Kinematic Unification of Gravitational and Electromagnetic Phenomena

Jody A. Geiger

Informativity Institute

Chicago, IL USA 60605

jodygeiger@informativity.org

ORCID: 0000–0001–5389–0447

ABSTRACT

We present Measurement Quantization (MQ) as a count-based kinematic framework in which physical observables arise from discrete Internal Frame measure and are realized through the Frames mapping into the System Frame. Within this formulation, electromagnetic and gravitational phenomena emerge from count-dependent realizations of a single action defined over integer measure counts, with continuum field descriptions recovered through coarse-graining. MQ reproduces relativistic and gauge invariant structures while introducing a frame dependent non-Lorentz length contraction governed by the Informativity differential 2Q_Ln_L, which modifies normalization of inferred quantities as a function of count separation. Across established benchmarks, MQ yields quantitative concordance with experiment at a cumulative significance of 8.74σ, with independent cross-channel tests exceeding 5σ and CODATA consistency achieved at the 3.94σ level for a common observable. Notably, the evolution of CODATA values for the gravitational constant aligns with distinct MQ realizations at residuals well below reported experimental uncertainties, consistent with shifts in statistically dominant measurement scales. MQ further yields closed-form relations linking electromagnetic and gravitational observables, including distinct finite-count and asymptotic realizations of G, and reduces all coupling constants to functions of fundamental measures and the invariant scalar θ_si, eliminating independent coupling parameters. Unlike conventional unification approaches based on symmetry extension or geometric postulates, MQ derives both geometry and interaction strength from a shared discrete kinematic origin. The framework is experimentally falsifiable through cross-channel comparisons of inferred quantities and predicted count-dependent normalization effects. These results establish MQ as a quantitatively constrained unification candidate grounded in first principles measure structure.

Keywords: Measurement Quantization, unification, gravity, electromagnetism, fundamental units, foundations of physics

1. INTRODUCTION

The objective of unification has historically been to demonstrate that apparently distinct physical interactions arise from a single underlying structure. In its strongest form, unification requires not only formal compatibility between theories, but the elimination of independent postulates, parameters, and sector-specific assumptions. The persistence of multiple interaction frameworks – most notably gravitation and electromagnetism – indicates that such a reduction has not yet been achieved within the standard theoretical paradigm.

This work approaches the unification problem from a strictly kinematic standpoint. Rather than beginning with fields, forces, or geometric curvature as primitives, it adopts a count-based description of physical separation and interaction. The implications of this shift are developed in subsequent sections. Here, we restrict attention to the historical development of the unification problem and the structural obstruction that has prevented its resolution.

1.1 The Unification Problem

The unification of physical interactions has been a central objective of theoretical physics since the nineteenth century. The first successful unification was achieved by Maxwell, who demonstrated that electricity and magnetism are manifestations of a single electromagnetic field described by a unified set of differential equations [1]. This result established a paradigm: distinct phenomena could be understood as different realizations of a single underlying structure.

The next major advance occurred with Einstein’s formulation of General Relativity (GR), in which gravitation is not treated as a force but as curvature of spacetime induced by energy–momentum [2]. While conceptually profound, this formulation introduced a structural divergence from electromagnetism. Electromagnetism is described as a gauge theory on a fixed spacetime background, whereas gravitation defines the geometry of spacetime itself. This distinction – geometric versus field-theoretic description – has remained a central barrier to unification.

Early attempts to bridge this divide focused on geometric extensions. The Kaluza–Klein framework introduced higher-dimensional spacetime in which electromagnetic degrees of freedom emerge from metric components [3,4]. Although elegant, these models required additional dimensions without empirical necessity and did not eliminate independent coupling structures. Later developments in gauge theory, particularly Yang–Mills theory, generalized the electromagnetic U(1) symmetry to non-Abelian groups, forming the basis of the Standard Model [5]. However, gravity remained external to this framework.

Efforts to incorporate gravity into a unified quantum framework led to string theory and related approaches, in which all interactions arise from excitations of fundamental extended objects [6,7]. While these theories achieve formal unification at high energies, they introduce a vast landscape of possible solutions and rely on structures not directly accessible to experiment. Loop quantum gravity and other non-perturbative approaches attempt to quantize spacetime itself [8], but do not unify gravity with the gauge interactions in a single dynamical object.

Across these approaches, a common pattern emerges: unification is pursued by adding structure – extra dimensions, additional symmetries, new fields, or quantization prescriptions. This strategy has achieved partial success but has not eliminated the fundamental separation between interaction sectors. In particular, independent coupling constants (e.g. e, G, ℏ) persist, distinct mathematical frameworks are retained for different interactions and the origin of these structures is not derived from a single primitive. This persistent separation suggests that the obstruction to unification is not merely technical, but conceptual.

1.1.1 Structural Obstruction in Classical and Modern Frameworks

The central obstruction can be stated as follows. Existing theories assume continuum primitives – fields, metrics, or wave functions – defined on a pre-existing manifold. Within this assumption, geometry (GR) and gauge structure (EM/SM) arise from different foundational objects. Couplings are introduced and phenomenologically, discrete physical measurement plays no fundamental role in defining structure.

Because these primitives are assumed rather than derived, attempts at unification must reconcile incompatible starting points. The result is either hybrid theories (GR + quantum field theory) or extended frameworks that preserve the underlying separation.

1.1.2 MQ Perspective: Bypassing the Obstruction

Measurement Quantization (MQ) departs from this paradigm by removing continuum primitives from the foundation of the theory. Instead, it begins with integer counts of fundamental measures ([9], Appx. W), from which both geometry and field behavior emerge. In this framework, spatial and temporal separations are defined by counts n_L and n_T, The invariant ratio c = l_f/t_f establishes relativistic structure without postulating spacetime geometry, and continuum descriptions arise only as coarse-grained representations of an underlying discrete count structure.

The key implication is that geometry and interaction are no longer distinct constructs. Both are consequences of the same Frames mapping between the Internal Frame and System Frame, as defined in the MQ formalism (see Appx. A) ([9], Appx. K). This shift bypasses the historical obstruction. No independent metric is assumed. That is, gravitational structure emerges from a common count geometry ([9], Appx. I, J). No independent gauge field is introduced. Rather, electromagnetic behavior emerges as a symmetry of count relabeling. And no separate coupling constants are required. Instead, interaction strengths derive from a single geometric scalar ([9], Appx. P,Q,AA). In this way, MQ does not unify existing theories by embedding them in a larger structure. Instead, it replaces their foundational assumptions, allowing both gravitational and electromagnetic phenomena to arise from a common kinematic origin.

1.2 MQ as a Count–Based Kinematic Framework

This section introduces the minimal structure required to define MQ, with emphasis on the geometric interpretation of the Frames Principle. We will gain an understanding of this principle through the right–triangle construction provided in Fig. 1.

First, the Frames Principle asserts that all physically meaningful measures arise from mappings between a discrete Internal Frame defined by integer count structure and a non-discrete System Frame. A frame is understood as a self-consistent realization of fundamental measures – length, time, and interaction – constructed from countable units within a configuration domain. Physical observables do not exist in isolation, but only as relational quantities established through admissible mappings between such frames. These mappings preserve invariant couplings while permitting systematic deformation in measured quantities, a property of the underlying discrete geometry. In this way, apparent continuous symmetries emerge as large-scale limits of count-preserving transformations, and geometric relations – such as the right–triangle construction introduced below – serve as the minimal representation of consistency between frames.

1.2.1 Fundamental Counts and Invariant Ratio

MQ begins with two primary quantities: the integer counts n_L and n_T, representing separations in units of the fundamental length l_f and fundamental time t_f, respectively. These counts define a discrete kinematic structure in which physical distance and duration are not continuous primitives but are constructed from countable units.

The invariant relation between these counts is given by c = l_f/t_f, where c is the invariant conversion factor linking temporal and spatial counts. Let ω denote the Internal Frame configuration domain and let ω′ be a relational region over which counts are defined. Physical realization proceeds through a mapping , where ℛ is the emergent System Frame relational structure in which geometric observables are defined. See Appx. A for formal definitions of all symbols.

This relation does not arise as an imposed postulate, but as a closure constraint required for mapping consistency. A fixed ratio l_f/t_f is the unique condition under which count-based transformations between temporal and spatial separations remain single-valued, independent of the sequence of count operations. Equivalently, admissible mappings must commute under composition of temporal and spatial count increments, ensuring that the realization is well-defined.

 Any deviation from this fixed ratio would introduce path dependence in the assignment of separations, leading to frame-dependent inconsistencies and violating the requirement that M produce a unique and invariant realization in the System Frame. In this sense, the relation c = l_f/t_f replaces the assumption of a spacetime metric with a kinematic closure condition on admissible mappings within a count geometry. The formulation aligns with the requirement that any kinematic framework reproduce Lorentz-invariant structure in the appropriate limit [10,11], while avoiding the introduction of a continuous spacetime manifold as a primitive object.

 1.2.2 The Internal Frame, System Frame, and Reference Frame

MQ defines physical relations through three distinct but interrelated constructs. The Internal Frame describes a discrete domain in which separations are defined strictly by integer counts n_L and n_T. The System Frame describes a continuous representation in which those counts are mapped to measurable quantities. And the Reference Frame describes the operational frame associated with measurement and comparison.

The Frames Principle states that all physical observables arise from the difference geometry between the Internal and System Frames, evaluated relative to a Reference Frame. This replaces the classical paradigm in which observables are defined solely relative to an observer or coordinate system.

Fig. 1. Count of l_f describing the separation between two phenomenon in the System Frame, with discrete length evaluated in the Internal Frame. Modified from [9].

1.2.3 Right–Triangle Construction and Count Geometry

The geometric content of the Frames Principle is captured by the right–triangle representation in Fig. 1. Side a represents the reference count and is always assigned a count of 1. Side b represents the Internal Frame count and when that count is multiplied by l_f it represents the distance associated with an interaction event. And the hypotenuse corresponds to the separation in the System Frame. This yields a count-based analog of the Pythagorean relation, where n_L is the count defined in the Internal Frame and n_L + Q_L encodes the discrete–continuous mapping difference. This construction is not merely illustrative. It defines the operational geometry from which all MQ expressions are derived. The orthogonality reflects the independence of the discrete count structure from its continuous representation, while their combination produces the measurable separation.

1.2.4 Emergence of the Discrete–Continuous Mapping

The mapping between the Internal and System Frame introduces a nontrivial geometric effect arising from the incompatibility between discrete count structure and continuous representation. This effect is encoded through the mapping component Q_L, which quantifies the deviation between count-defined separation in the Internal Frame and its realized measure in the System Frame. Because the discrete count structure cannot be represented exactly within a continuous domain, the mapping necessarily induces a correction to measured separation.

Unlike Lorentz contraction, which depends on relative velocity, this effect is purely kinematic and arises from the structure of the Frames mapping. Its magnitude depends on the count scale and vanishes in the large-count limit, ensuring consistency with classical continuum descriptions. In this sense, the mapping defines a scale-dependent deformation of measured distance that is intrinsic to the transition between frames.

This behavior is consistent with general constraints on discrete-to-continuum mappings in physical systems, where coarse-graining produces effective continuum behavior while retaining underlying discrete structure [12,13]. Within MQ, however, this effect is not an emergent statistical artifact but a deterministic consequence of the Frames Principle and therefore enters directly into the definition of measurable quantities.

A second consequence of the mapping arises from the evolution of count structure under the internal update index A. With each increment of the Internal Frame update index A, the effective relational region associated with an interaction undergoes a loss Q_L relative to the Internal Frame reference scale. Because this loss occurs below the reference count scale, it produces a cumulative displacement in the realized separation. At the level of the System Frame, this displacement manifests as curvature in the effective geometric description.

This kinematic origin of curvature can be expressed directly in terms of the mapping structure. In particular, the effective gravitational coupling associated with this displacement follows from the same count-based construction ([9], Appx. C):

where the factor 2Q_Ln_L represents the non-Lorentz length contraction term arising from the Frames mapping and θ_si is a radial scalar describing the radial rate at which the universe expands. Notably, the contraction term is significant at interaction scales that are small – most prominently in electromagnetic interactions – and diminishes with increasing separation, recovering classical gravitational behavior in the macroscopic limit.

The appearance of this term at the kinematic level is central to the MQ framework. It establishes that curvature and coupling are not introduced independently but arise from the same underlying mapping structure that governs the relation between discrete counts and continuous measurement. As will be shown in later sections, this same structure is responsible for the unified emergence of electromagnetic and gravitational behavior from a single action through count-dependent realization under the Frames mapping.

1.2.5 Kinematic Sufficiency of the Triangle Construction

The right–triangle construction provides a complete kinematic description within MQ. The invariant ratio ensures consistency between temporal and spatial counts. The geometric relation defines measurable separation. And the frame mapping encodes all deviations between discrete and continuous representations. No additional geometric primitives – such as a metric tensor or connection – are required at this stage. Those structures, if they arise, do so only after coarse-graining and are not fundamental inputs to the framework.

This distinguishes MQ from both GR and gauge field theories. In GR, the metric is primary and defines geometry [2]. In gauge theories, fields are defined on a fixed background manifold [14]. In MQ, both geometry and interaction emerge from the same count-based construction.

1.3 Criteria for a Successful Unification

A theory that claims to unify distinct physical interactions must satisfy a set of well-established criteria that have emerged across more than a century of theoretical development. These criteria are not arbitrary. Rather, they reflect structural requirements that have been repeatedly identified in successful physical theories and in prior unification attempts. The purpose of this section is to state these criteria precisely, justify their necessity, and establish the standard against which the MQ framework will be evaluated in subsequent sections.

1.3.1 Single Underlying Kinematic Structure

A necessary condition for unification is the existence of a single underlying kinematic framework from which all interactions arise. Historically, successful theories such as electromagnetism and GR achieve internal coherence by grounding all observable phenomena in a common geometric or field-theoretic structure [1,2]. Conversely, the coexistence of multiple independent kinematic primitives – such as a spacetime metric for gravity and gauge fields for electromagnetism – signals an incomplete unification.

The requirement of a single kinematic structure ensures that all observables are derived from a common set of primitives. One, no interaction requires an independent geometric or algebraic substrate. Two, cross-domain predictions are meaningful without additional assumptions. In this way the criteria are both necessary and restrictive such that they exclude hybrid constructions in which distinct sectors are combined without eliminating their foundational independence.

1.3.2 Single Dynamical Principle

Beyond kinematics, a unified theory must possess a single dynamical principle, typically expressed through an action functional whose variation governs all interactions. The role of the action principle in modern physics is foundational. It provides a unified mechanism from which equations of motion, conservation laws, and symmetry relations are derived [15,16].

Formally, this requirement can be stated as:

There exists a single action functional S such that all physically admissible field configurations satisfy δS=0 under appropriate variations.

This condition ensures that: gravitational and electromagnetic dynamics are not introduced separately, conservation laws arise from symmetries of the same functional, and that no sector-specific equations are postulated independently. The absence of such a principle leads to theories that are descriptive rather than explanatory, lacking a unifying origin for their dynamical content.

1.3.3 Recovery of Established Physical Limits

A viable unification must reproduce established physical theories within their respective domains of validity. This includes, at minimum, the recovery of Maxwell’s equations in the appropriate weak-field or small-perturbation limit [1,5], the recovery of geodesic motion and curvature-based dynamics in the macroscopic gravitational limit [2], and compatibility with Lorentz invariance in contexts where Special Relativity (SR) is experimentally confirmed [10,11].

This requirement reflects a fundamental principle of the theoretical community: new theories must extend, not contradict, empirically validated results. In practice, this imposes strong constraints on any proposed framework, as it must reproduce distinct observed behaviors through a single underlying structure without introducing independent dynamical sectors.

1.3.4 Symmetry and Conservation Consistency

A unified theory must exhibit internal consistency through its symmetry structure. In modern physics, symmetries are not optional features but determine conservation laws through Noether’s theorem [15]. Accordingly, a unified framework must ensure that energy–momentum conservation arises from invariance under coordinate transformations, gauge invariance yields conserved currents and that all conservation laws emerge from the same underlying dynamical structure. These requirements are critical because they link mathematical consistency to physical observability. A theory that fails to produce the correct conservation laws from its symmetries cannot be considered complete.

1.3.5 Reduction of Independent Parameters

A defining feature of unification is the reduction of independent coupling constants and free parameters. In the Standard Model and GR, fundamental constants such as G, e, and ħ are introduced independently. A successful unification should instead demonstrate that such quantities arise from a smaller set of underlying parameters or from a single geometric or kinematic structure [6,7].

This criterion is essential for two reasons. First, it increases predictive power by reducing arbitrariness. And secondly, it indicates that seemingly distinct interactions share a common origin. Without parameter reduction, a theory may provide a formal unification but fail to achieve explanatory unification.

1.3.6 Predictive and Falsifiable Structure

Finally, a unified theory must produce new, testable predictions that distinguish it from existing frameworks. The requirement of falsifiability is central to the scientific method and ensures that unification is not merely a reinterpretation of known results. Such predictions may include deviations from classical behavior at specific count scales, new relationships between observables, or measurable corrections arising from count-dependent normalization within the unified structure. This criterion ensures that unification has empirical content and can be subjected to experimental validation.

1.3.7 Why These Criteria are Definitive

The criteria listed above are not independent. They form a tightly coupled set of requirements that collectively define unification. A theory that satisfies only a subset of these conditions cannot be considered fully unified for three reasons. One, a single kinematic structure without a single dynamical principle lacks predictive completeness. Two, a single action without parameter reduction retains unexplained arbitrariness. And three, recovery of limits without new predictions reduces to consistency rather than unification.

Historically, attempts at unification have failed precisely because they satisfy some but not all of these conditions. For example, higher-dimensional theories achieve geometric unification but do not eliminate independent parameters [3,4], while quantum gravity approaches may provide dynamical consistency without reproducing all classical limits [8]. Thus, these criteria are both inclusive – capturing all necessary aspects of unification – and exclude frameworks that do not fully resolve the structural separation between interactions.

1.4 Scope of the Present Paper

The present work is structured to evaluate whether the MQ framework satisfies the criteria for physical unification established in Sec. 1.3, while maintaining strict separation between foundational construction, empirical validation, and full field-theoretic reconstruction. The scope is therefore deliberately constrained. This paper establishes the kinematic and statistical basis for unification and demonstrates high-significance empirical concordance, while deferring the complete derivation of the emergent field equations and their tensorial structure to the appendices and subsequent work.

1.4.1 Primary Objective: Empirical Validation of the Frames Principle

The central objective of this paper is to test the Frames Principle through a statistically rigorous concordance analysis across physically independent implementations. The analysis evaluates whether a single count-based kinematic framework – defined by the Internal and System Frames, and their associated Frames mapping (Appx. I) – can simultaneously reproduce gravitational, electromagnetic, and cosmological observables without introducing independent sector-specific parameters.

This objective is operationalized through the construction of a comprehensive σ–concordance analysis, which compares MQ derived expressions to experimentally measured quantities across multiple domains. The methodology follows standard statistical practice to quantify agreement between prediction and measurement [43–46].

Within this framework, each implementation of the Frames Principle constitutes a physically independent test. Independence is defined not only by distinct observables but also by the absence of shared systematic dependencies, ensuring that aggregate significance reflects genuinely independent evidence rather than repeated sensitivity to the same underlying effects.

1.4.2 Role of the Concordance Analysis

The σ–concordance analysis presented in Table I serves as the empirical core of the paper. It is designed to address a central question: can a single kinematic framework, without adjustable coupling constants, reproduce experimentally measured quantities across distinct physical scales with statistically significant agreement? The table reports a global concordance analysis across independent degrees of freedom, using a right–tail p–value converted into an agreement sigma. The global assessments are grouped by discipline and span a broad range of count separations, allowing the empirical performance of the Frames Principle to be evaluated both by implementation class and across CODATA publications. From one publication to the next, the results show only modest variation, while the Frames Principle assessment remains at high significance.

Table I is therefore not auxiliary but foundational. It provides a direct test of parameter reduction (Sec. 1.3.5), a quantitative assessment of cross-domain consistency, and an empirical benchmark against which classical (Planck-based) formulations can be compared. Importantly, where expressions include the MQ-predicted length contraction, the “w/o MQ” column omits this term. This comparison isolates the contribution of the discrete contraction structure and distinguishes MQ expressions from the corresponding classical expressions obtained when the contraction term is removed. Because removal of this term reduces MQ’s fundamental measures to their Planck-form counterparts, the contrast provided in Table I tests whether the observed concordance is attributable specifically to the Frames mapping rather than to shared dimensional analysis or unit construction.

The summary provided in Table I is complemented by the detailed implementation-level concordance analysis presented in Appx. H. There, the full 2022 CODATA breakdown, methodological notes, and implementation-specific exclusions are presented. In this way, the main text establishes the empirical significance of unification at a glance, while Appx. H provides the full audit trail underlying that result.

1.4.3 Scope of Theoretical Development

While the MQ framework is capable of supporting further extensions, the present work already establishes a unified field-theoretic structure at the level required for physical unification. In particular, a single action defined on the count-based kinematic substrate is constructed and shown to generate both electromagnetic and gravitational behavior as distinct realizations of the same underlying dynamics. Electromagnetic behavior arises from evaluation of the action at the fixed interaction count defined by the electromagnetic demarcation, yielding an emergent Abelian gauge structure with associated Maxwell-type equations, while gravitational behavior corresponds to count-resolved realizations in which the mapping produces an effective metric description with geodesic dynamics and a mapping-dependent gravitational coupling.

In addition, the core symmetry and consistency relations are not deferred but demonstrated within the main development. Variation of the action with respect to the emergent geometric descriptor yields the stress–energy tensor, invariance under admissible diffeomorphic transformations yields covariant conservation, and invariance of the derivative-based action under gauge re-parameterization ensures the consistency of the energy–momentum tensor. These structures arise as co-emergent consequences of a single action defined through the Frames mapping between the Internal Frame and the System Frame, rather than being imposed independently. The result is a closed consistency chain linking gauge structure, gravitational response, and conservation laws within a unified kinematic origin.

The role of the appendices is therefore not to introduce these structures, but to provide their detailed formal development and explicit construction in equation form. The main text establishes their existence, interdependence, and physical necessity, while the appendices supply the extended derivations, normalization conditions, and structure-specific refinements required for full technical completeness.

This staging reflects a methodological choice, but not a limitation of scope. Unlike prior unification programs in which formal structure is introduced independently of empirical validation [6–8], the present work integrates both simultaneously. A unified action-based framework is constructed and shown to satisfy the required symmetry and consistency conditions, while its physical validity is established through high-significance concordance with observational data. This dual requirement ensures that the unification is not only mathematically coherent but empirically grounded, consistent with the principle that physical theories must ultimately be judged by their predictive and observational adequacy [16–18].

1.4.4 Delineation of What Is and Is Not Claimed

To maintain clarity while accurately reflecting the scope of the present work, the claims established in this paper are delineated as follows. First, a single count-based kinematic framework defined through the Frames Principle is shown to generate a unified dynamical structure from which both electromagnetic and gravitational phenomena emerge as distinct realizations of the same underlying action. Second, this framework yields explicit expressions for physical observables across multiple domains, and these expressions achieve statistically significant agreement with experimental data when evaluated through independent implementations of the Frames mapping. Third, the inclusion of the non-Lorentz length contraction term arising from the Internal Frame is shown to be necessary to achieve this level of empirical concordance.

In addition to empirical agreement, the paper establishes a coherent theoretical closure at the level required for physical unification. A single action functional defined on the count-based substrate is constructed and shown to generate, under distinct realizations determined by count separation, an emergent Abelian gauge structure with associated Maxwell-type equations when evaluated at the fixed interaction count defined by the electromagnetic demarcation, and an emergent geometric description with metric structure and geodesic dynamics for count-resolved realizations. The associated symmetry and consistency relations are likewise demonstrated. Variation of the action yields the stress–energy tensor, invariance under admissible transformations yields covariant conservation, and gauge invariance is preserved within the energy–momentum tensor. These structures arise as coupled consequences of the same kinematic origin rather than as independent postulates.

At the same time, several elements remain outside the scope of what is claimed here. We do not establish a fully general, closed-form tensorial reconstruction of all field equations across arbitrary scales, nor do we provide a complete treatment of quantum field-theoretic extensions, non-Abelian generalizations, or full renormalization structure. In addition, while the framework is shown to reproduce the relevant limiting behavior consistent with established electromagnetic and gravitational theories, a comprehensive comparison with all competing unification programs at the level of quantization and ultraviolet completion is not undertaken.

These limitations reflect scope control rather than structural incompleteness. The appendices provide detailed derivations of the action, symmetry relations, and mapping structure, while the main text establishes their interdependence and physical consequences. The result is a unified framework that is both empirically validated and theoretically coherent at the level necessary to support the central claim of kinematic unification, with further extensions reserved for subsequent development.

2. METHODS

2.1 Internal–System Mapping and Count Geometry

The Frames Principle asserts that all physically meaningful observables arise through a mapping from a discrete count-based configuration domain to a realized relational structure. In MQ, the Internal Frame is the discrete relational domain ω, whose primitive content consists solely of conserved count configurations and adjacency relations. Observable quantities do not exist intrinsically within ω. They arise only after realization through the Frames mapping, which induces the emergent System Frame reference structure ℛ. Geometry, continuity, and measurement are therefore not assumed a priori but emerge from the mapping itself.

This construction differs fundamentally from formulations that begin with a pre-existing spacetime manifold. The primary kinematic object is the count structure of the Internal Frame, while the continuum structures employed in physical measurement arise as realized encodings of that count structure. The role of the mapping is therefore not merely representational; it defines the conditions under which observables become physically meaningful.

2.1.1 Frames and Mapping Structure

Let ω'⊂ω denote a relational region of the Internal Frame. The Frames mapping is defined by

where ℛ is the System Frame reference structure and Ω is the realized System Frame image of the Internal Frame region ω^'. A valid realization preserves the normalization, realization consistency, and holonomy constraints required by the Frames Principle. Consequently, physically observable quantities are associated with elements of Ω, not with elements of ω^' directly.

Equation (2) establishes the fundamental distinction between Internal Frame and System Frame descriptions. The Internal Frame contains only discrete relational count structure, while the System Frame contains the realized relational encoding through which measurement is defined. Observable quantities therefore arise only after application of the Frames mapping and are represented by realized configurations in Ω, rather than by elements of the underlying count substrate itself.

The Frames mapping induces a realization measure on Ω. The corresponding System Frame encoded availability density is

where μ_M is the measure induced by the Frames mapping M on the realized configuration Ω, and dV is the associated System Frame volume element. The density ρ_S represents the realized encoded availability obtained from conserved Internal Frame count structure under the mapping and serves as the fundamental System Frame scalar from which normalized probabilities, expectation values, and continuum observables are constructed.

Equations (2) and (3) provide the minimal kinematic framework required for the remainder of the analysis. Equation (2) specifies how discrete count configurations acquire realized relational meaning through the Frames mapping, while Eq. (3) defines the corresponding System Frame encoded availability density induced by that realization. Together, these relations establish the discrete-to-continuous correspondence required by the Frames Principle and provide the foundation from which both fixed interaction scale realizations and curvature resolved realizations emerge within MQ.

2.1.2 Right–Triangle Construction and Count Geometry

The geometric consistency of the mapping is captured by a minimal relational construction. Consider a normalized reference count and an Internal Frame count n_L. The mapped System Frame separation n_L + Q_L is defined through a constraint relation that enforces consistency of count composition:

This relation is not a geometric axiom but a consequence of consistency between discrete count structure and its continuous embedding. It provides the minimal closure condition required for a valid Frames mapping.

2.1.3 Discrete–Continuous Mapping and Contraction Structure

The mapping M introduces a systematic deviation between Internal Frame counts and System Frame separations. This deviation is captured through a mapping dependent contraction term expressed at the level of count realization:

where n_L is equal to the count of l_f along side b and n_L + Q_L describes side c, Q_L represents the non-whole count portion of the hypotenuse. It follows that

. This contraction is not Lorentzian. It does not arise from relative velocity but from the incompatibility between discrete count structure and continuous encoding. Unlike Lorentz contraction, which depends on kinematic state, MQ contraction depends on the radial scaler rate θ_si of the expanding frame of the universe, D_U = 2θ_siA_U ([9], Appx. AZ). Consequently, all System Frame observables inherit a scale dependence through θ_si = l_fm_f/2t_f, affecting inferred distances, coupling strengths, and geometric relations.

2.1.4 Kinematic Role of the Mapping

The Frames mapping simultaneously performs three roles. It embeds discrete count structure into a continuous representation. It introduces count-dependent length contraction via 2Q_Ln_L ([9], Appx. D). And it provides a unified origin for geometric and interaction structure, including the emergence of curvature as the cumulative realization of mapping-induced deformation over extended count separation. In this framework, structures typically introduced independently – such as metrics, curvature, or gauge fields – emerge from the properties of M, consistent with the derivation of gauge structure from underlying symmetries.

2.1.5 Relation to Classical Limits

In the large-count limit,

and the mapping reduces to a classical continuous description. In this limit discrete corrections vanish, the System Frame geometry becomes effectively smooth and standard relativistic and field-theoretic behavior is recovered. This is consistent with renormalization and continuum limit arguments in field theory.

2.1.6 Section Boundary

The contraction structure introduced here defines the kinematic origin of geometric scaling and interaction strength. Its formal realization in terms of curvature, metric normalization, and coupling emergence requires a tensorial treatment. These results are developed in Appx. D, where discrete–continuous mapping is extended to curvature scaling and gravitational coupling.

2.2 Fundamental Measures and the Role of θ

MQ is defined on a discrete substrate within the Internal Frame, where all physical quantities arise from integer-valued counts of the fundamental measures l_f , m_f and t_f. These measures constitute the primitive elements of the count structure and are defined on the configuration domain ω, with relational subsets ω^'⊂ω supporting finite count measures N_I(ω^',A). No geometric or continuum structure is assumed at this level. Instead, all continuum descriptions arise through the Frames mapping M:ω→ℛ, which induces the System Frame representation. The only non-trivial invariant parameter admitted by this structure is the dimensionless scalar

which encodes the relative scaling of the fundamental measures. This parameter is invariant under admissible transformations of the Internal Frame count assignments and therefore uniquely governs the mapping between discrete and non-discrete descriptions through the normalization constraints imposed by the Frames mapping.

Within this framework, all emergent physical scales are determined by how count densities and adjacency fluxes transform under M. In particular, the characteristic count length associated with electromagnetic interactions is fixed by the electromagnetic demarcation at n_L = 84.60055 ([9], Appx. AB, AE, AM). The count of l_f defines a distinguished scale in the Internal Frame at which boundary flux relations yield stable, repeatable interaction outcomes in the System Frame. This scale arises from the normalization constraints imposed by the Frames Principle and the requirement of consistent realization under repeated measurement mappings. While a calculation of this value yields slightly different results following the 5^th digit after the decimal, analysis shows that the value is determined geometrically by ([9], Appx. AC)

The parameter θ_si ([9], Appx. AD) is an intrinsic constant of the MQ count structure, but its numerical value is fixed through empirical input, most directly via the fine structure constant. While it may also be expressed in terms of the fundamental measures, such representations are not independent, as they require additional experimentally determined quantities (e.g., the electron mass and Bohr radius) and therefore do not constitute a first principles derivation. Its physical significance emerges only after the Frames mapping induces a non-discrete relational encoding, at which point effective dimensional quantities and interaction strengths appear as functions of θ_si. No reduction arguments are introduced at this stage. These follow from the mapping-dependent contraction structure and its extension to curvature and coupling, developed in Appx. D and the subsequent derivation of interaction structure in Sec. 3.

2.3 Why Gravity and Electromagnetism Are Expected to Share One Substrate

The preceding sections have established three essential elements of the MQ framework: the Frames Principle as the defining structure of measurement, the existence of an Internal Frame governed by discrete count relations, and a nontrivial Frames mapping that generates the observable System Frame. The question addressed in this section is whether these structures admit multiple independent dynamical substrates – one for gravitation and one for electromagnetism – or whether they necessarily imply a single underlying kinematic origin. Within MQ, the latter is not an assumption but a consequence of the construction.

2.3.1 Uniqueness of the Underlying Kinematic Substrate

The Internal Frame is defined as a configuration domain ω supporting admissible count assignments and a Frames mapping M:ω→ℛ into the System Frame structure ℛ. Valid realizations of the mapping exist on referenceable domains U⊂ω satisfying normalization, unique realization, and holonomy-free closure. By construction, physical observables arise exclusively through the realized image M(U). There is no independent geometric or dynamical structure in the System Frame beyond what is induced by M.

It follows that all interactions accessible to observation must be expressible as distinct realizations of the same mapping applied to the same underlying count structure. Introducing separate substrates for different interactions would require either multiple independent configuration domains ω, or multiple incompatible mappings acting on a shared domain.

The first possibility violates the requirement of a single referenceable universe domain U, which must satisfy normalization, unique realization, and holonomy-free closure across all admissible interactions. The second possibility introduces inconsistencies in the reference structure ℛ, since distinct mappings would in general fail to preserve common invariants on shared reference cycles γ, leading to observable contradictions.

Therefore, the Frames Principle enforces a single admissible mapping structure, and hence a single kinematic substrate, from which all interactions must emerge.

2.3.2 Electromagnetism and Gravitation as Distinct Mapping Realizations

Given a single substrate, the observed distinction between electromagnetic and gravitational phenomena arises from different interaction distances realized within the same Frames mapping, rather than from fundamentally different underlying structures. This perspective is consistent with established features of both interactions. Electromagnetism is characterized by gauge structure and local field variations, as formalized in Maxwell theory and its quantum extensions [1,5,17]. Gravitation is described by geometric relations associated with spacetime curvature and geodesic motion [2,18].

In MQ, these differences are reinterpreted as consequences of how the Frames mapping resolves discrete count structure at different interaction distances. For all known electromagnetic interactions—specifically blackbody radiation, charge couplings, and elementary charge—the interaction occurs at a fixed count separation given by the electromagnetic demarcation, n_L = 84.60055, the count of l_f at which electromagnetic interactions are realized. By contrast, gravitational interactions are evaluated at the actual interaction distance. The difference relevant to the non-Lorentz length contraction is therefore not that gravitation constitutes a fundamentally separate effect, but that the interaction distance entering the mapping differs.

The non-Lorentz contraction effect predicted by MQ is most significant at interaction distances below a few hundred counts of l_f and diminishes rapidly with increasing distance. Electromagnetic interactions probe a fixed finite-count distance at which this effect remains non-negligible, whereas gravitational interactions are typically realized at larger distances for which it is negligible. The distinction therefore arises from count separation within the same mapping structure, not from separate underlying substrates.

These descriptions are not independent sectors but complementary realizations of the same underlying count geometry. The distinction follows from interaction distance, normalization, and the relative magnitude of the Informativity differential within the Frames mapping.

2.3.3 Parameter Reduction as Evidence of a Shared Substrate

A central empirical indication that gravity and electromagnetism share a common origin is the reduction of independent constants achieved within MQ. As demonstrated in prior work, quantities traditionally treated as independent – such as the gravitational constant G, the fine structure constant α, and Planck-scale quantities – can be expressed in terms of shared fundamental measures and mapping relations [9,19]. This contrasts with conventional formulations, where electromagnetic and gravitational couplings are introduced independently and lack a unifying derivation [16,17]. The MQ framework instead yields interdependent expressions, indicating that both interactions arise from the same structural parameters.

The concordance results summarized in Table I support this conclusion. Independent measurements across gravitational and electromagnetic domains are simultaneously reproduced within a single parameter framework, without introducing interaction specific adjustments. This level of cross-domain agreement would be statistically improbable if the underlying substrates were distinct.

2.3.4 Empirical Cross-Consistency: The Hubble Tension

The resolution of the Hubble tension provides a concrete example of how a single substrate manifests differently across interaction channels. Electromagnetic distance ladder measurements and gravitational inferences yield systematically different expansion rates. In MQ, this discrepancy is not attributed to new physics in either description but to the fact that each probes a different realization of the Frames mapping.

Specifically, electromagnetic observations correspond to a mapping more directly tied to the Internal Frame count structure, while gravitational inferences reflect a mapping dominated by System Frame normalization at effectively macroscopic interaction distances. The resulting difference in MQ inferred distances yields a 3.35 sigma concordance, quantitatively consistent with the SPT-3G [20] and PANTHEON+ [21] measurements. A deeper explanation is provided in Ref. [22]. Importantly, this explanation is only coherent if both measurement channels originate from the same underlying substrate. If electromagnetism and gravitation were fundamentally independent, there would be no reason to expect a systematic, quantitatively predictable relation between their respective measurements.

2.3.5 Structural Constraint from Conservation and Invariance

The requirement of a shared substrate is further reinforced by conservation and invariance principles. In classical and quantum field theory, conservation laws arise from symmetries via Noether’s theorem [15]. In MQ, these symmetries are not imposed externally but emerge from the structural constraints of the Frames mapping, including normalization and holonomy-free closure.

Because these constraints apply globally to the mapping M, they must hold across all interaction realizations. Separate substrates would generically lead to incompatible symmetry structures, violating conservation laws when interactions are combined. The observed consistency of conservation principles across electromagnetic and gravitational phenomena therefore supports a unified origin.

2.3.6 Classical Limit and Consistency with Established Theories

In the appropriate limits, the MQ framework recovers the standard descriptions of both electromagnetism and gravitation. When the mapping-induced contraction becomes negligible, the System Frame representation approaches a continuous description in which Maxwell-type equations govern electromagnetic behavior and Einstein’s field equations govern gravitational dynamics. This ensures consistency with the well-tested predictions of both theories [1,2,16,18], while identifying their origin in a common kinematic structure.

The distinction between interaction channels arises through how the Informativity differential enters under the Frames mapping. Electromagnetic interactions are realized at a fixed interaction count defined by the electromagnetic demarcation n_L = 276/θ_si, for which the mapping-induced contraction is constant. This yields a stable normalization of the System Frame realization and supports the emergence of a local Maxwell-type description.

Gravitational interactions, by contrast, are realized at count separations corresponding to the physical distance between interacting bodies. The Informativity differential therefore varies with separation 2Q_Ln_L → 1, introducing a scale-dependent normalization while preserving the geometric structure of the field equations. The resulting dynamics are described by Einstein’s equations in the large-count limit.

3. RESULTS

3.0 Single Action Principle

The preceding sections established that all observable structure arises from a single Frames mapping M:ω→ℛ acting on a unified Internal Frame configuration domain (Appx. I). Equally important, Section 1.4.2 demonstrated that this mapping is not merely a formal construct, but one that satisfies a stringent empirical constraint: a global σ–concordance across eleven physically independent implementations spanning gravitational, electromagnetic, and cosmological scales (Table I; Appx. H). Any proposed dynamical formulation must therefore reproduce this cross-domain agreement without introducing independent sector-specific couplings.

The logical consequence of these combined structural and empirical results is that the dynamical description of physical systems must itself be unified. There must exist a single scalar functional whose stationary realizations generate all admissible System Frame observables and preserve the concordance structure already established. This object is referred to as the MQ action (Appx. J).

In contrast to conventional formulations – where distinct actions are postulated for gravitational and electromagnetic formulations – the MQ framework requires that all dynamics emerge from a single kinematic origin. This requirement aligns with the role of action principles in classical and quantum field theory, where stationary action selects physically realizable trajectories and field configurations [16,17]. However, in MQ the action is not defined on a pre-existing manifold. Rather, it is induced by the admissibility constraints of the Frames mapping itself. In this sense, the action is not an independent postulate, but the unique dynamical object consistent with both the closure structure of the mapping and the empirically observed concordance across physical domains.

3.1 The MQ Action as the Common Dynamical Object

We define the MQ action as a scalar functional over admissible System Frame encodings induced by the Frames mapping:

with the associated Lagrangian density written as

Here ρ_R is the System Frame encoded availability density defined on the realized configuration Ω=M(ω^')⊂ℛ, ∇ρ_R captures variation within the encoded continuum representation, l_f, m_f, and t_f are the MQ fundamental measures, and dμ is the induced measure on Ω. This notation is consistent with the current term definitions (Appx. A), in which ℛ denotes the relational codomain of the Frames mapping and physically realized configurations correspond to subsets Ω⊂ℛ.

The triplet of fundamental measures defines the primitive dimensional structure of the MQ framework. These measures are not independent in their physical role. They are linked by the invariant scalar combination θ_si = l_fm_f/2t_f, which constitutes the unique dimensionless parameter governing the relative scaling between the Internal Frame and the System Frame.

While θ_si does not appear explicitly in Eq. (9), it organizes the admissible structure of the Lagrangian density ℒ_MQ by constraining how the fundamental measures enter all dimensionless combinations. In this sense, θ_si functions as the structural invariant of the theory, whereas l_f, m_f, and t_f provide its dimensional realization. The MQ action is therefore specified at three distinct but connected levels.

At the primitive level, the dynamical structure is fixed by the admissible dependence of ℒ_MQ on the encoded availability variables and the fundamental measures, prior to any identification with empirical units. At the invariant level, all physically relevant dimensionless combinations are functions of θ_si. In particular, the electromagnetic demarcation and relative frame scalings depend only on θ_si (Appx. J), for example n_L = 276/ θ_si, which fixes the electromagnetic interaction scale in units of l_f. At the calibrated level, a physical realization of the theory is obtained by identifying the MQ measures against experimentally defined quantities already used elsewhere in the paper, so that numerical values for l_f, m_f, t_f and θ_si are fixed and conventional physical constants appear as composite quantities. This calibration and its parameter-reduction consequences are developed later in the parameter-reduction discussion of Sec. 3.5.5.

Crucially, no independent coupling constants are introduced at the level of the action. All such constants arise only after calibration as derived quantities from the fundamental measures and their invariant relation. The Lagrangian therefore encodes the dynamical content of the theory at the primitive level, while higher-level physical laws emerge through invariant structure and calibrated realization.

The admissible form of ℒ_MQ is constrained by the requirements of normalization of encoded availability, holonomy-free closure on contractible reference cycles γ, and invariance under frames automorphisms Φ. These constraints restrict admissible terms in the Lagrangian density, analogously to the way symmetry principles constrain Lagrangians in gauge theory [5,15].

A key structural feature is that the action depends only on quantities derived from the Frames mapping. No independent geometric metric or external gauge field is introduced at this stage. Metric and gauge structures arise only as emergent descriptions in appropriate limits. In this respect, the MQ action plays a formal role analogous to geometric actions in GR, while differing fundamentally in that curvature is not assumed a priori. Rather, geometric structure is treated as a derived consequence of the mapping between discrete count structure and continuum encoding.

3.1.1 Variational Principle and Admissible Field Variations

Physical realizations correspond to stationary points of the MQ action under admissible variations of the encoded availability density, δS_MQ=0. However, not all variations δρ_R are permitted. Admissible variations must preserve the defining constraints of the Frames mapping:

Normalization constraint: total encoded availability is conserved under variation,

Holonomy constraint: variations must not introduce nontrivial holonomy on contractible cycles,

Realization constraint: variations must preserve unique realization mappings, ensuring that outcome-to-record correspondence remains injective.

In addition, admissible variations must preserve the induced boundary–adjacency structure inherited from the Internal Frame, ensuring that flux relations across ∂Ω remain consistent with the underlying count continuity relations, and that the encoded availability density remains a valid non-negative measure on Ω.

Under these constraints, the Euler–Lagrange equation associated with Eq. (9) takes the form:

This equation defines the fundamental dynamical condition for the encoded availability field. Its structure mirrors standard variational formulations in field theory [16,17], but its interpretation is distinct. It governs admissible encodings of discrete count structure rather than the evolution of fields on a pre-existing spacetime manifold and is restricted to variations consistent with the Internal Frame continuity and boundary flux structure induced by the Frames mapping.

3.1.1.1 Emergent Temporal Structure from Internal Frame Ordering

In MQ, temporal structure is not introduced as a primitive coordinate but emerges from the ordered progression of Internal Frame configurations indexed by A. The mapping from Internal Frame configurations to System Frame observables is given by the Frames mapping M:ω→ℛ, with realized configurations Ω(A)=M(ω^'(A))⊂ℛ, where ω^'(A)⊂ω denotes the relational region at update index A. Temporal structure in the System Frame is therefore induced not by mapping alone, but by the ordered sequence of mapped configurations.

Temporal evolution in the System Frame therefore reflects a realized ordering of count configurations under M, rather than an independently defined continuous parameter. This construction is consistent with the Frames Principle, which requires that all observable quantities arise from relational structure rather than assumed background coordinates [15]. Consequently, temporal observables must be defined through measurable frame-dependent quantities.

Within this structure, temporal expansion is characterized through frame-resolved expansion rates. The System Frame expansion rate H_sys is the System Frame realization of the expansion parameter, while the Internal Frame expansion rate H_int is the Internal Frame realization associated with the ordering of count configurations over A. These quantities differ because the System Frame encodes interaction-constrained realizations – most notably through the electromagnetic demarcation – whereas the Internal Frame describes the full count evolution without observational constraint.

When accounting for these differences we find that H_sys = π/θ_siA_U and H_int = (2θ_si-π)/θ_siA_U ([22], Eqs. 84-85). The emergent cosmological term is therefore expressed in a frame-resolved manner as

These relations follow from the MQ formulation of cosmological expansion and preserve its standard dependence on the Hubble parameter in relativistic cosmology [16,23,24]. In MQ, however, the distinction between H_sys and H_int reflects the discrete–continuous structure induced by the Frames mapping, rather than observational uncertainty or model-dependent corrections.

A complete derivation of H_sys and H_int, including their reduction to MQ fundamental measures and their role in resolving discrepancies in cosmological observations, is provided in [22]. The present formulation emphasizes that temporal structure in MQ emerges from a frame-dependent realization of count evolution and is not defined through direct differential correspondence between A and an externally imposed time coordinate.

3.1.1.2 Consistency with the Variational Structure

Under this identification, differential operators appearing in the MQ action are not primitive structures but arise as System Frame representations of normalized variation in the underlying Internal Frame count configuration. In the Internal Frame, all variation is defined discretely through adjacency relations and count redistribution governed by the conservation structure. The Frames mapping, together with the normalization functional, induces a continuum encoding in which these discrete variations are represented by differential operators acting on System Frame fields such as the encoded availability density.

The Euler–Lagrange equation therefore simultaneously enforces Internal Frame admissibility—through variation of the action with respect to realizable count configurations – and System Frame dynamics – through the induced differential structure acting on the continuum-encoded fields. No independent temporal degree of freedom is introduced. Apparent time derivatives arise from the mapping of the Internal Frame update index into the System Frame representation and reflect the normalization of sequential count redistribution rather than an underlying spacetime primitive.

3.1.1.3 Conservation Laws

Conservation laws arise directly from invariance of the MQ action under admissible transformations. This is consistent with Noether’s theorem, which relates symmetries of the action to conserved quantities [15]. In MQ, these symmetries correspond to invariances of the Frames mapping itself, rather than to transformations of an external spacetime.

3.1.2 Scale Dependence and Emergent Realization under the Frames Mapping

The MQ action admits scale-dependent realizations determined by the relative contribution of derivative structure and mapping-induced deformation within the Frames mapping. Both non-Lorentz contraction and emergent curvature arise from this same mapping-induced deformation, with curvature representing the geometric manifestation of the relation between the Internal and System Frame. These behaviors arise as scale-dependent realizations of a single kinematic construction governed by the invariant scalar.

The non-Lorentz contraction associated with the discrete structure of the Internal Frame is formalized as the Informativity differential. This quantity characterizes the deviation between relational separation in the Internal Frame and its encoded representation in the System Frame, given by 2Q_Ln_L. It applies strictly to realized spatial separation, reflecting a mapping-induced effect in the three-dimensional relational structure of the System Frame. The Informativity differential is greatest at small count separations and decreases rapidly with increasing separation, becoming negligible beyond a few hundred counts of l_f. Mass and time, being singular in dimension, do not admit this form of mapping-induced contraction or curvature.

Notably, electromagnetic interactions are uniquely realized at a fixed count separation given by the electromagnetic demarcation. In contrast, interactions not fixed at the electromagnetic demarcation are realized across continuous count separations, including at small scales, where mapping-induced deformation remains locally significant. At larger count separations, such as those relevant to gravitational interactions, the Informativity differential becomes negligible, and the mapping approaches a smooth relational encoding. These realizations arise from the same action without the introduction of independent structures or coupling parameters. The distinction between electromagnetic behavior and gravitational curvature is therefore determined solely by how the Informativity differential enters through normalization at the electromagnetic demarcation versus its negligible contribution in count-resolved realizations across the physical domain.

3.1.3 Electromagnetic Limit

The electromagnetic demarcation corresponds to a fixed interaction count distance, which is resolved through three independent constructions. They are given by charge couplings, blackbody radiation, and elementary charge ([9], Appx. AB, AE, AO), respectively:

Each expression independently resolves the same interaction scale to seven digits. As such, we commonly use the first expression which offers the greatest precision, n_L = 84.60054567(93). Importantly, the correlated results indicate that the electromagnetic demarcation is not a free parameter but a constrained geometric property of the Frames mapping.

The origin of this expression is clarified through the discrete structure of quantization ratios ([9], Appx. AC). The admissible solutions are restricted by the requirement that products of count and the invariant scalar yield integer-resolved values in the Internal Frame. In particular, only one combination satisfies this condition within the relevant range: n_Lθ_si = 276 such that r_q = n_Lθ_si/⌈ n_Lθ_si⌋.

This constraint uniquely identifies the electromagnetic interaction scale. Neighboring integer combinations (Table II) do not satisfy the required integer closure under quantization ratios and therefore do not correspond to admissible realizations. Within the corresponding count range, only a finite number of integer-resolved configurations exist, and only the value associated with n_θ = 42 yields a solution consistent with the observed electromagnetic scale. This establishes the relation between the internal count n_θ = 42 and the measured interaction distance 84.60055l_f, where the latter represents the System Frame encoding of the former under the Frames mapping.

Table II. Quantization ratios r_q for n_L with respect to the Internal Frame.

The dynamics reduce to conservation of the discrete measurement availability count n_I(ξ,A) within the Internal Frame, expressed as

Here J_I denotes the Internal Frame adjacency count-flux object whose net contribution at relational cell ξ is evaluated by (div⁡_adjJ_I)(ξ,A). The relation expresses local conservation of measurement availability entirely within the discrete relational adjacency structure of the Internal Frame. The adjacency divergence operator measures the net count imbalance associated with a relational cell through admissible adjacency transfers and provides the local conservation structure governing redistribution of count across the relational domain.

Only after application of the Frames mapping does this conserved count structure admit realization within the System Frame. Under mapping normalization, the conserved count, adjacency flux, and associated count imbalance are realized as continuum quantities defined on the emergent relational structure. The resulting continuum conservation relations therefore do not constitute independent postulates but arise from the realization of conserved Internal Frame count under the Frames mapping.

Within MQ, electromagnetic and gravitational behavior originate from the same conserved count substrate governed by the MQ Action and constrained by the MQ variational principle. Electromagnetic behavior corresponds to realization at the fixed interaction scale defined by the electromagnetic demarcation, where the Informativity differential remains significant and fixed by count separation. Gravitational behavior corresponds to realization in the curvature-resolved regime, where the same mapping-induced structure is evaluated across varying count separations. Both descriptions therefore emerge from a common conservation framework and represent distinct realizations of the same underlying count structure under the Frames mapping.

3.2 Fixed Interaction Distance and Emergent Abelian Structure

At the electromagnetic demarcation, the MQ action for electromagnetic interactions reduces to a local form evaluated at a single count separation, such that the mapping-induced normalization – and therefore the Informativity differential – remains constant across the interaction domain. This behavior is specific to the electromagnetic demarcation and contrasts with the count-resolved realization of the gravitational case. In this limit, the Lagrangian density introduced in Eq. (9) simplifies to the effective local form

where 𝒦 is not an independent coupling parameter, but an effective normalization coefficient fixed by the underlying fundamental measures (l_f, m_f, t_f) and the electromagnetic interaction distance. Equation (18) therefore represents the reduction of the MQ action.

Applying the Euler–Lagrange equation to Eq. (18) gives ∇⋅(𝒦∇ρ_R)=0. For fixed 𝒦, as applies to the electromagnetic demarcation, this reduces to

up to normalization-preserving constraints. To connect with standard field representations, we define the System Frame log gradient availability field

This field encodes fractional variation in encoded availability density and provides the natural local connection associated with this fixed distance realization. Since 𝒜 is defined as the gradient of a scalar, the commutator of successive System Frame derivatives vanishes,

which is the local structural condition underlying Abelian behavior in this context.

The resulting field equations therefore exhibit linearity and locality. Within MQ, these properties arise because the Informativity differential is fixed by the interaction count at which electromagnetic interactions are realized, so the mapping-induced normalization enters only as a constant and does not contribute to variation across the interaction domain. No independent gauge structure is introduced. Instead, gauge-like behavior emerges from the invariance of the Frames mapping under admissible transformations. This establishes the electromagnetic realization as a fixed interaction-count limit of the MQ action.

3.2.1 Gauge Transformation and Commutativity

The encoded availability density ρ_R admits a multiplicative scaling freedom consistent with the System Frame realization of the Internal Frame count structure. Consider the transformation

where χ is a smooth scalar function on the System Frame domain ℛ. Under this transformation, the logarithmic gradient field that enters the MQ action transforms as

where A≡∇ln⁡ρ_R is defined by the derivative structure of the encoded availability density. The transformation law follows directly from the identity

and therefore, arises from the multiplicative scaling freedom of ρ_R. Because the transformation parameter χ enters additively and independently of A, successive transformations satisfy

Thus, the transformation group is Abelian.

This establishes that the emergent gauge structure is U(1), consistent with classical electromagnetism [1,17]. Crucially, this structure is not imposed axiomatically but follows from the intrinsic scaling of the encoded availability density under the Frames mapping. Therein, the Frames mapping demonstrates that gauge symmetry arises as a kinematic consequence of the construction.

3.2.2 Noether Current and Maxwell-Type Equations

The invariance of the derivative-only MQ action under the transformation (Eq. 22) implies the existence of a conserved current via Noether’s theorem [15]. The associated current is obtained by considering infinitesimal variations χ→ϵχ, yielding

where δρ_R=χ ρ_R follows from Eq. (22), and ℒ_ρ denotes the derivative-only Lagrangian density expressed in terms of ρ_R and its derivatives. Using the log gradient structure 𝒜≡∇ln⁡ρ_R, and the derivative dependence of ℒ_ρ, one obtains

which is fully determined by the encoded availability density and its induced gradient field. This current satisfies the continuity equation

consistent with conservation arising from the underlying Internal Frame count structure under the Frames mapping.

To establish correspondence with electromagnetic field theory, define the field strength tensor

with 𝒜_μ≡(ϕ,𝒜) constructed from the log gradient field. The resulting field equations take the form

which are the Maxwell-type equations governing an Abelian gauge field [1,25].

Thus, within MQ, Maxwell’s equations emerge as the Euler–Lagrange and Noether consequences of the derivative-only bound of the single MQ action.

3.2.3 Interpretation of the Electromagnetic Demarcation

The realization of electromagnetic interactions corresponds to a well-defined condition on the discrete count structure determined by the Frames mapping. Within MQ, electromagnetic interactions occur at a fixed Internal Frame count scale

which defines the electromagnetic demarcation. This scale follows from the invariant scalar structure governing the mapping between the Internal and System Frame, as established in Sec. 2.3.

At this count scale, the contraction term 2Q_Ln_L is fixed. As a result, it contributes as a constant normalization to the MQ action and does not enter into local variation of the encoded availability density. The Euler–Lagrange structure therefore reduces to a form governed entirely by local variation, with the contraction structure implicitly encoded through normalization.

This produces an effective field description characterized by commutative structure and linear superposition, consistent with the emergence of an Abelian gauge structure. The resulting equations are therefore of Maxwell type in the appropriate limit, in agreement with the established role of gauge symmetry in electromagnetic phenomena [1,5,17].

3.3 Gravitational Limit

The gravitational limit of the MQ framework arises at large Internal Frame count separations n_L. In this limit, the non-Lorentz contraction term 2Q_Ln_L approaches its asymptotic value and ceases to contribute to variation in the Frames mapping. This behavior follows directly from the discrete–continuous mapping structure, such that contraction effects decay rapidly with increasing count scale. As a result, gravitational interactions are realized at count separations where the mapping can be effectively evaluated near its upper bound. We often omit the term from leading-order expressions without loss of accuracy. The dynamics are then governed by global constraints imposed on the encoded availability density ρ_R through normalization and holonomy-free closure of the Frames mapping.

Variations in ρ_R across extended domains produce curvature-like effects that are naturally interpreted as a metric structure. This interpretation is consistent with the geometric description of gravitation in GR, where curvature arises from global consistency conditions on spacetime [2,18]. To formalize this emergence, the coarse-grained mapping admits an effective metric tensor

defined implicitly through the normalization and relational consistency conditions of the encoded density field. The resulting dynamics are governed by geodesic propagation in this emergent geometry, further developed in Appx. E.

Importantly, this does not represent a distinct dynamical sector. Both electromagnetic and gravitational behavior arise from the same MQ action functional, with the contraction magnitude determined solely by count separation. At the fixed interaction count associated with the electromagnetic demarcation, the Informativity differential is also fixed. At larger separations, its magnitude becomes negligible. The distinction between field-theoretic and geometric descriptions therefore reflects how the Frames mapping is evaluated across count separation as opposed to the introduction of separate physical mechanisms.

3.3.1 Coarse-Graining of Count Geometry

Coarse-graining of count geometry is defined through the realization of conserved Internal Frame count structure under the Frames mapping M:ω→ℛ, which maps relational count configurations defined on the configuration domain ω into the realized System Frame relational structure ℛ. Here, ω^'⊂ω denotes a bounded relational region composed of adjacency-linked relational cells ξ, with realized configuration Ω=M(ω^')⊂ℛ.

In the fine-grained description, the primitive quantity is the discrete measurement availability count n_I(ξ,A), defined on the relational adjacency structure of the Internal Frame. Redistribution of count between neighboring relational cells is described by the adjacency count-flux object J_I, and local conservation of measurement availability is governed by the Internal Frame continuity relation

Here J_I denotes the Internal Frame adjacency count-flux object, while (div⁡_adjJ_I)(ξ,A) measures the net count imbalance associated with relational cell ξ arising from admissible adjacency transfers. No geometric, coordinate, vector, or continuum structure is assumed. Equation (32) expresses conservation entirely within the discrete relational structure of the Internal Frame.

For relational regions ω^'⊂ω sufficiently large compared with the fundamental length scale l_f, a coarse-grained description becomes admissible. The local Internal Frame count density is defined by adjacency averaging of the primitive count field,

where Adj⁡(ξ) denotes the set of admissible neighboring relational cells associated with ξ, and ∣Adj⁡(ξ)∣ is its cardinality.

The total conserved count associated with a relational region is

which defines the regional measurement availability contained within ω^'.

Summing Eq. (32) over the relational region yields the regional conservation relation

where the boundary count-flux is

The quantity ϕ_I(∂ω^',A) represents the net outward transport of measurement availability across the adjacency boundary ∂ω^'. It is an exact discrete count-flux and does not correspond to a geometric surface integral.

The coarse-grained density ρ_I therefore introduces no new physical structure. It provides a locally averaged representation of conserved Internal Frame count while preserving the underlying conservation relations. Through the Frames mapping, the conserved count structure encoded by n_I, N_I, J_I, and ϕ_I admits a mapping normalized realization within the System Frame, where continuum observables are defined. The resulting continuum description is therefore not an independent postulate but a realization of conserved count under the Frames mapping, consistent with the Frames Principle

Nontrivial structure emerges when the Frames mapping cannot be globally trivialized across extended relational regions. In such cases, aggregation of local mappings introduces residual incompatibility in the closure of reference information around relational cycles. Let γ⊂ω be a closed reference cycle. The corresponding holonomy functional is ℋ(γ). A mapping defect is present whenever ∃ γ⊂ω^' such that ℋ(γ)≠0.

More generally, the defect class associated with the relational region is

Equation (37) identifies the simultaneous presence of persistent boundary count-flux and nontrivial holonomy as the condition for a genuine mapping obstruction. The defect is therefore not merely a local imbalance in adjacency transfers but a global obstruction to extension of a holonomy-free Frames mapping across the relational domain.

Under coarse-graining, these discrete defects are encoded in the spatial variation of ρ_R, providing the precursor to emergent geometric structure in the System Frame. At this stage, however, no metric or curvature object has yet been introduced. The result is solely a nontrivial organization of encoded availability across the relational domain induced by the conservation, flux, and closure properties of the underlying Internal Frame count structure.

Thus, coarse-graining serves as the mechanism by which discrete count structure becomes compatible with continuum representation while preserving the essential mapping constraints that give rise to emergent geometric behavior in subsequent sections.

3.3.2 Emergent Metric Structure

The emergence of a metric structure follows from the requirement that the System Frame encoding support consistent comparison of encoded availability across neighboring relational regions. Following Sec. 3.1, the coarse-grained encoded availability density ρ_S defines the primary scalar field on the realized relational domain. From this, we construct the System Frame log gradient field

which is well-defined under the continuum encoding induced by the Frames mapping. This object arises from the mapped representation of Internal Frame count structure and has no counterpart in the Internal Frame. In regions where the Frames mapping is locally trivializable, the field 𝒜_μ is integrable, and its circulation over any contractible loop vanishes,

This corresponds to a regime in which the coarse-grained structure admits a flat representation, consistent with the absence of mapping defects. However, when the mapping fails to be globally trivializable, the circulation becomes nonzero over closed relational cycles,

indicating the presence of residual mismatch inherited from the discrete adjacency structure. This non-integrability defines a connection structure in the System Frame.

The associated antisymmetric field strength is

which encodes the local failure of integrability of the log gradient field. This object arises directly from the mapping of Internal Frame defects and is therefore a derived quantity.

To construct a metric structure, we consider second-order variations in realized separation. The natural quadratic form governing local variations is defined on a local coordinate chart x^μ of the emergent System Frame as

where the metric tensor

characterizes the local realization of count resolved separation under the Frames mapping.

The continuum description admits a direct count based realization. In MQ (see Fig. 1), the fundamental realized separation is determined by

where n_L is the Internal Frame count separation and Q_L is the mapping-induced residual count contribution. The corresponding continuum line element is obtained through coarse-grained realization,

establishing the connection between realized count separation and the emergent metric description.

The metric may therefore be expressed as the coarse-grained second-order response of the count resolved separation functional,

providing an explicit bridge between discrete count structure and continuum geometry. Using the count coordinate correspondence

the metric becomes

The associated connection coefficients are therefore determined by the same count-resolved structure,

and inherit their form from the realized metric rather than from an independently postulated spacetime geometry. The resulting curvature tensors follow the standard differential geometric construction,

Thus, the metric, connection, and curvature arise as successive realizations of count-resolved separation under the Frames mapping. Equations (43–51) make this dependence explicit, demonstrating how the standard geometric structures of gravitation emerge from the same count-based kinematic framework that governs the realized physical descriptions obtained from MQ. This establishes the geometric description of MQ as a consequence of Internal Frame count structure while preserving the conventional continuum formulation used in gravitational physics [2,18].

3.3.3 Geodesic Dynamics from the Same Action

Particle dynamics in the gravitational limit follow directly from the same MQ action functional (Sec. 3.1). The distinction between electromagnetic and gravitational behavior arises solely from how the Frames mapping is evaluated across count separation. In the continuum encoding, trajectory evolution is governed by the effective geometric structure induced by the coarse-grained encoded availability field ρ_R and the emergent metric

(Sec. 3.3.2). The action is therefore understood as the evaluation of the MQ action along an admissible trajectory x^μ(λ), with λ an affine parameter consistent with the variational structure (Sec. 3.1.1). Variation of this action with respect to admissible trajectory perturbations yields the Euler–Lagrange equation

which is identical in form to the variational structure obtained in Appx. E (Eqs. E.4–E.8), where the covariant divergence and volume-form identities yield the Euler–Lagrange condition under admissible variations of encoded availability. This establishes that the trajectory equations here are not independently introduced but are the direct specialization of the general MQ variational principle under coarse-grained evaluation of the Frames mapping. Upon rearrangement using metric compatibility, Eq. (52) takes the standard geodesic form

where the effective connection coefficients are given by

These equations demonstrate that geodesic motion is not introduced as an independent postulate but follows from extremization of the MQ action after coarse-graining of conserved count structure after the Frames mapping has induced the effective metric

and its associated connection. Because

is derived from coarse-grained Internal Frame count structure rather than assumed as a primitive spacetime object, the connection coefficients

inherit their form from the same underlying adjacency relations. Curvature and trajectory evolution therefore arise from a common count-based origin within the Frames mapping.

This establishes that the geometric description of gravitation – traditionally introduced through an independent spacetime metric – is recovered here as a secondary construct arising from count-based kinematics. The resulting structure is formally equivalent to the geodesic formulation of GR [2,18], but differs fundamentally in that both the metric and the connection are derived rather than postulated.

Importantly, this derivation closes the unification loop. The same MQ action functional governs both electromagnetic and gravitational behavior without the introduction of independent dynamical structures. The observed distinction between interactions is therefore not fundamental but reflects how a single underlying Frames mapping is evaluated at a fixed interaction count in the electromagnetic case and across continuous count separations in the gravitational case.

3.3.4 Emergent Gravitational Coupling from MQ Scaling

The gravitational coupling constant G is not introduced as an independent parameter but arises from Frames mapping normalization of Internal Frame count structure into the System Frame, with its effective value determined by count-dependent contraction governed by the Informativity differential 2Q_Ln_L (Secs. 3.3.1–3.3.3). At the level of MQ primitives the normalization of geometric response yields

where the dependence on l_f, m_f, t_f and 2Q_Ln_L is explicit and derived directly from the count-based action (Sec. 3.1). This result aligns with an electromagnetic determination of G as recommended in the 2010 CODATA publication [19]. The role of the contraction term is now made precise (Sec. 3.1.2). For gravitational interactions, often measured at macroscopic separations, the contraction term rapidly approaches its upper bound with no contribution to an observable variation,

Consequently, gravitational coupling appears constant in macroscopic measurements, consistent with time-of-swing determinations discussed in the 2014 CODATA publication of G [20]. The coupling constant is therefore understood not as fundamental, but as the large-count limit of a more general MQ scaling relation. Notably, within MQ, we do not italicize terms resolved at the upper count limit. All other terms are italicized, as is the practice for any measured variable.

This behavior contrasts with the electromagnetic realization, where interactions occur at the fixed count separation defined by the electromagnetic demarcation, n_L=84.60055l_f (Eq. 31). At this count, the non-Lorentz length contraction remains significant. Electromagnetic interactions therefore probe a fixed finite-count separation within the mapping, while gravitational interactions correspond to larger separations for which the contraction approaches unity. Distinction between coupling structures arises from count separation within the same underlying Frames mapping.

The same normalization procedure extends directly to the scalar structure governing large-scale expansion. Defining the geometric scalar through the MQ scaling relation ([9], Appx. BB) and the System Frame representation of the Hubble constant ([22], Eq. 84), respectively

one obtains

which links the effective cosmological term to the same count-based normalization that determines gravitational coupling. In this formulation, the cosmological contribution is not an independent parameter, but a direct consequence of geometric scaling under the Frames mapping. This establishes that both gravitational coupling and cosmological dynamics arise from a single kinematic origin. The distinction between local gravitational interaction and global expansion is therefore not fundamental but reflects different manifestations of the same normalization structure applied across scales.

Importantly, because the contraction term becomes nontrivial only at small count scales, MQ predicts that deviations from constant G should appear only at count separations approaching the electromagnetic demarcation. At macroscopic distances, the suppression of 2Q_Ln_L ensures agreement with classical gravitational measurements, while at short distances the framework allows for systematic, albeit small, deviations tied directly to the underlying discrete measure structure.

3.4 Unified Symmetry and Consistency Structure

The preceding sections established that the MQ action yields both the fixed-count electromagnetic realization and the coarse-grained geometric gravitational description as count-dependent realizations of a single underlying Frames mapping. The remaining requirement for unification is internal consistency. The framework must show that the source structure, conservation relations, and gauge compatibility of these emergent descriptions arise from the same variational origin rather than being imposed independently.

In MQ, this requirement is especially stringent because none of the central continuum objects are fundamental. The effective metric descriptor, the associated connection structure, and the encoded matter variables all arise only after admissible coarse-graining of Internal Frame count structure into a System Frame representation. It follows that any acceptable stress–energy definition, conservation statement, or gauge consistency condition must be formulated with respect to these emergent descriptors and must remain subordinate to the admissibility conditions of the Frames mapping itself. Those admissibility conditions – normalization, unique realization, and holonomy-free closure on admissible cycles – were already introduced as part of the variational structure and remain the governing constraints here.

Accordingly, this section does not introduce three disconnected consistency principles. Rather, it shows that the MQ framework supports a single consistency structure with three mutually linked manifestations. First, variation with respect to the emergent metric descriptor defines the appropriate stress–energy response object for the geometric realization. Second, invariance of the effective action under admissible geometric reparameterizations yields the corresponding covariant conservation law. Third, compatibility of the derivative structure with the same action guarantees that gauge structure does not conflict with the emergent energy–momentum description.

The purpose of the present section is therefore limited and precise. It is not to reconstruct the full gravitational field equations, nor to anticipate the parameter-reduction program of Sec. 3.5. Instead, the goal is to demonstrate that once the effective geometric realization has emerged, the MQ framework retains the same level of structural discipline expected of any viable unified theory: a well-defined source term, a conservation law derived from invariance, and consistency between geometric and gauge descriptions. In this way, Secs. 3.4.1–3.4.4 complete the logical transition from emergent dynamics to unified consistency, thereby preparing the ground for the parameter-reduction analysis that follows.

3.4.1 Stress–Energy from Metric Variation

Within MQ, the analogue of stress–energy is defined only after an admissible Frames mapping has produced an effective local System Frame relational descriptor

. This object is not fundamental. It is the emergent symmetric rank-2 descriptor encoding local relational geometry induced from Internal Frame count structure. Its role parallels the metric in GR, but its ontological status differs. It is not a primitive field but a derived encoding of count geometry [2,18,26,27]. Let the matter action be written as

where ψ denotes the encoded non-geometric fields arising from the Frames mapping, and ℛ is the admissible System Frame region corresponding to a mapped subset of the Internal Frame configuration domain. The dependence on

is entirely induced through the mapping and its associated covariant structure.

Variation is taken with respect to admissible deformations of

generated by the Frames mapping. These variations are constrained to preserve normalization, unique realization, and holonomy-free closure, ensuring that the perturbed configuration remains within the admissible class defined by the MQ variational principle. The first variation of the matter action defines the MQ stress–energy tensor,

which yields the explicit definition (Appx. E)

This equation is the MQ analogue of the Hilbert definition of stress–energy. It defines the response of encoded matter to admissible variations of the emergent relational geometry. Importantly, the variation is not applied to a fundamental spacetime field, but to a descriptor induced by the Frames mapping. Thus, T_μν measures the sensitivity of encoded matter to changes in the mapping-induced geometry rather than to perturbations of a primitive continuum background. Because the variation is taken with respect to a symmetric rank-2 descriptor, the resulting tensor is symmetric, T_μν=T_νμ. This symmetry is not imposed but follows directly from the structure of admissible variations.

The validity of Eqs. (61,62) and the upcoming Eq. (63) require that the matter action depends on

only through the induced covariant structure and not through independent derivatives of the metric treated as separate matter variables. Under this condition, the tensor defined above is the unique symmetric response object appropriate for the emergent gravitational realization [18,26,27,28].

For completeness, the same construction may be written at the level of the effective action density (Appx. E). When the encoded matter Lagrangian takes the local form ℒ_ℳ𝒬, the stress–energy tensor admits the equivalent explicit representation

This form does not introduce a new definition. It is the local density form of Eq. (62) and makes explicit how the encoded matter responds to admissible metric variation in the emergent System Frame description.

The conceptual distinction from GR is therefore precise. In GR, stress–energy arises from variation with respect to a fundamental spacetime metric. In MQ, the identical variational structure is retained, but the varied object is an emergent descriptor derived from Internal Frame count geometry. This preserves the mathematical structure of the Hilbert construction while replacing its ontological foundation.

The conservation law associated with this definition is not assumed here. It follows from invariance properties of the effective action under admissible transformations of the emergent geometry and is derived explicitly in Sec. 3.4.2, consistent with the MQ analogue of Noether’s theorem developed in Sec. 3.2.2 [15,28].

3.4.2 Covariant Conservation from Diffeomorphism Invariance

This section establishes that the conservation structure associated with the MQ action is fully consistent with the covariant conservation laws of continuum field theory. The expressions presented below follow the standard variational treatment of diffeomorphism invariance and are included primarily to demonstrate that no additional assumptions or inconsistencies arise when the Frames mapping is evaluated in its coarse-grained geometric form. In particular, the resulting conservation law is not introduced independently but follows directly from admissible variations of the MQ action, ensuring compatibility with the established geometric formulation while preserving the count-based origin of the framework. The equations are therefore provided for completeness and reference, rather than as new structural input.

3.4.2.1 Variation of Fields under Diffeomorphism Invariance

Consider an infinitesimal diffeomorphism generated by a smooth vector field ξ^μ defined on the System Frame realization ℛ. The corresponding infinitesimal coordinate transformation is

the associated co-vector defined by

. The induced variation of the emergent metric is then given by the Lie derivative in covariant form,

where ∇_μ is the Levi–Civita covariant derivative compatible with

.

Crucially, admissibility in MQ requires that ξ^μ generate deformations that preserve normalization of the Frames mapping, unique realization, and closure consistency on contractible reference cycles. Thus, Eq. (65) represents not an arbitrary coordinate variation, but the System Frame encoding of admissible Internal Frame deformations.

To connect this admissible diffeomorphism-induced variation with the MQ variational definition, Eq. (65) provides a restricted but sufficient class of metric variations for evaluating the action variation in Eq. (61). Substituting the induced variation into Eq. (61) and using the definition of T^μν in Eqs. (62–63), the resulting variation depends linearly on ξ^μ. Integration by parts transfers derivatives onto T^μν, yielding a divergence term proportional to ∇_μT^μν. Since ξ^μ is arbitrary within the admissible class, invariance of the action under these transformations implies ∇_μT^μν=0, establishing the conservation law within the MQ variational framework.

The invariance considered here must be interpreted consistently with the MQ variational framework. In contrast to GR, where diffeomorphism invariance is defined over arbitrary smooth variations of the metric, the MQ action is varied only over admissible configurations generated by the Frames mapping. These variations are constrained to preserve normalization, unique realization, and holonomy-free closure, as established in Appx. B. However, this restriction does not weaken the invariance argument. The admissible variations form a closed class under infinitesimal transformations of the emergent System Frame geometry and therefore include the subset required to generate the conservation identity. In this sense, diffeomorphism invariance is not imposed as an independent symmetry but arises as the effective invariance of the emergent geometric descriptor under admissible mapping-induced deformations.

3.4.2.2 Variation of the Action

Variation of the matter action under this transformation as presented in Eq. (61) is

Substituting the admissible diffeomorphism-induced metric variation from Eq. (65), and using the symmetry of T^μν, one obtains

where index lowering is performed with

. The reduction to Eq. (67) and the covariant integration-by-parts step leading to Eq. (68) follow from standard identities associated with metric compatibility and are derived explicitly in Eqs. E.4–E.9. Substituting Eq. (65) yields

Applying covariant integration by parts and using metric compatibility,

where boundary terms vanish under standard compact-support or asymptotic falloff conditions.

3.4.2.3 Covariant Conservation Law

Diffeomorphism invariance requires δS_MQ=0 for arbitrary admissible ξ^μ. Therefore,

This is the MQ covariant conservation law. Its formal structure matches that of GR [18,24], but its interpretation differs fundamentally.

3.4.2.4 Internal Frame Origin and Continuum Realization of Conservation

In MQ, Eq. (69) is the realized System Frame representation of an underlying Internal Frame count conservation law defined over relational regions ω^'⊂ω (Appx. K). The primitive Internal Frame quantity is the discrete measurement availability count n_I(ξ,A), where ξ labels a relational cell and Ais the Internal Frame update index. The regional count is defined as

The conservation law is fundamentally relational and takes the form of an adjacency-balance relation between the local count n_I(ξ,A) and the adjacency count-flux J_I, defined on oriented relational links of the configuration domain ω. At the relational-cell level,

Summing Eq. (72) over all ξ∈ω^'yields the regional conservation law

where ∂ω^'denotes the adjacency boundary and ϕ_I(∂ω^',A)is the Internal Frame boundary count-flux,

Using Eq. (71), Eq. (73) may be written compactly as

For closed relational regions satisfying

the total Internal Frame count is conserved,

Only after application of the Frames mapping does this conservation structure admit a realized System Frame continuum representation. Under that realization, the conserved Internal Frame count induces an encoded availability density ρ_S(r,t) together with the corresponding realized conservation relations of the System Frame. The continuum continuity relation therefore belongs to the realized System Frame, while Eqs. (67–73) remain the primitive discrete conservation structure from which that continuum representation is derived.

The conservation laws appearing in the realized System Frame are not independent postulates. Rather, they arise from the realization of conserved Internal Frame count under the Frames mapping and the associated mapping normalization functional. The discrete adjacency-balance relation of Eq. (73) therefore provides the primitive conservation statement, while the continuum conservation laws represent its realized geometric encoding.

Within the continuum description, conservation is expressed through the stress–energy tensor T_μν, whose covariant conservation follows from the metric realization of the MQ action and the diffeomorphism invariance of the emergent geometric description. The resulting conservation law is

Equation (78) is therefore the realized System Frame expression of conservation, whereas Eqs. (62–68) describe the underlying Internal Frame count conservation from which that continuum relation ultimately derives. Covariant conservation in the System Frame is thus the continuum realization of conserved count under the Frames mapping. Equation (72) provides the primitive adjacency-balance condition, while Eq. (78) represents the corresponding conservation law in the emergent metric description.

3.4.2.5 Significance

As established by Eqs. (72–78), the discrete adjacency conservation law of the Internal Frame is carried into the continuum as the covariant conservation statement ∇_μT^μν=0. Thus, Eq. (72) is the continuum image of count conservation, while the discrete adjacency relations in the Internal Frame provide the fundamental kinematic origin.

The result places covariant conservation precisely within the logical structure of the framework. It follows from the stress–energy definition (Sec. 3.4.1), precedes gauge structure (Sec. 3.4.3), and ensures compatibility with established relativistic field theory while eliminating the need for an independent conservation postulate. Accordingly, MQ reproduces the conservation structure required by modern field theory [15,18,24], but reinterprets it as a consequence of the Frames Principle applied to a discrete count-based substrate.

3.4.3 Gauge Invariance of the Energy–Momentum Tensor

At the fixed interaction count defined by the electromagnetic demarcation, where the mapping-induced normalization is constant (Sec. 3.2), the emergent Abelian gauge structure is encoded in the System Frame through a potential field defined on the realized relational structure. Gauge transformations correspond to reparameterizations of this encoded representation,

where χ is a scalar field on the System Frame. Under this transformation, the field strength tensor becomes

demonstrating invariance under Abelian gauge transformations through the commutativity of partial derivatives. Observable gauge quantities must therefore be constructed from the gauge invariant field tensor F_μν together with the emergent metric descriptor

, consistent with standard gauge invariant field theory [15,17,18,27].

Let the gauge contribution to the MQ Action be defined as the realization of the action at the fixed interaction distance corresponding to the electromagnetic demarcation in the System Frame,

such that the MQ Action admits distinct realizations determined by count separation under the Frames mapping. In this formulation, the gauge contribution corresponds to evaluation at a fixed interaction distance, while gravitational behavior corresponds to count-resolved realizations in which the same action yields an effective metric structure in the System Frame.

The associated stress–energy tensor (Sec. 3.4.1) is obtained through variation with respect to the emergent metric descriptor,

Because ℒ_gauge depends on 𝒜_μ only through the gauge invariant tensor F_μν, its variation under the transformation (79) vanishes identically. It follows directly that

This establishes that the energy–momentum tensor derived from the MQ action is gauge invariant without requiring an independent postulate. In local density form, for a gauge invariant Lagrangian depending only on F_μν and

,

Thus, the stress–energy tensor may be written equivalently as

Since the Lagrangian depends only on the gauge invariant field strength, the tensor inherits gauge invariance directly,

. This is not an independent definition, but the explicit local realization of the variational statement in Eq. (82).

Within the MQ framework, gauge invariance must be interpreted at the correct structural level. Gauge transformations are not symmetries of the Internal Frame, which is governed by discrete count structure and the Frames mapping, but arise only after encoding into the System Frame, where derivative-based representations become admissible. They express that distinct System Frame encodings related by 𝒜_μ→𝒜_μ+∂_μλ correspond to the same underlying count configuration. Observable quantities are therefore restricted to invariant combinations, and the stress–energy tensor inherits this invariance directly. Gauge symmetry is thus a redundancy of representation, not a fundamental symmetry of the Internal Frame.

Therein, the MQ action satisfies co-emergent consistency conditions: metric variation yields the stress–energy tensor (Sec. 3.4.1), diffeomorphism invariance yields covariant conservation (Sec. 3.4.2), and gauge invariance ensures an invariant energy–momentum tensor. These are not imposed independently but arise from a single underlying action defined on the MQ kinematic substrate. Electromagnetic gauge structure, gravitational stress–energy response, and conservation laws therefore emerge as unified consequences of the same count-based action principle, establishing internal consistency across all symmetry structures required for unification.

The gauge invariant form of

has a direct consequence for the coupling structure of the unified theory. Because the stress–energy tensor is obtained entirely from variation of the same MQ action, no independent coupling parameter is introduced at the level of the electromagnetic interactions. All coupling information is therefore encoded through the Frames mapping and the emergent metric descriptor

. The source term entering the gravitational limit is therefore obtained from the same unified functional,

where S_MQ contains both derivative- and curvature-dominated contributions determined by count separation. Because the electromagnetic contribution satisfies

, the source entering the emergent gravitational description inherits gauge invariance automatically. Thus, no additional compatibility condition between electromagnetic and gravitational descriptions is required: both arise from a single variational structure. This establishes that the absence of independent coupling constants is not an imposed constraint, but a structural consequence of the MQ action.

3.4.4 Why these Structures Matter for Unification

The results of Secs. 3.4.1–3.4.3 establish a single closed consistency chain within the MQ framework: variation of the MQ Action with respect to the emergent geometric descriptor yields the stress–energy tensor (Appx. E.2.3–E.2.4), invariance under admissible diffeomorphic deformations – enforced by holonomy-free closure on contractible cycles – yields covariant conservation (Appx. E.3.1–E.3.2), and invariance of the derivative-based action under Abelian gauge reparameterization – arising from admissible re-scalings of encoded availability – leaves the corresponding energy–momentum tensor unchanged (Appx. E.2.5). As emphasized at the conclusion of Sec. 3.4.3, all contributions to the stress–energy tensor arise from a single MQ Action, so that symmetry, conservation, and sourcing are not independently specified but share a common variational origin.

The significance of this chain lies in the fact that these structures are not introduced independently but arise from a single underlying condition: the admissibility of the Frames mapping between the Internal and System Frame.

In conventional continuum formulations, these structures are implemented through formally distinct mechanisms. The stress–energy tensor is obtained by variation of the matter action with respect to a metric, covariant conservation follows from diffeomorphism invariance, and gauge invariance is imposed through symmetry of the relevant field description. These constructions are mutually consistent but logically independent. In contrast, within MQ, all three arise from the same kinematic requirement: that the Frames mapping preserves normalization, ensures unique realization, and admits holonomy-free closure on contractible cycles. This relocation of structure eliminates a layer of independently specified structure that has historically obstructed unification [15,18,27,28].

The preceding subsections further show that both the emergent geometric realization and the emergent Abelian gauge structure are derived from the same action functional defined on the MQ substrate. The gravitational limit is obtained when the coarse-grained mapping admits an effective geometric description, yielding dynamics consistent with geodesic motion and metric variation. The electromagnetic limit is obtained in the electromagnetic demarcation, where the admissible encoding produces an Abelian gauge structure governed by the field strength tensor. The critical point is that these structures are not constructed separately and later combined; rather, they are different realizations of the same variational structure relative to the count distance. This establishes that the apparent distinction between geometric and gauge interactions is not fundamental but emergent from a deeper kinematic origin [17,18,24,29].

A direct consequence of this unified origin is a restriction of admissible theory space. Because stress–energy, conservation, and gauge invariance must all arise from the same Frames mapping, any candidate realization that violates normalization, uniqueness, or closure conditions cannot produce a consistent physical theory within MQ. Thus, the framework does not expand the space of allowable models through independent structural freedoms; instead, it constrains that space by requiring that all interaction descriptions satisfy a single set of kinematic admissibility conditions. This constraint-driven structure is a defining feature of a viable unification framework, as it prevents agreement in one description from being obtained at the expense of inconsistency in another [15,17,18,30].

The broader implication is that the familiar symmetry structures of continuum physics – gauge invariance and diffeomorphism invariance – are not fundamental postulates within MQ but emergent properties of the encoded System Frame. The underlying ontology resides in the discrete count structure of the Internal Frame, with the Frames mapping acting as the mechanism through which continuum representations arise. In this sense, MQ does not unify pre-existing theories by imposing additional symmetries or higher-dimensional constructs, as in Kaluza–Klein or string-theoretic approaches [3–7]. Instead, it derives both geometric and gauge structures from a single kinematic substrate, shifting the unification problem from external synthesis to internal derivation [18,24,29,31].

For the purposes of the present section, the unificatory content is therefore precisely defined. MQ establishes that the stress–energy tensor, its covariant conservation, and gauge invariance of the energy–momentum tensor all arise from one variational principle applied to a single count-based kinematic structure. This does not yet exhaust all aspects of gravitational and electromagnetic phenomena, but it provides the minimal structural closure required of any candidate unified theory: a common origin for the symmetry and conservation principles that govern both interaction descriptions.

3.5. Internal Frame Origin of Continuum Conservation Laws

In conventional formulations, gravitational and electromagnetic interactions are specified by independent constants introduced axiomatically, including G, e, and ħ. These constants are not derived from a common structure and therefore represent independent inputs to the theory. In MQ, the relevant conservation structure is defined first within the Internal Frame through discrete count measures and adjacency relations, together with the Frames mapping M:ω→R, which induces the realized System Frame representation. The distinct interaction behaviors observed in the continuum arise as different realizations of this underlying count structure, governed by the invariant parameter θ_si introduced in Sec. 2.2.

The key structural result is that covariant conservation in the System Frame is not a primitive postulate, but the mapped realization of count-conserving relations defined on the Internal Frame. Let n_I(ξ,A) denote the Internal Frame discrete count at relational cell ξand update index A. The corresponding regional count is

At the discrete level, conservation is expressed as an adjacency-balance relation between the local count and the adjacency count-flux. This relation is defined purely on the relational adjacency structure and carries no geometric or directional interpretation in the continuum sense, consistent with the Internal Frame definitions. At the relational-cell level,

Summing Eq. (88) over the relational region ω^'⊂ω gives the regional conservation law

where ∂ω^' denotes the adjacency boundary of ω^', consisting of admissible adjacency links connecting relational cells inside the region to cells outside the region. The quantity ϕ_I(∂ω^',A) is the Internal Frame boundary count-flux,

representing the exact net outward transport of measurement availability across the adjacency boundary. This quantity is a discrete count-flux defined entirely on the relational adjacency structure of the Internal Frame. It is not a geometric surface integral and carries no dependence on metric structure, coordinate systems, normal vectors, or continuum geometry.

Using Eq. (87), the regional conservation law immediately implies

Equation (91) states that when no net count-flux crosses the adjacency boundary, the total measurement availability contained within the relational region remains invariant under Internal Frame evolution. Regional conservation therefore follows directly from the local continuity relation and the adjacency-flux structure without introducing any geometric assumptions.

Equations (87–91) are entirely Internal Frame statements. They describe conservation of discrete measurement availability through redistribution of count across the relational adjacency structure. The quantities n_I, N_I, J_I, and ϕ_I are count-based objects defined exclusively on the relational domain ω, while div⁡_adj is a combinatorial operator acting on admissible adjacency transfers. No continuum density, coordinate system, geometric current, surface integral, or differential manifold structure is required.

Only after application of the Frames mapping does this conserved count structure admit realization within the System Frame. Under mapping normalization, conserved count, adjacency flux, and regional count transport are realized as continuum quantities defined on the realized relational structure Ω=M(ω^'). The resulting continuum conservation relations therefore do not constitute independent postulates but arise from the realization of conserved Internal Frame count under the Frames mapping. In this sense, conservation in the realized System Frame is inherited directly from the discrete conservation structure encoded by Eqs. (87–91).

The corresponding continuum representation supplies the conservation structure required by the MQ Action and the MQ variational principle. In the realized continuum description, mapping-normalized count conservation induces the conserved current structure associated with encoded availability and yields the covariant conservation condition satisfied by the emergent stress-energy tensor, ∇_μT^μν=0.

This relation is therefore not imposed at the primitive count level. Rather, it is the realized System Frame conservation law obtained through application of the Frames mapping to the underlying discrete conservation structure of the Internal Frame.

3.5.1 Mapping of Interaction Structure

This section establishes that all interaction structure originates from a single invariant count parameter and the conservation laws of the Internal Frame. Specifically, the absence of independently introduced coupling constants implies that interaction strength is determined by the same invariant parameter governing the Frames mapping. In MQ, this role is played by the dimensionless parameter θ_si, which controls the discrete-to-continuum realization of admissible configurations.

Dependence enters through the normalization of conserved flux under the mapping. For electromagnetic interactions, this normalization yields a fixed interaction distance at the electromagnetic demarcation. For gravitational phenomena, the same conserved quantity is realized at the relative distance between two phenomena (Appx. D). In both cases, interaction strength is determined by count-resolved realizations of the same conserved structure. Thus, both interaction strengths arise from a common relation governed by θ_si, with their distinction reflecting different realizations of the same conserved structure under the Frames mapping.

Because interaction strength is fixed by the normalization, the effective couplings are not free parameters (Appx. F). Independent constants are reduced. This reduction imposes cross-domain constraints, whereby measurements in one interaction realization restrict the admissible realization of others, providing a necessary condition for unification within MQ. Explicit derivations of standard physical constants are given in Appx. H.

3.5.2 Geometric Origin of Coupling Structure

The relation between gravitational and electromagnetic quantities in MQ follows from the common normalization structure governing realized fields in the System Frame. No independent coupling constants appear at the level of the MQ Action,

and all interaction structure arises from the realization of invariant count relations under the Frames mapping.

The passage from discrete count structure in the Internal Frame to continuum quantities in the System Frame is governed by the mapping normalization functional (Secs. A.3, A.7). This functional maps admissible regional counts N_I(ω^',A) into a normalized continuum measure, from which the encoded availability density ρ_ℛ is defined, enforcing conservation and realization consistency while providing the operative link between invariant count measures and the realized field quantities entering Eq. (92). No additional degrees of freedom are introduced.

Within this structure, electromagnetic and gravitational quantities correspond to distinct realizations of the same normalized count substrate. For electromagnetic interactions, normalization occurs at the electromagnetic demarcation, where the count separation n_L = 276/θ_si, sets the fixed interaction scale and thereby determines the scale of the emergent Abelian gauge structure. For gravitational interactions, the same normalized count structure is realized through coarse-graining, yielding the effective metric

and its associated geometric scaling (Sec. 3.3.4, Appx. D). Accordingly, both phenomena are linked through their dependence on θ_si and the normalization structure of the Frames mapping. Their distinction is realized as a function of count separation.

The Internal Frame provides the invariant count substrate, while the System Frame realizes measurable quantities through normalization. The discrete count separation n_L, defined as the count of l_f between a system and a center of mass, characterizes the realized interaction scale, but it should be noted that each is a distinct realization of the Frames mapping. As a consequence, constants associated with electromagnetic and gravitational interactions do not appear independently at the level of the action. Electromagnetic quantities arise from boundary-flux normalization at the fixed count separation n_L, while gravitational quantities arise from curvature scaling of the same normalized count structure. Both are determined by the invariant parameter θ_si through the normalization functional. This establishes that the relation between gravitational and electromagnetic quantities in MQ is not a reduction of one interaction to the other, but the elimination of their independence at the level of first principles.

3.5.3 Fundamental, Realized, and Direct Measurement of Phenomena

The distinction between fundamental and directly measured quantities follows from the Frames Principle, which asserts that all observables arise through the Frames mapping from the Internal Frame to the System Frame . Fundamental measures are defined within the Internal Frame as invariant count-based quantities, while experimentally accessible values correspond to their realized forms in the System Frame. The mismatch between these values is therefore not anomalous, but a direct consequence of mapping-induced realization.

The MQ fundamental measures are resolved in Ref. ([9], BN5, BN7, and BN10),

Notice, that both ħ and G are not italicized indicating that we should use the macroscopically determined value for G as found in the 2014 CODATA publication and the calculated value for ħ as resolved at the upper count limit. Multiplying Eq. (93) by Eq. (95) gives

and multiplying Eq. (94) by Eq. (95) gives

These results show that the MQ fundamental measures satisfy the identities

The result is structurally significant. The Compton-form relations emerge directly from the MQ fundamental definitions without introducing any additional assumptions. In standard treatments, the Compton wavelength is defined with respect to a particle mass introduced independently. Here, the same structure arises as a necessary identity of the fundamental measures themselves, demonstrating that Compton-scale relations are intrinsic to the MQ kinematic framework.

The transition to experimentally accessible quantities follows from realization in the System Frame. As established in Appx. A, all observables are defined only after mapping and normalization . Consequently, the physically realized description is determined by the mass scale at which measurement is instantiated. For electromagnetic observations, this description is defined by the electron mass m_e. Substituting the realized mass scale m_e into Eqs. (98–100) yields the realized Compton relations

These relations define the first directly accessible length and time scales associated with electromagnetic measurement and correspond to the reduced Compton scale of the electron [32]. The appearance of the electron mass reflects the realized measurement scale imposed by the interaction count [50,51], consistent with the definition of the electromagnetic demarcation.

Evaluating these expressions using the measured electron mass yields the realized quantities reported in Table III, with the “Realized (EM)” calculations corresponding to the electromagnetic determination of m_e [33]. Notably, the directly measured displacement l_direct is consistent with the predicted realized length l_realized [34] within experimental uncertainty.

Importantly, this analysis addresses the observation that fundamental mass is many orders of magnitude larger than directly measured phenomena such as the mass of the electron. Since the fundamental measures satisfy l_fm_f = 2θ_sit_f, it follows that determination of a length larger than the smallest realizable length must be correlated with a mass larger than the smallest realizable mass. The true minimal measurable values for these quantities, however, remain experimentally unresolved.

The notion of a smallest measure is complicated by the Frames mapping process. Notably, fundamental measures are mapping constructions between the Internal and System Frame representations of the three dimensions. Up to this point, the mapping – identified through the metric differential – differs across the resolved physical constants. Because of the fundamental role of three-dimensional structure, it remains unclear whether the mapping values correspond directly to minimal realizable measurements in the System Frame or instead reflect normalization relative to those minima. In this sense, the metric differential has not been uniquely fixed. The Compton scale may provide guidance in this direction, but at present the analysis identifies structural constraints rather than definitive minimal values.

3.5.4 Relation of Gravitational and Electromagnetic Quantities through θ

The invariant scalar θ_si organizes the relation between invariant count structure and measurable physical quantities. It is invariant under Internal Frame representation while admitting distinct dimensional realizations under System Frame encoding.

3.5.4.1 Electromagnetic Realization: Explicit Dependence

The dependence of electromagnetic quantities on θ_si is explicit within the MQ term structure. By example, the inverse fine structure constant is given by ([9], Appx. AA)

This expression follows directly from the count-based structure of the Internal Frame and its realization under the Frames mapping [19,25,35]. It establishes that a quantity conventionally treated as an independent coupling constant is fully determined within the MQ framework by the invariant scalar θ_si. The same scalar defines the electromagnetic demarcation n_L = 276/θ_si, where n_L is the count of fundamental lengths l_f between a system and a reference center of mass. Together, Eq. (104) and the definition of the electromagnetic demarcation establish that the electromagnetic interactions corresponds to a fixed realization of θ_si at a specific count scale, while ∇_μT^μν=0 (Eq. 78) gives the covariant conservation law satisfied by the corresponding stress–energy tensor.

3.5.4.2 Gravitational Realization: Mapping-Dependent Realization

Gravitational behavior is governed by the same invariant scalar θ_si, but its realization differs fundamentally from the electromagnetic case in that it is not associated with a fixed interaction scale. Instead, gravitational observables emerge through mapping-dependent realization over a relational separation characterized by the count distance n_L.

In MQ, no independent gravitational constant is introduced at the level of the action. Gravitational coupling arises from the normalization of Internal Frame count structure under the Frames mapping, together with the scaling behavior induced by relational separation. This leads directly to a mapping-dependent realization of the gravitational coupling ([9], Appx. C, D)

where 2Q_Ln_L encodes the Informativity differential and n_L denotes the count separation between the realized system and the associated center of mass. This expression follows directly from the MQ scaling relations.

Unlike electromagnetic interactions, which are characterized by a fixed interaction scale n_L = 276/θ_si, gravitational behavior admits realization over arbitrary count separations. Consequently, the effective gravitational coupling inherits an explicit dependence on n_L. The conventional gravitational constant is recovered only in the asymptotic limit of large count separation. Taking the upper-count limit yields

demonstrating that the macroscopic gravitational constant is itself a derived quantity, fixed entirely by the MQ fundamental measures. This formulation also makes clear that gravitational coupling in MQ is not a universal constant in the conventional sense, but a realization-dependent quantity determined by θ_si, the fundamental measures l_f, m_f, t_f, and the relational count distance n_L.

Accordingly, the apparent constancy of G in classical physics reflects the large-scale limit of the theory, while deviations at shorter count scales arise naturally from the underlying count geometry. This replaces the notion of a fundamental gravitational constant with a derived, scale-dependent quantity emerging from first principles MQ structure.

3.5.5 Comparison with Classical Formulations

Classical physics treats electromagnetism and gravitation as fundamentally distinct theoretical structures. Electromagnetism is formulated through Maxwell’s equations and the associated electromagnetic constants. Gravitation is formulated through Einstein’s field equations and the gravitational constant G [1,2]. In each case, the theory is introduced with its own primitive structure. This separation is part of the standard framework and is reflected both in the mathematics and in the independent status granted to the corresponding constants.

Within MQ, the comparison is different because these descriptions do not begin with separate primitives. As established in Secs. 3.5.1 and 3.5.2, both are resolved from the same Frames mapping, the same fundamental measures, and the same invariant scalar θ_si. The contrast with the classical picture can therefore be stated precisely.

(i) Independence vs. derivation of constants

In the classical description, constants entering electromagnetic and gravitational theory are independent inputs. In MQ, by contrast, measurable quantities are resolved from the common count-based structure defined by the fundamental measures l_f, m_f, t_f, the associated count structure n_L and θ_si. Equation (95) makes this explicit for the inverse fine structure constant. The point is not that every constant has already been rederived in this subsection, but that the framework no longer grants them independent foundational status.

(ii) Distinct vs. unified geometric structure

In GR, geometry is primary. In classical electromagnetism, the theory is formulated on a spacetime background through a gauge field description. MQ does not begin with either as a primitive structure. The Internal Frame, System Frame, and Frames mapping supply a single underlying kinematic structure from which different physical descriptions are realized. Thus, the classical duality of metric and field descriptions is replaced by a single relational substrate.

(iii) Scale separation vs. scale-dependent realization

Classically, electromagnetism and gravitation are distinct across all scales by construction. In MQ, the realized description depends on count separation. Electromagnetic interactions are tied to the fixed electromagnetic demarcation introduced in Eqs. (14–16) and expressed in Sec. 3.5.4 through n_L = 276/θ_si, whereas gravitational quantities are resolved relative to n_L, the count distance to a center of mass. The distinction is therefore scale-dependent and count-dependent, not foundational. This is consistent with the role assigned earlier to count distance and contraction structure, while avoiding the introduction of any new derivation here.

(iv) Parameter proliferation vs. parameter reduction

Many unification programs extend the standard framework by adding dimensions, fields, or new independent parameters [3–8]. MQ proceeds in the opposite direction. It reduces the number of independent inputs by requiring that physical quantities be expressed through the same invariant count structure. In this respect, MQ does not merely re-describe two classes of physical behavior. It restricts admissible physical descriptions by denying them separate primitive origins.

(v) Conceptual implication

The conceptual implication is that the classical distinction between electromagnetism and gravitation is retained as an effective difference in realization, but not as a difference in foundation. In MQ, the distinction reflects how the Frames mapping resolves the underlying count structure in the System Frame under different conditions, not the existence of two unrelated primitive interactions.

3.6 Observable Structure of Coupling and Frame-Dependent Differentiation

A physically meaningful unification of gravitation and electromagnetism requires that each phenomenon be expressed in terms of a common set of physically distinct quantities whose definitions are independent of the phenomena being compared. The comparison must isolate the terms responsible for differences between the interactions such that these differences correspond to well-defined physical effects rather than independent coupling assumptions. To establish this, we consider the relationship between the gravitational constant G and the electric constant ɛ₀. Given that the electric and magnetic constants satisfy ɛ₀ = 1/(u₀c²), it is sufficient to compare G and ɛ₀.

Using the MQ representations for G and ɛ₀ (Appx. F), we arrange both expressions such that dimensional terms appear explicitly and in comparable form, preserving the decomposition into fundamental measures and dimensionless factors,