition under which a globally consistent Frames mapping can exist.

4.0.4 Corollaries and a Testable Early-Universe Consequence

Corollary 4.0.1. Universe initiation is a structural condition within the Internal Frame configuration domain, requiring the simultaneous satisfaction of three independent orthogonal closure constraints.

Corollary 4.0.2. Universe initiation does not violate determinism. It corresponds to a measure-nonzero subset of deterministic Internal Frame initial conditions satisfying the persistent defect predicate of the Frames mapping.

Corollary 4.0.3. The quantum epoch corresponds to a structural relaxation regime of the initiation defect rather than a temporally defined interval. This regime terminates when the √3 referencing criterion is satisfied and a globally consistent Frames mapping becomes possible, yielding the first stable System Frame (Appx. D.4; Appx. BF; Appx. BG).

Corollary 4.0.4. Because the quantum epoch extent and the total mass–energy encoding during that regime are MQ-native—determined by Internal Frame geometry and the Frames Principle rather than by inflationary reheating, recombination physics, or stochastic initial conditions—the present-day CMB temperature becomes a quantitatively testable consistency condition of the MQ description of early universe structure (Eqs. H.5–H.12; Table 1).

4.0.5 Testable Early-Universe Consequence: The Cosmic Microwave Background

The Universe Initiation Theorem establishes a necessary and sufficient geometric condition for the emergence of a referenceable universe, but by itself it remains a purely structural result. To serve as a physical theory of cosmogenesis, MQ must provide at least one direct, quantitative observational consequence that follows uniquely from the initiation mechanism and the associated early universe structure.

Such a consequence is provided by the finite quantum epoch implied by the initiation defect. Within MQ, this regime is defined by the progressive satisfaction of mapping constraints culminating in the √3 closure condition, rather than by an externally defined temporal parameter. During this regime, quantum referencing permits the mapping-consistent encoding of count-derived mass–energy in the absence of classical thermalization. Here, thermalization denotes the establishment of local thermodynamic equilibrium through classical many-body interactions, which requires a globally referenceable domain and well-defined macroscopic observables.

The √3 closure condition,

R_{\min} = \sqrt{3} l_{f}

(S3)

, derived from the minimal orthogonal construction (Appx. D.4; [5], Appx. BF), defines the termination of this regime – the quantum epoch – and the emergence of the first globally valid System Frame. Prior to this condition, no globally meaningful observables exist. Following its satisfaction, classical referencing becomes well-defined.

Building on this result, [5], Appx. BG demonstrates that the scale associated with this transition is uniquely determined by the same closure condition, with θ_si as the only measured input parameter. The √3 factor appears explicitly as the defining structural modifier of this transition, fixing the mapping-consistent scale at which global referenceability is achieved.

Within MQ, the mass–energy content associated with this regime is not arbitrary. It is fully determined by the structural geometry of the Frames mapping and the √3 termination condition. This fixes the total mapping–consistent mass–energy allocation in Internal Frame units independent of stochastic initial conditions or reheating models.

It follows that the mass–energy defined by the mapping structure,

M_{acr} = θ_{si}^{4} / l_{f}

(S4)

, ([5], Appx. BE), subsequently redshifts into a homogeneous and isotropic radiation field, observed today as the CMB. Once the quantum epoch terminates, no additional mass–energy contributes to the CMB radiation density. Allocation during the subsequent expansionary regime instead manifests as net mass accretion within an increasing comoving volume, with the normalization fixed by the same radiation energy scale encoded in the observed CMB temperature.

In Ref. [5], Appx. BH, the mapping-consistent mass–energy allocation and the post-transition expansion jointly determine the present-day radiation temperature. Using the Planck 2018 measured age of the universe [17], this construction yields T_CMB = 2.72509 ± 0.00341 K, in quantitative agreement with the Fixsen measurement within experimental uncertainty [16]. The corresponding right-tail concordance lies within observational bounds, providing a direct empirical consistency test of the √3 initiation criterion without the introduction of adjustable parameters. This result is not obtained by fitting, introducing inflationary reheating, or postulating initial thermal conditions. It follows from a hierarchy of well-defined theoretical and empirical inputs.

At the structural level, the prediction is fully determined by MQ-native principles: the persistent open-boundary mapping defect, the √3 closure condition for global referenceability, conservation of count under the Frames mapping, and the quantum-to-classical referencing transition.

At the physical level, the numerical evaluation depends only on independently measured constants, including the fine structure constant α, which determines half of the fundamental momentum θ_si ([5], Appx. AD), the speed of light c, used to resolve the fundamental mass m_f ([5], Appx. BM), the Planck 2018 measurement of the cosmic age A_U [17], and the exact CODATA 2022 values [33] of the radiation constant a, Planck’s constant h, and the Boltzmann constant k_B.

The CMB temperature therefore serves as a quantitatively constrained empirical anchor linking the structural initiation condition to an observable prediction, providing a direct test of the MQ cosmogenesis framework with no additional model freedom.

4.0.6 Empirical Anchors of the MQ Framework

Universe initiation in MQ is determined by the geometric condition established in Theorem 4.0: a persistent open-boundary mapping defect satisfying orthogonal closure under the Frames mapping. This construction, developed from the deterministic count structure of the Internal Frame and its encoding into the System Frame, fully specifies the conditions for referenceability.

The resulting framework admits direct connection to empirical observables, which are developed as follows:

Frames Mapping: The physical relevance of the Frames mapping is supported by the resolution of the Hubble tension against the SPT-3G [14] and PANTHEON+SH0ES [15] measurements yielding a 3.35σ concordance. The AIP Advances analysis [13] shows that the discrepancy between early- and late-universe expansion rates arises naturally from frame dependent reference structure, providing empirical support for distinct Internal Frame and System Frame encodings.

Discrete Internal Frame Measure: The discreteness of the Internal Frame is supported by precision measurements of the gravitational constant. The systematic divergence between electromagnetic and macroscopic determinations of G ([5], Appx. K; Table II) is captured in the MQ relation,

G = 2 Q_{L} n_{L} \frac{c^{3} t_{f}}{m_{f}},

(24)

which incorporates the MQ non-Lorentz length contraction 2Q_Ln_L arising from the Frames mapping. This relation follows directly from the discrete count structure and finite measurement capacity of the Internal Frame ([5], Appx. D), and its physical significance is evaluated through concordance with precision measurements of G.

Using the CODATA 2010 and 2014 recommended values as independent benchmarks, the MQ formulation yields a global concordance of 3.94σ, reflecting consistency between the MQ prediction and the observed spread in measured values of G without dataset-specific adjustment. The result depends only on θ_si, fixed independently by the fine structure constant, and on the geometric structure of the Frames mapping, with no phenomenological correction introduced.

The observed dispersion in measurements of G is therefore consistent with a discrete Internal Frame, making Eq. (12) a testable consequence of MQ whose concordance provides empirical support for the Frames Principle.

Later CODATA adjustments (2018, 2022), reflecting evolving experimental weighting and methodology, are discussed in the MQ review paper ([5], following Table II), while the earlier datasets provide the most direct test by preserving the underlying experimental divergence.

√3 Referenceability Criterion: The √3 global referenceability criterion defines the minimal orthogonal closure radius required for a uniquely realized Frames mapping (Eqs. B.23, C.3), establishing the geometric threshold at which a referenceable universe first becomes possible.

Mass Allocation/Accretion: The MQ kinematic law of mass allocation/accretion ([5], Appx. BE)

M_{acr} = \frac{θ_{si}^{4}}{l_{f}},

(25)

applies across both the quantum and expansionary epochs at a fixed kinematic rate. In MQ this is not a thermodynamic production law, but a count-based allocation law defined by the Internal–System Frame relation. The total mass–energy assigned during the pre-temporal quantum epoch is then carried forward under autonomous expansion. The present day CMB energy density and temperature follow from that allocated content together with dilution under expansion and the standard radiation relation ρ=aT⁴. The explicit MQ expressions are given in Appx. E and compared with measurement in Table 2. The resulting first principles prediction for T_CMB is in quantitative agreement with Fixsen [16].

A dimensional clarification is useful here. The parameter θ_si is dimensionless in the Internal Frame and does not introduce units under algebraic multiplication. The initial construction of M_acr is a ratio of a count of m_f to a count of t_f, so its primitive dimensional form is m_f/t_f, i.e. kg s^-1. Any subsequent multiplication by factors of θ_si does not alter the units. Dimensional realization occurs only through the Frames mapping into the System Frame. Accordingly, M_acr should be interpreted as a mass allocation rate. Total allocated mass is obtained only after multiplication by the appropriate duration. A more detailed understanding of this property is provided in Appx. J.

Quantum Epoch: The existence and termination of a finite quantum epoch is supported by the emergence of physical time. Quantum epoch dynamics and their termination are summarized in Eqs. (H.3, H.4), which define the transition from quantum to classical referencing in the absence of any external or pre-existing temporal parameter. Physical time is shown to emerge only after this transition as an operational construct internal to the System Frame.

Quantum Epoch Duration: The extent of the quantum epoch and its relation to the expansionary epoch are determined by MQ internal geometry through the parameter θ_si, independent of cosmological observation or external temporal input. In MQ the quantity A_qe appears algebraically as the exponential scaling

A_{qe} = e^{(\sqrt{3} θ_{si}^{3} / 2)},

(26)

which represents a dimensionless geometric scaling relation.

As shown in Appx. D, the derivation begins with the MQ allocation relations for the quantum epoch, which leads to a characteristic expansion velocity

v^{'} = \frac{2 l_{f}}{θ_{si}^{3} A_{qe}} .

(27)

The expression attains physical units of m s^-1 only after substitution of the MQ age relation and cosmological density relations (Sec. D.5). Integrating this velocity over the quantum epoch yields the radius relation

R_{qe} = \frac{2 l_{f}}{θ_{si}^{3}} \ln (A_{qe}),

(28)

(Eq. D.11), which defines the geometric growth of the referenceable domain during the quantum epoch. Importantly, there is no physically realized elapsed time associated with the quantum epoch. The quantum epoch corresponds to a pre-referenceable regime in which System Frame time is not yet defined.

Accordingly, A_qe should be interpreted as a scaling parameter arising from MQ internal geometry rather than as a temporal coordinate. The SI-equivalent extent associated with the quantum epoch is obtained only after the boundary condition

R_{qe} = \sqrt{3} l_{f}

(S5)

is imposed and the expansion kinematics are resolved. Solving the MQ scale hierarchy gives the closed-form expression in Eq. (12) which corresponds, after kinematic resolution relative to the later realized expansionary epoch, to the encoded quantum epoch extent A_qe = 1.14652655409 × 10¹³ s = 363,312 years.

The expansionary epoch scale is related to the quantum epoch through the MQ radial expansion relation ([5], Appx. BG-BH)

A_{ee} = A_{qe} {(2 θ_{si})}^{1 / 3} .

(29)

Unlike Eq. (14), which expresses the exponential scaling factor arising from the MQ hierarchy, Eq. (17) represents the physically realized expansionary scaling of the System Frame and therefore carries the dimensional interpretation of elapsed cosmic time once the quantum epoch terminates. Accordingly, the quantum epoch scale and the subsequent expansionary epoch scale follow directly from MQ internal geometry and the Frames mapping, with dimensional realization obtained only through the kinematic relations summarized in the MQ-native parameter appendix.

To assess the physical significance of this construction, we evaluate the predicted CMB temperature using the Planck 2018 (TT,TE,EE+lowE+lensing) determination of the cosmic age, A_ee = 13.797 ± 0.023 Gyr. This provides a direct comparison between the MQ-derived relation and the observational determinations of Fixsen [16] and Planck [17] as a function of the MQ expansionary scaling. Under this input, the MQ framework yields agreement with the observed CMB temperature at the 0.32σ level.

Alternatively, using the MQ-inferred cosmic age A_ee = 13.7943 ± 0.0039 Gyr, derived independently within the MQ framework, the resulting CMB temperature agrees with observation at the 6.81σ level. This larger concordance should be interpreted as a self-consistency check of the MQ kinematic relations rather than as an independent observational test, because the age input is itself MQ-derived.

These results demonstrate that the MQ expression provides a quantitatively accurate prediction of the CMB temperature, with concordance limited by the precision and independence of the cosmic age. Current empirical age determinations are either model-dependent or insufficiently precise to fully discriminate between frameworks, implying that a decisive test of the Frames Principle requires an independent, high-precision measurement. Nevertheless, the existence of a parameter-free MQ calculation achieving observational agreement at this level provides measurable support for the framework and reduces reliance on inflationary reheating as the sole explanation for the observed CMB temperature.

Early Universe Chronology: MQ also provides a forward and inverse mapping between CMB temperature and cosmic age. Using the measured CMB temperature and the MQ universe radius definition, the reconstructed cosmic age is ([5], Appx. BI),

A_{U} = {(\frac{3 θ_{si} {(2 θ_{si})}^{1 / 3} e^{(\sqrt{3} θ_{si}^{3} / 2)}}{4 π ac l_{f} T_{CMB}^{4}})}^{1 / 3} .

(30)

This value agrees quantitatively with the Planck 2018 determination of the cosmic age. The resulting T_MQ should not be treated as an independent determination of cosmic age, since present age estimates are themselves model-dependent and, in CMB analyses, partly inferred from the same thermal history being evaluated. The relevant claim is therefore internal consistency: the MQ temperature relation maps the assumed age scale, quantum epoch allocation, and expansionary dilution into the observed CMB temperature without adding a separate reheating parameter. This supports the coherence of the initiation-to-CMB derivation, while the independent empirical tests remain the forward predictions that do not reuse the CMB temperature or its model-derived age constraints.

Observational Domains: Distinct observational domains predicted by MQ are supported by ΛCDM measurements of the CMB power spectrum. Appx. J demonstrates that MQ fractional count domains – fundamental, observable, unobserved, and dark – map onto the measured cosmological density parameters of standard cosmology. This correspondence establishes a structural mapping between MQ geometric domains and observational cosmological components without introducing new dynamical fields or fitted parameters.

Taken together, these results indicate that the foundational elements used in the MQ cosmogenesis framework have measurable empirical counterparts. MQ therefore does not rely solely on unconstrained assumptions. Its primitive concepts are tied to measurement-accessible structures through the Frames mapping.

The universe initiation mechanism is thus not merely mathematically consistent but empirically anchored. The deepest structural features of cosmology – length, time, mass, expansion, observational limits, cosmological tensions, and the CMB – can be interpreted as manifestations of a single underlying geometric principle: the Frames mapping between discrete Internal Frame and the emergent System Frame encoding.

4.1 Structural Cosmogenesis and Global Reference Topology

This section reformulates universe initiation and multiplicity as properties of the Frames mapping, treating origin as a global constraint condition and multiplicity as a classification of inequivalent referenceable System Frame realizations within the configuration domain of the Internal Frame.

In this formulation, “cosmogenesis” does not represent a dynamical process but the structural resolution of a global obstruction to a consistent Frames mapping. Universe multiplicity correspondingly reflects the set of inequivalent mappings satisfying the global constraints, rather than any stochastic or temporal generation mechanism. The objective is therefore to determine the admissible set of referenceable realizations under MQ constraints and to identify the conditions under which this set is restricted.

4.1.1 Absence of External Time and the Meaning of “Change”

Prior to the realization of a referenceable System Frame, the configuration domain of the Internal Frame does not admit a global temporal coordinate. The Frames mapping must simultaneously satisfy global uniqueness, closure, and consistency conditions (Eqs. 4, B.1, B.3–B.4, C.19). When these constraints cannot be satisfied globally, a mapping obstruction arises.

This obstruction is not produced through temporal evolution but reflects the non-existence of a globally extendable mapping satisfying the imposed constraints. The resulting realization corresponds to the minimal domain over which consistency can be achieved.

Accordingly, universe initiation is not a dynamical event but the structural condition under which a locally self-consistent referenceable domain becomes admissible. This condition follows deterministically from the geometry of the configuration domain and the constraints imposed on the Frames mapping.

Temporal ordering emerges after the establishment of a valid System Frame, at which point time appears as an internal coordinate of the realized reference structure (Sec. 2.2, Appx. A.5, C.6). The notion of “change” is therefore reduced to the existence or non-existence of a globally consistent mapping under fixed constraints, rather than evolution with respect to an external parameter.

4.1.2 Constraint Classes, Universe Multiplicity, and Coexistence

As established throughout Sections 3–4, any referenceable universe U⊂ω under the Frames mapping must satisfy a finite set of structural constraint classes, defined as collections of logically related constraints that operate at the same scope under the Frames mapping, and which are jointly necessary for the existence of a valid System Frame realization, but not all of which generate inequivalent global structure. These constraint classes correspond to mapping-consistent predicates on admissible subdomains U satisfying normalization, unique realization, and holonomy-free closure (Sec. A.3). They govern distinct aspects of referenceability, and their relevance depends on whether one is addressing existence, simultaneous extension over a common domain, or classification.

They are not introduced as an external mathematical axiom system. They collect the minimal MQ-derived requirements for a referenceable universe domain U⊂ω: finite count structure in the Internal Frame, admissible realization under the Frames mapping, and global closure/defect structure required for universe initiation. The ten constraint classes required for admissibility under the Frames mapping are:

Nonnegative count measure: Internal Frame measurement availability count and any derived density representations must be nonnegative.

Finite additivity over disjoint relational regions: counts assigned to disjoint Internal Frame relational regions must combine additively.

Local finiteness of mapped availability: every finite realized System Frame region induced by M(U) must correspond to finite encoded availability.

Closed-region count conservation: for any relational region closed under the adjacency relation, total Internal Frame count is conserved under deterministic updates.

Mapping normalization: conserved Internal Frame count must admit a finite normalized encoded availability distribution after application of the Frames mapping.

Single-valued realization: each admissible interaction context must instantiate exactly one realized System Frame outcome.

Repeatable record encoding: realized outcomes must map injectively to stable macroscopic records that remain distinguishable under repeated interrogation.

Persistent open-boundary obstruction: universe initiation requires a non-removable boundary condition supporting persistent count flux and anchoring a mapping defect class D.

Holonomy-free contractible closure: every contractible reference cycle γin the regular domain must close without residual holonomy under the Frames mapping.

Orthogonal closure of independent generators: the admissible domain must support closure across the independent count generators required for a globally consistent reference structure.

These ten classes separate naturally into four categories:

Measure-admissibility constraints: (1–4), fixing the finite conserved count structure.

Mapping-realization constraints: (5–7), fixing normalization, outcome uniqueness, and record stability.

Initiation-obstruction constraint: (8), fixing the persistent defect condition required for nontrivial universe initiation.

Global closure constraints: (9–10), fixing holonomy-free and orthogonal consistency of the reference structure.

A physically admissible universe corresponds to a global configuration satisfying all ten constraint classes simultaneously. Let the finite MQ constraint set be C_MQ={C₁,C₂,...,C₁₀}. A referenceable universe domain U⊂ω is admissible if and only if C_MQ(U)=true and D(U)≠0, where D(U) denotes the mapping defect class present on U, identifying the existence of at least one nontrivial initiation obstruction anchoring a non-extendable realization of the Frames mapping (Appx. A.3). The defect condition does not define admissibility itself but ensures nontrivial realization consistent with universe initiation (Theorem 4.0).

The question of universe multiplicity is therefore not one of individual admissibility but of joint admissibility within a shared configuration domain. In this setting one must determine which constraint classes remain inequivalent under the equivalence relation generated by global Frames automorphisms Φ, defined as structure-preserving transformations on U (Appx. A.3).

As shown in Appx. E, only the topological initiation constraint and the geometric closure constraints contribute to the classification of inequivalent universes. These correspond respectively to invariants of the mapping defect class D and the holonomy functional evaluated over reference cycles γ (Appx. A.3). All remaining constraint classes are invariant under the Frames automorphism equivalence relation and therefore do not distinguish distinct universes.

4.1.3: Constraint Reduction, Equivalence, and Universe Counting

Not all physically admissible universes necessarily correspond to inequivalent realizations within a single configuration domain. Within MQ, coexistence requires that two realizations M(U₁) and M(U₂) arise from subdomains that can be jointly embedded within a common Internal Frame configuration domain ω without violating the subset of constraints that must hold globally across embeddings, namely: unique realization, holonomy–free closure on contractible reference cycles, and consistency of the global Frames mapping.

These conditions induce an equivalence relation on the set of maximal referenceable domains U. Specifically, U₁∼U₂ if and only if there exists a frames automorphism Φ:U→U, with defect class D fixed, that preserves normalization, repeatable realization, and holonomy–free closure, and maps U₁ to U₂. This relation is reflexive, symmetric, and transitive, and therefore partitions the admissible domains into equivalence classes. Only inequivalent classes correspond to physically distinct universes.

An apparent tension now arises. Although all ten constraint classes are required for the existence of an admissible universe, only a subset contributes to the classification of inequivalent universes. The resolution follows from the constraint–reduction argument established in Appx. E (Sec. E.2).

Suppose two domains U₁ and U₂ differ only in measure–theoretic quantities, such as total count, density distribution, or conserved quantities. By Lemma E.2.1 and Proposition E.2.2, all differences governed by Constraints 1–4, together with their normalization under Constraint 5, can be absorbed by an invertible frames automorphism Φ. These constraints therefore do not modify the holonomy functional on reference cycles γ or the defect class D, and hence do not distinguish equivalence classes.

Similarly, by Lemma E.2.3 and Proposition E.2.4, all differences governed by Constraints 6–7 are likewise removed by automorphisms preserving realization and normalization. Variations in record encoding and realization mappings do not alter the holonomy structure or admissible reference cycles and therefore do not produce inequivalent universes.

By contrast, Constraint (8) introduces a globally protected obstruction through persistent boundary flux associated with the defect class D, while Constraints (9–10) fix the admissible structure of independent reference cycles γ. These structures are invariant under all admissible frames automorphisms and therefore define the discrete classification labels of inequivalent universes.

Under the Universe Initiation Theorem (Appx. C), a minimal referenceable domain admits exactly three independent non-contractible reference cycle generators and at least one nontrivial defect class D. Under the equivalence relation ∼, the only invariant degrees of freedom are: (i) which independent generator carries nontrivial holonomy, and (ii) the sign of the associated boundary flux.

Thus, the classification reduces to the product of these invariant labels, {defect-supporting generator} × {boundary flux sign}. Assuming three independent generators and two admissible flux orientations, the number of inequivalent classes is

| U_{distinct} | = 3 \times 2 = 6 .

(31)

This result defines a classification on the quotient set of referenceable domains under frames automorphism equivalence and does not imply the simultaneous physical realization of multiple universes.

Corollary 4.1.1: Referenceability Stability and Flux Polarity

The classification result above exhausts the set of physically distinct globally referenceable universes permitted by the MQ constraints. No further reduction arises from additional structural assumptions. However, the persistence of referenceability over extended Internal Frame update indices introduces an interpretive refinement of the classification.

A universe that remains referenceable must support persistent macroscopic records, a non-vanishing domain of realizable states, sustained availability of realizable configurations, and the continued existence of a globally consistent System Frame. These conditions are not independent constraints but follow as structural consequences of the initiation geometry and mapping consistency conditions already classified.

In particular, the polarity of the persistent open-boundary flux distinguishes between universes that are globally reference-expanding and those that are globally reference-contracting. Universes with outward-directed Internal Frame count flux admit sustained referenceability, while those with inward-directed flux undergo progressive global depletion of realizable states and therefore fail to maintain a viable System Frame over extended Internal Frame update indices.

Defining a flux–polarity (stability) predicate Σ(U) such that

\sum (U) = {\begin{array}{l} + 1, foroutwardopen - boundarycountflux \\ + 1, forinwardopen - boundarycountflux \end{array},

(32)

one may interpret the classification result as identifying three structurally stable and three structurally unstable equivalence classes, corresponding to the two flux polarities across the three orthogonal generator choices.

Importantly, this refinement does not alter the universe count derived above. It merely clarifies which equivalence classes correspond to long-lived referenceable domains. All stable universes share the same kinematic structure and governing relations under the Frames mapping, but their realized physical scales and coupling strengths depend on the value of θ_si. Consequently, while dimensionality and the form of the governing equations are preserved, the realized magnitudes of fundamental measures – and therefore expansion rate, curvature response to mass, and interaction scales – vary with θ_si. Only a restricted subset of such realizations is expected to support stable composite structure, including bound states analogous to atoms, implying that physically stable universes represent a measure–restricted subset of the admissible classes.

Finally, MQ admits well-defined conditions under which a universe domain U may lose referenceability and cease to admit a valid System Frame realization. Such loss is not a physical event within spacetime, but a global failure of the Frames mapping to satisfy normalization, unique realization, and macroscopic record stability. While MQ does not presently impose a finite upper bound on universe longevity, the existence of our Universe as a realized reference structure at finite Internal Frame update index implies that the configuration domain cannot support an unbounded accumulation of simultaneously referenceable domains, as such accumulation would violate the normalization and closure conditions required by the Frames Principle.

Accordingly, the framework permits a conceptual lifecycle of initiation, autonomous reference expansion, and eventual de-referenceability, understood as a structural consequence of Frames mapping geometry. In this context, the emergence of a universe supporting stable matter and persistent records can be interpreted as a selection effect over the space of admissible θ_si-realizations, consistent with an anthropic conditioning on referenceable observers, without introducing any additional dynamical assumptions beyond those already specified by MQ.

4.1.4 Empirical Prediction and Observability

Although the MQ universes classified in Section 4.1.3 are locally indistinguishable and satisfy the same ten constraint classes, they need not be physically identical. Constraints 8–10 fix the global topological and geometric closure class of each universe, and these structures do not permit either local or global change of class under any Frames automorphism of the Frames mapping M. However, class invariance does not imply trivial realization. A globally fixed reference structure may still admit nontrivial holonomy, defined as a mapping-induced closure mismatch arising from relational comparison along a closed reference cycle γ within the Internal Frame configuration domain ω, which itself carries no geometric structure.

In the minimal defect class underlying the 3×2 classification in Section 4.1.3, the Universe Initiation Theorem implies three independent orthogonal generators together with a single nontrivial global obstruction. Under these conditions, Constraint 8 (persistent open-boundary flux) together with the closure Constraints (9–10) produces a distinguished generator label – not a metric anisotropy – such that mapping-consistent reference comparison along closed cycles associated with that generator class fails to satisfy holonomy-free closure in M. More precisely, there exists at least one non-contractible reference cycle γ⊆U for which the associated holonomy functional ℋ(γ) (Appx. F) is nonzero, while all contractible cycles remain holonomy-free, The persistence of such nonzero holonomy is fixed by the defect class D (Appx. E).

Let the transport closure observable be defined by Δℋ(γ) ≠ 0, where Δℋ(γ) denotes the closure offset accumulated under mapping-consistent relational comparison along the closed cycle γ. In a holonomy-free universe this quantity vanishes for all admissible cycles, whereas in an obstructed universe at least one generator class admits cycles for which Δℋ(γ) ≠ 0. The mismatch may manifest as a discrepancy in synchronized reference counts, accumulated quantum phase, or other transported relational standards defined within the System Frame. Importantly, this effect reflects a global property of the Frames mapping realization and does not alter mapping-consistent local observables.

Mathematically, this phenomenon is analogous to well-known holonomy effects such as Berry phase and Aharonov–Bohm transport [34,35]. The analogy is structural only: in those cases, local field equations remain unchanged while a globally transported reference accumulates a path-dependent offset. In the MQ framework the source of the holonomy is not an external gauge field but the defect-and-closure class of the global Frames mapping M. The obstruction arises from the persistent open-boundary flux required by Constraint 8 together with the orthogonal closure conditions of Constraints 9–10.

The consequence is that universes within the same local physical class may nevertheless carry distinct global reference structures. While mapping-consistent local observables in the System Frame remain isotropic and identical across the equivalence classes identified in Sec. 4.1.3, the universe possesses a discrete global label specifying which orthogonal generator carries the obstruction and the polarity of the associated open-boundary flux. This label is encoded in the holonomy structure of the Frames mapping. Because of this, MQ yields a concrete empirical prediction.

If the Universe belongs to one of the obstructed classes identified in Section 4.1.3, then sufficiently large, closed reference cycles should exhibit nonzero transport closure offsets Δℋ(γ). Detecting such a mismatch would identify the obstructed generator and therefore determine which of the three minimal MQ topologies our Universe instantiates.

Several classes of experiment could in principle probe such effects. One possibility involves ultra-long-baseline mapping-consistent comparison of synchronized reference systems around closed relational networks constructed through distributed protocols. A second possibility involves extreme-baseline interferometric measurements capable of detecting accumulated phase offsets produced by large-scale relational cycles. A third approach involves global synchronization consistency tests across widely separated reference networks, designed to detect closure failures when different comparison paths are evaluated. Finally, cosmological observations sensitive to large-scale correlation structure could in principle reveal constraints consistent with the presence of a single obstructed generator.

Operationally, the prediction is not that one spatial direction exhibits altered local physics. Rather, one generator of the global reference structure carries a non-removable closure inconsistency detectable only through global relational comparisons. Local measurements confined to small regions of the System Frame therefore remain isotropic and consistent with mapping-realized observables.

In this sense MQ provides a potential observational handle on the discrete global classification introduced in Sec. 4.1.3. The multiplicity of universes identified there is not an unobservable metaphysical extension of the framework but a discrete structural property of global referenceability. At least in principle, this property is accessible through sufficiently sensitive tests of global closure consistency arising from the Frames mapping, as required by the Frames Principle.

4.2.0 Long-Term Universe Outcomes and Interaction Dynamics

This section classifies the long-term structural outcomes for referenceable universes within the MQ framework. Rather than treating universes as members of a stochastic population evolving in an external temporal parameter, the analysis is formulated entirely in terms of global mapping compatibility, referenceability, and geometric coexistence within the configuration domain of the Internal Frame. The central question addressed here regards the structural conditions under which distinct referenceable domains can coexist, interact, dominate, or lose referenceability under the deterministic constraints imposed by the Frames mapping.

Within MQ, a universe corresponds to a maximal referenceable domain U⊆ω on which a valid Frames mapping M:U→Ω exists satisfying normalization, unique realization, and holonomy-free closure on all contractible reference cycles. The existence of such a domain is governed by the mapping-normalized realization of conserved Internal Frame count together with the associated boundary count-flux over any relational region ω^'⊂U. Where 𝒩[ω^'] denotes the mapping normalization functional evaluated over the relational region ω^'⊂ω, then

N [ω^{'}] \equiv N [N_{I} (ω^{'}, A), ϕ_{I} (\partial ω^{'}, A)],

(33)

which explicitly aggregates the conserved count defined in Eq. (1) and the boundary flux functional defined in Eq. (5), with their consistency enforced by the continuity relations of Eqs. (3) and (6). The normalization structure introduced through Eq. (11) therefore links Internal Frame conservation to admissible System Frame realization. Here 𝒩[ω^'] depends jointly on the regional count measure N_I(ω^',A) and the boundary count-flux ϕ_I(∂ω^',A), thereby connecting conservation of measurement availability within the Internal Frame to admissible realization under the Frames mapping.

For clarity, in expanded form using Eqs. (1) and (5), this becomes

N [ω^{'}] \equiv N [\int_{ω^{'}} ρ_{I} (ξ, A) d μ (ξ), - \sum_{(ξ, ξ^{'}) \in \partial ω^{'}} J (ξ \to ξ^{'}, A)] .

(34)

Equation (34) makes this dependence explicit by expressing the normalization functional directly in terms of the Internal Frame count density representation ρ_I(ξ,A) and the associated boundary count-flux. Consequently, admissibility of a referenceable universe is governed jointly by conservation of count, consistency of boundary transfer, and holonomy-free closure under the Frames mapping.

Together with the holonomy constraint H(γ)=0 for all contractible γ⊂U, these relations constitute the conservation and closure conditions required of an admissible realization. They define the validity of a mapping M through the joint satisfaction of count conservation (Eqs. 1, 3), boundary flux consistency (Eqs. 5, 6), and closure under transport.

The long-term structural behavior of universes is therefore determined not by dynamical evolution in an external time parameter but by the topology of the configuration domain and by the extension properties of mappings satisfying the normalization condition above. When multiple candidate domains U₁,U₂⊆ωare considered, their coexistence is governed by whether their associated mappings preserve this normalization structure under joint extension.

In this formulation, the term interaction dynamics does not refer to physical collisions or causal exchanges occurring within spacetime. Instead, it denotes structural relations between referenceable domains arising from the admissibility or incompatibility of their simultaneous realization within the same underlying count medium of the Internal Frame. Such relations arise when no extension mapping M^* exists on U₁∪U₂ for which the normalization functional 𝒩[ω^'] and holonomy constraints remain simultaneously satisfied.

Formally, coexistence requires an extension M^*:U₁∪U₂→Ω, such that for all ω^'⊆U₁∪U₂, 𝒩[ω^'] is preserved and H(γ)=0 for all contractible γ, with defect class Dunchanged. Failure of this condition defines structural incompatibility.

The possible outcomes of such relations are therefore purely structural resolutions of mapping constraints. Four broad classes of long-term outcomes arise naturally from the MQ framework.

Autonomous reference expansion, in which a universe admits a sequence of nested domains U⊂U^'⊂ωwith corresponding extensions of the mapping M→M^', such that the normalization functional 𝒩[ω^''] remains invariant under extension for all ω^''⊂U^', and holonomy-free closure is preserved on all contractible cycles. In this case, conserved count and flux structure admit consistent realization over increasing domains.

Asymptotic structural isolation, in which two universes correspond to domains U₁,U₂⊆ω with U₁∩U₂=∅, and no extension mapping exists that preserves the normalization functional across U₁∪U₂. Each domain independently satisfies the admissibility conditions, but the combined domain does not admit a consistent normalization structure, preventing coexistence through extension.

Mapping dominance, in which one universe domain U₁ admits extension to a larger domain U^*⊃U₁∪U₂ such that the normalization functional is preserved globally under M^*, while no such extension exists for U₂. In this case, only one mapping satisfies the fundamental normalization condition over the maximal admissible domain, and the competing structure fails to remain consistent with conserved count realization.

De-referenceability, in which a universe domain U ceases to admit a valid mapping because the normalization condition fails, i.e., 𝒩[ω^'] cannot be satisfied for some ω^'⊂U, or equivalently, contractible cycles develop nonzero holonomy, ∃γ⊂U such that H(γ)≠0. Either condition represents a breakdown in the realization of conserved count under the Frames mapping, eliminating the associated System Frame structure.

These outcomes should not be interpreted as events occurring within a pre-existing temporal background of the System Frame. Instead, they represent geometric and informational resolutions of global mapping constraints within the atemporal configuration domain of the Internal Frame, consistent with the Frames Principle.

In this sense the MQ framework replaces the conventional stochastic multiverse picture with a deterministic structural classification of referenceable domains. Long-term universe outcomes are governed not by probabilistic branching or dynamic competition in external time, but by preservation or failure of the mapping normalization functional, the holonomy structure of reference cycles, and the extension properties of admissible mappings under the Frames mapping.

4.2.1 The Configuration Domain and the Meaning of “External”

In conventional physical theories, anything described as external to a system is implicitly assumed to exist within some background space or spacetime. Within MQ this intuition does not apply, since no System Frame exists prior to the realization of a valid Frames mapping M (Sec. 4.1.1; Appx. E).

The configuration domain ω is therefore not a physical space but a purely relational domain defined by admissible Internal Frame count assignments subject to MQ constraint systems (Sec. 3.6; see also Appx. A for formal definitions). It admits no spatial or temporal structure and cannot be interpreted as a container in which universes are located. Instead, ω denotes the total domain of admissible relational configurations on which candidate Frames mappings M may be defined.

Accordingly, ω possesses no metric or intrinsic geometry and carries no physically interpretable topology. It has no inherent notion of distance, direction, adjacency, or locality. Statements referring to universes as being spatially separated, embedded, or located within ω therefore have no meaning within MQ.

Within MQ, a universe is defined as a maximal referenceable relational structure corresponding to a maximal subdomain U⊂ω supporting a valid Frames mapping M. Formally, U is maximal under inclusion such that M:U→Ω exists and satisfies the MQ conditions of unique realization, normalization, and holonomy-free closure on contractible cycles (Appx. G.1; see also Sec. 4.0.2).

Distinct universes correspond to maximal subdomains U_i⊂ω whose associated Frames mappings M_i cannot be extended simultaneously while preserving global referenceability constraints imposed by the Frames Principle (Sec. 4.2.0; Sec. 2.1; cf. Sec. 2.3). Their distinction does not arise from spatial separation but from incompatibility of constraint structures. Two universes are therefore distinct precisely when no single Frames mapping can encode both relational structures while satisfying the Frames Principle.

The notion of what lies “outside” a universe is therefore not another region within ω, but the absence of any relational configuration that admits a valid Frames mapping consistent with MQ constraints.

4.2.2 Autonomous Expansion and Asymptotic Isolation

For an isolated universe defined by a maximal referenceable domain U⊂ω, possessing a persistent mapping initiation defect and no competing open-boundary constraints in the surrounding configuration domain ω, the total Internal Frame count within any finite relational subset Ω⊂U evolves according to the MQ mass allocation law (Eq. 13).

In this regime, the Frames mapping M remains uniquely realizable and normalizable under the global referenceability constraints imposed by the Frames Principle (Sec. 2.1) as the maximal referenceable domain U increases in accordance with the MQ mass allocation relations. The universe therefore undergoes autonomous expansion of its referenceable structure within its emergent System Frame.

As expansion proceeds, the geometric influence of the initiation defect boundary becomes increasingly diluted. Mass allocation continues at the fixed MQ rate (Eqs. D.3–D.5; Appx. E), while the effective mass accretion density decreases with increasing referenceable volume. Thus, the total allocated mass increases monotonically even while the effective allocation density decreases with increasing referenceable volume.

This regime corresponds to asymptotic dominance of configurations that preserve global mapping consistency, such that the Frames mapping M remains extendable without violation of normalization and closure constraints.

The universe remains indefinitely referenceable while the set of Internal Frame configurations compatible with its Frames mapping M grows relative to configurations that would violate the global referenceability constraints.

Because ω possesses no spatial interpretation, isolation is not defined by distance from other universes. Instead, isolation refers to the existence of a maximal domain U for which the associated Frames mapping M can be extended without encountering incompatible constraint structures (Sec. 4.2.0).

4.2.3 Outcomes and Frames Mapping Consistency

When two universes correspond to candidate Frames mappings M₁ and M₂ whose maximal domains U₁ and U₂⊂ω overlap as subsets of the configuration domain ω, the resulting outcome is determined entirely by the compatibility of those mappings.

A merged System Frame requires the existence of a single Frames mapping M that remains uniquely realizable, normalizable, and repeatable on the union U₁∪U₂ while satisfying the global constraints of the Frames Principle (Sec. 2.1).

However, within the MQ formalism such joint mappings are generically excluded by holonomy obstruction and orthogonal closure incompatibility (Appx. F–G).

More precisely, joint realizability fails when no single mapping exists whose induced holonomy class and closure algebra simultaneously preserve consistency across both domains, thereby violating the global referenceability constraints required for a valid System Frame (Sec. 4.2.0).

Consequently, overlap of the domains U₁ and U₂ within ω does not imply the existence of a single consistent System Frame. Differences in defect-carrying loop generators, holonomy class, or closure algebra typically prevent the existence of a globally consistent joint Frames mapping. What is resolved in such situations is therefore not spatial overlap but the compatibility or incompatibility of global reference structures under the Frames Principle.

4.2.4 Configuration-Space Coexistence without Mapping Compatibility

If two universes correspond to maximal domains U₁⊂ω and U₂⊂ω for which no valid Frames mapping exists on the union U₁∪U₂, then the universes remain mutually unreferenceable. In this case no shared Frames mapping exists under the Frames Principle (Sec. 2.1), and therefore no physical interaction or observational coupling can be defined between them.

When there is no observational coupling and no shared reference geometry between the two universes, the domains U₁ and U₂ may intersect as abstract subsets of the configuration domain ω. Such intersection has no physical or geometric interpretation because geometry itself arises only after a valid Frames mapping establishes a System Frame (Sec. 4.2.0).

From within either realized System Frame, the other universe is not merely unobservable but not representable within the admissible relational structure defined by the existing Frames mapping. Its relational configurations cannot be encoded without violating the global referenceability constraints required for the existence of that System Frame.

4.2.5 Mapping Dominance and Loss of Referenceability

In exceptional configurations occurring during the pre-closure quantum referencing regime – prior to the formation of a globally valid System Frame (Appx. E) – the configuration domain ω may admit more than one locally consistent candidate extension of a Frames mapping over overlapping regions. This reflects a temporary non-uniqueness of admissible mappings on partially referenceable domains before global unique realization and holonomy-consistent closure have been established.

If two candidate mappings impose incompatible defect-holonomy invariants, then at most one mapping can remain simultaneously holonomy-consistent, uniquely realizable, and normalizable when extended to the union U₁∪U₂ of their candidate domains. The mapping that continues to satisfy the Frames Principle (Sec. 2.1) and global referenceability constraints (Sec. 4.2.0) on the shared domain remains extendable, whereas the competing mapping necessarily fails at least one constraint and therefore cannot realize a valid System Frame.

Once the quantum referencing epoch terminates and reference expansion begins, the global equivalence class of the realized mapping becomes fixed up to Frames automorphism. Constraints 6–7 enforce persistence of unique realization and stable records, while Constraints 8–10 fix the defect holonomy and orthogonal closure structures governing global loop consistency. Accordingly, the selection of a single globally extendable mapping is completed prior to the establishment of a mature System Frame, and no further competition between mappings is admissible thereafter.

4.2.6 Disappearance via De-Referenceability

A universe ceases to exist as a referenceable physical system when no Frames mapping can be extended over its maximal domain U⊂ω while simultaneously satisfying the global referenceability constraints. Within MQ this condition is both necessary and sufficient for the existence of a realized System Frame (Sec. 4.2.0).

Within MQ three structurally distinct failure modes can produce this outcome.

Defect-class non-realizability (Constraint 8, initiation): the mapping defect class that anchors global referenceability ceases to remain realizable within the accessible configuration domain ω.

Open-boundary predicate failure (Constraint 8): the persistent open-boundary count flux required for initiation becomes globally neutralized, rendering the open-boundary condition unsatisfiable.

Holonomy-closure inconsistency (Constraints 9–10): the growing set of global reference cycles imposes mutually incompatible holonomy constraints such that the associated loop functionals cannot simultaneously satisfy closure conditions for all admissible cycles (Appx. F).

In each case what fails is not local physical law but the existence of a globally consistent reference structure. When no Frames mapping exists that satisfies Constraints 6–10 on the full domain U, the corresponding System Frame cannot be defined.

This disappearance does not correspond to annihilation of matter, energy, or intrinsic count density. The underlying Internal Frame count structure persists as part of the configuration domain ω, even though no valid Frames mapping remains defined on it.

Consequently, no spacetime event represents this transition. Because the System Frame itself ceases to exist, there is no temporal ordering within which the disappearance can be represented. What vanishes is not the underlying count structure but the possibility of a referenceable spacetime itself.

MQ does not presently specify a deterministic Internal Frame evolution law governing these failure modes. Nevertheless, the existence of a finite initiation boundary for our Universe implies that the configuration domain ω must admit structural conditions under which prior referenceable domains become de-referenceable.

5. APPENDICES

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

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.

System Frame

The System Frame is the non–discrete, relational encoding of Internal Frame structure induced by the Frames mapping. It does not possess an independent fundamental measure but instead represents observable quantities as continuous relational separations derived from discrete count structure. All physically meaningful observables are defined in the System Frame.

Frames mapping

The Frames mapping is the kinematic transformation that relates discrete count configurations in the Internal Frame to continuous relational structure in the System Frame. This mapping preserves realization consistency and enforces closure conditions on relational configurations. Observable quantities arise only after application of the Frames mapping, and all geometric and interaction structure emerges from its properties.

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.

MQ Action

The MQ Action is the fundamental kinematic functional defined over the Internal Frame count structure. It governs the evolution and consistency of relational configurations under the Frames mapping. The action admits distinct realizations determined by count separation: fixed interaction–scale realizations, corresponding to electromagnetic behavior at the electromagnetic demarcation, and count–resolved realizations, corresponding to gravitational behavior through the mapping–induced metric structure. In both cases, observable dynamics arise from the normalization and realization of conserved count under the mapping. All emergent physical behavior, including electromagnetic interactions and gravitational curvature, therefore arises from this single action without the introduction of any independent dynamical components.

MQ variational principle

The MQ variational principle states that physically realized configurations in the System Frame correspond to stationary points of the MQ action functional defined over admissible realizations of encoded availability derived from the Internal Frame, thereby replacing the independent postulation of dynamical laws with a single extremization condition from which all effective interactions emerge. Formally, let M:ω→ℛ denote the Frames mapping, and let ω^'⊂ω be a relational region of the Internal Frame with realized configuration Ω=M(ω^')⊂ℛ. Let ρ_R denote the corresponding coarse–grained encoded availability field. The MQ action is defined as the functional

S [ρ_{R}] = \int_{R} L (ρ_{R}, \partial ρ_{R}, \dots) d μ,

(S6)

where ℒ is constructed from count–based relations induced by the mapping M, and dμ is the induced measure on ℛ. The MQ variational principle asserts that admissible realizations satisfy δS=0 under variations consistent with the constraints of the Frames mapping and the conservation relations defined on ω^' (cf. Appx. E).

Informativity differential

The Informativity differential is the non–Lorentz length contraction associated with the discrete count structure of the Internal Frame. It arises from the normalization of conserved count under the Frames mapping and quantifies the difference between relational separation in the Internal Frame and its realized representation in the System Frame. The magnitude of the effect is count–dependent, increasing at small separations and diminishing toward the upper–count limit. The Informativity differential is distinct from curvature and represents one of the two geometric consequences of realization in the System Frame, the other being the emergent curvature associated with count loss under successive updates.

Metric differential

The metric differential characterizes the discrete–continuous mismatch that arises when conserved Internal Frame count structure is realized as a continuous observable in the System Frame under the Frames mapping. It does not represent a physical deformation in itself but rather the necessity of expressing integer–resolved count relations as normalized, non–discrete quantities.

For a fundamental measure defined by a count n in the Internal Frame, realization in the System Frame requires application of a mapping normalization functional. This process preserves count conservation while enforcing continuity, introducing a representation–level mismatch between the discrete count structure and its continuous encoding. The metric differential refers to this mismatch as a structural consequence of the mapping.

The metric differential is therefore distinct from the Informativity differential. While the Informativity differential describes a non–Lorentz length contraction in realized separations, the metric differential captures the underlying requirement that discrete count relations be expressed in a continuous form. In this sense, it provides the count–level representation of the discrete–continuous correspondence enforced by the Frames mapping, whereas the Informativity differential describes its realized geometric manifestation in the System Frame.

Electromagnetic demarcation

The electromagnetic demarcation is the fixed interaction count distance n_L = 276/θ_si, corresponding to approximately 84.60055l_f, at which known electromagnetic interactions are realized in MQ. At this scale, the effect of the Informativity differential is still significant. Its influence increases with decreasing interaction distance and rapidly diminishes with increasing distance, becoming physically negligible beyond a few hundred counts of l_f. The electromagnetic demarcation is the fixed electromagnetic interaction scale at which that discrete measure effect is realized.

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.

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 non–Lorentz 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.

Fixed interaction scale

A fixed interaction scale is a count–defined interaction distance at which a physical interaction is realized under the Frames mapping. For electromagnetic interactions, this occurs at the electromagnetic demarcation n_L = 276/θ_si, where the interaction is evaluated at a single, fixed count separation. At this scale, both the Informativity differential (non–Lorentz length contraction) and the corresponding curvature are realized with fixed values determined by the mapping. The interaction therefore reflects a specific point in the discrete count structure rather than a limiting regime.

Curvature–resolved regime

A curvature–resolved regime refers to the dependence of realized geometric structure on count separation under the Frames mapping. In this description, curvature is a direct consequence of the mapping evaluated at a given count distance. Different separations correspond to different realized curvature values in the System Frame, providing a geometric description of interaction that varies with count scale. This behavior characterizes gravitational phenomena, where measurements probe a range of separations rather than a fixed interaction scale.

Terms

A.1 Primitive definitions

This subsection defines the irreducible count-based quantities that constitute the Internal Frame. These objects form the non-geometric substrate of MQ and serve as the primitive inputs to all subsequent constructions under the Frames mapping.

ℕ₀ — the set of nonnegative integers {0,1,2, …}, used as the codomain for discrete count variables (e.g., n_I, n_S prior to density normalization).

ℤ — is the set of integers, used as the codomain for signed Internal Frame count-transfer quantities such as adjacency flux J_ij. Positive and negative signs denote orientation of transfer along an admissible adjacency link and do not imply geometric direction in the System Frame.

ℝ³ — three–dimensional System Frame configuration space for single–subsystem spatial coordinates r.

ℝ^3N — System Frame configuration space for an N–subsystem (or N–particle) encoded state Ψ(r₁,…,r_N,t).

A.2 Discrete Internal Frame primitives

This subsection establishes the primitive combinatorial and relational structures of the Internal Frame count substrate. These objects define admissible count configurations, adjacency relations, conservation structure, and regional count evolution prior to realization under the Frames mapping.

A — The Internal Frame update index is a discrete ordering parameter labeling successive configurations of the Internal Frame count medium. It indexes deterministic updates of relational count assignments n_I(ξ,A) across the configuration domain ω, with regional evolution defined on subsets ω^'⊂ω through conservation and adjacency relations. The update index A is not a physical time coordinate and carries no metric, geometric, or causal structure. It does not correspond to System Frame time t, nor to any observable temporal parameter. Instead, it provides an intrinsic ordering of count redistribution steps within the Internal Frame. Evolution with respect to A defines the deterministic conservation and redistribution of count via adjacency relations, as expressed in Internal Frame conservation laws of the form

\frac{\partial ρ_{I} (ξ, A)}{\partial A} + {div}_{adj} J_{I} (ξ, A) = 0

(S7)

. The mapping between the Internal Frame update index A and System Frame time t is established only through the Frames mapping M, and is generally nontrivial. Consequently, A should be interpreted as a kinematic sequencing parameter intrinsic to the discrete count substrate, rather than as a temporal observable.

n_I(ξ,A) — is the Internal Frame discrete measurement availability count at relational cell ξ and internal update index A; integer–valued with n_I∈ℕ₀. (No System Frame coordinates allowed.)

ρ_I(ξ,A) — is the Internal Frame count density at relational cell ξ and update index A. The quantity ρ_I represents the local distribution of measurement availability within the discrete relational structure of the Internal Frame. It is defined exclusively on the relational domain ω and carries no geometric, metric, coordinate, or continuum interpretation. The local count density ρ_I is constructed from the underlying count variable n_I(ξ,A) and serves as the local conserved quantity appearing in Internal Frame conservation relations and adjacency–flux evolution equations. The evolution of ρ_I is governed by the discrete continuity relation

\frac{\partial ρ_{I} (ξ, A)}{\partial A} + {div}_{adj} J_{I} (ξ, A) = 0

(S8)

, or equivalently,

\frac{\partial ρ_{I} (ξ, A)}{\partial A} + \sum_{ξ^{'} \in \partial Ω (ξ)} J_{I} (ξ \to ξ^{'}, A) = 0

(S9)

. The quantity ρ_I remains an Internal Frame object and must not be interpreted as a continuum mass density, probability density, or geometric field. Under the Frames mapping M, conservation of ρ_I provides the count structure from which realized System Frame observables and encoded availability measures are constructed.

N_I(ω^',A) — total Internal Frame count in a relational region Ω at internal update index A, typically

N_{I} (ω^{'}, A) = \sum_{ξ \in ω^{'}} n_{I} (ξ, A)

(S10)

(or the appropriate internal measure analogue).

ξ — is a discrete relational cell label indexing elements of the Internal Frame configuration domain ω. Each ξ∈ω identifies a primitive micro–configuration at which Internal Frame count variables n_I(ξ,A), density ρ_I(ξ,A), and adjacency flux J_I(ξ→ξ^',A) are defined. It may be described as

ξ \in ω, ω^{'} \subset ω, \sum_{ξ \in ω^{'}} n_{I} (ξ, A)

(S11)

. Key properties are that it is a pure index / label, not a coordinate. It carries no metric, geometric, or spatial meaning. It is defined only through membership in ω, adjacency relations ξ∼ξ^', and serves as a summation index and as a domain element for count density and flux.

ω — The Internal Frame configuration domain ω is an abstract relational domain containing all admissible Internal Frame count assignments and constraint systems on which candidate Frames mappings may be defined. It carries no metric, geometric, or temporal structure and does not correspond to any System Frame coordinate domain. A referenceable universe corresponds to a maximal subdomain U⊂ω on which a Frames mapping exists satisfying the global referenceability constraints (unique realization, normalization, and holonomy–free closure). Regional subsets ω^'⊂ω are used to define finite count measures, conservation laws, and boundary–flux relations within the Internal Frame. The corresponding realized System Frame configurations are given by Ω=M(ω^')⊂ℛ.

ω^' — Is a relational region of the Internal Frame, with ω^'⊂ω. This term denotes a finite subset of the Internal Frame relational adjacency domain used to define regional count measures, conservation laws, and boundary–flux relations. It carries no geometric or metric structure and does not correspond to a System Frame coordinate domain. Regional Internal Frame counts are written N_I(ω^',A).

∂ω — Adjacency boundary of an Internal Frame consisting of oriented adjacency links connecting cells in ω^' to cells outside ω^'. It appears in Internal Frame conservation relations through boundary–flux terms such as ϕ_I(∂ω^',A).

Ω₁,Ω₂ — Internal Frame relational subsets used in finite additivity relations. They satisfy Ω₁∩Ω₂=∅ and N_I(Ω₁∪Ω₂,A)=N_I(Ω₁,A)+N_I(Ω₂,A).

div_adj — is the adjacency divergence operator acting on the Internal Frame count–flux J_I. It is defined by

{div}_{adj} J_{I} (ξ, A) : = \sum_{ξ^{'} \in \partial Ω (ξ)} J_{I} (ξ \to ξ^{'}, A)

(S12)

, where ∂Ω(ξ) denotes the set of admissible neighboring relational cells of ξ. The operator returns the net count imbalance associated with relational cell ξ during update step A by summing count–flux contributions over all admissible adjacency links incident on ξ. It is a purely combinatorial operator defined on the adjacency structure of the Internal Frame and carries no metric, geometric, directional, vector, or differential structure. The operator measures local count imbalance within the conserved count network and appears in the Internal Frame continuity relation

\frac{\partial ρ_{I} (ξ, A)}{\partial A} + {div}_{adj} J_{I} (ξ, A) = 0

(S13)

. It is a purely combinatorial operator defined on the adjacency structure of the Internal Frame and carries no metric, geometric, directional, vector, or differential structure. The operator measures local count imbalance within the conserved count network and appears in the Internal Frame continuity relation

\frac{\partial ρ_{I} (ξ, A)}{\partial A} + {div}_{adj} J_{I} (ξ, A) = 0

(S14)

. Equivalently,

\frac{\partial ρ_{I} (ξ, A)}{\partial A} + \sum_{ξ^{'} \in \partial Ω (ξ)} J_{I} (ξ \to ξ^{'}, A) = 0

(S15)

. Unlike a continuum divergence, div⁡_adj does not act on a vector field and does not involve derivatives. It operates exclusively on count transfers between admissible neighboring relational cells. Under the Frames mapping, normalized realizations of the count–flux J_I and adjacency divergence div⁡_adjJ_I correspond to the conserved current and its divergence in the System Frame, respectively.

Adj(ξ) — is the set of all relational cells ξ′ connected to ξ by an admissible adjacency link within the Internal Frame configuration domain ω. The adjacency neighborhood is a purely combinatorial object defined on the relational structure of the Internal Frame. It carries no metric, geometric, coordinate, vector, or continuum interpretation. Adj(ξ) specifies the relational cells that participate in local count redistribution through the adjacency count-flux J_I(ξ → ξ′, A). It therefore provides the neighborhood structure used in the definitions of the adjacency divergence operator div_adj, the local conservation law, and boundary-flux relations. Under the Frames mapping, realizations derived from Adj(ξ) may contribute to emergent System Frame structures, but Adj(ξ) itself remains an Internal Frame object and is not a geometric neighborhood in the System Frame.

J_I — is the Internal Frame adjacency count–flux object defined on oriented adjacency links ξ→ξ^' within the configuration domain ω. For each admissible link, J_I(ξ→ξ^',A) specifies the signed count transfer between relational cells during update step A. The object J_I is the discrete flux structure associated with the conserved count field n_I and provides the local transfer mechanism appearing in the Internal Frame continuity relation Δ_An_I(ξ,A)+div_adjJ_I(ξ,A)=0. J_I is a purely combinatorial object defined exclusively on the adjacency structure of ω. It carries no metric, geometric, directional, vector, or differential structure. The relational cell labels ξ and ξ^' denote adjacency relationships only and do not represent positions in a geometric space. Consequently, J_I is not a vector field and no geometric differential operators act upon it. The associated adjacency divergence operator is defined by

{div}_{adj} J_{I} (ξ, A) : = \sum_{ξ^{'} \in \partial Ω (ξ)} J_{I} (ξ \to ξ^{'}, A)

(S16)

, where N(ξ) denotes the set of admissible neighboring relational cells of ξ. This operator returns the net count imbalance associated with relational cell ξ during update step A and measures the local departure from count balance produced by adjacency transfers. Under the Frames mapping M, J_I provides the discrete count-flux structure whose mapping-normalized realization defines the corresponding conserved current in the System Frame. The mapping does not reinterpret J_I as a geometric vector. Rather, continuum current representations emerge through normalization of conserved count flow under the Frames mapping.

ϕ_I(∂ω^' ,A) — is the Internal Frame boundary count–flux associated with a relational region ω^' ⊂ω at update index A. It is defined as the net outward signed sum of adjacency flux contributions across all oriented boundary links connecting cells in ω^' to cells in ω∖ω^' ,

ϕ_{I} (\partial ω^{'}, A) = \sum_{(ξ, ξ^{'}) \in \partial ω^{'}} J (ξ \to ξ^{'}, A)

(S17)

, where ∂ω^' consists of all admissible oriented adjacency links with ξ∈ω^' and ξ^'∉ω^' . This quantity represents the exact discrete boundary transport of measurement availability and has units of counts per update step. It is not a geometric surface integral and involves no notion of area, normal vectors, or metric structure. The boundary flux is related to the adjacency divergence of the count–flux through the discrete divergence relation, expressing regional conservation as the sum of local count imbalances over ω^' . Under the Frames mapping, preservation of ϕ_I constrains the mapping normalization functional, thereby fixing how conserved Internal Frame flux is realized as a System Frame conserved current.

𝓁: an oriented adjacency link (edge) in the Internal Frame relational adjacency domain; elements of a closed loop γ are sequences of such links.

A.3 Frames Mapping and Realization Structures

This subsection defines the relational structures associated with admissible Frames mappings and referenceable universe domains. These objects govern normalization, realization, holonomy structure, record encoding, and the consistency conditions required for emergent System Frame observables.

U — is a referenceable universe domain when a maximal subdomain U⊂ω of the Internal Frame configuration domain on which a valid Frames mapping M exists satisfying normalization, unique realization, and holonomy–free closure on all contractible reference cycles. The corresponding System Frame reference structure is given by the relational image M(U)⊂ℛ. The domain U is maximal in the sense that no proper extension U^'⊃U admits a Frames mapping satisfying the same global referenceability constraints. Distinct universes correspond to maximal subdomains whose associated Frames mappings cannot be simultaneously extended over their union without violating at least one global constraint.

γ — A reference cycle is a closed admissible relational path within a referenceable domain U⊂ω∖D used to evaluate loop functionals and consistency conditions associated with the Frames mapping. A reference cycle consists of a finite ordered sequence of admissible adjacency links returning to its origin. Contractible cycles must satisfy holonomy–free closure, while non–contractible cycles may carry defect-induced holonomy. Reference cycles are used to probe closure structure, measure consistency, and global referenceability conditions within admissible universe domains.

𝒞 — The count functional is a scalar functional acting on conserved Internal Frame count structure, mapping N_I(ω^',A) (and, where required, its associated density and flux structure) to a System Frame–compatible scalar prior to normalization. It encodes how relational count configurations contribute to realized observables independent of mapping normalization. More explicitly, it may be expressed as 𝒞:{ρ_I, J_I, ∂ω^'} → ℝ⁺, with the following constraints: i) consistency with conservation laws, ii) invariance under admissible Internal Frame relabelings, and iii) independence from System Frame coordinates.

f(r) — A System Frame observable function defined on the realized spatial coordinate domain r∈ℝ³. The function f(r) represents an arbitrary measurable quantity evaluated on a realized configuration and is used to construct expectation values over the normalized encoded availability density ρ_P(r,t). It carries no primitive ontological status and does not correspond to an Internal Frame count variable, update index, or conservation quantity. Observable averages are obtained through expressions of the form

⟨ f ⟩ = \int_{R^{3}} f (r) ρ_{P} (r, t) d^{3} r

(S18)

, or equivalently through the wave function encoding representation. The symbol fserves as a generic test function for realized System Frame observables and is not restricted to any particular physical quantity.

𝒩[ω^' ]–The mapping normalization functional is a normalization operator induced by the Frames mapping that converts conserved Internal Frame count measures into System Frame observables by enforcing consistency with boundary–flux conservation and holonomy constraints. It acts on conserved count structure over a relational region ω^'⊂ω and determines the normalization applied in the realization of M(ω^'). Formally, it is given as

N [ω^{'}] = 1 / Z (ω^{'}) with Z (ω^{'}) = \sum_{ξ \in ω^{'}} n_{I} (ξ, A)

(S19)

. Modified under discrete effects this is 𝒩[ω^' ] → 𝒩[ω^' ]⋅(2Q_Ln_Lr).

ℒ_R[Ω] — Realized separation functional is a System Frame observable defined as a functional of conserved Internal Frame count structure over a relational region ω^'⊂ω, obtained via the Frames mapping M. It assigns a measurable separation to a configuration of count density and adjacency relations without presupposing metric structure in the Internal Frame. The realized configuration is given by Ω=M(ω^')⊂ℛ. Formally, it is described as ℒ_R[Ω] = ℱ(ρ_I(ξ,A), J_I(ξ→ξ^',A), ∂ω^'), with realization ℒ_R[Ω] ≡ 𝒩[ω^']⋅𝒞(N_I(ω^',A)) where 𝒩 is the normalization induced by mapping and 𝒞 is the count functional consistent with conservation laws.

ℛ — The System Frame realization structure ℛ is the relational image induced by the Frames mapping M:ω→ℛ, where ω is the Internal Frame relational domain. The structure ℛ comprises all admissible realized relational configurations consistent with the Frames mapping and the Frames Principle. It is not a pre-existing geometric manifold and does not constitute primitive ontology. Rather, it is the realized relational encoding through which observable System Frame quantities are represented. A realized physical system corresponds to a realized configuration Ω=M(ω^')⊆ℛ, where ω^'⊆ω is an admissible Internal Frame relational region. Realized coordinates, observables, realization manifolds, and encoded probability measures arise only within ℛ.

ℜ — is a realization manifold. ℜ⊆ℛ is the set of admissible System Frame realizations compatible with a specified Internal Frame configuration and any applicable realization constraints. Realization manifolds are induced by the Frames mapping and need not be unique. They may arise from constraints imposed by count separation, conservation laws, boundary conditions, contextual restrictions, or other admissible realization conditions. Probability measures are assigned only after normalization over the admissible realization manifold.

M — is the Frames mapping is a relational mapping M:ω→ℛ that assigns Internal Frame count configurations in the relational domain ωto elements of the uniquely realizable relational structure ℛdefining the emergent System Frame. A valid realization exists only on referenceable subdomains ω^'⊆ω that satisfy normalization, repeatable realization, and holonomy-free closure on all contractible reference cycles. For any admissible subdomain ω^', the corresponding realized configuration is Ω=M(ω^')⊆ℛ. The mapping establishes consistent reference relations from which System Frame coordinates, observables, and realization structures are derived. Observable quantities arise only after application of the Frames mapping under the Frames Principle.

ℛ_ξ∈ℛ — is a System Frame realization element ℛ_ξ is an individual realized relational configuration associated with an admissible Internal Frame configuration ξ under the Frames mapping M. ℛ_ξ∈ℛ. The collection {ℛ_ξ} forms the realization structure ℛ. Realization elements encode relational information only and do not represent independent physical entities. Observable quantities arise from relations among realization elements within admissible realized configurations.

R_i — is the observer-specific macroscopic record state space associated with observer i. It consists of stable, observer-accessible realized states encoding measurement records and contextual outcomes. r∈R_i. Elements of R_i correspond to persistent realized record states that remain invariant under repeated observation and admissible evolution. Record states arise only after contextual restriction and normalization within the System Frame and therefore belong to the empirical layer of description rather than the primitive Internal Frame count structure.

E — is the admissible interaction set associated with a referenceable universe domain U. Elements x∈E denote admissible interaction configurations supporting realization under the Frames mapping. The set E is defined relationally within the Internal Frame and carries no continuum coordinate, metric, or geometric structure. Realization mappings assign outcome labels through R:E→O. The set E specifies the admissible interaction structure on which realization and record encodings are defined.

O — is the outcome label space associated with a referenceable Frames mapping. Elements o∈O are abstract realization labels assigned by the realization mapping R:E→O to admissible interactions x∈E. The set O is not a geometric or physical state space and carries no metric, coordinate, or dynamical structure. It indexes realizable outcomes within the System Frame encoding and serves as the domain of record encoding mappings ρ:O→ℛ. The realized outcome subset is O_R:=R(E)⊆O, consisting of outcomes actually realized on a given admissible interaction structure.

τ — is a record relabeling bijection induced by an admissible outcome relabeling under a Frames automorphism. The map τ:ℛ_a→ℛ_b acts on realized macroscopic record states and preserves observable equivalence under relabeling of realized outcomes. For an outcome relabeling σ:O_R→O_R, the induced record relabeling satisfies τ(ρ_a(o))=ρ_b(σ(o)),o∈O_R. The map τ preserves stable record distinguishability and introduces no change in geometric, closure, or measurable structure.

ρ_i:O→R_i — Record encoding mapping from outcome space O to macroscopic record states R_i. The mapping is injective on the realized outcome set O_R, ensuring that distinct realized outcomes correspond to distinct observable records.

ℋ(γ) — A holonomy functional is the closure mismatch obtained by transporting reference data around a closed reference cycle γunder the Frames mapping. For holonomy–free mappings, ℋ(γ)=0 on all contractible cycles.

D — A mapping defect class is a global obstruction in the Frames mapping preventing extension of a holonomy–free mapping across the full configuration domain. Persistent defects anchor universe initiation and determine the holonomy class of the resulting reference structure.

Φ — A frames automorphism is an invertible transformation Φ:U→U acting on a referenceable universe domain U⊂ω that preserves the structural constraints of the Frames mapping M, including normalization, repeatable realization, and holonomy–free closure on contractible cycles. Two universe realizations related by Φ are physically equivalent descriptions of the same System Frame structure.

𝒞_s — is the set of admissible Internal Frame configurations that correspond to a single System Frame state s under a valid Frames mapping. It represents the pre-image of s and defines the domain over which encoded availability weights are assigned and normalized.

A.4 Emergent System Frame Continuum Structures

This subsection establishes the continuum relational structures induced through the Frames mapping. These objects define encoded availability densities, emergent coordinate representations, continuum observables, and associated differential structures within the realized System Frame.

Ω — It denotes the System Frame image of a relational region ω^'⊂ω under the Frames mapping M, such that Ω=M(ω^'). The set Ω consists of encoded relational configurations corresponding to physically realized systems and carries the continuous relational structure in which observables are defined. It is not an Internal Frame domain and does not support count–based conservation laws directly.

𝒜 — System Frame log–gradient availability potential. Vector field defined on the System Frame continuum encoding by 𝒜 ≡ ∇ln⁡ρ_R, where ρ_R is the System Frame encoded availability density. This field measures the fractional spatial variation of encoded availability and represents the natural connection structure associated with multiplicative rescaling of ρ_R. It emerges from the Frames mapping of the Internal Frame count measure and has no counterpart in the Internal Frame. Under a multiplicative rescaling of encoded availability,

ρ_{R} \to ρ_{R}^{'} = e^{χ} ρ_{R}

(S20)

, the field transforms as 𝒜 → 𝒜^'=𝒜+∇χ, establishing its role as an emergent Abelian gauge potential in the System Frame representation.

n_k(t)— encoded availability in partition A_k

n_{k} (t) = \int_{A_{k}} ρ_{S} (r, t) d^{3} r

(S21)

. Represents the pre–normalized encoded availability in region A_k. Used to construct conditional probabilities via normalization:

P (k) = n_{k} (t) / N_{1} (t)

(S22)

ρ_S(r,t) — System Frame encoded availability density

ρ_{S} (r, t) = \frac{d N_{S}}{d^{3} r} = \lim_{Δ V \to 0} \frac{n_{S} (Δ V, t)}{Δ V}

(S23)

.
Continuum density obtained from the deterministic Frames mapping applied to the Internal Frame count measure. It is the Radon–Nikodym derivative of the induced System Frame count measure with respect to Lebesgue volume. Non–negative and integrable over finite regions. The total encoded availability is N₁(t)=∫ρ_S(r,t) d³r. Not normalized in general. Represents encoded measurement availability prior to probabilistic interpretation. Serves as the fundamental System Frame scalar from which ρ_pand all observable statistics are derived.

n_S(r,t) — discretized System Frame availability count field. Discrete encoded availability count associated with a System Frame cell (index j) or region A at time t, obtained via the Frames mapping from Internal Frame availability. Integer–valued when discretized. Used for counts and partitions (e.g., n_S,j(t) or n_S(A,t)), not as the argument of continuum differential operators.

g(r,t) — System Frame log–gradient availability field (dimension

1 / length

(S24)

), defined by

g (r, t) = \nabla \ln ρ_{S} (r, t) = \frac{\nabla ρ_{S} (r, t)}{ρ_{S} (r, t)};

(S25)

measures fractional spatial variation of encoded availability density.

r — System Frame spatial coordinate vector in ℝ³.

t — System Frame time coordinate used in the continuum encoding and Schrödinger evolution.

r_A,r_B — System Frame spatial coordinate vectors for subsystems A and B in a bipartite (entanglement/Bell) configuration.

A.5 Realized separation and metric structure

This subsection introduces the realized separation functional and its continuum encoding in the System Frame. The emergent metric is defined as the tensor that reproduces the mapped separation relations under coarse-graining of the discrete count structure.

N₁(t) — total encoded availability count in the single–subsystem System Frame normalization, defined by the inline equation

N_{1} (t) = \int_{R^{3}} ρ_{S} (r, t) d^{3} r

(S26)

N₂(t) — total encoded availability count in the bipartite (two–subsystem) System Frame normalization, defined by the inline equation

N_{2} (t) = \int_{R^{3}} \int_{R^{3}} ρ_{S} (r_{A}, r_{B}, t) d^{3} r_{A} d^{3} r_{B}

(S27)

ρ_P(r,t) — normalized System Frame probability density

ρ_{p} (r, t) = \frac{ρ_{S} (r, t)}{N_{1} (t)}, \int ρ_{p} d^{3} r = 1

(S28)

.
Dimensionless normalized density derived from encoded availability. Defines outcome probabilities and detection statistics. The MQ wavefunction satisfies

Ψ_{MQ} = \sqrt{ρ_{p}} e^{i ϕ}

(S29)

, with ∣Ψ_MQ∣²=ρ_p. Probability in MQ arises exclusively from normalization of encoded availability ρ_S, not from primitive Internal Frame randomness. All probabilistic and field–theoretic constructions in the System Frame are defined with respect to ρ_p.

ρ_P1(r,t),ρ_P2(r,t) — marginal probability densities for subsystems 1 and 2 obtained by integrating the joint density over the other subsystem’s coordinates.

𝒪_out — The System Frame output domain O_out⊂ℛ is the realized relational region associated with measurement readout, whose measurable partition {A_k} defines distinct macroscopic record classes. These subsets satisfy

⋃_{k} A_{k} = O_{out}

(S30)

and A_i∩A_j=∅for i≠j. The domain O_out exists only in the System Frame as a partitioned subset of the realized relational structure and has no counterpart in the Internal Frame. Probabilities associated with outcomes are obtained by normalization of encoded availability over the partition elements A_k, and distinct subsets correspond, via the record encoding mapping, to distinct macroscopic record states.

A_k — is subset A_k⊂O_out⊂ℛ is a measurable region of the System Frame output domain corresponding to a distinct macroscopic record class or measurement outcome channel. The collection {A_k} forms a mutually exclusive and exhaustive partition satisfying A_i∩A_j=∅ for i≠j and

⋃_{k} A_{k} = O_{out}

(S31)

. Probabilities are obtained by normalization of encoded availability over these subsets, with outcome weights given by

n_{k} (t) = \int_{A_{k}} ρ (r, t) d^{3} r

(S32)

, where ρ is the System Frame encoded availability density. The subsets A_kexist only within the realized System Frame encoding and have no counterpart in the Internal Frame. Distinct subsets correspond, via the record encoding mapping, to distinct macroscopic record states. Indicator functions associated with A_k define restriction and conditioning operations within the encoded Hilbert space representation.

χ_{A_{k}} (r)

(S33)

— is associated with each subset A_k is the indicator function

χ_{A_{k}} (r)

(S34)

, defined by

χ_{A_{k}} (r) = 1

(S35)

for r∈A_k and 0 otherwise, which implements restriction and conditioning of encoded availability and wavefunction support to the corresponding measurement region.

Ψ(r,t) — encoded System Frame wave function in position representation; Ψ∈L²(ℝ³).

∣Ψ⟩ — encoded System Frame state vector in Hilbert space; used for basis expansions, projection updates, and expectation values.

ϕ(r,t) — System Frame phase field of the encoded MQ wave function; defines interference and velocity/momentum fields via gradients.

A.6 Differential and operator structure

This subsection defines the operators acting on realized quantities in the System Frame. These include mapping-consistent derivative and update structures that enable the construction of curvature and field equations from the emergent metric.

∇ — System Frame gradient operator acting on continuum–encoded fields.

∇⋅ — System Frame divergence operator acting on geometric vector fields.

H — System Frame Hamiltonian operator governing unitary Schrödinger evolution (in the encoded continuum representation), typically

H = - ħ^{2} / (2 m) \nabla^{2} + V (r, t)

(S36)

Ĥ — System Frame Hamiltonian operator governing unitary Schrödinger evolution in the encoded continuum representation. For a non-relativistic encoded state,

\hat{H} = - (ħ^{2} / 2 m) \nabla^{2} + V (r, t)

(S37)

. The operator Ĥ is the continuum realization of the admissible discrete redistribution generator H_d obtained under the Frames mapping. It acts on encoded System Frame wave functions Ψ(r,t) and generates Schrödinger-form evolution through

i ħ \frac{\partial Ψ}{\partial t} = \hat{H} Ψ

(S38)

. The operator represents redistribution dynamics of normalized encoded availability under closed evolution. Under the admissibility and boundary conditions, Ĥ is Hermitian and generates unitary evolution of the encoded state.

U(t) — unitary time–evolution operator in the System Frame Hilbert representation:

U (t) = e^{- iHt / ħ}

(S39)

V(r,t) — potential energy function in the System Frame Schrödinger equation; may include external fields and interactions as encoded in the continuum representation.

S(r,t) — phase action field used in polar (Madelung) decomposition

Ψ = \sqrt{ρ_{P}} e^{iS / ħ}

(S40)

; related to phase by S=ħϕ when ϕ is dimensionless.

v(r,t) — System Frame velocity field in Madelung form:

v (r, t) = (ħ / m) \nabla ϕ (r, t)

(S41)

p(r,t) — momentum field in the System Frame phase–gradient (Madelung) representation; p(r,t)=ħ∇ϕ(r,t), with p=mv when velocity is defined.

p̂ — momentum operator in position representation: p̂=-iħ∇.

J_ρ(r,t) — System Frame probability current density defined with respect to the normalized density ρ_p. It satisfies the continuity equation ∂_tρ_p+∇⋅J_ρ=0. In Madelung form,

J_{ρ} = ρ_{p} v, v = \frac{ħ}{m} \nabla ϕ .

(S42)

Represents conserved probability flow in the System Frame. This current is not defined for ρ_S, ensuring consistency between conservation laws and normalized observables.

Δ_d — Discrete Laplacian (difference–operator Laplacian) defined on the lattice/adjacency structure of the Internal Frame. It is the discrete analog of the continuum operator ∇², constructed from adjacency relations ξ∼ξ^' and acting on count–defined fields. In the continuum limit, it approximates the standard Laplacian via

\nabla^{2} \approx \frac{Δ_{d}}{(Δ x)^{2}}

(S43)

. This operator is used in the lattice/adjacency discretization of dynamical equations, including Schrödinger–type evolution, and encodes second–order relational variation across adjacent Internal Frame configurations.

H_d — discrete Hamiltonian operator used in lattice/discretized dynamics, typically

H_{d} = - ħ^{2} / (2 m) Δ_{d} / (Δ x)^{2} + V

(S44)

, with Δ_d the discrete Laplacian defined on the chosen adjacency/lattice and Δx the spatial step.

Q(r,t) — quantum potential (Madelung form)

Q (r, t) = - \frac{ħ^{2}}{2 m} \frac{\nabla^{2} \sqrt{ρ_{p} (r, t)}}{\sqrt{ρ_{p} (r, t)}}

(S45)

Defined strictly in terms of the normalized density ρ_p. Encodes quantum corrections to the Hamilton–Jacobi–like equation arising from the encoded availability structure after normalization.

A.7 Fundamental constant and measures

This subsection collects the invariant constants and normalization measures that appear in the MQ framework. These quantities fix the scale of mapping between discrete counts and continuum observables without introducing independent phenomenological parameters.

l_f,m_f,t_f — are the fundamental measures, more precise expressions for Planck’s units–length, mass, and time–that consider the effects of length contraction associated with discrete measure.

2Q_Ln_Lr — also known as the Informativity differential describes the length contraction associated with discrete measure. It is a dimensionless normalization modifier arising from the discrete count structure of the Internal Frame, representing the deviation between integer–resolved count measures and their realized System Frame encoding under the Frames mapping M:ω→ℛ. It quantifies how conserved Internal Frame count N_I(ω',A) is redistributed and normalized when expressed as a System Frame observable.

G — is the gravitational constant, G=6.67407794(17) × 10^–¹¹ m³ kg^–¹ s^–² [60] such that its value considers the effects of length contraction associated with discrete measure at the upper count limit. Italicized G identifies a measure (e.g., G=6.67384483(17) × 10^–¹¹ m³ kg^–¹ s^–² [59] at the blackbody demarcation ([13], Appx. AB)).

θ_si — can be measured as the polarization angle of quantum entangled X–rays at the degenerate frequency of a maximal Bell state. As an angle θ_si=3.26239 rad ± 2 μrad [36]. As a momentum θ_si = 3.262390305(89) kg m s^–¹. And with respect to the System Frame, θ_si has no units.

c is the speed of light which may also be written as c=n_Ll_f/n_Tt_f =299,792,458 m/s [29] such that n_L=n_T=1 is physically significant.

H₀ describes the rate of expanding space between galaxies. MQ expands this nomenclature as a function of two parameters. Expansion can be described either as a function of the local rate H (km s^–¹ Mpc^–¹) or the scalar rate H (i.e. magnitude). Terms describe either observed expansion (subscript 0) or metric expansion (no subscript).

H_CMB describes the rate of expansion in km s^–¹ Mpc^–¹ as resolved using CMB data, also known as the Hubble constant H₀.

H_cep describes the rate of expansion between galaxies in km s^–¹ Mpc^–¹ as resolved using cepheid data, also known as the Hubble constant H₀.

H_sys represents a System Frame description of the Hubble constant as would be consistent with a CMB study. This calculation describes the metric expansion of space relative to the non–discrete System Frame.

H_int represents an Internal Frame description of the Hubble constant as would be consistent with a cepheid study. This calculation describes the metric expansion of space relative to the discrete Internal Frame.

H_b represents the baseline expression which falls at the midpoint of H_sys and H_int.

H describes the scalar rate of expansion associated with the increasing radius of the universe. For our universe, this non–dimensional scalar is described by, H=θ_si.

H₀ describes the observed scalar rate of expansion associated with the increasing radius of the universe. There is no measure of this value at this time.

l_p, m_p and t_p are Planck’s units for length, mass, and time.

n_L, n_M and n_T are physically significant discrete counts of l_f, m_f and t_f respectively.

n_L describes the count of l_f representative of a change in position of an observable measured with respect to the observer’s frame of reference.

n_Tu is a count of t_f equal to the age of the universe.

n_Lu is a count of l_f equal to the diameter of the universe.

n_Lc describes the count of l_f representative of a change in position of light measured with respect to the observer’s frame of reference.

n_θ is the System Frame count of θ_si resolved as a function of the Internal Frame demarcation adjusted for the metric differential (i.e., ⌊84.60055/2⌉ = 42).

h is Planck’s constant.

ħ is the reduced Planck constant, 1.054571817 10^–³⁴ m² kg s^–¹. When accounting for the Informativity differential at the upper count bound, this term is not italicized (i.e., ħ=1.054534984(26) 10^–³⁴ m² kg s^–¹).

A_ee is the dilated age of the universe as measured from the current expansionary epoch. Its value is a function of expansion with respect to the total elapsed time representing the quantum epoch.

A_qe is the non–dilated age of the universe corresponding to the quantum epoch.

V_qe describes the volume of the universe at the end of the quantum epoch.

Ω_vis is that domain of the universe that is presently visible.

Ω_obs is that domain of the universe that is presently visible or will be visible given infinite elapsed time.

Ω_dk is that domain of the universe that will never be visible.

Ω_uobs is that domain of the universe that will eventually be visible but is not presently visible.

Ω_tot is the entire domain that makes up the universe (i.e., 1 or 100%).

Ω_f is that domain of the universe that is fundamental. Its physical significance describes a pivot point with respect to the remaining domains.

M_tot is the total mass of the universe, including all forms of energy.

M_acr is the rate of mass accretion in the universe.

m_e is the mass of an electron.

a₀ is the ground state orbital of an atom, also known as the Bohr radius.

α_f^–¹, α_p^–¹, and α_c^–¹ are the fundamental, Planck–like, and classical forms of the inverse fine structure constant. α_f^–¹=42θ_si is defined with respect to the System Frame; the remaining two against the Internal Frame such that α_c^–¹ differs from α_p^–¹, accounting for the length contraction associated with discrete measure.

γ is a collection of terms that describe the geometry between the System and Internal frames.

Λ describes Einstein’s cosmological constant.

ɛ₀ is the electric constant.

u₀ is the magnetic constant.

k_e is Coulomb’s constant.

e— is elementary charge.

E_CMB is the total energy associated with the CMB.

E_ρ:cmb is the energy density associated with the CMB.

k is the Boltzmann constant.

ρ is the energy density accumulated at a given age of the universe.

T_CMB is the temperature of the Cosmic Microwave Background.

A.8 Expansion and cosmological scalars

This subsection defines the scalar quantities governing expansion in MQ, including mapping-derived normalization factors and cosmological terms. These quantities describe the homogeneous expansion as it emerges from count-based structure under the Frames mapping.

t — denotes the global ordering parameter labeling the one-parameter family of realized configurations in the System Frame. It provides the monotonic parameterization of expansion as defined by the evolution of the mapping-realized separation functional and does not represent a primitive temporal coordinate in the Internal Frame.

a(t) — denotes a derived System Frame scale function used to parameterize the evolution of realized separations under mapping normalization. It is strictly increasing and piecewise differentiable but does not constitute a fundamental degree of freedom; rather, it provides a convenient parametrization of expansion extracted from the underlying separation functional.

l_* — denotes an invariant internal reference scale associated with the mapping-normalized separation structure. It remains constant under expansion and serves to fix the reference normalization relating discrete count structure to realized System Frame separations, without introducing an independent dynamical scale.

Q_L — denotes the mapping-induced residual count contribution entering the realized separation functional. In the right-triangle count construction, n_L gives the Internal Frame count separation, while n_L + Q_L gives the realized hypotenuse whose continuum encoding defines the System Frame separation. Q_L therefore replaces the older Internal-to-System scale-ratio notation by expressing expansion and curvature through count-resolved residual structure rather than through a global ratio.

Λ_MQ — denotes the MQ-derived cosmological term obtained from the scalar contribution after mapping-consistent normalization. It is not introduced phenomenologically in the action but is fixed by the expansion structure and realized separation relations used to construct the metric and field equation.

Λ_sys — denotes the System Frame realization of the MQ cosmological term, evaluated using the System Frame expansion rate. It represents the continuum-frame scalar contribution entering comparison with CMB-inferred cosmological determinations.

Λ_int — denotes the Internal Frame realization of the MQ cosmological term, evaluated using the Internal Frame expansion rate. It represents the discrete-frame scalar realization associated with the corresponding mapping-normalized expansion structure.

A.9 Mapping and normalization symbols

This subsection collects symbols associated with mapping normalization and the single-action realization of dynamical quantities under the Frames mapping. It includes the functions and parameters that convert dimensionless scalar quantities into their observable, unit-resolved forms and govern admissible dynamical variation.

ρ_f— denotes an invariant, dimensionless internal geometric normalization factor used in the mapping of homogeneous expansion. It is not the FRW scale factor and does not define the general emergent metric. Its role is limited to fixing the reference normalization through which discrete count evolution is related to continuum expansion in the homogeneous limit.

Δ_H — denotes the mapping-induced normalization offset between System Frame and Internal Frame realizations of the scalar expansion parameter. It is defined by Δ_H=θ_si-π and represents a frame-resolution offset under the Frames mapping, not an independent dynamical parameter or a global count-expansion mismatch.

H— denotes the dimensionless scalar rate of radial expansion defined at the level of the System Frame realization of the Frames mapping. It represents the intrinsic expansion magnitude governing the growth of realized separations and is independent of observational units. Within MQ, H = θ_si, so that H is not a measured local rate but the mapping-defined scalar controlling the expansion geometry and associated mapping-consistent scalar relations.

H₀— denotes the dimensionful expansion rate expressed in observational units (e.g., km s^-1
Mpc^-1). It is obtained through mapping-consistent normalization of the scalar H, converting the dimensionless expansion into a measurable quantity. It does not constitute an independent parameter, but rather the observational representation of H under a specified measurement convention at the present epoch.

H_sys— is the System Frame realization of the expansion rate, obtained by expressing the scalar H under mapping normalization consistent with the non-discrete relational encoding. It corresponds to expansion values inferred from CMB-based measurements and represents the continuum realization of the underlying scalar expansion.

H_int— is the Internal Frame realization of the expansion rate, obtained by expressing the scalar H under mapping normalization consistent with discrete count structure. It corresponds to expansion values inferred from local distance-ladder measurements and differs from H_sys by a mapping-induced normalization offset rather than by an independent physical parameter.

H — denotes the dimensionful local expansion rate expressed in observational units (e.g., km s^-1 Mpc^-1), prior to specification of a particular frame realization. It represents the normalized, unit-resolved form of the scalar expansion parameter H and serves as the general observable expansion rate from which specific realizations such as H₀, H_sys, and H_int are defined.

S — denotes the single MQ action functional defined over the count structure and its coarse-grained realization under the Frames mapping. It supplies the common variational origin of the metric, curvature, source, coupling, and scalar contributions, so no independent gravitational or matter action is fundamental at the primitive level.

S_g— denotes the geometric contribution to the realized action when the single MQ action is expressed in continuum form. It is a derived representation of the curvature constructed from the emergent metric, not an independently postulated Einstein-Hilbert action.

S_m— denotes the matter contribution to the realized action when the single MQ action is expressed in continuum form. It defines the stress-energy tensor through variation with respect to the emergent System Frame metric and is not an independently specified matter description. The paper states that the stress-energy tensor is obtained by varying the same action with respect to the emergent metric.

Φ— denotes the coarse-grained dynamical field encoding deviations of the count configuration under realization. It enters the MQ action as the realized field variable whose admissible variations are constrained by the Frames mapping and conserved count structure.

ℒ— denotes the realized Lagrangian density appearing in the MQ action. It is constructed from mapping-induced quantities and contains no independent gravitational coupling or separate primitive decomposition.

ℒ_m— denotes the matter Lagrangian density in the realized continuum representation. It is constructed from encoded availability, its derivatives, and the emergent metric, with admissible terms fixed by the MQ variational principle rather than by an independently postulated matter description.

A.10 Field equation components

This subsection defines the geometric and source terms entering the MQ field equation. These quantities are constructed from the emergent metric and mapping-consistent action, ensuring that curvature, stress-energy, and coupling arise from a single unified framework.

ρ_tot— the total density

ρ_{tot} = 3 H_{sys}^{2} / 8 π G

(S46)

inferred from the MQ System Frame expansion rate corresponds to the observationally inferred cosmic energy density when the MQ kinematic expansion law is interpreted within the standard gravitational framework.

ρ_c— the critical density

ρ_{c} = 3 H^{2} / 8 π G

(S47)

is the total energy density required for spatial flatness in the standard Friedmann–Lemaître cosmological model. It provides the conventional normalization scale against which fractional density components are defined.

ρ_Λ— the vacuum density

ρ_{Λ} = Λ c^{2} / 8 π G

(S48)

is the effective energy density associated with the cosmological constant term appearing in Einstein’s field equations. In the MQ framework, this quantity corresponds to the dynamical realization, within general relativity, of the geometric scalar derived from invariant measure relations.

ρ_m— the matter density ρ_m=(1-Ω_Λ)ρ_c represents the non–scalar contribution to the total cosmic energy density. Under the assumption of spatial flatness, it corresponds to the complement of the scalar fraction and includes both baryonic and dark matter components.

ρ_MQ— the MQ geometric density

ρ_{MQ} = \sqrt{Λ} c^{2} / 2 l_{f}

(S49)

is the density scale arising directly from invariant fundamental measure normalization in the Internal Frame. It depends only on the geometric scalar Λand the fundamental length measure l_f, and does not involve the gravitational coupling constant. This quantity represents the kinematic density associated with the induced scalar prior to any dynamical interpretation.

Ω_Λ— The fractional scalar density parameter

Ω_{Λ} = ρ_{Λ} / ρ_{c} = Λ c^{2} / 3 H^{2}

(S50)

specifies the proportion of the total cosmic energy density attributed to the scalar contribution. Within MQ it is determined entirely by invariant internal relations.

Ω_b— baryonic matter fraction (ΛCDM comparison bookkeeping).

Ω_c— cold dark matter fraction (ΛCDM comparison bookkeeping).

g_μν — The quantity g_μν denotes the emergent metric tensor in the System Frame. In standard general relativity, g_μν is a fundamental field defining spacetime geometry. In MQ, it is a derived object obtained through coarse–graining of the realized separation functional induced by the Frames mapping. Its normalization is fixed by mapping constraints and not freely specified.

G_μν— The tensor G_μν is the curvature tensor constructed from g_μν through second–order differential structure. In MQ, it represents the continuum limit of the discrete second–difference operator acting on count displacement in the Internal Frame. Its divergence–free property follows from mapping–induced invariance, not from an independent geometric postulate.

T_μν — The tensor T_μν denotes the stress–energy tensor in the System Frame. In MQ, it is defined variationally from the matter contribution to the MQ action,

T_{μ ν} = - \frac{2}{\sqrt{- g}} \frac{δ S_{matter}}{δ g^{μ ν}},

(S51)

and therefore encodes the response of normalized count density to variations in the emergent metric. It is not independently specified and inherits its structure from encoded availability and mapping normalization.

𝒦_MQ — denotes the MQ gravitational coupling factor appearing in the emergent field equation. It defines the proportional coupling between the realized curvature sector and the mapping-induced stress-energy structure in the System Frame. Unlike standard GR, 𝒦_MQ is not introduced as an independent phenomenological constant, but is derived from the MQ primitive measures and the Frames mapping. In the realized continuum representation,

K_{MQ} = 8 π G / c^{4} = 4 π t_{f} / θ_{si}

(S52)

. The quantity therefore encodes the mapping-normalized gravitational response implied by the conserved Internal Frame count structure under coarse-grained realization.

Λ — The scalar Λ denotes the cosmological term appearing in the continuum field equation. In MQ, it is not a fundamental constant but a derived scalar arising from the same realized separation structure that defines the emergent metric

g_{μ ν}^{sys}

(S53)

. It reflects the large-scale normalization of the System Frame geometry induced by the Frames mapping and is therefore fixed by the underlying count geometry, not introduced as an independent parameter.

APPENDIX B: TOPOLOGICAL ORIGIN OF MAPPING DEFECTS IN DETERMINISTIC COUNT MEDIA

This appendix establishes the topological origin of mapping defects arising in deterministic count media subject to the closure requirements of the Frames mapping. The analysis is formulated entirely within the relational adjacency structure of the Internal Frame and assumes the count conservation relations established in Appx. C. No manifold, metric structure, geometric embedding, or stochastic dynamics are introduced. The objective is to determine when local holonomy-free closure can fail to extend globally and to identify the resulting obstruction as a mapping defect class.

Let ω denote the global configuration domain of the Internal Frame count medium. Let ω^'⊂ω be a relational region and let U⊂ω denote a referenceable universe domain containing ω^'. Let M:U→Ω denote the Frames mapping, where Ω denotes the realized relational image in the System Frame corresponding to U. Let ϕ_I(∂ω^',A) denote the Internal Frame boundary count-flux across the adjacency boundary of a relational region ω^'⊂ω, and let Δ(γ) denote the holonomy functional measuring closure mismatch around a reference cycle γ. A valid Frames mapping on a referenceable universe domain must satisfy unique realization, holonomy-free closure on contractible reference cycles, and structural persistence against incompatible boundary constraints.

Closed versus open domains

If U is closed with respect to the relational adjacency structure of the Internal Frame, then admissible boundary data are determined by internal loop closure under the Frames mapping M. Holonomy-free closure requires that transporting reference data around any contractible reference cycle returns without closure mismatch. The local count-conservation and boundary flux relations entering this obstruction analysis are

Δ A ρ_{I} (ξ, A) + \sum_{ξ^{'} \in Adj (ξ)} J (ξ \to ξ^{'}, A) = 0,

(B.1)

ϕ_{I} (\partial ω^{'}, A) = \sum_{(ξ, ξ^{'}) \in \partial ω^{'}} J (ξ \to ξ^{'}, A),

(B.2)

Δ (γ) = 0, \forall γ \subset U, contractible.

(B.3)

Equations (B.1–B.2) are the local conservation relations proved in Appx. C and are included here only to identify the count-flux quantities entering the obstruction analysis. Equation (B.3) states the local holonomy-free closure condition required for unique realization under the Frames mapping.

Now suppose the domain carries a global boundary constraint that cannot be generated by contractible interior cycles. In MQ this condition corresponds to the existence of a non-zero preserved boundary count-flux across the adjacency boundary ∂U, inducing non-trivial holonomy under the Frames mapping. Let ℋ(∂U) denote the holonomy functional evaluated on the boundary adjacency structure ∂U. This functional measures the net closure mismatch induced by boundary-constrained transport under the Frames mapping. The obstruction condition is

H (\partial U) \neq 0 .

(B.4)

A non-vanishing holonomy functional evaluated on ∂U indicates that the closure constraints imposed by the boundary adjacency structure cannot be generated entirely from contractible interior cycles. Equivalently, local closure may hold for every internal contractible loop,

Δ (γ) = 0 for all internal contractible loops,

(B.5)

while the boundary-induced cycle class carries non-zero net holonomy,

\sum_{γ \in Γ_{\partial U}} Δ (γ) ≢ 0 (mod 2π),

(B.6)

where Γ_∂U denotes the set of admissible boundary-induced reference cycles generated by the adjacency structure of ∂U.

Equations (B.3–B.6) define a global obstruction: holonomy-free closure holds locally on contractible cycles yet fails globally because the preserved boundary count-flux induces non-trivial holonomy. Therefore, the Frames mapping cannot be extended to a globally consistent realization over all of U without localizing the obstruction.

Defect localization

A holonomy structure that is trivial on all internal contractible loops cannot support non-trivial boundary holonomy as a globally smooth extension of the same closure condition. The inconsistency between internal closure and boundary holonomy must therefore localize as a defect set.

In MQ this localization corresponds to the emergence of a non-empty mapping defect class D⊂U, defined as the support on which the obstruction to extension of a holonomy-free Frames mapping is carried. The defect is not a failure of deterministic count conservation. It is a failure of global Frames mapping consistency induced by the incompatibility between local closure and boundary-constrained transport.

Equivalently, the obstruction defined by Eqs. (B.2–B.4) implies the existence of a non-empty mapping defect class D⊂U such that the Frames mapping fails to satisfy unique realization or holonomy-free closure when extended across D, while remaining consistent on U∖D. The detailed realization constraints satisfied by D, including the minimal orthogonal closure condition and associated

\sqrt{3}

(S54)

realization threshold, are established in Appx. C.

This incompatibility arises from the conjunction of holonomy-free closure with a non-zero boundary flux constraint defined on the Internal Frame adjacency structure. It is therefore a structural consequence of the Frames mapping, not an independent dynamical mechanism. These conditions prevent extension of a globally consistent realization across the entire configuration domain.

General obstruction principle

The preceding construction gives the MQ realization of the obstruction principle: incompatibility between local closure conditions and global boundary constraints induces a non-trivial mapping defect class D carrying the obstruction. When a relational mapping is required to satisfy mutually incompatible global constraints, consistency failure localizes as a defect set within the domain. MQ realizes this principle for the Frames mapping defined on the discrete configuration domain ω, without invoking an underlying geometric manifold.

Defects and universe initiation

In MQ a mapping defect class D is the localized support of the obstruction preventing global extension of a holonomy-free Frames mapping. Appx. C demonstrates that persistent boundary count-flux together with localized defect support provides the necessary and sufficient conditions for the existence of a referenceable universe domain U⊂ω. A referenceable universe corresponds to a maximal referenceable universe domain U⊂ω containing a structurally persistent mapping defect class D that supports non-zero open-boundary count-flux.

The obstruction defined in Eqs. (B.2–B.6) is therefore not a dynamical creation mechanism. It is the topological condition requiring the existence of localized defect support whenever local holonomy-free closure and non-trivial holonomy are jointly imposed. The minimal orthogonal realization of that defect support, including the

\sqrt{3}

(S55)

referencing criterion, is derived in Appx. C.

Thus, cosmogenesis in MQ corresponds to the existence of a non-removable mapping defect that satisfies the conditions for referenceability under the Frames mapping, consistent with the Frames Principle. The mapping defect class D, constrained by minimal orthogonal closure, carries the global obstruction and anchors the realized relational structure Ω, thereby fixing the minimal scale at which consistent referenceability is achieved.

The significance of the present appendix is therefore topological rather than dynamical. Equations (B.4–B.6) establish that non-trivial holonomy cannot be removed while preserving local closure. The resulting obstruction must localize into a mapping defect class D, and this defect provides the topological support required for the universe initiation construction developed rigorously in Appx. C.

APPENDIX C: FULL PROOFS OF THE UNIVERSE INITIATION THEOREM

This appendix provides the complete mathematical proof of the Universe Initiation Theorem using only deterministic count redistribution on the Internal Frame configuration domain, adjacency-constrained conservation structure, and the Frames Principle [5,6]. No stochastic postulates, inflationary dynamics, or phenomenological assumptions are invoked.

All arguments are formulated within the configuration domain ω of the Internal Frame. No external spatial manifold or physical time parameter is assumed. The labels introduced below serve only to index relational cells, adjacency aggregation, and conservation constraints. They carry no independent geometric or temporal meaning.

C.1 Preliminaries and Definitions

Let ω denote the Internal Frame configuration domain. Let ω^'⊂ω denote a bounded relational region. Let U⊂ω denote a maximal referenceable universe domain when such a domain exists. The corresponding realized System Frame image is

M : U \subset ω \to Ω = M (U) \subset R .

(C.1)

The realized configuration Ω belongs to the System Frame realization structure ℛ and therefore does not constitute an Internal Frame conservation domain. Conservation statements throughout this appendix are defined exclusively on subsets of ω.

Let the Internal Frame count structure be described by the deterministic count density ρ_I(ξ,A), together with the adjacency count-flux J_I(ξ→ξ^',A). These quantities describe redistribution of measurement availability between admissible neighboring relational cells. Local count conservation is governed by the primitive adjacency conservation law

Δ A ρ_{I} (ξ, A) + \sum_{ξ^{'} \in Adj (ξ)} J (ξ \to ξ^{'}, A) = 0 .

(C.2)

The quantity ρ_I(ξ,A) denotes the local Internal Frame count density, while J_I(ξ→ξ^',A) denotes the signed count transfer along an admissible oriented adjacency link connecting relational cells ξ and ξ^'. The set Adj(ξ) contains all admissible neighboring relational cells of ξ. The update index A labels successive deterministic redistributions of count within the Internal Frame and is not a physical time coordinate.

The adjacency count-flux is purely combinatorial. It carries no metric, geometric, coordinate, vector, or continuum interpretation. Consequently, Eq. (C.2) is not a geometric continuity equation but rather a conservation law defined on the relational adjacency structure of the Internal Frame.

For any bounded relational region ω^'⊂ω, the total Internal Frame count is

N_{I} (ω^{'}, A) = \int_{ξ \in ω^{'}} ρ_{I} (ξ, A) .

(C.3)

This quantity represents the total measurement availability contained within the relational region ω^' at update index A.

To analyze conservation on finite relational regions, define the adjacency boundary

d ω^{'} = {(ξ, ξ^{'}) | ξ \in ω^{'}, ξ^{'} \notin ω^{'}, ξ^{'} \in Adj (ξ)} .

(C.4)

The boundary ∂ω^' consists of all oriented adjacency links connecting cells contained within ω^' to cells outside ω^'. This object is purely relational and does not correspond to a geometric surface.

The associated boundary count-flux is

ϕ_{I} (\partial ω^{'}, A) = \sum_{\begin{array}{l} (ξ \to ξ^{'}) \in \partial ω^{'} \\ ξ \in ω^{'}, ξ^{'} \notin ω^{'} \end{array}} J_{I} (ξ \to ξ^{'}, A) .

(C.5)

The quantity ϕ_I represents the net outward transport of measurement availability across the adjacency boundary and provides the exact discrete flux associated with the bounded relational region.

Summing Eq. (C.2) over all relational cells contained in ω^' causes all adjacency transfers between cells that both lie within ω^' to cancel pairwise. Only boundary-crossing contributions remain, yielding

Δ_{A} N_{I} (ω^{'}, A) = - ϕ_{I} (\partial ω^{'}, A) .

(C.6)

Equation (C.6) is the regional conservation law associated with deterministic count redistribution on the Internal Frame. It follows from local adjacency conservation and requires no continuum approximation.

A valid Frames mapping M on a maximal referenceable universe domain U⊂ω requires the conserved Internal Frame count structure to satisfy the global referenceability constraints introduced earlier in this paper. The conserved count must admit finite normalization under the mapping normalization functional 𝒩_M; each admissible interaction must correspond to a unique realized System Frame outcome; and every contractible reference cycle γ⊂U must satisfy holonomy-free closure, ℋ(γ)=0. These conditions ensure that deterministic Internal Frame count redistribution admits a consistent realized System Frame representation.

Only after application of the Frames mapping do continuum objects such as coordinates, vector fields, divergence operators, and continuous conserved currents become meaningful. Consequently, Eq. (C.2) constitutes the primitive conservation law from which all subsequent System Frame conservation descriptions are derived.

C.2 Proof of Necessity: Closed Regions Cannot Initiate Universes

Lemma C.1

If a relational region ω^'⊂ω admits zero net count-transfer across its adjacency boundary, then the total Internal Frame count within the region is conserved.

Proof: Consider a bounded relational region ω^'⊂ω composed of relational cells ξ. Let ∂ω^' denote the adjacency boundary consisting of oriented adjacency links connecting cells inside ω^' to adjacent cells outside the region. The net outward count-transfer across this boundary is measured by the boundary count-flux ϕ_I(∂ω^',A), defined as the signed sum of adjacency count-flux contributions crossing the boundary during Internal Frame update step A.

For a strictly closed relational region, no count-transfer occurs across the adjacency boundary. The closure condition is therefore

ϕ_{I} (\partial ω^{'}, A) = 0 .

(C.7)

We begin from the local conservation relation established in Eq. (C.2), which expresses conservation of Internal Frame count measure at each relational cell ξ∈ω^'. Summing Eq. (C.2) over all cells of the bounded region yields

\sum_{ξ \in ω^{'}} Δ_{A} ρ_{I} (ξ, A) + \sum_{ξ \in ω^{'}} \sum_{ξ^{'} \in Adj (ξ)} J (ξ \to ξ^{'}, A) = 0 .

(C.8)

The first term represents the update change of the total Internal Frame count measure contained within the region. Using the regional Internal Frame count measure

N_{I} (ω^{'}, A) = \int_{ω^{'}} ρ_{I} (ξ, A) d μ (ξ)

(S56)

, consistent with Eq. (C.3), this term becomes

\sum_{ξ \in ω^{'}} Δ_{A} ρ_{I} (ξ, A) = Δ A N_{I} (ω^{'}, A) .

(C.9)

We next evaluate the adjacency-transfer term. For any adjacent pair of relational cells ξ,ξ^'∈ω^', the corresponding transfer contributions enter with opposite orientation,

J (ξ \to ξ^{'}, A) + J (ξ^{'} \to ξ, A) = 0 .

(C.10)

Consequently, all adjacency transfers between cells entirely contained within ω^' cancel pairwise when summed over the region.

The only remaining contributions are adjacency links for which one cell lies inside ω^' and the adjacent cell lies outside. These surviving terms collectively define the net outward boundary count-flux,

\sum_{\begin{array}{l} ξ \in ω^{'} \\ ξ^{'} \notin ω^{'} \end{array}} J (ξ \to ξ^{'}, A) = ϕ_{I} (\partial ω^{'}, A) .

(C.11)

Substituting Eqs. (C.9) and (C.11) into Eq. (C.8) gives the regional conservation relation

Δ_{A} N_{I} (ω^{'}, A) + ϕ_{I} (\partial ω^{'}, A) = 0 .

(C.12)

Applying the closed-region condition of Eq. (C.7) immediately yields

ϕ_{I} (\partial ω^{'}, A) = 0 \Rightarrow Δ A N_{I} (ω^{'}, A) = 0 .

(C.13)

Thus, the total Internal Frame count measure within a strictly closed relational region remains invariant across successive Internal Frame update steps. ∎

A closed relational region ω^', characterized by vanishing boundary count-flux, cannot exhibit net growth of Internal Frame count measure. Therefore, a closed region cannot supply the nonzero boundary count-transfer required for the open-boundary condition established in Theorem 4.0 (Universe Initiation Theorem). Any realizable universe must consequently arise from a relational system whose boundary condition is not globally closed, so that sustained boundary count-transfer can support nonconstant evolution of the Internal Frame count measure N_I(ω^',A). The persistence and geometric consequences of such boundary violation are established in the subsequent propositions.

C.3 Necessity: Closed Regions Cannot Initiate Universes

Proposition C.1 ■
A strictly closed relational region ω^'⊂ω cannot initiate a universe.

Proof: By Lemma C.1, the total Internal Frame count within ω^' is conserved. Thus, N_I(ω^',A) remains invariant under updates of the Internal Frame update index A. Since a strictly closed relational region satisfies ϕ_I(∂ω^',A)=0, no net count transfer occurs across ∂ω^', and the regional count content available to the Frames mapping remains finite.

Let R_i(ω^') denote the observer-specific macroscopic record state space associated with realizable configurations within ω^', and let ρ_i:O→R_i denote the record encoding mapping, which must remain injective on the realized outcome set O_R under a valid Frames mapping. Because N_I(ω^',A) is finite and conserved, the admissible mapped realization content supported entirely within ω^' is finite. Deterministic evolution on a finite realization set implies eventual recurrence. Therefore, there exist update indices A₁≠A₂ such that

M (ξ, A_{1}) = M (ξ, A_{2}) \Rightarrow (ξ, A_{1}), (ξ, A_{2}) \in R_{M} (ω^{'}) .

(C.14)

Finite mapping normalization is still formally available for the closed region,

N_{M} [N_{I} (ω^{'}, A), ϕ_{I} (\partial ω^{'}, A)] < \infty .

(C.15)

However, recurrence in a strictly closed finite realization set prevents the mapped record structure from supporting indefinitely distinguishable realized outcomes. Distinct Internal Frame update states are forced into the same realized macroscopic record state, so injectivity of ρ_i on O_R cannot be maintained for universe initiation. This violates the repeatable record-encoding condition required for a valid Frames mapping on any referenceable universe domain U⊂ω.

Therefore, no referenceable universe domain U can be supported entirely within a strictly closed relational region ω^'. ■

C.4 Necessity: Persistence of Boundary Violation

Proposition C.2
A transient boundary count–flux violation is insufficient for universe initiation.

Proof: Suppose non-zero boundary count–flux exists only over a finite interval of Internal Frame updates. Let K∈ℕ denote the finite number of consecutive update steps during which the boundary count–flux remains non-zero. Then

\exists K \in N suchthat ϕ_{I} (\partial ω^{'}, A) \neq 0 for \forall A \in {A_{0}, A_{0} + 1, . . ., A_{0} + K} .

(C.16)

For all updates outside that finite interval,

ϕ_{I} (\partial ω^{'}, A) = 0 .

(C.17)

From the regional conservation relation,

Δ_{A} N_{I} (ω^{'}, A) = - ϕ_{I} (\partial ω^{'}, A),

(C.18)

it follows that once the finite interval terminates,

ϕ_{I} (\partial ω^{'}, A) = 0 \Rightarrow Δ_{A} N_{I} (ω^{'}, A) = 0,

(C.19)

Thus, after the transient boundary event, N_I(ω^',A) again becomes update-invariant. The relational region therefore reduces to the strictly closed case of Proposition C.1 with respect to subsequent realization. A finite-duration boundary violation may alter the regional count content, but it does not provide the structurally non-removable open-boundary condition required to prevent recurrence and sustain a referenceable universe domain.

Thus, a transient boundary count–flux violation cannot initiate a universe. Persistent open-boundary count–flux is required. ■

C.5 Holonomy and Mapping Defects

Let γ denote a closed reference cycle in a referenceable universe domain U⊂ω under the Frames mapping M. The holonomy functional ℋ(γ) denotes the closure mismatch obtained by transporting reference data around γ under M. For a valid mapping on U, holonomy-free closure requires

H (γ) = 0, \forall γ \subset U is contractible.

(C.20)

Lemma C.2
Unique realization of the Frames mapping on U requires holonomy-free closure on all contractible reference cycles.

If Eq. (C.20) fails on a contractible cycle, the returned realization depends on the path used to transport reference data. The same relational configuration would then admit inequivalent realized reference assignments, violating unique realization. Thus, holonomy-free closure on contractible cycles is a necessary condition for a valid Frames mapping.

A mapping defect class D is supported within a bounded relational region ω^'⊂ω when closure consistency cannot be extended across the full region. The obstruction condition is

\exists_{γ} \subset ω^{'}, such that H (γ) \neq 0 .

(C.21)

The corresponding defect criterion is

D (ω^{'}) \neq 0 \Leftrightarrow ϕ_{I} (\partial ω^{'}, A) \neq 0 \land \exists γ_{i} \subset ω^{'} : H (γ_{i}) \neq 0 .

(C.22)

Equation (C.22) identifies the joint condition required for a mapping defect class in this appendix: non-zero boundary count–flux prevents reduction to the closed-region case, while non-zero holonomy identifies the obstruction to extending holonomy-free closure across the bounded relational region. The defect is therefore a failure of global Frames mapping consistency, not a failure of deterministic Internal Frame count conservation.

The deterministic count-conservation law remains intact. Mapping defects arise from the incompatibility between boundary-supported count transfer and global closure under the Frames mapping. This establishes the obstruction structure required for the sufficiency result in Proposition C.3.

C.6 Sufficiency: Defects Generate Referenceable Domains

Proposition C.3.
A persistent mapping defect D⊂ω that supports sustained non-zero boundary count–flux and satisfies holonomy-free closure on a maximal referenceable subdomain is sufficient to initiate a referenceable universe.

Proof: Let D⊂ω denote a mapping defect class supported within a bounded relational region ω^'⊂ω. By Eq. (C.22), the defect condition is the joint presence of non-zero boundary count–flux and non-vanishing holonomy on at least one defect-supported reference cycle.

Thus, D identifies the localized support of the obstruction to extending holonomy-free closure across the full relational region. The defect is not a failure of deterministic count conservation, but a failure of global Frames mapping consistency. This interpretation is consistent with the definition of a mapping defect as a global obstruction to extension of a holonomy-free Frames mapping.

Suppose there exists a finite persistence interval of Internal Frame updates such that

\exists Δ A > 0, such that ϕ_{I} (\partial ω^{'}, A) \neq 0 for A \in [A_{0}, A_{0} + Δ A] .

(C.23)

From the regional conservation relation established in Lemma C.1, persistent non-zero boundary count–flux implies that the regional count content evolves throughout the persistence interval. In particular,

Δ_{A} N_{I} (ω^{'}, A) \neq 0 .

(C.24)

Consequently, the defect changes the count content available for realization within ω^' and prevents reduction to the closed-boundary recurrence condition established in Proposition C.1.

On the maximal referenceable subdomain U⊂ω, holonomy-free closure is satisfied for every contractible reference cycle while the defect condition requires at least one reference cycle contained in ω^' with non-vanishing holonomy. The defect therefore supplies boundary-supported count transfer, while the maximal referenceable subdomain U remains holonomy-free on contractible reference cycles and admits consistent realization.

The induced encoded availability under the Frames mapping admits normalization,

\int_{Ω} {| Ψ |}^{2} d μ_{Ω} = 1,

(C.25)

and there exists a maximal referenceable subdomain U⊂ω on which both injective realization through the record encoding mapping ρ_i:O→R_i and holonomy-free closure are satisfied. A valid Frames mapping therefore exists on U, and its System Frame image Ω=M(U) defines a referenceable universe. ∎

C.7 Minimal Radius and the

\sqrt{3}

(S57)

Threshold

The minimal realization scale follows from orthogonal closure of count separation along independent relational directions. Let l_f denote the fundamental realized length scale in the System Frame, obtained through the Frames mapping from the underlying Internal Frame configuration. The smallest count vector admitting simultaneous closure along three independent adjacency directions is

n_{\min} = (1, 1, 1),

(C.26)

where n_min denotes the minimal closure count vector.

The corresponding minimal orthogonal configuration is

U_{\min} = {γ_{1}, γ_{2}, γ_{3}}, γ_{i} ⊥ γ_{j} (i \neq j) .

(C.27)

where U_min denotes the minimal orthogonal reference-cycle configuration composed of three mutually orthogonal reference cycles.

The realized separation associated with this configuration is therefore

R_{\min} = ‖ n_{\min} ‖ l_{f} = \sqrt{3} l_{f} .

(C.28)

where R_min denotes the minimal realized separation associated with U_min.

Equation (C.28) follows from the mapped norm of the minimal three-direction count vector. It represents the smallest realized separation capable of simultaneously satisfying orthogonal closure constraints under the Frames mapping, without assuming a pre-existing metric geometry in the Internal Frame.

Below this scale, at least one closure direction is absent and a complete three-direction orthogonal realization cannot be formed. At R_min, all minimal closure constraints are satisfied simultaneously, establishing the smallest realized radius admitting holonomy-free closure.

C.8 Completion of the Universe Initiation Theorem

Propositions C.1–C.3, together with the minimal orthogonal closure condition established in Sec. C.7, provide the necessary and sufficient conditions for the existence of a referenceable universe within the Internal Frame configuration domain.

Proposition C.1 establishes that a strictly closed relational region cannot generate a referenceable universe. Proposition C.2 establishes that transient boundary count–flux is insufficient. Proposition C.3 establishes that a persistent mapping defect supporting non-zero boundary count–flux, together with holonomy-free closure on a maximal referenceable subdomain, is sufficient for realization. Section C.7 identifies the minimal realized closure scale required for that realization.

Under these conditions, a valid Frames mapping exists on a maximal referenceable subdomain U⊂ω, and its System Frame image Ω=M(U) defines a referenceable universe.

Therefore, the Universe Initiation Theorem holds. ∎

APPENDIX D: DEFECT GEOMETRY, ORTHOGONALITY, AND GRAPH-THEORETIC CLOSURE

This appendix establishes the relational, orthogonal, and graph-theoretic structure associated with deterministic mapping defects in the Internal Frame configuration domain. The analysis is formulated entirely within ω, with bounded relational regions ω^'⊂ω used for local closure analysis and maximal referenceable domains U⊂ω used only where a valid Frames mapping exists. No embedding in external spacetime is assumed, and realized geometric quantities appear only after application of the Frames mapping.

The objective is twofold: first, to identify how deterministic mapping defects arise from incompatible closure and boundary constraints, and second, to establish the minimal algebraic structure required for simultaneous closure under a self-consistent Frames mapping.

D.1 Discrete Count Geometry

The geometric content of the Frames Principle is rendered explicit through the right-triangle construction shown in Fig. 2. This construction is not a geometric structure inside ω. Rather, it is the realized representation obtained after conserved count structure is normalized through the Frames mapping. The labels a, b, and c denote the sides of the illustrative right-triangle construction (Fig. 2) and are not independent MQ variables.

Within this representation, Side a specifies the unit reference count, thereby fixing the normalization scale used for count comparison prior to the introduction of any continuous realized geometry. Side b corresponds to the Internal Frame count separation n_L; following application of the Frames mapping, the corresponding realized separation is expressed in units of the fundamental realized length scale l_f.

Side c, the hypotenuse, represents the realized separation in the System Frame. This hypotenuse corresponds to the mapping-consistent realized separation functional in which the Internal Frame count contribution is combined with the mapping-induced residual count contribution Q_L, such that the realized separation is represented by the count combination n_L+Q_L (Appx. A.8). The quantity Q_L therefore represents the mapping-induced residual count contribution entering the realized separation functional, not an additional primitive measure in the Internal Frame.

This construction produces a count-resolved relation consistent with the right triangle configuration. The orthogonality of Side a and Side b reflects algebraic independence between the reference normalization and the count separation, while the hypotenuse encodes the realized separation following mapping-induced normalization. The separation measured in the System Frame is thus a mapping-normalized realization of conserved count, rather than a direct projection of discrete count, and necessarily includes the residual contribution Q_L.

The purpose of this representation is operational rather than merely illustrative. It isolates the discrete-continuous mismatch inherent in the realized separation functional and supplies the realized geometric structure employed in the subsequent development of closure and defect arguments.

D.2 Closure and Loop Consistency in Count Graphs

A valid Frames mapping is defined on a referenceable domain U⊂ω only when reference data transported around every contractible reference cycle return without closure mismatch. Let γ⊂U be a closed reference cycle and let ℋ(γ) denote the holonomy functional associated with the Frames mapping. Holonomy-free closure requires

H (γ) = 0, \forall γ \subset U .

(D.1)

Equation (D.1) uses the holonomy functional because γ is a reference cycle and the relevant MQ object is the closure mismatch under the Frames mapping, not a signed increment sum over an undefined edge functional.

Lemma D.1 (Loop Closure Condition)

A necessary condition for a uniquely realized Frames mapping on a referenceable domain U is that Eq. (D.1) hold for all contractible reference cycles.

Proof: If ℋ(γ)≠0 for some contractible γ⊂U, transporting reference data around γ returns a closure mismatch. The realized reference then depends on the path used to compare the same relational configuration, so M is not uniquely realized. Conversely, if ℋ(γ)=0 for all contractible cycles, then any two admissible paths whose concatenation forms a contractible loop induce the same realized reference relation, establishing path independence on the contractible part of U. ∎

D.3 Orthogonality and Independent Loop Requirements

Consider a bounded relational region ω^'⊂ω used to test whether a valid maximal referenceable domain U⊂ω can exist. To support a valid Frames mapping, the eventual referenceable domain must admit independent closed reference cycles required for global closure.

Lemma D.2 (Independent Loop Requirement)

If a candidate relational region fails to support one of the independent closure cycles required by the minimal closure structure, then no valid referenceable domain U containing that region can satisfy holonomy-free closure for all contractible reference cycles associated with that closure structure.

Proof: If one required independent closure cycle cannot be formed, then two admissible relational comparisons can connect the same reference data while differing by transport along that independent closure cycle. Their concatenation produces a contractible reference cycle with nonzero closure mismatch, ℋ(γ)≠0, contradicting Eq. (D.1). ∎

Thus, any minimal referenceable domain must support three independent closure cycles, consistent with the minimal orthogonal construction developed below.

D.4 Minimal Orthogonal Construct

The smallest structure satisfying the independent loop requirement is the minimal orthogonal closure configuration, defined by one unit of count contribution associated with each of three independent closure cycles. Define the minimal closure count vector by n_min=(1,1,1), where n_min is a symbol denoting the smallest count configuration capable of supporting the minimal orthogonal closure structure.

Define R_min as the minimal realized closure radius associated with the mapped realization of n_min. After application of the Frames mapping, the corresponding realized center-to-vertex separation is

R_{\min} = \sqrt{1^{2} + 1^{2} + 1^{2}} l_{f} = \sqrt{3} l_{f} .

(D.2)

Here l_f is the realized length scale used to express the mapped separation in the System Frame. The Internal Frame contribution is the count vector n_min, not a pre-existing spatial length. Equation (D.2) is therefore the realized form of the minimal closure condition.

This radius represents the minimal realized scale at which orthogonal closure can be satisfied simultaneously. Below this scale, at least one required closure cycle is absent and complete closure cannot be formed. At

R_{\min} = \sqrt{3} l_{f}

(S58)

, all minimal closure constraints are satisfied simultaneously, establishing the smallest realized separation admitting holonomy-free closure under the Frames mapping.

D.5 Deterministic Origin of Defects

Mapping defects arise when local holonomy-free closure constraints are incompatible with global boundary conditions on an Internal Frame relational region ω^'. In graph-theoretic language, this is a frustration condition: the local closure equations cannot all be satisfied under the imposed boundary data.

Lemma D.3 (Deterministic Defect Formation)

In a finite relational adjacency structure with admissible boundary conditions on ∂ω^', there exists a non-empty class of boundary conditions for which at least one holonomy-free closure constraint cannot be simultaneously satisfied. This follows from standard results in topological defect theory, graph frustration, constraint satisfaction, and discrete gauge theory. When the independent closure constraints imposed by reference cycles exceed the admissible boundary conditions available to satisfy them, frustration is unavoidable [25,29,39]. Thus, mapping defects are not stochastic fluctuations; they are deterministic consequences of incompatible closure and boundary constraints under the Frames mapping.

D.6 Persistence of Defects

A mapping defect persists when the closure inconsistency cannot be removed by admissible local reassignment of boundary conditions. In MQ, persistence corresponds to a non-removable obstruction supported by sustained boundary count-flux across ∂ω^'.

Persistence occurs when the independent holonomy-free closure constraints cannot be simultaneously satisfied by the admissible boundary conditions. In this regime, no admissible reassignment of boundary conditions can satisfy all closure constraints, and the inconsistency cannot be eliminated by local deformation of the adjacency structure.

This condition is consistent with established results in constraint satisfaction, graph frustration, and discrete gauge theory, where over constrained systems generically admit no global solution and therefore exhibit persistent obstruction [40].

In this case, the obstruction cannot be removed by local deformation of the adjacency and boundary conditions. The resulting persistent defect class can then support the conditions associated with the Universe Initiation Theorem by preventing extension of a globally holonomy-free Frames mapping across the full configuration domain while permitting a valid restriction on a maximal referenceable domain U.

D.7 Relation to Known Mathematical Results

The preceding results align with established mathematical and physical structures: topological defect formation in deterministic lattices and spin systems [25], frustration and constraint-satisfaction theory [39], and holonomy or loop consistency in discrete gauge theories [29]. These references do not make MQ a lattice gauge theory or a spin model. Rather, they establish that the mathematical relation between local closure, boundary constraints, and persistent obstruction is standard across well-developed mathematical and physical settings.

MQ adds the physical interpretation supplied by the Frames Principle: the obstruction is not a defect in spacetime but a failure of global mapping consistency within ω, and a universe corresponds to a maximal subdomain whose realized System Frame image satisfies normalization, unique realization, and holonomy-free closure.

This connection grounds the physical claim. The mathematical constraints identify when a consistent reference structure can exist. The Frames mapping supplies the operational bridge from those constraints to observable physical structure. Consequently, deterministic defect formation is not merely an abstract graph property but the MQ condition under which a referenceable physical world becomes possible.

APPENDIX E: QUANTUM EPOCH – FRAMES MAPPING PRIOR TO UNIVERSE FORMATION

This appendix establishes the MQ early universe construction connecting the pre-referenceable quantum epoch to the first realized System Frame and to the present-day CMB. Its purpose is not to introduce additional assumptions, but to show that the definitions of the Internal Frame, System Frame, Frames mapping, the fundamental measures, and the observational domains jointly imply a measurable consequence: the present-day CMB density and temperature.

Before the Frames mapping becomes globally valid, the quantum epoch corresponds to a pre-referenceable state associated with a candidate referenceable domain U. In this state, physical length, elapsed time, mass, temperature, density, and spacetime volume are not yet defined as observables. Such quantities acquire physical meaning only after realization into the System Frame image Ω=M(U). Accordingly, the SI-equivalent quantities appearing below are interpreted as mapped encodings of Internal Frame count structure and not as observables existing within the pre-referenceable quantum epoch itself. MQ therefore does not describe a classical thermodynamic universe preceding referenceability. Rather, it describes a count-structured allocation whose realized System Frame encoding is identified with the physical quantities associated with the CMB after realization.

E.1 Count-Resolved Curvature and the Role of θ_si

The first step is to connect MQ count geometry to a curvature scale that becomes physically meaningful only after realization. Within the Internal Frame, the non-whole count contribution Q_L represents the residual count structure arising from realization of discrete count relations under the Frames mapping. When expressed through the fundamental measures and subsequently encoded within the realized radial description, the following spatial curvature is obtained:

\frac{Q_{L} l_{f}}{n_{L} t_{f}^{2}},

(E.1)

\frac{Q_{L} l_{f}}{n_{L} t_{f}^{2}} (\frac{c}{θ_{si}}) = \frac{Q_{L} c^{2}}{n_{L} t_{f} θ_{si}} = \frac{Q_{L} l_{f} c^{2}}{n_{L} l_{f} t_{f} θ_{si}} = \frac{Q_{L} c^{3}}{r θ_{si}},

(E.2)

\frac{Q_{L} c^{3}}{r θ_{si}} \approx \frac{G}{r^{2}} .

(E.3)

G = \lim_{n_{L} \to \infty} 2 Q_{L} n_{Lr} c^{3} \frac{t_{f}}{m_{f}} = c^{3} \frac{t_{f}}{m_{f}} .

(E.4)

These expressions show that θ_si is not introduced as an independent phenomenological parameter. Rather, within MQ it functions as the mapping-normalization quantity relating residual count structure to realized curvature measures. The comparison with

G / r^{2}

(S59)

is introduced only after realization, when the count construction acquires a System Frame interpretation.

Relative to the 2010 and 2014 CODATA recommended values, MQ yields a 3.94σ concordance (Appx. K). Similar behavior is obtained when the 2018 and 2022 CODATA adjustments are included, with the corresponding MQ analysis discussed following Table II of Ref. [5]. The concordance should be interpreted as an empirical consistency test of the MQ construction rather than as an independent derivation of the CODATA adjustment procedure.

We further draw attention to the 2Q_Ln_L term, a geometric consequence of the discrete Internal Frame count structure. Within MQ this term produces the Informativity differential, a non-Lorentz length contraction associated with discrete measure ([5], Appx. D). The effect enters calculations involving the gravitational constant and other physical constants. Its magnitude increases at small count separation and diminishes rapidly with increasing separation.

Accordingly, MQ distinguishes measures evaluated below the upper-count limit from their macroscopic realizations by notation. Italicized quantities denote count-resolved measures, while non-italicized quantities denote macroscopic quantities or quantities evaluated in the upper-count limit.

At present, two physically relevant realization regimes have been identified. Electromagnetic interactions occur at the fixed count separation 84.60055 l_f ([5], Appx. AB, AE, AO), where the Informativity differential remains significant. Most other presently accessible measurements occur at substantially larger count separations, where the effect is negligible within current experimental precision.

E.2 Quantum Epoch Termination and First Mapping Realization

As established in Appx. C and Appx. D, a globally valid Frames mapping requires simultaneous orthogonal closure across three independent count directions. The corresponding minimal closure condition is

\sqrt{1^{2} + 1^{2} + 1^{2}} = \sqrt{3}

(S60)

. This quantity does not represent a physical radius during the quantum epoch. Rather, it represents the minimum count-closure condition required for global referenceability. Below this threshold, no externally referenceable realization can be consistently formed. Consequently, the realized System Frame image Ω does not yet exist and physical observables remain undefined within the framework.

Upon satisfaction of this closure condition, the Frames mapping becomes globally valid, Ω=M(U) is realized, and quantities such as length, mass, time, density, and temperature become well-defined System Frame observables. The

\sqrt{3}

(S61)

threshold should therefore be interpreted as a referenceability condition associated with realization rather than as a physical size scale of a pre-existing spacetime.

E.3 Observational Domains as Mapping Geometry

The observational domains introduced below are mapping-derived geometric partitions. They are not independent cosmological components added to MQ but arise as geometric consequences of the Frames mapping applied to an expanding realized frame.

The visible domain, Ω_vis, corresponds to the portion of the realized universe presently referenceable within the current expansion epoch. The observable domain, Ω_obs, corresponds to the total referenceable portion of the realized universe and therefore includes both the presently visible domain and the presently unobserved portion that becomes referenceable at later epochs. The unobserved domain, Ω_uobs, is the difference between the observable and visible domains and represents regions that are not presently referenceable but become referenceable as additional elapsed time accumulates. The dark domain, Ω_dk, is the complement of the observable domain and represents regions that remain permanently non-referenceable as a consequence of the realized expansion geometry.

Unlike the preceding domains, the fundamental domain, Ω_f, is not an additional spatial partition of the universe. Rather, it is a geometric invariant of the domain construction whose value constrains and determines the relationships among the observable partitions. In this sense, Ω_f serves as the normalization pivot of the partition geometry, while Ω_vis, Ω_uobs, Ω_obs, and Ω_dk describe the realized referenceability structure of the universe.

Ω_{tot} = Ω_{obs} + Ω_{dk} .

(E.5)

Ω_{dk} + Ω_{obs} = 1 .

(E.6)

Ω_{obs} = 2 θ_{si} Ω_{vis} .

(E.7)

Ω_{uobs} = Ω_{obs} - Ω_{vis} .

(E.8)

All expressions resolve the domain fractions directly from the radial expansion scalar θ_si, showing that the same mapping-normalization quantity appearing in the realization construction also determines the observable partitioning. Furthermore, the partition is not specified independently of the realization geometry. Rather, the visible, observable, unobserved, and dark domains emerge as mutually constrained components of a single geometric construction whose invariant structure is encoded by Ω_f.

This relationship is significant because the same parameter governing realization and count-resolved curvature also fixes the observational partition employed in the subsequent mass allocation calculation.

2 θ_{si} Ω_{vis} = Ω_{obs},

(E.9)

\frac{2}{θ_{si}^{2}} = 4 \frac{Ω_{vis}}{θ_{si} Ω_{obs}} = Ω_{f},

(E.10)

Ω_{f} = \frac{2}{θ_{si}^{2}} .

(E.11)

2 θ_{si} Ω_{vis} = Ω_{obs},

(E.12)

2 θ_{si} Ω_{vis} \frac{θ_{si}^{2}}{2} - 1 = Ω_{obs} \frac{θ_{si}^{2}}{2} - 1,

(E.13)

θ_{si}^{3} Ω_{vis} - 1 = Ω_{obs} \frac{θ_{si}^{2}}{2} - 1 .

(E.14)

Using Ω_dk + Ω_obs = Ω_tot = 1, the dark domain follows as

Ω_{dk} = (Ω_{tot} - Ω_{dk}) \frac{θ_{si}^{2}}{2} - 1 = (1 - Ω_{dk}) \frac{θ_{si}^{2}}{2} - 1 .

(E.15)

Ω_{dk} = (1 - Ω_{dk}) \frac{θ_{si}^{2}}{2} - 1 .

(E.16)

Ω_{dk} = \frac{θ_{si}^{2} - 2}{θ_{si}^{2} + 2} .

(E.17)

Such that the dark is the total (i.e., 1) minus the observable, then

1 - Ω_{obs} = Ω_{obs} \frac{θ_{si}^{2}}{2} - 1 .

(E.18)

2 = Ω_{obs} \frac{θ_{si}^{2}}{2} + Ω_{obs},

(E.19)

Ω_{obs} = \frac{4}{θ_{si}^{2} + 2} .

(E.20)

Ω_{obs} = 2 θ_{si} Ω_{vis} = \frac{4}{θ_{si}^{2} + 2},

(E.21)

Ω_{vis} = \frac{2}{θ_{si} (θ_{si}^{2} + 2)} .

(E.22)

Ω_{uobs} = Ω_{obs} - Ω_{vis} = \frac{4 θ_{si} - 2}{θ_{si} (θ_{si}^{2} + 2)} .

(E.23)

The connection to the CMB power spectrum should be understood through parameter sensitivity rather than through literal identification of each MQ domain with an isolated acoustic peak. In the standard treatment, the CMB temperature power spectrum is produced by acoustic oscillations of the photon–baryon fluid before recombination, modified by gravitational forcing, baryon loading, radiation driving, diffusion damping, projection geometry, and later lensing. The relative locations and amplitudes of the acoustic peaks constrain combinations of baryon density, cold dark matter density, total matter density, curvature, expansion history, and dark-energy effects. Thus, the peaks do not represent separate material domains. They encode the response of the coupled early universe plasma and gravitational potentials to the density content and geometry of the universe.

Within MQ, the proposed significance of Table 1 is therefore not that Ω_vis, Ω_uobs, Ω_obs, Ω_dk, and Ω_f are themselves acoustic oscillations. Rather, the claim is that the same mapping-derived fractions that partition referenceability also reproduce the fractional density structure inferred from the CMB acoustic hierarchy. The visible domain corresponds numerically to the baryonic fraction, whose observational imprint in the CMB is associated with baryon loading and the relative enhancement or suppression of acoustic peak amplitudes. The unobserved domain corresponds numerically to the cold dark matter fraction, whose imprint appears through gravitational potential structure, matter radiation evolution, and the relative heights of the higher acoustic peaks. The observable domain corresponds to the combined matter fraction, Ω_c+Ω_b, which controls the matter contribution to the expansion and gravitational response inferred from the full spectrum. The dark domain corresponds numerically to the complementary dark energy fraction, whose effect is not a primary acoustic oscillation but enters through the angular diameter distance to last scattering, late-time expansion, the integrated Sachs–Wolfe contribution, and lensing.

The fundamental domain, Ω_f, requires a distinct interpretation. Because Ω_f is not a spatial partition, it should not be identified with baryonic matter, cold dark matter, dark energy, or the observable matter fraction. Its role is instead geometric. It fixes the invariant normalization relation from which the other domain fractions are obtained. In the CMB spectrum, such a quantity would not appear as an additional density component in the ΛCDM parameter inventory. Its possible observational role is as a normalization constraint reflected in the organization of the acoustic hierarchy itself. Thus, Ω_f may be interpreted as the MQ geometric pivot whose value is indirectly reflected in the peak structure through the derived fractions Ω_vis, Ω_uobs, Ω_obs, and Ω_dk, rather than as an additional cosmological substance.

While these quantities are not directly equivalent to the density parameters appearing in ΛCDM, they provide geometric constraints that map onto the density fractions inferred from the CMB power spectrum. Their numerical values and resulting partition structure are summarized in Table 1.

Table 1. Cosmological Density Parameters: Planck 2018 vs. MQ Predictions.

Parameter	MQ Domain	Planck 2018 (ΛCDM) [%]^a	MQ Fraction [%]
Dark energy Ω_Λ	Dark Ω_dk	68.34±0.84%	68.36241612(24)
Dark Matter Ω_c	Unobserved Ω_uobs	26.73±0.82%	26.78874899(19)
Ω_c+Ω_b	Observable Ω_obs	31.66±0.84%	31.63758388(24)
Baryonic Ω_b	Visible Ω_vis	4.97±0.82%	4.848834891(60)
-	Fundamental Ω_f	-	18.79135772(17)

^a 2018 Planck collaboration results using TT,TE,EE+lowE [11].

At the phenomenological level, this geometric partitioning suggests that cosmological observables sensitive to inferred mass–energy distributions, including matter-density estimates, acoustic peak amplitudes, angular diameter distance constraints, and weak lensing reconstructions, may contain signatures associated with mapping defined observational domains. This does not require treating the MQ domains as additional substances. Instead, it treats the CMB spectrum as the observational transfer function through which the underlying referenceability partition becomes visible in the fitted cosmological density fractions.

Consequently, observationally inferred mass distributions could exhibit signatures associated with the underlying mapping structure in addition to contributions arising from particulate matter. The strongest version of this claim is not that the MQ domains replace the acoustic physics of the CMB, but that the domain fractions provide a geometric origin for the same fractional structure that standard cosmology extracts from acoustic oscillations, projection effects, and lensing.

The existence of these geometric constraints should not be interpreted as a complete description of cosmological phenomenology. Interaction specific dynamics, structure formation, lensing amplitudes, recombination physics, and early universe perturbation evolution require additional physical modeling beyond the domain partition relations derived here. Accordingly, the Frames mapping is proposed as a foundational geometric constraint on cosmological structure rather than a complete cosmological model in itself.

E.4 Quantum-Epoch Mass Allocation and Expansion-Epoch Mass Accretion

During the quantum epoch, MQ does not describe the production of physical mass within a pre-existing spacetime. Because no realized physical observables yet exist the relevant quantity is mass allocation: a count-based assignment that becomes realized as System Frame mass-energy only when a valid Frames mapping first exists. Using the observational domain relations and the MQ mapping between the quantum and expansionary epochs, we resolve

\frac{M_{obs}}{M_{f}} - 1 = 1 - M_{obs},

(E.24)

M_{tot} = M_{f} (\frac{2}{M_{obs}} - 1),

(E.25)

M_{tot} = n_{Tu} m_{f} θ_{si} (\frac{2}{1} \frac{θ_{si}^{2} + 2}{4} - 1) = n_{Tu} m_{f} \frac{θ_{si}^{3}}{2} .

(E.26)

And such that A_U = n_Tut_f, then

M_{tot} = A_{U} \frac{θ_{si}^{4}}{l_{f}} .

(E.27)

The corresponding rate of mass allocation is

M_{acr} = \frac{n_{Mu}}{n_{Tu}} = \frac{θ_{si}^{3}}{2} = 17.3611207370 (89) units m_{f} per unit t_{f} .

(E.28)

M_{acr} = \frac{θ_{si}^{4}}{l_{f}} = 7.00887947539 (16) 1 0^{36} kg s^{- 1} .

(E.29)

Prior to realization, these relations determine the count allocation associated with the future realized mass-energy content of the universe. After realization, the same fixed relation is interpreted as the mass accretion law of the expansionary epoch. The allocation fixed during the quantum epoch therefore determines the realized mass-energy content later associated with the CMB.

E.5 SI-Equivalent Quantum Epoch Scale Prior to Mapping Realization

Before the count radius exceeds the

\sqrt{3}

(S62)

threshold, external reference is not possible. The object under consideration is therefore not yet a physical universe in the ordinary sense, but a pre-realized candidate domain U that has not yet admitted a globally valid Frames mapping. Consequently, the effective rates below are not physical velocities in spacetime. They are SI-equivalent encodings of the mapping depth required to reach the first referenceable Ω=M(U).

\frac{n_{Lu} l_{f}}{n_{Tu} t_{f}} = \frac{2 θ_{si}}{n_{Mu} m_{f}},

(E.30)

\frac{n_{Lu'} l_{f}}{n_{Tu} t_{f}} = \frac{2 θ_{si}}{n_{Mu} m_{f}},

(E.31)

v' = \frac{n_{Lu'} l_{f}}{n_{Tu} t_{f}} = \frac{2 θ_{si}}{n_{Mu} m_{f}} = \frac{2 θ_{si}}{M_{U}} = \frac{2 θ_{si}}{V_{U} ρ_{c}} = \frac{2 θ_{si}}{(4 / 3) π c^{3} θ_{si}^{3} A_{qe}^{3} ρ_{c}},

(E.32)

v' = \frac{3}{2 π c^{3} θ_{si}^{2} A_{qe}^{3}} \frac{1}{ρ_{c}} = \frac{3}{2 π c^{3} θ_{si}^{2} A_{qe}^{3}} \frac{8 π G}{3 H_{f}^{2}} = \frac{4}{c^{3} θ_{si}^{2} A_{qe}^{3} H_{f}^{2}} \frac{l_{f} c^{3}}{2 θ_{si}},

(E.33)

v' = \frac{1}{H_{f}^{2}} \frac{2 l_{f}}{θ_{si}^{3} A_{qe}^{3}} = A_{qe}^{2} \frac{2 l_{f}}{θ_{si}^{3} A_{qe}^{3}} = \frac{2 l_{f}}{θ_{si}^{3} A_{qe}} = 8.1195732436 {(21)}^{- 50} m s^{- 1},

(E.34)

R_{qe} = \frac{2 l_{f}}{θ_{si}^{3}} \int (\frac{1}{A_{qe}}) dt = \frac{2 l_{f}}{θ_{si}^{3}} \ln (A_{qe}),

(E.35)

A_{qe} = e^{\sqrt{3} θ_{si}^{3} / 2} = 1.14652655409 (32) 1 0^{13} s = 363, 312.34127 (22) y .

(E.36)

The numerical value of A_qe is therefore not an elapsed time experienced within a realized universe. It is an SI-equivalent scale associated with the mapping threshold and the count allocation required for first realization. Its significance is that it fixes the allocation subsequently realized at the onset of the expansionary epoch.

E.6 Emergence of Length, Mass, and Time

Once the

\sqrt{3}

(S63)

condition is satisfied, length, mass, and time become realized properties of Ω. MQ then resolves the fundamental measures by relating the count resolved curvature expression to the standard gravitational form. This is the point at which l_f, m_f, and t_f acquire their physical interpretation as realized measures rather than pre-realization count scales.

\frac{Q_{L} c^{3}}{r θ_{si}} = \frac{G}{r^{2}},

(E.37)

l_{f} = \frac{G θ_{si}}{Q_{L} n_{L} c^{3}} = \frac{2 G θ_{si}}{c^{3}} .

(E.38)

We use G and c from the 2014 CODATA [10] because this result corresponds to the MQ macroscopic measure of G at the upper count limit. The calculation is anchored by θ_si, which is resolved using the most current recommended value for the fine structure constant ([5], Appx. AD).

l_{f} = \frac{2 G θ_{si}}{c^{3}} = \frac{26.674081 0^{- 11} 3.26239}{(299792458)^{3}} = 1.616201 0^{- 35} m,

(E.39)

t_{f} = \frac{l_{f}}{c} = \frac{2 G θ_{si}}{c^{4}} = \frac{26.674081 0^{- 11} 3.26239}{(299792458)^{4}} = 5.391061 0^{- 44} s,

(E.40)

m_{f} = \frac{2 θ_{si}}{c} = t_{f} \frac{c^{3}}{G} = \frac{23.26239}{299792458} = 2.176431 0^{- 8} kg .

(E.41)

E.7 CMB Formation from Quantum Epoch Allocation

With the mapping threshold and fundamental measures fixed, MQ resolves the mass-energy content associated with the realized CMB. Within MQ, the CMB is not introduced as an independent phenomenological input. Rather, it is the realized radiation field associated with the quantum epoch allocation once U first admits a valid Frames mapping and becomes represented by Ω=M(U).

R_{U} = \frac{2 l_{f}}{θ_{si}^{3}} \int (\frac{1}{A_{U}}) dt = \frac{2 l_{f}}{θ_{si}^{3}} \ln (A_{U}),

(E.42)

A_{qe} = e^{\sqrt{3} θ_{si}^{3} / 2} = 1.14652655409 (32) 1 0^{13} s,

(E.43)

A_{ee} = A_{qe} {(2 θ_{si})}^{1 / 3} = 2.14242727551 (18) 1 0^{13} s = 678, 894.236415 (88) y .

(E.44)

M_{CMB} = n_{Tu} m_{f} \frac{θ_{si}^{3}}{2} = 1.50161 0^{50} kg,

(E.45)

ρ = \frac{M_{CMB} c^{2}}{V_{U}} = 4.17501 0^{- 14} J m^{- 3},

(E.46)

a = \frac{4 σ}{c} = \frac{8 π^{5} k_{B}^{4}}{15 c^{3} h^{3}} = 7.5657332501 0^{- 16} J m^{- 3} K^{- 4},

(E.47)

T = {(\frac{ρ}{a})}^{1 / 4} = 2.7255 K .

(E.48)

Compared with Fixsen’s CMB temperature measurement T_CMB=2.7248(57) K [16], the MQ result lies within experimental uncertainty. Within the MQ construction, the same Frames mapping structure that establishes referenceability also fixes the quantum epoch allocation, the realized mass-energy content, and the resulting CMB temperature.

Table 2. Comparison of MQ predicted CMB values with measurements.

	A_qe [yrs]	A_ee [Gyr]	ρ_CMB [Jm^-3]	T_CMB [K]
MQ	363,312(20)	13.797	4.1723(21) 10^-14	2.72509 ± 0.00341
MQ A_ee=f(T_CMB)	363,312(20)	13.7943	4.1747(21) 10^-14	2.72548 ± 0.00341
Planck 2018		13.797^b	4.17 10^-14^c	2.72548 ± 0.00057^a

^a A consolidated study of CMB temperature measurements prior to 2009 [16].

^b Microwave (WMAP) All-Sky Survey [41].

^c Corresponds to about 411 photons/cm³ [42].

Table 2 compares the MQ derived quantum epoch termination scale A_qe, CMB mass-energy density ρ_CMB, and temperature T_CMB with observational determinations as a function of the present-day System Frame cosmic age A_ee. The predicted temperature follows from the MQ geometric construction together with θ_si, which is resolved from the CODATA recommended value of the fine structure constant [12]. The calculation additionally employs the radiation constant and the observed near-flatness of spatial curvature. No inflationary parameters, recombination parameters, or additional adjustable parameters enter the MQ derivation.

E.7 Effects of Universe Age on Concordance

As one example, resolving the CMB energy density requires the present System Frame age of the universe, A_ee, because the density depends explicitly on the present expansion volume. For this purpose, we adopt the Planck 2018 (TT,TE,EE+lowE+lensing) determination A_ee=13.797±0.023 Gyr [17]. This yields an MQ prediction of T_CMB=2.72509±0.00058 K, in agreement with the observational value 2.7248±0.0057 K at the 0.32σ level.

It is important to note that current determinations of A_ee remain model dependent and rely on assumptions regarding the inferred expansion history. The uncertainty in the MQ comparison therefore reflects, in part, the limited precision and framework dependence of present cosmic age determinations rather than uncertainty in the MQ relations themselves. MQ additionally provides an alternative age construction using only the measured CMB temperature, θ_si, the radiation constant a, and the fundamental length l_f, the latter itself derived from θ_si together with measurements of the electron mass and Bohr radius.

A_{ee} = {(\frac{3 θ_{si} {(2 θ_{si})}^{1 / 3} e^{(\sqrt{3} θ_{si}) / 2}}{4 π ac l_{f} T_{CMB}^{4}})}^{1 / 3} .

(E.49)

Using the temperature resolved from the Fixsen study [16], this inversion yields an age estimate A_ee=13.794±0.0039 Gyr without reference to inflationary priors or recombination era modeling. The difference between this value and the Planck determination therefore reflects a framework level difference in inferred expansion history rather than an internal inconsistency within MQ.

A complementary approach uses the System Frame measurement construction implemented in the SPT-3G study [18], which reports a Hubble rate of 68.3±1.5 km s^-1 Mpc^-1. Using the MQ relation,

A_{ee} = \frac{π}{θ_{si} H_{sys}} \frac{km / Mpc}{s / y 1 0^{9}},

(E.50)

the corresponding age estimate is approximately 13.8 Gyr, with an allowed range spanning roughly 13.5–14.1 Gyr. The MQ construction is therefore consistent with current observational uncertainty. Further improvement in concordance will require more precise and less model dependent determinations of the cosmic age.

APPENDIX F: IMPOSSIBILITY OF JOINT SYSTEM FRAME CONSTRUCTION

This appendix establishes that two inequivalent referenceable universes, i.e., universes distinguished by non-removable global referenceability invariants, cannot be jointly realized within a single System Frame. Universes differing only by encoded availability assignments, normalization structure, realization labels, or record encodings (Constraints 1–7) remain frames automorphism equivalent and therefore represent alternative descriptions of the same referenceable universe domain. The obstruction addressed here applies only to universes distinguished by persistent defect structure and inequivalent closure structure under the Frames mapping. The result is not a statement about physical interaction, spatial proximity, or dynamical collision, but about the structural non-composability of inequivalent Frames mappings into a single globally consistent relational realization.

Within MQ, universes are not physical objects embedded in an external space, but maximal referenceable universe domains defined within the Internal Frame configuration domain ω. A universe is specified by a referenceable universe domain U together with its realized relational structure Ω=M(U). All physically meaningful observables arise only within such a realization and possess no independent meaning outside the corresponding Frames mapping.

F.1 Underlying Structure of a System Frame Construct

The results established in Appx. E provide the conditions under which a referenceable universe domain admits realization under the Frames mapping. Once realization occurs, conserved Internal Frame count structure is represented as observable relational structure within the System Frame. The purpose of this section is to identify the minimal realization structures required for a System Frame construct and to establish the structures that must be preserved under admissible mappings.

The construction begins with conserved Internal Frame count. For a bounded relational region ω^'⊂ω, the total count measure is defined by Eq. (1).

N_{I} (ω^{'}, A) = \sum_{ξ \in ω^{'}} n_{I} (ξ, A) .

(F.1)

Here n_I(ξ,A) is the discrete measurement availability count associated with relational cell ξ at update index A. Deterministic evolution of the count medium is governed by the adjacency conservation law given by Eq. (2).

Δ_{A} n_{I} (ξ, A) + (di v_{adj} J_{I}) (ξ, A) = 0 .

(F.2)

Here J_I is the Internal Frame adjacency count-flux object and div_adj is the adjacency divergence operator acting on that flux structure. Summation over a bounded region gives the corresponding regional balance relation,

Δ_{A} \sum_{ξ \in ω^{'}} n_{I} (ξ, A) + ϕ_{I} (\partial ω^{'}, A) = 0,

(F.3)

where the boundary count-flux is defined by Eq. (4).

ϕ_{I} (\partial ω^{'}, A) = \sum_{\begin{array}{l} ξ \in ω^{'} \\ ξ^{'} \notin ω^{'} \end{array}} J_{I} (ξ \to ξ, A) .

(F.4)

For a region closed under the adjacency relation, the boundary flux vanishes and the regional count is conserved,

ϕ_{I} (\partial ω^{'}, A) = 0 \Rightarrow Δ_{A} N_{I} (ω^{'}, A) = 0 .

(F.5)

Equations (1–5) define the conserved relational structure from which realizable observables are obtained. A System Frame is therefore not introduced as an independent geometric arena. It is the realized relational structure obtained when conserved Internal Frame count admits a valid Frames mapping.

For a bounded Internal Frame relational region ω^', realization is determined by the mapping normalization functional,

R_{M} (ω^{'}) = N_{M} [C_{I} (ω^{'}), ϕ_{I} (\partial ω^{'}, A)] .

(F.6)

Equation (F.1) expresses the transition from conserved count structure to realized observables. Here 𝒞(ω^') is the count functional associated with ω^', 𝒩_M is the mapping normalization functional induced by the Frames mapping M, and ℛ_M(ω^') is the corresponding realized quantity.

Realization induces a mapping-dependent measure on the emergent System Frame. Let μ_M denote the measure induced on the realized relational structure Ω by the Frames mapping. The corresponding encoded availability density is given by

ρ_{sys} (x, t) = \frac{d μ_{M}}{dV},

(F.7)

which provides the continuum encoding of conserved count following realization. Observable probabilities arise only after normalization of this encoded availability,

P (x, t) = \frac{ρ_{sys} (x, t)}{\int_{Ω} ρ_{sys} (x, t) d^{3} x} .

(F.8)

Probability is therefore a derived System Frame quantity obtained from normalized encoded availability rather than a primitive feature of the Internal Frame. The same realization structure that determines observable measures also determines the admissible distribution of outcomes within the realized domain.

F.2 Definition F.2.1 (Frames automorphism equivalence)

Let Φ denote an invertible Frames automorphism acting on admissible referenceable universe domains and preserving normalization, unique realization, holonomy-free closure on contractible reference cycles away from defect cores, orthogonal closure structure, and defect structure. Define the equivalence relation ≡ on the collection of admissible referenceable universes by U_a≡U_b if and only if there exists Φ such that Φ(U_a)=U_b.

F.2 Lemma F.2.1 (Normalization Equivalence under Mapping Preservation)

Let U_a and U_b be two admissible referenceable universe domains defined on the same Internal Frame configuration domain ω with defect set D⊂ω. Let D denote the subset of configurations at which the Frames mapping fails to satisfy local holonomy-free closure on contractible reference cycles or orthogonal closure conditions. Define the regular domain ω∖D as the subset of configurations on which the Frames mapping remains locally well-defined and admits standard reference cycle and orthogonal closure structure.

Assume that U_a and U_b agree on all reference cycle closure structure, orthogonal closure structure, realization structure, and defect structure on ω∖D, and differ only in normalization-compatible encoded availability structure permitted by the measure-admissibility and mapping-normalization requirements of Constraints 1–5.

Proof. Because U_a and U_b differ only through encoded availability structure satisfying the same normalization, conservation, finite additivity, and local finiteness requirements, their difference may be represented by an admissible positive rescaling that leaves invariant the underlying realization structure, reference cycle closure relations, orthogonal closure structure, and defect structure.

The resulting transformation therefore defines an invertible Frames automorphism Φ preserving all structural constraints while mapping U_a to U_b. Hence differences arising solely from Constraint 1–5 structure do not distinguish inequivalent frames automorphism classes. ∎

Remark. This lemma formalizes the statement that Constraint 1–5 structure does not constitute distinguishing structural data. Persistent defect structure and closure invariants remain unchanged under admissible normalization preserving transformations.

Proposition F.2.2 (Non-Discriminability under Frames Automorphisms)

Let ≡ denote the equivalence relation on admissible referenceable universe domains generated by invertible frames automorphisms preserving normalization, unique realization, orthogonal closure, and holonomy-free closure on the regular domain ω∖D.

Let U_a,U_b⊂ω be admissible referenceable universe domains defined on the same Internal Frame configuration domain ω with defect set D⊂ω. Define the regular domain ω∖D as the domain supporting local reference-cycle closure and unique realization.

If U_a and U_b possess identical reference-cycle closure structure, orthogonal closure structure, realization structure, and defect structure on ω∖D, and differ only through structure governed by Constraints 1–5, then U_a≡U_b. Consequently, Constraints 1–5 do not distinguish inequivalent frames automorphism classes. Only persistent defect structure and closure invariants remain as distinguishing structural data.

Proof. Under the stated assumptions, U_a and U_b differ only through structure satisfying identical normalization and conservation requirements while agreeing on all reference-cycle closure structure, orthogonal closure structure, realization structure, and defect structure over the regular domain ω∖D. By Lemma F.2.1, such differences admit an invertible frames automorphism Φ preserving normalization, realization, reference-cycle closure, orthogonal closure structure, and defect structure such that Φ(U_a)=U_b.

Therefore, by Definition F.2.1, U_a≡U_b. Accordingly, all differences governed solely by Constraints 1–5 may be absorbed by admissible frames automorphisms and do not distinguish inequivalent equivalence classes. ∎

Lemma F.2.3 (Realization–Record Equivalence under Frames Automorphisms)

Let U_a and U_b denote admissible referenceable universe domains defined on the same Internal Frame configuration domain ω with shared defect structure D⊂ω. Assume the two domains agree on all reference-cycle closure structure, orthogonal closure structure, realization structure, and defect structure over the regular domain ω∖D, differing only in their realization mappings, record encodings, and associated outcome labels while preserving the single-valued realization and repeatable record-encoding requirements of Constraints 6–7.

M_{i} : U_{i} \to Ω_{i}, i \in {a, b} .

(F.9)

The admissible interaction set is denoted by E_i, and the associated outcome label space by O_i.

R_{i} : E_{i} \to O_{i},

(F.10)

O_{R_{i}} = R_{i} (E_{i}) \subseteq O_{i},

(F.11)

Constraint 6 guarantees that each admissible interaction possesses a unique realized outcome label, although multiple interaction configurations may correspond to the same outcome. Record states are represented by the observer specific macroscopic record state space ℛ_i.

ρ_{i} : O_{R_{i}} \to R_{i},

(F.12)

assigns realized outcomes to stable macroscopic record states and remains injective on

O_{R_{i}}

(S64)

. Consequently, distinct realized outcomes correspond to distinct observable records.

σ (R_{a} (x)) : = R_{b} (x), x \in E_{a} .

(F.13)

Here σ denotes an outcome relabeling operator acting on realized outcomes and should not be confused with the Frames mapping M_i.

τ (ρ_{a} (o)) : = ρ_{b} (σ (o)), o \in O_{R_{a}} .

(F.14)

τ (ρ_{a} (R_{a} (x))) : = ρ_{b} (σ (R_{a} (x))), x \in E .

(F.15)

The map τ:ℛ_a→ℛ_b therefore provides a one-to-one correspondence between realized macroscopic record states preserving observable equivalence under outcome relabeling.

Statement. There exists an invertible Frames automorphism Φ such that Φ(U_a)=U_b, with realization and record structures satisfying

R_{b} = σ \circ R_{a} on E, ρ_{b} \circ σ = τ \circ ρ_{a} on O_{R_{a}} .

(F.16)

Proof. Constraint 6 guarantees that the outcome relabeling map σ is well defined and bijective on the realized outcome subset. Injectivity of the record encoding mappings guarantees that the induced record relabeling τ is likewise well defined and bijective on realized record states. Consequently, all differences between U_a and U_b are confined to outcome labels and their associated record labels.

Because the automorphism Φ preserves the underlying Internal Frame configuration domain ω, the defect structure D, normalization, unique realization, reference cycle closure, and orthogonal closure conditions remain unchanged. The realized System Frame structures therefore differ only by admissible relabelings and not by any reference cycle closure, orthogonal closure, defect, realization, or measurable structure. Hence the two universe descriptions are structurally equivalent,

U_{a} \equiv U_{b} \Leftrightarrow \exists Φ, σ, τ .

(F.17)

Observable structure is therefore invariant under realization and record relabeling, and all differences between U_a and U_b remain representation-level rather than structural. ∎

Remark (scope). This lemma establishes equivalence only at the level of realization and record representation. It does not imply that arbitrary modifications of R_i or ρ_i are admissible. The result applies only to relabelings preserving the single-valued realization and repeatable record encoding requirements embodied in Constraints 6–7 on a common admissible interaction structure.

Proposition F.2.4 (Realization–Record Non-Discriminability)

Let U_a and U_b be admissible referenceable universe domains agreeing on Constraints 1–5 and on the reference cycle closure structure, orthogonal closure structure, realization structure, and defect structure governing admissible realization, while differing only in realization mappings R_i and record encodings ρ_i subject to Constraints 6–7. Then U_a≡U_b. Consequently, Constraints 6–7 do not introduce additional invariants distinguishing inequivalent frames automorphism classes.

Proof. The result follows directly from Lemma F.2.3. All differences in realization mappings and record encodings consistent with the single-valued realization and repeatable record encoding requirements of Constraints 6–7 are removable through an admissible frames automorphism preserving normalization, realization consistency, holonomy-free closure, orthogonal closure structure, and defect structure.

Hence U_a≡U_b. ∎

F.3 Lemma F.3.1 (Non-Additivity of Mapping Invariants)

There exists no admissible joint extension of Frames mappings capable of simultaneously realizing two independently referenceable universe domains whose closure structure assigns incompatible holonomy evaluations to at least one admissible reference cycle.

To make this statement precise, let M_i denote admissible Frames mappings defined on referenceable universe domains U_i, each satisfying the realization, closure, and defect requirements established by Constraints 6–10. Let

H_{M_{i}} (γ)

(S65)

denote the holonomy functional associated with mapping M_i, evaluated on an admissible reference cycle γ. As defined in Appx. A, the holonomy functional measures the closure mismatch obtained by transporting reference data around a closed reference cycle under a Frames mapping. For admissible contractible cycles, holonomy-free closure requires vanishing holonomy.

The scalar value θ_si fixes the admissible orthogonal closure structure of a realized System Frame and thereby determines the corresponding behavior of the holonomy functional on admissible reference cycles. Distinct values of θ_si may therefore correspond to distinct realization structure whenever their induced holonomy evaluations differ on an admissible reference cycle.

Any admissible extension simultaneously realizing two independently referenceable universe domains would require a single closure structure capable of reproducing the holonomy relations associated with both mappings. If the two mappings assign incompatible holonomy evaluations to the same admissible reference cycle, no single admissible joint closure structure can reproduce both. Accordingly, admissible closure structure is non-additive under attempted joint extension. No admissible joint closure structure exists that simultaneously reproduces incompatible holonomy evaluations. ∎

F.4 Lemma F.4.1 (Invariant Determinacy)

For any admissible Frames mapping M, the value θ_si, together with the admissible reference cycle structure of the domain, determines the corresponding holonomy functional ℋ_M(γ) on admissible reference cycles γ⊂ω∖D.

Proof. The value θ_si fixes the orthogonal closure structure associated with admissible realization and therefore fixes the closure relations satisfied by admissible reference cycles in the corresponding domain. Since the holonomy functional measures closure mismatch under transport around a reference cycle, the resulting holonomy evaluations are determined once θ_si and the admissible reference cycle structure are specified.

Consequently, two admissible Frames mappings whose closure data induce different holonomy evaluations on an admissible reference cycle cannot be represented by the same admissible closure structure. ∎

F.4 Theorem F.4 (Non-Composability of Frames Mappings)

There exists no admissible Frames mapping jointly realizing two referenceable universe domains whose holonomy functionals assign incompatible values to a common admissible reference cycle. Consequently, distinct referenceable universe domains possessing incompatible holonomy structure cannot be jointly realized within a single System Frame realization structure.

Proof. Let M_a and M_b denote admissible Frames mappings defined on referenceable universe domains U_a and U_b, respectively. By definition of a referenceable universe domain, each mapping satisfies normalization, unique realization, repeatable record encoding, and holonomy-free closure on all contractible reference cycles.

Assume, for contradiction, that there exists an admissible joint mapping extending both mappings into a common realization structure,

M_{joint} : U_{a} \cup U_{b} \to Ω_{joint} .

(F.18)

Because M_joint is assumed admissible, every contractible reference cycle contained in the joint domain must satisfy holonomy-free closure,

H_{M_{joint}} (γ) = 0, ​ ​ ​ γ \subset U_{a} \cup U_{b}, γ contractible .

(F.19)

For each constituent mapping M_i,

H_{M_{i}} (γ) = 0, γ \subset U_{i} \ D_{i}, γ contractible, i \in {a, b} .

(F.20)

Suppose there exists an admissible reference cycle γ on which the induced holonomy structures of M_a and M_b are incompatible. Then

H_{M_{a}} (γ) \neq H_{M_{b}} (γ) \Rightarrow ∄ M_{joint} .

(F.21)

Since M_joint extends both mappings, restriction of the joint mapping to either universe domain must reproduce the corresponding constituent mapping,

M_{joint} |_{U_{i} = M_{i}} \Rightarrow H_{M_{joint}} (γ) = H_{M_{i}} (γ), i \in {a, b} .

(F.22)

Equations (F.21) and (F.22) cannot simultaneously hold. The joint mapping would be required to assign a unique holonomy value to the same admissible reference cycle while reproducing two distinct holonomy evaluations inherited from M_a and M_b. This violates the uniqueness required of an admissible Frames mapping.

Therefore, no admissible joint mapping extending both universe domains can exist. The assumption of M_joint leads to contradiction, implying ∄ M_joint. Hence, referenceable universe domains possessing incompatible holonomy structure are not jointly realizable within a single System Frame realization structure. ■

F.5 Structural Outcomes of Mapping Incompatibility

By Theorem F.4, joint realization within a single realized System Frame structure is impossible whenever the corresponding Frames mappings possess incompatible holonomy evaluations. The possible structural outcomes are therefore restricted to three classes.

Independent referenceability. Multiple mappings remain individually admissible and continue to define distinct referenceable universe domains, but no admissible extension exists to a common realized System Frame structure.

Mapping dominance. One admissible classification outcome in which a single Frames mapping remains globally referenceable while another fails the closure conditions required for admissible realization.

Mutual de-referenceability. One admissible classification outcome in which competing mappings fail the closure conditions required for admissible realization and neither remains referenceable.

These outcomes are structural consequences of realization consistency within the Internal Frame configuration domain. They do not correspond to physical collision, merger, or interaction between realized universes. Rather, they describe the admissibility or inadmissibility of extending distinct Frames mappings to a common realized structure.

Accordingly, MQ does not admit a mathematically meaningful notion of universe merger. What is excluded is not the combination of physical objects, but the existence of a single globally consistent realization structure arising from incompatible Frames mappings.

APPENDIX G: INTERACTION CLASSES, REFERENCEABILITY AND THE CONDITIONS FOR UNIVERSE DEATH

This appendix defines the structural conditions under which a referenceable universe exists within MQ and establishes the necessary and sufficient conditions under which referenceability fails. Universe death in MQ does not correspond to thermodynamic collapse, material destruction, or dynamical cessation. It denotes failure of the global validity conditions required for a valid Frames mapping.

G.1 Definition: Referenceable Universe

A referenceable universe in MQ is defined as a maximal admissible relational domain U⊂ω on which a valid Frames mapping M:U→Ω exists satisfying the global referenceability constraints below.

Geometric and topological consistency

Holonomy–free closure: For all admissible contractible reference cycles γ⊂U∖D, ℋ(γ)=0. Equivalently, the induced relational structure is path–independent on all contractible cycles.

Orthogonality consistency: Transport induced under the Frames mapping preserves admissible orthogonal closure structure across all admissible relational regions.

Defect compatibility: The mapping defect class D⊂ω is well defined, and the induced realization structure remains globally consistent on U∖D.

Loop closure consistency: All admissible contractible reference cycles in U∖D close consistently under the induced realization structure of the Frames mapping.

Structural admissibility

Normalization consistency: The Frames mapping admits finite normalization over admissible relational regions, thereby permitting finite encoded availability and finite referenceable realization structure.

Domain consistency: The mapping is globally defined on the maximal referenceable universe domain U⊂ω, with admissible realization structure restricted to U∖D.

Realization and record consistency

Unique realization (Constraint 6): Each admissible interaction configuration admits exactly one realized outcome.

Stable record encoding (Constraint 7): Realized outcomes admit injective encoding into stable macroscopic record states through ρ:O_R→ℛ, thereby ensuring persistent distinguishability of realized outcomes under repeated admissible observation.

These conditions are jointly necessary and sufficient for validity of the Frames mapping. Failure of any one condition renders the mapping non-referenceable.

Accordingly, a referenceable universe exists precisely when a single globally consistent Frames mapping can be constructed that simultaneously:

admits holonomy–free closure on all contractible reference cycles,

permits finite normalization of conserved count structure,

assigns unique realizations to admissible interaction configurations, and

admits stable injective record encoding of realized outcomes.

The relational structure Ω is not assumed as a primitive continuum manifold. It is induced from admissible Internal Frame count structure through the Frames mapping and therefore inherits its consistency conditions from admissibility of the mapping itself.

Maximality means that no proper extension U^'⊃U admits a valid extension of the same Frames mapping while preserving normalization, unique realization, and holonomy–free closure.

Referenceability in MQ is therefore not defined through matter content, thermodynamic state, metric manifold assumption, or external temporal evolution. It is defined entirely through the existence of a globally admissible Frames mapping satisfying the required realization and consistency conditions.

Loss of referenceability (“universe death”) denotes failure of the admissibility conditions required for a globally valid Frames mapping. It does not imply destruction of an underlying count substrate, but rather loss of the conditions necessary for consistent realized observables.

G.2 Holonomy and Unique Realization Mappings

For every admissible contractible reference cycle γ⊂U, the holonomy functional must satisfy ℋ(γ)=0. This is the holonomy–free closure condition on the Frames mapping M:U→Ω. The condition applies only to admissible contractible reference cycles contained entirely within the valid realization domain U∖D, while non-contractible cycles may consistently carry defect-induced holonomy associated with the mapping defect class D.

Holonomy–free closure guarantees that transport of reference structure around any admissible contractible cycle produces no residual closure mismatch. Equivalently, the induced System Frame relational structure remains path–independent on admissible contractible cycles and therefore admits a single globally consistent realization structure for each admissible interaction configuration.

If ℋ(γ)≠0 on a contractible admissible cycle, then realization assignments become transport-dependent and no globally consistent realization structure can be constructed. The obstruction is purely structural rather than dynamical and arises from incompatibility of admissible closure assignments under the Frames mapping.

Accordingly, the condition ℋ(γ)=0 is not an independent postulate, but the explicit cycle-level realization of the global referenceability constraints required for validity of the Frames mapping.

G.3 Theorem: Geometric Conditions for Non–Referenceability

A referenceable universe becomes non–referenceable if any one of the following conditions holds.

Holonomy failure: There exists a contractible reference cycle γ⊂U such that ℋ(γ)≠0.

Normalization failure: The Frames mapping fails to admit finite normalization over the referenceable universe domain U.

Record failure: No injective stable record encoding exists on the realized outcome set O_R. Equivalently, no mapping ρ:O_R→ℛ exists assigning distinct realized outcomes to distinct stable macroscopic record states.

These conditions constitute the complete admissibility failure classes presently admitted within the MQ referenceability framework. Specifically, ℋ(γ)≠0 is failure of holonomy–free closure, normalization failure is the loss of finite referenceability and failure of ρ is the loss of stable repeatable realization encoding.

Accordingly, a universe ceases to exist in the MQ sense precisely when at least one defining validity condition of the Frames mapping fails. This theorem is therefore the appendix-level restatement of the global referenceability criterion introduced in Sec. G.1 and is consistent with the realization–record framework established throughout the paper and with the non-composability structure developed in Appx. F.

Proof: If there exists a contractible reference cycle γ⊂U such that ℋ(γ)≠0, then the induced realization structure is path-dependent on a contractible cycle and the Frames mapping fails holonomy–free closure. The mapping is therefore invalid.

If normalization fails, finite encoded availability cannot be consistently assigned over admissible relational regions, so the realization structure ceases to admit finite referenceability.

If no injective stable record encoding exists on O_R, then distinct realized outcomes cannot be persistently distinguished through admissible macroscopic records, violating repeatable realization consistency.

Each condition therefore independently destroys global referenceability. Since these are precisely the defining validity conditions imposed on admissible Frames mappings, the list is exhaustive. ∎

Non-referenceability does not require disappearance of the Internal Frame configuration domain ω. Rather, it signifies that no admissible realization structure satisfying the global mapping constraints can be consistently induced from it. For this reason, the theorem concerns the existence of valid realization structure, not ontological elimination of the underlying count substrate.

G.4 Mathematical Condition for Mapping Dominance

Consider two referenceable universe domains U₁,U₂⊂ω with valid Frames mappings M₁:U₁→Ω₁ and M₂:U₂→Ω₂. Although M₁ and M₂ may each independently satisfy the referenceability conditions on their own maximal admissible domains, they may nevertheless fail to admit a common jointly extendable realization structure.

Mapping dominance denotes asymmetric failure of joint extendibility between otherwise admissible Frames mappings. A sufficient structural indicator is the existence of an admissible overlapping contractible reference cycle γ⊂U₁∪U₂ such that no joint Frames mapping M_joint:U₁∪U₂→Ω_joint can simultaneously satisfy ℋ_joint(γ)=0 while restricting consistently to both M₁and M₂. Equivalently, if every attempted joint extension forces ℋ₂(γ)≠0 for some admissible contractible reference cycle γ⊂U₂, while M₁ remains holonomy–free, then M₂ is structurally subordinate and mapping dominance is said to occur.

This condition is the asymmetric specialization of the non-composability structure established in Appx. F, where incompatibility of admissible realization structure prevents existence of a single globally coherent extension. Dominance therefore refers solely to preservation of admissibility under attempted joint extension and does not imply energetic superiority, causal suppression, collision dynamics, or physical force interaction. Rather, mapping dominance is a structural failure of joint extendibility under the global closure and realization constraints imposed by the Frames mapping formalism.

G.5 Interaction Classes of Universes

The following classes exhaust the presently defined structural compatibility relations between referenceable universe domains within MQ.

Coexistence: Two Frames mappings remain independently admissible on their respective maximal domains and no attempted joint extension produces admissibility failure in either mapping.

Dominance: A joint compatibility attempt forces one mapping to lose validity, typically through violation of holonomy–free closure on an admissible contractible reference cycle, while the competing mapping remains referenceable.

Mutual failure: No admissible jointly extendable realization structure exists and attempted extension destroys admissibility conditions for both mappings.

Reopening: A previously obstructed realization structure again satisfies the admissibility conditions for a valid Frames mapping. Reopening therefore denotes restoration of global referenceability conditions rather than recreation of material structure.

These interaction classes are structural categories determined by the existence, compatibility, and joint extendibility of globally valid Frames mappings M:U→Ω on maximal referenceable universe domains U⊂ω. Accordingly, coexistence, dominance, mutual failure, and reopening describe admissibility relations between relational realization structures rather than dynamical, thermodynamic, causal, or material processes. What is excluded within MQ is not coexistence of abstract mathematical structures as such, but construction of a single globally holonomy–free, normalized, uniquely realizing, repeatably recordable Frames mapping from structurally incompatible relational configurations.

APPENDIX H: DEFECT FORMATION AND OPEN BOUNDARY THEOREM

This appendix establishes the geometric and kinematic conditions under which a referenceable universe domain forms through persistent open-boundary mapping defects in the Frames mapping. The construction follows from the Frames Principle, orthogonal closure constraints, and the topological persistence conditions established in Appx. B–D.

From Appx. B and Theorem 4.0, a referenceable universe domain exists if and only if a persistent open-boundary mapping defect forms in the Frames mapping M:U→Ω, where U⊂ω is a maximal referenceable subdomain of the Internal Frame configuration domain ω, and Ω=M(U) is the corresponding realized System Frame relational structure.

This condition is expressed by

Φ_{\partial ω^{'}} \neq 0

(S66)

, where ω^'⊂ω denotes a finite relational subset of the Internal Frame adjacency domain used to define regional count measures, ∂ω^' is the associated adjacency boundary, and

Φ_{\partial ω^{'}}

(S67)

represents the net signed transport of discrete measurement availability across that boundary, as defined in Appx. A and Sec. 3.1. The corresponding realized System Frame relational region is given by Ω=M(ω^').

A nonzero value of

Φ_{\partial ω^{'}}

(S68)

implies that the Internal Frame regional conservation relation cannot be globally satisfied on the full configuration domain ω while simultaneously preserving normalization, unique realization, repeatable realization, and holonomy-free closure on admissible contractible reference cycles. The non-vanishing boundary count-flux therefore produces a persistent mapping obstruction defining a mapping defect class D, representing a global incompatibility between Internal Frame count conservation and globally consistent realization under the Frames mapping.

The defect is topological in the sense that no continuous deformation of an admissible mapping can remove the obstruction while preserving the global referenceability constraints. Consequently, the resulting referenceable universe domain corresponds to the maximal admissible subdomain U⊂ω on which the Frames mapping remains globally self-consistent despite the persistent open-boundary incompatibility.

The defect is not an independent dynamical source or geometric object, but a structural property of admissible Frames mappings under the global referenceability constraints established in Secs. 3.2–3.6 and Appx. C–F. Because the obstruction persists under admissible extension of the mapping domain, the realized System Frame structure cannot be globally extended to a holonomy-free closed realization on the full Internal Frame configuration domain ω.

APPENDIX I: PHYSICAL SIGNIFICANCE OF THE FRAMES PRINCIPLE AND THE HUBBLE TENSION

A central requirement of any foundational framework is that it admit direct and falsifiable confrontation with observation. Within MQ, one of the clearest empirical consequences of the Frames Principle arises from cosmic expansion itself. Because MQ distinguishes between the Internal Frame and the System Frame as physically distinct realizations of the Frames mapping, quantities associated with large-scale metric expansion will not resolve identically when evaluated relative to different frame structures.

This distinction is not a coordinate transformation, gauge choice, or reparameterization of an equivalent geometry. The Internal Frame and System Frame correspond to distinct realizations of referenceability within the same underlying count structure and therefore encode different operational realizations of distance, duration, and expansion. In this limited sense, the frame dependence introduced by MQ is analogous to the role of inertial frames in special relativity, where measurable quantities such as time intervals and lengths depend on physical frame structure rather than coordinate convention alone. The analogy is restricted to the existence of physically distinct frame realizations and does not imply Lorentz equivalence between relativistic inertial transformations and the Frames mapping.

In a recent AIP Advances publication [13], the MQ formulation of cosmological expansion was evaluated explicitly relative to both frames and compared directly with contemporary observational datasets. Within this construction, CMB-derived expansion measurements and local distance ladder determinations resolve relative to opposite realizations of the scalar expansion parameter θ_si, separated by a geometric differential ΔH. This separation is termed a metric differential because it arises directly from the topology of the Frames mapping and parallels the frame-resolved differentials previously identified for fundamental physical measures.

To distinguish the corresponding frame realizations observationally, we denote the System Frame expansion realization by H_sys, consistent with globally averaged radiation-anchored observables such as CMB analyses. Likewise, H_int denotes the Internal Frame realization associated with locally calibrated matter-anchored observables including Cepheid and Type Ia supernova distance ladder measurements. The baseline scalar expansion realization is denoted H, with the metric differential given by

Δ_{H} = θ_{si} - π .

(I.1)

Such that H = n_Lul_f/2n_Tut_f = m_fl_f/2t_f = θ_si and such that m_f = nLu/n_Tu ([13], Eq. 61) establishes the baseline expression H=θ_si and such that CMB calculations are resolved relative to the non-discrete System Frame, then such calculations fall below the baseline expression. We calculate H_sys by subtracting ∆_H from H. Conversely, cepheid measurements are resolved relative to the discrete Internal Frame. We calculate H_int by adding ∆_H to H.

H_{sys} = f (H - Δ_{H}) .

(I.2)

H_{int} = f (H + Δ_{H}) .

(I.3)

Such that H=θ_si and ∆_H=θ_si-π, then

H_{sys} = f (H - Δ_{H}) = f (θ_{si} - (θ_{si} - π)) = f (π) .

(I.4)

We can combine the first two expressions and reduce to resolve H_int in terms of θ_si and π.

H_{int} - f (Δ_{H}) = H_{sys} + f (Δ_{H}),

(I.5)

H_{int} = H_{sys} + f (2 Δ_{H}) = f (π) + f (2 θ_{si} - 2 π) = f (2 θ_{si} - π) .

(I.6)

We now resolve the baseline expression such that H=(km/Mpc)/A_U and A_U=D_U/2θ_si. Then,

H = \frac{1}{A_{U}} \frac{km}{Mpc} = \frac{2 θ_{si}}{D_{U}} \frac{km}{Mpc} .

(I.7)

We can then replace the scalar rate θ_si using either π or 2θ_si-π. The corresponding calculations are,

H_{sys} = \frac{π}{θ_{si} A_{U}} \frac{km / Mpc}{s / y 1 0^{9}} = 68.259 (19) km s^{- 1} Mp c^{- 1},

(I.8)

H_{int} = \frac{2 θ_{si} - π}{θ_{si} A_{U}} \frac{km / Mpc}{s / y 1 0^{9}} = 73.508 (21) km s^{- 1} Mp c^{- 1} .

(I.9)

where θ_si is the System–Internal Frame mapping parameter derived independently within MQ, and A_U is the cosmic age expressed in System Frame units. Note, A_U = A_ee. The numerical pre-factors arise directly from the topology and geometry of the Frames mapping and are not introduced through observational fitting.

Evaluating these expressions using the MQ-derived value of θ_si, obtained from the CODATA 2022 recommended value of the fine structure constant together with the Planck 2018 cosmic age [17], yields H_sys = 68.259(19) km s^-1 Mpc^-1 and H_int = 73.508(21) km s^-1 Mpc^-1. These values correspond, respectively, to the two dominant observational determinations of the Hubble constant: the CMB-anchored inference from SPT-3G (68.3±1.5 km s^-1 Mpc^-1 [14]) and the local distance ladder inference from PANTHEON+SH0ES (73.5±1.1 km s^-1 Mpc^-1 [15). Their combined concordance corresponds to a right-tail agreement significance of 3.35σ, numerically matching the observed separation commonly referred to as the Hubble tension.

Within MQ, the observed separation emerges as a structural consequence of resolving cosmological expansion relative to different frame realizations. The System Frame corresponds naturally to globally averaged radiation-anchored observables such as the CMB, whereas the Internal Frame corresponds naturally to locally calibrated matter-anchored observables including Cepheid and Type Ia supernova distance ladder measurements. No free parameters are introduced: the mapping parameter θ_si is independently constrained through MQ relations involving the fine structure constant, and the resulting frame-resolved expansion observables follow uniquely from the topology of the Frames mapping. The magnitude of the observed Hubble separation therefore emerges directly from the MQ geometric construction rather than from phenomenological fitting.

This result provides a direct physical realization of the Frames Principle. A single cosmological realization, when evaluated relative to different admissible frame structures, yields systematically different yet internally self-consistent expansion observables. In this sense, MQ replaces a single absolute realization of cosmological expansion with frame-dependent geometric expansion determined by the topology of the Frames mapping.

APPENDIX J: FRAME–DEPENDENT DIMENSIONAL ANALYSIS IN MQ

In the Internal Frame, the structural parameter associated with the fundamental measures is defined as a dimensionless count relation,

S_{MQ} [Ф] = \int d^{4} x L_{MQ} (Ф, \partial_{μ} Ф, g_{μ ν}^{sys}, θ_{si}),

(J.1)

where n_L, n_M, and n_T denote the upper bound counts associated with length, mass, and time, respectively. By example, the upper bound count for length is 1/l_f. The respective bounds describe the maximum count of a measure relative to itself, but when compared with time, we resolve the maximum count of a measure per increment of t_f. This becomes vitally important, for instance, in a description of dark matter ([5], Appx. AK) in that it provides an effective upper bound to mass per increment of elapsed time t_f. Through the Frames mapping, this relation is realized in the System Frame by substitution of the fundamental measures via their upper bound count relations. This yields

θ_{sys} = \frac{l_{f} m_{f}}{2 t_{f}},

(J.2)

which now carries the dimensions of momentum. Under this realization, the fundamental measures take their standard physical units. This does not imply that the Internal Frame is itself unit-bearing. Rather, the dimensionful form arises only through the Frames mapping, which converts the underlying count relation into its physical realization. Thus, θ_int represents a single structural quantity admitting dual representation: dimensionless in the Internal Frame and dimensionful in the System Frame.

By contrast, the System Frame realization θ_s_ys, numerically identical to θ_si which is used throughout MQ, is dimensionful and corresponds to half of the fundamental momentum under realization through (l_f, m_f, t_f). Internal Frame quantities specify count relations. Physical dimensions enter only when those relations are realized through (l_f, m_f, t_f) in the System Frame. A dimensionless representation arises only when the same structure is expressed as a normalized ratio under the discrete–continuous mapping, as demonstrated by the Unity Expression ([5], Appx. AX),

{(\frac{1}{2 θ_{si}})}^{2 / 3} + {(\frac{n_{L}}{n_{Lc}})}^{2} = 1 .

(J.3)

Thus, the frame of reference must always be considered when carrying out a unit analysis. A second consideration arises from substitutions involving θ_si = l_fm_f/2t_f, where mixing structural and realized forms can introduce spurious dimensional factors. Consider, for example, its use as a scalar relating the age of the universe to its diameter, D_U = 2θ_siA_U. θ_si serves as a scalar, yet the substitution formally introduces the relation L/T → ML/T. Without careful interpretation, such substitutions can incorrectly introduce mass dimensions into purely kinematic relations. The most notable examples occur in descriptions of the Hubble constant.

H_{sys} = \frac{π}{θ_{si} A_{U}} (\frac{km / Mpc}{s / y 1 0^{9}}) .

(J.4)

In reducing expressions similar to the Hubble constant one may replace A_U = D_U/2θ_si. This changes the units from seconds to seconds per kilogram, 1/(LM/T)T = 1/LM. This apparent inconsistency reflects the improper treatment of θ_si as a dimensional quantity rather than a mapping-normalized scalar.

In our description of the cosmological constant, we find the first instance applies:

Λ_{MQ} = 3 {(\frac{π}{θ_{si} A_{U} c} \frac{km / Mpc}{s / y 1 0^{9}})}^{2} \frac{θ_{si}^{2} - 2}{θ_{si}^{2} + 2} .

(J.5)

In this example we can focus on the denominator. That is, (LM/T)T(L/T) = L²M/T. However, because θ_si enters as a System Frame scalar, its apparent units are not applicable and θ_si should be treated as dimensionless. Therein, we recognize that this expression should have been written as T(L/T) = 1/L, giving us length squared in the numerator when considering the ratio is in parenthesis squared. With this correction, the expression yields the expected dimensional form for the cosmological constant, Λ ∼ L⁻².

With regard to nomenclature, it is sometimes necessary to suppress the explicit subscript “si” when clarifying labels such as “int” and “sys” are needed. MQ imposes no rigid rule for which frame labels take priority. The choice is made to prevent ambiguity in the surrounding expressions.

Historically, the subscript “si” has been typeset in italics. While we preserve that convention here, italics can occasionally obscure meaning when θ_si appears inside field equations. Future presentations may prefer a non-italic subscript for clarity.

It is also important to clarify the origin of the “si” designation. The value of θₛᵢ arises from experimental measurements of entangled X-ray signal–idler pairs at their degenerate frequency, as reported by Shwartz and Harris [36-38]. In a specific configuration, the signal and idler polarization angles are equal, and their measured value, θₛᵢ ≈ 3.26239 radians, corresponds directly to half of the fundamental momentum. Within the summary MQ publication [5], this identification is made explicit in Appx. A, where the signal, idler, and pump angles are expressed in the MQ formalism, and in Appx. S, where the fundamental expression – a parallel construct to the Planck units that includes non-Lorentz length contraction – demonstrates that angular measure and momentum are equal in value and physically correlated at the bound. Thus, θₛᵢ reflects a measured physical quantity whose angular and momentum representations are dual descriptions of the same underlying structure. The dimensionless or dimensionful character of θₛᵢ therefore depends on whether the relation is expressed as a normalized mapping (Internal Frame) or realized through (l_f, m_f, t_f) in the System Frame. Failure to distinguish these roles leads to incorrect dimensional interpretations within MQ.

APPENDIX K: CONCORDANCE ANALYSIS

Table 3: One–tail (Right–tail) p–value with Corresponding Agreement Sigma for 11 Implementations of the Frames Principle (2022 CODATA)*[12]

Phenomenon

Exp.

Calculation

Measure

MQ Calculated

Measure

p–value

Pull

σ_agree

w/oMQ

G = \frac{Q_{L} r c^{3}}{θ_{si}}

(S69)

Gravitational Constant (CODATA 2010) m³ kg⁻¹ s⁻²

αm_ea₀c

6.673 844 83(17) 10^–11

6.673 84(80) 10^–11

0.995

6.04 10^–3

2.59

0.16

Gravitational Constant ( CODATA 2014) m³ kg⁻¹ s⁻²

αm_ea₀c

6.674 077 94(10) 10^–11

6.674 08(31) 10^–11

0.995

6.65 10^–3

2.56

0.00

Gravitational Constant ( CODATA 2018) m³ kg⁻¹ s⁻²

αm_ea₀c

6.674 311 06(16) 10^–11

6.674 30(15) 10^–11

0.960

5.04 10^–2

1.75

0.00

Gravitational Constant ( CODATA 2022) m³ kg⁻¹ s⁻²

αm_ea₀c

6.674 311 05(16) 10^–11

6.674 30(15) 10^–11

0.960

5.04 10^–2

1.75

0.00

Global Concordance

0.9999

3.70

0.00

H_{sys} = (\frac{2 π}{2 θ_{si} A_{U}}) (\frac{km/Mpc}{s/y 10^{9}})

(S70)

Global Concordance w/o 2018 & 2022

0.99996

8.98 10^–3

3.94

0.41

Hubble Constant (SPT–3G) km s^–1 Mpc^–1

αm_ea₀k_BhcT_CMB

H_CMB

68.259(19)

68.3(15)

0.978

2.74 10^–2

2.02

n/a

H_{int} = (\frac{2 (2 θ_{si} - π)}{2 θ_{si} A_{U}}) (\frac{km/Mpc}{s/y 10^{9}})

(S71)

Hubble Constant (Pantheon+SH0ES) km s^–1 Mpc^–1

αm_ea₀k_BhcT_CMB

H_cepheids

73.508(21)

73.5(11)

0.994

7.47 10^–3

2.51

n/a

2 Q_{L} n_{Lr} (84 θ_{si} - ⌊ 42 θ_{si} ⌋)

(S72)

Global Concordance

0.9996

2.84 10^–2

3.35

n/a

Fine Structure Constant (Classical)

ħε₀ec

137.035 999 2(30)

137.035 999 177(21)

0.9997

3.47 10^–4

3.45

0.00

Global Concordance

0.9997

3.47 10^–⁴

3.45

0.00

Half Fundamental Momentum kg m s^–1

θ_{si} = (^{\frac{ħ c^{3}}{4 G}) \frac{1}{2}} \frac{l_{f} m_{f}}{2 t_{f}}

(S73)

αm_ea₀c

ħGc

3.262 390 305(89)

3.262 390(23)

0.9999999997

3.45 10^–10

6.20

4.79

Reduced Planck Constant J s

\frac{l_{f} θ_{si}}{Q_{L} n_{Lr}}

(S74)

αm_ea₀c

1.054 571 818(26) 10^–34

1.054 571 817(13) 10^–34

0.9998

3.04 10^–4

3.49

0.00

Magnetic Constant N A^–2

\frac{2 π γ m_{f}}{c^{4} l_{f}}

(S75)

αm_ea₀ε₀c

µ₀

1.256 637 063(64) 10^–06

1.256 637 061 27(20) 10^–6

0.9730

3.39 10^–²

1.93

0.00

G = \frac{2 π γ}{μ_{0} c^{2}}

(S76)

Gravitational Constant – unification f(μ₀) m³ kg⁻¹ s⁻²

αμ₀c

6.674 077 93(33) 10^–11

6.67408(31) 10^–11

0.995

6.69 10^–3

2.55

0.00

Global Concordance

0.995

2.55

0.00

(\frac{t_{f}}{2 Q_{L} n_{Lr} α m_{e}}) (^{\frac{ħ c^{3}}{G}) \frac{1}{2}}

(S77)

Bohr Radius (gravitational) m

αm_ea₀ħGc

αm_ea₀c

5.291 772 11(15) 10^–11

5.291 772 10544(82) 10^–11

0.9998

2.74 10^–4

3.52

0.00

\frac{4 π ε_{0} ℏ^{2}}{m_{e} e^{2}}

(S78)

Bohr Radius (electromagnetic) m

ε₀ħm_ee

αm_ea₀c

5.291 772 11(41) 10^–11

5.291 772 10544(82) 10^–11

0.99999

8.36 10^–⁶

4.35

0.00

Global Concordance

0.99999996

2.74 10^–4

5.38

0.00

MQ Tension Solutions (G, H₀)

0.9999999999

3.05 10^–2

6.63

0.00

MQ Tests of Length Contraction (𝐺, 𝛼)

0.9999998

8.99 10^–3

5.08

0.00

MQ Tests of Unification (G with μ₀, a₀, θ_si)

0.99999992

6.69 10^–3

5.24

0.00

MQ Physical Constant Solutions (𝐺, 𝛼, 𝜃_si, ħ, 𝜀₀, 𝜇₀, 𝑎₀)

1–(7.97 10^–1⁷)

3.57 10^–2

8.25

0.00

MQ Frames Principle (Includes all 11 implementations)

1–(1.1 10^–1⁹)

4.56 10^–2

8.74

0.00

* Gray text indicates implementations not included in the Frames Principle assessment.

The MQ concordance analysis is constructed on a statistically rigorous, traceable framework in which all comparisons are evaluated using one-tailed χ² right-tail tests, with p-values interpreted as probabilities of observing deviations at least as large as measured under the assumption that MQ is correct. Agreement is quantified via σ = Φ⁻¹(p), while the pull statistic provides signed deviation in units of experimental uncertainty, with |pull| ≥ 2 indicating potential tension. Degrees of freedom reflect only physically independent measurements, ensuring that overlapping systematics do not inflate significance. All reported quantities are derived directly from experimental inputs using standard NIST conventions, preserving traceability between “Calculation” and “Measure” columns. A key structural element is the explicit separation between MQ-corrected and classical (w/o MQ) formulations, where removal of the discrete frame length contraction term (2Q_Ln_L) reduces MQ units to Planck-based composites, establishing a controlled baseline for comparison.

Within this framework, concordance testing adheres to the Frames Principle, requiring that only physically independent realizations of invariant couplings be compared. Thus, Planck-derived CODATA values are excluded where they are not independent. Distinct experimental pathways (e.g., electromagnetic versus macroscopic gravitational realizations) are treated as separate validations of the same underlying invariant structure. Parameters without direct measurement (e.g., θ_si, ħ in certain regimes) are treated as calculated quantities with uncertainties propagated from experimentally grounded inputs, and all evaluations are carefully distinguished between electromagnetic-scale and macroscopic limits. MQ further enforces a consistent evaluation scale, typically at the upper interaction distance limit, to eliminate systematic bias introduced by distance-dependent length contraction. Finally, modern SI definitions are fully incorporated, recognizing that electromagnetic constants now inherit uncertainty through α, and all derived quantities (e.g., a₀, G, α) are evaluated in a manner that preserves internal consistency while testing cross-domain unification.

Internal Frame and Frames mapping overview System Frame encoding and measurement outcomes overview

STATEMENTS AND DECLARATIONS

I declare that the authors have no competing interests as defined by Springer, or other interests that might be perceived to influence the results and/or discussion reported in this paper.

DATA AVAILABILITY

I do not have any research data outside the submitted manuscript file.

FUNDING

This research has received no external funding.

ACKNOWLEDGEMENTS

The author thanks Hoyt Hudson for more than a decade of thoughtful discussion, careful reading, and candid critique across numerous drafts and related works. These ongoing exchanges have played an important role in shaping the presentation and internal consistency of the framework developed here. Responsibility for all conclusions rests solely with the author.

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