General Relativity as an Emergent Geometry of Discrete Measurement

Jody A. Geiger

Informativity Institute

Chicago, IL USA 60605

Editorial Board Member for Scientific Reports/Nature

jodygeiger@informativity.org

ORCID: 0000–0001–5389–0447

ABSTRACT

This work presents a first principles derivation of the constituent terms of Einstein’s field equation within the Measurement Quantization (MQ) framework. General relativity successfully encodes gravitational dynamics through a continuum field equation, yet its central objects—the metric, curvature tensor, coupling constant, and cosmological term—are introduced at the level of the continuum rather than derived from a more primitive structure. MQ addresses this gap by replacing the assumption of a pre–existing spacetime manifold with a kinematic construction in which all observables arise through the Frames mapping between the discrete Internal Frame and the relational System Frame. Starting from a count–based primitive ontology defined by fundamental measures and conservation relations, we construct a single action functional and corresponding MQ variational principle. Coarse–graining of conserved count structure yields an emergent System Frame metric, while discrete second–difference relations induce curvature. The stress–energy tensor is obtained variationally from the same action, and conservation follows from mapping–induced symmetry structure without independent postulate. The gravitational coupling and cosmological term are derived from mapping–consistent count scaling and global normalization constraints, respectively, rather than introduced phenomenologically. These results assemble into a field equation structurally equivalent to Einstein’s equation in the macroscopic limit, but with all constituent terms arising from a unified kinematic substrate. The formulation therefore provides a reduction of the metric, curvature, coupling, and cosmological contributions to a single underlying action, establishing a coherent first principles origin for gravitational dynamics within the MQ framework.

Keywords: Measurement Quantization, general relativity, cosmological constant, fundamental measures, Planck units, foundations of physics

1. INTRODUCTION

1.1 The conceptual gap in Einstein’s field equation

Einstein’s field equation [1] provides a remarkably successful continuum description of gravitation, encoding the relation between geometry and matter through

where g_μν defines spacetime structure, G_μν encodes curvature, T_μν represents matter–energy, and the constants G and Λ (Appx. A.7) set the strength and large–scale behavior of the interaction. This equation is uniquely constrained by general covariance and local energy–momentum conservation, and its structure can be derived from the Einstein–Hilbert action [1,2] under standard variational principles. However, this derivation operates entirely within a continuum framework in which the metric field and associated geometric objects are specified at the level of the continuum rather than realized from a more primitive construction.

This leads to a well–recognized conceptual limitation: while the field equation is dynamically consistent and empirically successful, its constituent terms are not reduced to a more fundamental underlying structure. The metric g_μν is postulated as a differentiable field on a manifold, the Einstein tensor G_μν is constructed from this field through differential geometry, and the coupling 8πG/c⁴ and cosmological term Λ enter as independent parameters fixed by observation rather than derived from underlying structure. In this sense, the theory specifies the evolution of geometric structure without a kinematic account of the origin of the metric, coupling normalization, or cosmological contribution.

From a foundations perspective, this gap is distinct from questions of quantization or unification. It concerns the absence of a derivation of the geometric and coupling sectors from a more primitive description of physical measure. Existing approaches – ranging from geometric unification attempts [3,4] to emergent–gravity programs [5–8] – have explored whether spacetime and its dynamics can arise from more primitive structures, but a derivation of all terms in Eq. (1) from a single kinematic construction remains unresolved.

The objective of the present work is to address precisely this point. Within the Measurement Quantization (MQ) framework [9], the aim is not to modify the macroscopic form of Einstein’s equation, but to derive each of its constituent terms from a single kinematic construction based on the Internal Frame, the System Frame, and their connecting Frames mapping, in accordance with the Frames Principle (Appx. A). In this formulation, the metric, curvature tensor, coupling constant, and cosmological term are developed from the coarse–grained realization of conserved count structure under the Frames mapping, thereby providing a first principles origin for the Einstein field equation.

1.2 Prior programs in the literature

The question of whether the gravitational field equation can be reduced to a more primitive structure has a long history, with several distinct programs attempting to unify or derive its geometric content. Early efforts focused on extending the geometric framework of General Relativity (GR) itself. In the Kaluza–Klein construction [3,10], a higher–dimensional metric is introduced such that the five–dimensional Einstein–Hilbert action,

yields both the five–dimensional Einstein field equation and Maxwell’s equations under dimensional reduction [10]. Here R₍₅₎ denotes the Ricci scalar [11] of the five–dimensional metric. This construction demonstrates that gravitational and electromagnetic dynamics can be embedded within a single geometric action but does so by extending the dimensional structure rather than providing a reduction of the primitive geometric inputs.

Weyl’s earlier program [4] sought unification within four dimensions by extending the notion of metric compatibility. In Weyl geometry, the connection satisfies

introducing a gauge field A_λ identified with electromagnetism. While conceptually influential as an early gauge–based unification attempt, the resulting non–metricity leads to non–integrable length transport, preventing agreement with observed matter behavior and therefore precluding a viable physical realization.

More recent approaches have shifted toward gauge–theoretic and emergent descriptions of gravity. In gauge formulations [12,13], gravity is treated as a connection theory associated with local symmetry groups, while emergent–gravity programs attempt to derive spacetime dynamics from statistical, thermodynamic, or quantum information principles. A common criterion across these efforts is whether the full structure of GR – including diffeomorphism invariance, equivalence principle behavior, and the field equation – can be realized without introducing independent postulates. In many cases, aspects of this structure are reproduced, but a complete reduction of all constituent terms to a single underlying construction remains unresolved.

The present work is distinguished from these programs in that it does not extend dimensionality, alter metric compatibility, or enlarge gauge symmetry, but instead seeks a reduction of the constituent terms to a count–based kinematic construction defined through the Internal Frame, the System Frame, and their connecting Frames mapping, consistent with the Frames Principle.

1.3 What counts as a successful first principles presentation

A first principles derivation of the gravitational field equation requires more than reproducing its continuum form. The central requirement is that each constituent term arise from a well–defined primitive structure governed by a single action. In particular, the metric, curvature, coupling, and cosmological contributions must not be independently introduced, but must instead emerge from a unified construction in which their interrelations are uniquely fixed a priori. The stress–energy tensor must follow from variation of the same action that governs the geometry, and its conservation must arise from mapping–consistent symmetry and admissible realization structure rather than from an external condition. Finally, the complete field equation must be recovered in the macroscopic limit without the insertion of additional parameters or postulates.

These requirements reflect the standard criteria applied across geometric, gauge–theoretic, and emergent approaches to gravity, where the essential question is whether the Einstein field equation can be derived rather than assumed. Within this context, the MQ framework is constructed to satisfy these conditions through a single kinematic structure. Its primitive content consists of the discrete count domain of the Internal Frame and its realization in the System Frame through the Frames mapping, consistent with the Frames Principle. No metric or geometric structure is specified at the primitive level; only the discrete count domain, adjacency relations, conservation of count, and admissibility conditions on the Frames mapping are taken as primitive. All observable quantities, including those appearing in the field equation, arise only after application of the mapping.

To make this structure explicit, the following subsections develop the mathematical content of the Frames Principle and its consequences for the emergence of measurable quantities.

1.3.1 The Internal Frame, System Frame, and Reference Frame

The MQ framework distinguishes three interrelated constructs through which physical relations are defined. The Internal Frame is a discrete domain in which physical relations are expressed solely in terms of integer counts n_Land n_T. The System Frame is the realized representation in which these counts are mapped to continuous, measurable quantities. The Reference Frame provides the operational context within the System Frame in which measurements are compared and interpreted and does not constitute an independent primitive structure.

The Frames Principle asserts that physical observables do not reside in any one of these structures independently. Rather, they arise from the mapping-induced relation between the discrete structure of the Internal Frame and its continuous realization in the System Frame, with measurement defined through mapping-consistent relational encoding relative to a Reference Frame. This replaces the conventional assumption that observables are defined directly within a continuum manifold or coordinate system. In MQ, measurable quantities are therefore relational observables obtained through the Frames mapping acting on conserved count structure rather than attributes defined on a pre-existing geometric domain.

Fig. 1. Count of l_f describing realized separation in the System Frame, with discrete count evaluated in the Internal Frame. Modified from [b].1.3.2 Right–Triangle Construction and Count Geometry

The count structure underlying the Frames Principle is made explicit through the right–triangle construction shown in Fig. 1. In this representation, Side a defines the reference count and is fixed to a value of unity, establishing the normalization scale for all comparisons. Side b corresponds to the Internal Frame count n_L. When multiplied by l_f, it defines the discrete count separation associated with an interaction event within the count domain.

Side c, the hypotenuse of the triangle, represents the realized separation in the System Frame. The hypotenuse corresponds to the mapping-consistent realized separation functional in which the Internal Frame count contribution is expressed together with the residual count contribution Q_L, such that the realized separation is given by the count combination n_L+Q_L (Appx. A.8). The term Q_L therefore represents the residual count contribution, measured in units of l_f, arising from the transition between discrete count structure and continuous realization.

This construction yields a count-resolved relation consistent with the right–triangle configuration, where the orthogonality of Side a and Side b reflects the independence of reference normalization from the discrete count structure, while the hypotenuse encodes realized separation through mapping-consistent normalization. The hypotenuse therefore represents measurable separation as a composite of these independent count contributions together with the mapping-induced remainder.

The role of this representation is not illustrative but operational. It defines the count structure through which MQ expressions are constructed. The separation measured in the System Frame is therefore a mapping-normalized realization of conserved count rather than a direct projection of discrete count and necessarily incorporates the residual contribution Q_L. This correction encodes the discrete–continuous mismatch inherent in the Frames mapping, and it is this mismatch that ultimately gives rise to the geometric structure developed in subsequent sections.

1.3.3 Emergence of the Discrete–Continuous Mapping

The mapping between the Internal Frame and System Frame introduces a necessary structural effect arising from the incompatibility between discrete count structure and continuous representation. This effect is encoded through the realized separation structure involving n_L and the residual contribution Q_L, which together quantify the mapping-induced deviation between count-defined separation and realized measure. Because a discrete count cannot be represented exactly within a continuous domain, the mapping induces a correction to measured separation.

This correction is purely kinematic. It does not depend on relative motion and is therefore distinct from conventional relativistic contraction effects [14,15]. Its magnitude depends on count scale and diminishes as counts increase, ensuring agreement with continuum descriptions at macroscopic scales. The mapping thus defines a scale-dependent deformation of measured distance intrinsic to the transition between discrete and continuous representations.

A second effect arises from the evolution of count structure under the internal update index A. With each increment of the update index, the realized relational region undergoes a residual count loss Q_L relative to the Internal Frame reference scale. Because this loss occurs below the reference scale, it manifests as a progressive displacement in realized separation. Under the Frames mapping, this mapping-induced displacement contributes to the curvature realized in the System Frame.

This kinematic origin of curvature can be expressed directly through the mapping structure. Such that

(Sec. 2.7), the gravitational coupling follows from the same count-based construction

and represents the mapping-induced contribution associated with the Informativity differential (Appx. A; [9], Appx. D). Here θ_sidenotes the mapping-consistent scalar normalization governing count realization. Although often written using the symbol θ_si, this quantity is context dependent. In specific limiting constructions it admits an angular interpretation ([9], Appx. A), while more generally it corresponds to a normalized half-momentum scale in System Frame units ([9], Appx. Q) and is dimensionless in the Internal Frame ([9], Appx. AX). These representations are fixed by the same underlying mapping normalization and therefore share a single invariant magnitude. A detailed treatment of this identification and its unit structure is given in Appx. F. Importantly, the contraction term is dominant at small count separations and diminishes with increasing separation, recovering classical gravitational behavior in the large-scale limit, consistent with a 3.94σ concordance (Appx. H) against the 2018 and 2022 CODATA recommended values [a].

The significance of this result is structural. Both curvature and coupling arise from the same mapping-induced relation between discrete and continuous descriptions. No independent geometric or dynamical sector is introduced. This establishes the mapping-consistent origin of both curvature and coupling within a single kinematic construction, providing the basis for deriving all terms of the field equation from the MQ action.

1.3.4 Single–action structure and dynamical closure

Because the MQ framework is constructed at the level of first principles, the resulting continuum expressions take forms already familiar from classical theory. The field equations themselves are therefore not altered in their final representation. What changes is the origin and definition of their constituent terms. Rather than being introduced independently at the continuum level, each component emerges from the Frames mapping acting on the underlying count structure. The novelty of the framework therefore lies not in modifying the final form of the equations, but in establishing that all contributing quantities – geometric, dynamical, and cosmological – arise from a common kinematic origin.

To make this origin explicit throughout the remainder of the paper, quantities realized through the Frames mapping are identified by the upright superscript label sys. This notation denotes a System Frame realization obtained from the underlying count structure and should not be interpreted as an independently postulated continuum quantity. Thus, symbols such as

, H^sys, Λ^sys, and related observables refer specifically to quantities derived through realization of the count configuration under the Frames mapping. The superscript sys therefore serves a dual role: it distinguishes realized System Frame quantities from their corresponding Internal Frame descriptions and identifies their derivational origin within the MQ framework.

Accordingly, whenever a quantity appears with the superscript sys, its continuum form should be understood as the realized representation of an underlying count-based configuration. In particular, metric structure, curvature, stress–energy, coupling, and cosmological contributions are not specified independently, but emerge from the same underlying count relations associated with the right-angle realization structure of Fig. 1. The continuum description is therefore interpreted as a realized representation of a more primitive count-based structure rather than as the starting point of the theory.

With this interpretation established, the dynamical content of the theory is governed by a single action functional defined over the count structure. Denoting the dynamical field by Φ, defined over the coarse-grained realization of the count structure, the action takes the form

where the Lagrangian is constructed from mapping–induced quantities and contains no independently specified gravitational coupling or sectoral decomposition. The MQ variational principle then requires

from which all admissible dynamics follow (see Appx. A for the definition of the MQ variational principle and Appx. C for its implementation within the MQ count functional framework).

The metric emerges through coarse-graining of the count structure under the Frames mapping, yielding a realized metric

in the System Frame without independent specification. Variation of the same action with respect to this emergent metric defines the stress–energy tensor by

Invariance of the action under diffeomorphisms implies the conservation law (Sec. 2.6),

In the macroscopic limit, where the discrete count structure admits a smooth representation, variation of the realized action S^sys yields a field equation of the form

with the cosmological term Λ^sys determined by the mapping-consistent expansion structure developed in Sec. 2.8 and Appx. E. Equation (9) is structurally identical to Einstein’s expression (Eq. 1), but with each term now defined by the MQ construction: the metric from the realized separation functional, curvature from second-difference structure (Sec. 2.4), stress–energy from variational response (Sec. 2.5) and coupling from the same fundamental measures. The metric, curvature tensor, coupling constant, and cosmological contribution therefore arise from a single mapping-consistent kinematic construction. The distinguishing feature relative to standard GR is not a modification of the continuum equation, but a derivation of its constituent terms from the realized separation structure of the underlying count geometry.

2. METHODS

2.1 Primitive kinematic structure

The MQ formulation begins with a minimal discrete ontology in which no continuum geometric objects are assumed. The primitive structure consists of the Internal Frame, defined as the domain of integer-valued measurement counts, and the System Frame, defined as the realized relational structure in which these counts become continuous observables through the Frames mapping, consistent with the Frames Principle. All physical quantities arise only through this mapping. Neither a metric nor a differentiable manifold is taken as fundamental.

The Internal Frame is specified by a discrete count structure built from three irreducible measures (l_f, m_f, t_f), which define the minimal units of spatial separation, mass assignment, and temporal update, respectively. Physical evolution is represented not by differentiable trajectories but by ordered updates of conserved count on the configuration domain ω, indexed by the Internal Frame update parameter A, which provides a purely kinematic ordering of configurations and does not correspond to a physical time coordinate.

The primitive count state is therefore described by the integer-valued availability count assignment n_i(A) on relational cells i∈ω, together with the corresponding regional count

on finite relational regions ω^'⊂ω, as defined in Appx. A.

The dimensionless parameter θ_si enters at this primitive level as a fixed normalization scalar for count separation. In particular, the count n_L satisfies

so that n_L = 276/θ_si ([9], Appx. AC). This relation defines the fixed count scale associated with the electromagnetic demarcation (Appx. A), but its role here is restricted to specifying normalization of primitive count separation. The discrete count configuration is accumulated over relational regions ω^'⊂ω through the regional count

which is the basic conserved quantity from which later coarse-grained observables are constructed. At this stage no metric structure is introduced, and no continuum derivative acts on the count variables. The count-to-continuum relation enters only through the Frames mapping,

which assigns Internal Frame count configurations to elements of the realized System Frame relational structure. Under this mapping, conserved regional count provides the primitive input from which measurable separations are later realized in a manner consistent with conservation and closure constraints.

These elements – discrete count assignments n_i(A), regional count

, the fundamental measures (l_f, m_f, t_f), the normalization scalar θ_si, and the Frames mapping M – constitute the complete primitive kinematic specification. No independent metric field, connection, or coupling constant is introduced at this level. All continuum geometric structure arises only after realization of the Internal Frame into the System Frame, establishing the foundation for the subsequent variational construction.

2.2 The MQ action and variational principle

The dynamical content of the MQ framework is governed by a single action functional defined on the realized coarse-grained representation of the Internal Frame count structure under the Frames mapping. Let Φ denote the dynamical field encoding realized deviations of the conserved count configuration and let

denote the realized metric structure associated with the continuum description. The action depends on Φ, its realized derivatives, and the metric, with normalization fixed by the mapping parameter θ_si, which enters through mapping-consistent normalization rather than as an independently specified coupling parameter.

The defining statement of the theory is that physically realized configurations correspond to stationary configurations under admissible variations. These variations are taken with respect to both Φ and

, subject to the constraint that they arise from consistent realizations of conserved count under the Frames mapping, as required by the MQ variational principle.

Variation with respect to the dynamical field Φ yields the Euler–Lagrange equations in realized form,

which govern the evolution of the coarse-grained encoded availability field. The derivative operator appearing here is the realized continuum limit of the System Frame of mapping-induced Internal Frame adjacency relations rather than an independently postulated continuum structure. Indices μ,ν label realized System Frame coordinates x^μ.

Restricting attention to variations of the realized configuration with respect to the realized metric structure, the first variation of the action takes the form

which defines the stress–energy tensor

as the functional derivative conjugate to the realized metric structure. This definition is not postulated independently but follows directly from stationarity under admissible variations consistent with the Frames mapping constraints.

This construction enforces a strict single action structure. The degrees of freedom conventionally associated with matter are already contained within the encoded availability structure governed by the MQ Action. Consequently, no independently specified primitive matter action exists, and no fundamental additive decomposition of the form S=S_g+S_m is admitted. While such a separation may appear in the realized continuum limit of the System Frame, it does not correspond to independent fundamental sectors.

The full first variation, combining the realized Euler–Lagrange variation with the metric response, may therefore be written as

establishing both the field equations and the source tensor as consequences of a single variational structure. The measure

is the realized volume element over System Frame coordinates x^μ, induced by the mapping normalization acting on conserved count.

At this stage no independent geometric structure, coupling constant, or cosmological term has been introduced. The MQ Action provides a complete dynamical specification in terms of a single functional. The geometric sector – including connection, curvature, and its relation to the stress–energy tensor – emerges only in subsequent sections from the realized metric structure.

2.3 Emergent metric from coarse–graining

We begin with the fundamental observation that the Frames mapping assigns to each discrete configuration a continuous representation in which separations between configurations are preserved under coarse–graining. In the Internal Frame, separation is defined purely as a count relation through the right-triangle count construction illustrated in Fig. 1. The realized hypotenuse count is first given by Side c equal to n_L+Q_L, or equivalently c²=(n_L+Q_L)². Dimensional separation enters after System Frame realization, where the corresponding finite measurable separation is

where n_L denotes the Internal Frame count separation and Q_L is the mapping-induced residual contribution.

Under the Frames mapping, this realized separation admits a continuum representation. Let dl² denote the local differential realization of separation in the System Frame, where the symbol d denotes the differential operator acting on continuum coordinates. By construction, this quantity is expressible as a bilinear form in the coordinate differentials,

which defines the emergent metric tensor

as the continuum tensor reproducing the mapped local separation functional. The construction provides the local differential encoding of the realized separation defined in Eq. (16) in the continuum limit.

Within the local quadratic approximation, the components of

follow from the second variation of the realized quadratic separation functional l²(x) with respect to coordinate displacements under the Frames mapping. In standard differential form,

For the count-resolved separation considered here, the realized coordinates are obtained from discrete count increments through the fundamental length measure, so that

Because the continuum coordinates arise from discrete count increments through x^μ=n^μl_f, derivatives transform as ∂/∂x^μ=(1/l_f)∂/∂n^μ. Applying two coordinate derivatives therefore introduces a factor

. The quantity differentiated in Eq. (18) is not the linear separation l, but the quadratic separation l² from Eq. (16),

. Thus Eq. (18) evaluates the Hessian of the squared separation functional. Since

is constant with respect to the count variables, it factors outside the derivatives, while the remaining second derivative acts only on the quadratic count term

. The factor

introduced by the two coordinate derivatives therefore cancels the explicit

multiplying Eq. (16), leaving only the second derivative of the count-resolved quadratic separation. Substituting Eq. (16) into Eq. (18) therefore yields the MQ first principles realization,

Equation (20) provides the count-resolved realization of the metric. Its structure reflects the mapping-consistent realization of the right-triangle count construction, with unit normalized reference side, Internal Frame count separation n_L, and realized hypotenuse count separation n_L+Q_L. The defining count relation is therefore

This relation encodes the mapping between discrete count structure and realized System Frame separation. The metric in Eq. (20) therefore captures the second-order response of the mapping-realized separation functional to coordinate displacement, providing the continuum representation of the underlying count structure without introducing an independent metric field.

2.4 Curvature sector and Einstein tensor

Section 2.3 establishes the emergent metric structure as a derived geometric structure built from the emergent metric within the realized System Frame. Although

is a System Frame tensor field, its dependence remains entirely determined by the realized separation functional l=(n_L+Q_L)l_f. Through Eq. (19), continuum derivatives of the metric remain traceable to the underlying count structure. Using Eq. (20), differentiation of the metric gives

Resolving the coordinate derivative through Eq. (19),

where l_f is the fundamental length introduced in Sec. 1.5 and Eq. (19). Thus, derivatives of the metric correspond to higher-order variation of the realized separation functional under the continuum realization induced by Eq. (19).

The Levi-Civita connection is defined as the torsion-free metric-compatible connection associated with the emergent metric structure. Written in component form, it is

Using Eq. (23), the connection becomes

Substituting Eq. (23) into Eq. (24) expresses the connection entirely in terms of the count-resolved metric:

This expression shows that the connection is generated by third-order variation of the realized separation functional under coarse–graining.

The curvature tensor is obtained from the non-commutativity of the covariant derivative defined by Eq. (24). Using the same connection, the Riemann tensor is

The Ricci tensor is the contraction of the Riemann tensor,

The scalar curvature is then obtained by contraction with the inverse emergent metric,

The Einstein tensor is then given by

Substituting Eqs. (28) and (29) into Eq. (30) gives the explicit curvature reduction,

Equations (22–31) complete the reduction from the emergent metric structure to the Einstein tensor. Since the metric is fixed by Eq. (20), every term in Eqs. (22–31) is constructed from

, its inverse, and its derivatives. The tensor

is therefore fully determined by the metric generated from the realized separation functional.

The divergence-free identity follows from standard differential-geometric properties of the Levi-Civita connection constructed from Eq. (24). Its contraction gives

This identity completes the curvature construction and establishes the mapping-consistent origin of the geometric structure.

2.5 Stress–energy tensor from the MQ action

The source term in Einstein’s field equation must be derived from the same kinematic structure that produces the emergent metric and curvature. In the MQ construction, encoded matter content enters through the realized System Frame structure induced by conserved Internal Frame count under the Frames mapping. Accordingly, the tensor

is not introduced as an independently postulated source term but is derived as the response of the matter contribution to admissible variation of the emergent metric obtained in Sec. 2.3.

The relevant action is the matter contribution to the MQ variational structure. In the realized continuum limit this action is

Here the encoded matter content is defined on the System Frame through the encoded availability induced by the Frames mapping, and

denotes the emergent metric generated by the count-resolved separation functional. Variation is therefore taken with respect to the emergent metric structure induced by the mapping, rather than an independently postulated geometric background.

Variation of the matter action with respect to this emergent metric defines the stress–energy tensor,

This is the Hilbert stress–energy definition written for the emergent MQ geometry. The distinction from standard GR is not the variational form itself, but the origin of the metric and encoded matter content entering the variation. In MQ, both arise from the same mapping-induced realization structure.

Equivalently, the first variation of the matter action may be written as

This identifies

as the tensor conjugate to admissible metric variation in the realized matter structure. The same emergent metric entering the curvature construction in Sec. 2.4 therefore determines the source contribution.

The determinant variation entering the derivation is

Using the determinant variation in Eq. (36), the matter-action variation reduces to the standard metric-response form already given in Eq. (35). Comparison of Eqs. (34) and (35) fixes the stress–energy tensor as the variational response of the matter action.

For a local matter Lagrangian whose dependence on the metric is algebraic, this may also be written in explicit density form as

Equation (37) is not a separate definition, but the local density form of Eq. (34) under the stated assumption that the matter Lagrangian contains no independent derivatives of the metric.

To complete the reduction, the matter field entering L_m must be specified in terms of MQ primitives. In the present framework, this field is constructed from the System Frame realization of encoded availability derived from conserved Internal Frame count under the Frames mapping. The primary object is the encoded availability density ρ_sys(x), obtained by mapping-normalized realization of count structure over a relational region.

Normalization of ρ_sys(x) defines the normalized probabilistic description, but the action is constructed from the underlying encoded availability prior to normalization. A convenient representation of the realized matter field is obtained by introducing a phase field S(x), associated with the conserved current structure, and defining the wavefunction form

This representation encodes the mapping-induced density and current structure and does not introduce additional dynamical postulates. The phase S(x) encodes the conserved current induced by the Frames mapping, while ρ_sys(x) encodes realized measurement availability.

The matter Lagrangian L_m is therefore a scalar density constructed from ψ(x), its derivatives, and the emergent metric

. Its explicit form is constrained by the requirement that variation under the MQ variational principle reproduces the continuity and current structure induced by the mapping.

To make this dependence explicit, the stress–energy tensor may be written directly in terms of the realized matter field representation. Substituting Eq. (38) into the matter Lagrangian, the stress–energy tensor assumes the form

This makes explicit that dependence on matter degrees of freedom is carried through ψ(x), and therefore through the encoded availability density ρ_sys(x) and phase S(x).

Thus

is identified as the tensor describing how mapping-realized encoded matter content responds to variation of the emergent metric. The geometry construction (Sec. 2.4) and the matter structure are therefore built from the same underlying realization structure, ensuring that the field equation contains no independently postulated source term.

Notably, the wavefunction representation in Eq. (38) does not introduce quantum dynamics into the gravitational description. The quantities ρ_sys(x) and S(x) arise deterministically from conserved Internal Frame count under the Frames mapping, while ψ(x) serves only as a compact encoding of the resulting density and current structure. Probabilistic interpretation emerges only after normalization and does not modify the underlying variational construction.

2.6 Conservation law from diffeomorphism invariance

The conservation law follows from the symmetry of the same MQ Action that generates the field equations. No independent conservation postulate is introduced.

Because the total action is constructed as a scalar functional over the System Frame induced by the Frames mapping, it is invariant under admissible diffeomorphisms of the System Frame coordinates. Under an infinitesimal transformation x^μ→x^μ+ξ^μ, invariance of the action implies δS_sys=0.

Decomposing the total action into geometric and matter contributions, S_sys=S_g+S_m, and applying the MQ variational principle under admissible variations of the emergent metric

, the variation of the matter contribution yields

consistent with the Hilbert variational definition established in Sec. 2.5.

Under the diffeomorphism-induced variation of the metric,

substitution into the variation of the matter action gives

Integrating by parts and using the arbitrariness of ξ^μ within the admissible class of variations defined by the Frames mapping yields

. This condition follows directly from invariance of the MQ Action under admissible coordinate transformations.

The same conservation law follows from the geometric construction. The Einstein tensor constructed from the emergent metric satisfies the contracted Bianchi identity,

, which, together with the field equation assembled in Secs. 2.4–2.5,

and equivalently,

ensures consistency between curvature and source under admissible evolution.

Using c=l_f/t_f and the dimensional relation l_fm_f=2θ_sit_f, these expressions show that the coupling between curvature and source is fixed by the same mapping-induced normalization that defines the metric. The Einstein tensor

is determined by the emergent metric and its curvature, while

is determined by variation of the matter contribution to the MQ Action. The field equation therefore relates two quantities derived from the same underlying conserved count structure.

Within MQ, both relations arise from preservation of conserved count under admissible realizations of the Frames mapping. Covariant conservation is therefore the continuum realization of Internal Frame count conservation.

Importantly, because the stress–energy tensor is defined as the variational response of the matter action constructed from encoded availability (Sec. 2.5), its conservation reflects invariance of the same underlying count structure under admissible mappings. No additional symmetry or conservation principle is introduced beyond those already required by the MQ variational principle.

Thus, covariant conservation of stress–energy in the System Frame is identified as the macroscopic expression of discrete count conservation in the Internal Frame, completing the consistency chain linking mapping structure, variational dynamics, and observable conservation laws.

2.7 Derivation of the gravitational coupling

The gravitational coupling in MQ is not introduced as an external parameter but is fixed by dimensional closure of the fundamental measures together with the continuum normalization of geometric structure generated by the Frames mapping.

The derivation begins with the right-angle construction introduced in Fig. 1, where realized separation in the System Frame is represented by the hypotenuse

. Within this construction, the quantity Q_L represents the residual count generated by the mapping between the discrete Internal Frame and the realized continuum description. The ratio Q_L/n_L therefore provides the finite-count measure of mapping-induced deformation associated with the realized separation.

In the curvature-resolved regime, this deformation is realized geometrically as gravitational curvature in the System Frame. This identification follows directly from the count geometry of the right-angle construction and the realization dependent structure of mapping-induced deformation.

This identification is supported independently by experiment. As shown in Appx. H, analysis of the non-Lorentz length contraction predicted by the Informativity differential yields a concordance of 5.08σ with measurement. Likewise, comparison of the electromagnetic and macroscopic determinations of Newton’s constant reported in the 2010 and 2014 CODATA adjustments yields a concordance of 3.94σ. These results support the interpretation that the mapping-induced deformation represented by Q_L/n_L corresponds to a physically realized geometric effect.

At finite count, the MQ expression carrying the dimensions of Newton’s constant is therefore written as

which follows from the finite-count curvature measure together with the primitive dimensional scales and expansion normalization. The quantity Q_L/n_L supplies the local curvature contribution, while

compliments the radial scaling and c supplies the expansion normalization. The momentum normalization scale θ_si closes the dimensional structure.

Using the MQ fundamental relation θ_si=l_fm_f/2t_f, the finite-count curvature expression may be rewritten as

which places the result in a form directly comparable to the Newtonian curvature relation G/r². Here θ_si acts as the normalization parameter connecting radial expansion, momentum, and gravitational curvature through dimensional closure of the fundamental measures.

At this stage it is useful to distinguish the notation used for Newton’s constant. Within MQ, the symbol G denotes the upper-count limiting value, G=c³(t_f/m_f), which is independent of realization scale. When finite-count effects are retained, the gravitational coupling includes the realization-dependent factor 2Q_Ln_L, corresponding to the Informativity differential and associated non-Lorentz length contraction. In this case the quantity is written in italics as G_sys to indicate realization dependence.

Substituting r=n_Ll_f then

where all factors are expressed in terms of the primitive measures l_f, m_f, and t_f, together with the normalization scale θ_si. This isolates the explicit realization dependence of the finite-count gravitational coupling.

Taking the upper count limit removes dependence on discrete realization scale,

which is finite, realization independent, and consistent with the MQ Planck-scale relation G=c³t_p/m_p (Appx. B). The gravitational coupling is therefore fixed uniquely by the fundamental measures and carries the correct dimensions, [G]=L³/MT², without introducing an independent gravitational parameter.

Moreover, the geometric construction and the encoded matter content arise from the same underlying MQ framework and are realized on the same emergent System Frame metric. The geometric contribution defines the normalization of curvature through

while the matter contribution produces

through metric variation of the MQ Action. In this representation, Eq. (49) is the continuum form of the geometric contribution obtained after coarse-graining of the MQ Action.

Accordingly, the coupling between curvature and the stress–energy tensor is determined entirely by dimensional closure of the fundamental measures established by the upper count limit in Eq. (48). No additional normalization or adjustment is required. The gravitational coupling appearing in the field equation is therefore a direct consequence of the MQ construction.

2.8 Derivation of the cosmological term

The cosmological term in MQ arises from the global evolution of conserved count under the Frames mapping. In contrast to the standard approach, where Λ is introduced phenomenologically in the Einstein–Hilbert action, the MQ construction derives the scalar contribution directly from invariant properties of the expanding count geometry. The scalar contribution is expressed in terms of the fundamental expansion parameter θ_si (Appx. G) as

where this identification follows from the mapping-consistent scalar expansion definition and preserves normalization under the invariant expansion condition. The corresponding expansion rate is given by (Appx. D)

where H_sys denotes the System Frame realization of the expansion rate fixed by mapping normalization. This expression represents the unit-resolved form of the scalar expansion H=θ_si ([18], Eq. 61) under mapping-consistent normalization and a spatially flat cosmology (k=0) as required by the invariant expansion condition and the mapping-consistent realization of the metric. The implied expansion rate is in 3.35σ concordance with observational constraints (Appx. H), specifically the SPT-3G analysis [16], the Pantheon+ cosmological analysis [17], and the MQ Hubble tension analysis [18].

Within the standard FLRW parametrization, the MQ scalar may be identified with the cosmological density parameter for comparison with observational cosmology through

with

, where G includes the MQ-predicted non-Lorentz length contraction. This relation provides a comparison bridge to standard cosmological notation and does not constitute an independent derivation of the MQ scalar.

Combining Eqs. (50–52) yields

Thus, within MQ, the cosmological term is fully determined by mapping-consistent quantities and contains no independent scalar parameter. Unit conversions are retained explicitly for clarity of the derivation. This is discussed further in Appx. F.

The scalar therefore enters the gravitational description as a consequence of the expansion structure defined by θ_si and realized through H_sys. The normalization factors arising from the Frames mapping cancel identically in the construction, fixing the scalar contribution through the same closure conditions that determine the geometric and matter contributions.

This construction resolves the status of Λ within MQ. The cosmological term is not introduced at the level of the action nor fixed phenomenologically. Instead, it is a derived scalar determined by invariant properties of the Internal Frame count structure under the Frames mapping. Its value is fixed for a given realization of the mapping, while its origin is kinematic and tied to the expansion structure encoded in the mapping.

Although Λ_sys appears constant at the level of the realized field equation description, consistency with the conservation law derived from diffeomorphism invariance (Sec. 2.6), ∇_μT^μν=0, requires that any evolution in the realized scalar contribution be balanced by corresponding evolution in stress-energy under expansion.

The construction is therefore structurally closed: the cosmological term is determined within MQ by Ω_Λ, H_sys, and the FLRW density identification in Eq. (52), with its magnitude fixed by Eq. (53). Its embedding in the gravitational description follows from the mapping-consistent realization of the metric and stress-energy tensor developed in Secs. 2.3–2.6.

2.9 Assembly of the field equation

We now assemble the MQ field equation in terms of the fundamental measures (l_f,m_f,t_f) and θ_si, presenting its fully reduced form. At this stage, the metric is already established as an emergent, coarse-grained System Frame object induced by the Frames mapping. The curvature follows from this metric as developed in Secs. 2.3–2.4, while the stress-energy tensor is defined variationally with respect to it, ensuring a consistent geometric and dynamical structure. The gravitational coupling (Sec. 2.7) and the cosmological term (Sec. 2.8; Appx. E) have likewise been derived from the same fundamental measures. With these components fixed, the field equation follows by direct composition.

We begin by introducing the geometric tensor entering the MQ field equation. The curvature tensors entering the field equation are constructed from the emergent metric

derived in Sec. 2.3 using the standard Levi–Civita connection [19]. The Riemann tensor

[20,21] is formed from this connection, with the Ricci tensor [22] and scalar curvature given by

and

, respectively. Indices are raised and lowered using

. Then,

which expresses the geometric tensor entirely in terms of the primitive measures and the normalization scalar θ_si. Because

is fixed by the coarse-grained realized count structure defined in Sec. 2.3 from Eq. (20),

the connection, Riemann tensor, Ricci tensor, scalar curvature, and Einstein tensor

defined in Eq. (54) are all determined by successive derivatives and contractions of the same mapping-realized count structure.

The stress-energy tensor entering the field equation is defined variationally by

This tensor arises from the count–geometry coupling under the MQ variational principle. The cosmological contribution is likewise fixed within MQ. From Appx. E, particularly Eqs. (E.1–E.7),

Using Eq. (48) together with the dimensional closure relation l_fm_f=2θ_sit_f (Eq. B.1), the Einstein field equation coupling becomes

where the reduction uses the MQ expression for the gravitational constant

, and the fundamental expression l_fm_f=2θ_sit_f, eliminating all intermediate scale dependence. Substituting Eqs. (54–58) then yields the assembled MQ field equation

For compact notation, define the MQ coupling by

Then, Eq. (59) may be written equivalently as

This completes the first principles assembly of Einstein’s field equation within MQ. The curvature term is constructed from the emergent System Frame metric, the scalar term is fixed by the mapping-consistent expansion structure, the source term is obtained variationally from the matter action, and the coupling is reduced to the fundamental measures. No independent Newton constant, cosmological parameter, or external stress-energy source is introduced.

Because Λ_sys depends on the expansion scale A_U (Appx. A.6; Appx. D), Eq. (61) implies that any realized evolution in the scalar contribution must be accompanied by compensating evolution in

, while G remains fixed kinematically by (l_f,m_f,t_f,θ_si). The geometric tensor

likewise cannot arbitrarily adjust to offset such evolution.

This behavior is consistent with the covariant conservation law, ∇_μT^μν=0, which requires evolution in the realized scalar contribution to be balanced by corresponding evolution of the stress-energy content. Because all remaining quantities in Eq. (61) are fixed by kinematics, the required decrease in

implies dilution of energy density under expansion. Within MQ, this excludes static realizations and implies expansion from a finite realization origin consistent with the realized count structure.

3. RESULTS

3.1 Term–by–term reduction of Einstein’s equation

The assembly of the MQ field equation in Sec. 2.9 establishes more than a formal correspondence with GR, providing a term-by-term reduction of Einstein’s field equation in which each constituent term is realized from a common kinematic origin defined by conserved Internal Frame count under the Frames mapping. This resolves the central gap identified in Sec. 1.1: while standard GR is dynamically complete, its constituent terms are introduced at the continuum level rather than derived from a shared primitive structure. Within MQ, Einstein’s equation is not modified at the macroscopic level but reinterpreted structurally, with all terms emerging as mapping-induced realizations of conserved count. In contrast to the Einstein–Hilbert formulation, which derives field equations from a specified action once the metric, coupling, and matter content are defined, MQ derives the origin of these terms (Table I) prior to assembling the field equation, establishing closure within the MQ framework in which no term entering Eq. (61) is independently introduced.

The metric

arises as the coarse-grained realization of the separation functional induced by the Frames mapping, ensuring consistency with its definition as a relational encoding rather than a fundamental manifold. Curvature follows as the continuum realization of discrete second-difference structure induced by adjacency relations in the Internal Frame. This ensures that geometric curvature reflects accumulated mapping-induced deformation. The stress-energy tensor is defined variationally from the MQ Action, preserving its standard functional role while eliminating the need for an externally supplied source term. The gravitational coupling is fixed by dimensional closure of the fundamental measures (l_f, m_f, t_f, θ_si), and the cosmological contribution is determined by the expansion parameter θ_si through the associated scalar density fraction Ω_Λ.

Several longstanding structural questions regarding Einstein’s equation are addressed by this reduction. First, the origin of geometry is recast as mapping-induced rather than fundamental, with spacetime structure arising from the realization of discrete count relations in the System Frame. Second, conservation of the stress-energy tensor follows directly from mapping-consistent invariance of the MQ Action, ensuring that geometric and matter contributions are jointly constrained by a single variational structure. Third, the gravitational coupling is no longer introduced as an independent phenomenological parameter. Fourth, the cosmological contribution is fixed by the same kinematic structure governing expansion, addressing the ambiguity in interpreting Λ as either vacuum energy or a geometric constant.

The completed Eq. (61) therefore represents a closure statement: the full Einstein field equation is obtained without introducing an independent spacetime manifold, independent matter source, free gravitational coupling, or phenomenological cosmological parameter. All terms are derived from the MQ Action under the constraints of the Frames mapping, and each admits a mapping-consistent realization in the System Frame. In this sense, the apparent equivalence between MQ and GR at the level of field equations reflects a deeper reduction in which the continuum description emerges as a coarse-grained limit of discrete kinematics.

This result does not alter the empirical content of GR. Instead, it provides a first principles account of why Einstein’s equation takes its observed form. The standard field equation remains the macroscopic continuum description, while MQ specifies the underlying structure from which that description emerges. Any measurable departure from GR must therefore originate from count-resolved realizations encoded in the mapping and become observable only outside the large-scale continuum regime.

3.2 Recovery of the standard macroscopic form

With the MQ field equation assembled in Sec. 2.9, recovery of the standard macroscopic form consists in demonstrating that System Frame quantities converge to their general relativistic counterparts in the continuum limit defined by large count separation under the Frames mapping. No additional assumptions are required beyond those already encoded in the MQ Action and its variational structure.

The metric

, defined as the coarse-grained realization of the separation functional, satisfies the smoothness and differentiability conditions required for continuum geometry when evaluated over regions large compared to the fundamental count scale. In this regime, the discrete adjacency structure admits a consistent coarse-grained limit, and the metric becomes operationally equivalent to a continuum spacetime metric at observational resolution.

The curvature follows accordingly. The Einstein tensor

, constructed as the continuum realization of the discrete second-difference operator on the Internal Frame adjacency structure, reduces to the standard curvature tensor of differential geometry. Its divergence-free property,

, is recovered as the continuum realization of mapping-induced conservation constraints, consistent with the contracted Bianchi identity [22].

The stress-energy tensor

, defined variationally from the MQ Action, retains the standard conservation law,

, as a direct consequence of mapping-consistent invariance.

The coupling constant K_sys=8πG/c⁴, derived from dimensional closure of the fundamental measures, numerically converges to the empirically inferred gravitational constant in the macroscopic regime in which measurements probe separations far above the fundamental count scale.

The cosmological contribution likewise retains its standard form. The scalar term

, fixed by the mapping-consistent expansion structure, enters the field equation identically to the general relativistic cosmological term. At macroscopic scales, any residual dependence on discrete structure is suppressed below observational sensitivity, leaving only the fixed effective value of Λ_sys.

The recovery is therefore exact at the level of the continuum field equation. Any deviation between MQ and GR arises only through count-resolved corrections encoded in the Frames mapping, which are parametrically suppressed in the large-scale limit relevant to classical gravitational phenomena.

This establishes that Einstein’s field equation is realized as the continuum limit of the MQ kinematic framework rather than introduced as an independent postulate, while preserving its empirical content in the macroscopic regime.

3.3 Internal consistency checks

The recovery of the macroscopic field equation in Sec. 3.2 establishes structural equivalence with the Einstein formulation. The present section verifies that this equivalence is dynamically and kinematically self-consistent by demonstrating that the internal constraints of GR arise from the Frames mapping acting on conserved count.

The geodesic limit may be stated equivalently through the standard point-particle action evaluated on the emergent metric constructed in Sec. 2.3.

or, for an affine parameter

Variation with fixed endpoints gives

These relations establish that extremization of the mapping-realized action yields the geodesic equation associated with the emergent metric and its metric-compatible connection derived from

. Since

is constructed in Sec. 2.3 and the connection is fixed in Sec. 2.4, this identifies localized motion in the coarse-grained System Frame with extremal curves of the realized metric.

The first requirement is the geodesic limit, which in MQ follows from conservation. In GR, free motion follows from covariant conservation of the stress-energy tensor together with the contracted Bianchi identity. In MQ, the corresponding result follows from conservation of adjacency flux in the Internal Frame, which under mapping normalization induces a conserved current in the System Frame, yielding

as the continuum realization of count conservation. For localized distributions, this conservation condition implies geodesic motion in the emergent metric.

The second requirement is compatibility between conservation and the field equation. In GR, the identity ∇^μG_μν=0 ensures consistency with energy–momentum conservation. Within MQ, this identity arises from invariance of the mapping-normalized metric under admissible transformations, so that the Einstein tensor constructed from

satisfies the same divergence-free condition.

The third requirement is invariance structure. The Einstein equation is invariant under diffeomorphisms of the spacetime manifold. In MQ, the continuum analog arises from automorphisms of the mapping domain that preserve normalization, repeatable realization, and closure. These transformations induce a symmetry on realized configurations equivalent to diffeomorphism invariance in the continuum limit.

The fourth requirement is frame-scaling consistency. The Frames mapping induces a normalization relating Internal Frame count representation to its System Frame realization. In the homogeneous limit this normalization reduces to a constant background factor, while more generally it is fixed by mapping constraints over the relational region. Physical observables must remain invariant under this transformation. Substitution into the MQ field equation shows that all normalization factors cancel identically, yielding a representation independent relation.

Taken together, these results establish that geodesic motion, conservation, invariance, and scaling consistency arise from preservation of conserved count under admissible mappings, reproducing the internal consistency structure required of GR in the continuum limit without introducing additional dynamical assumptions.

4. DISCUSSION

4.1 What is genuinely reduced to first principles

The preceding sections establish that the macroscopic field equation obtained from MQ is structurally equivalent to the Einstein equation, while its constituents arise from a single underlying kinematic construction. The present section isolates the structures reduced to first principles within this construction and distinguishes them from the elements that remain primitive.

The fundamental statement is not a field equation but the variational condition on the MQ Action, defined entirely on the Internal Frame configuration domain and its admissible Frames mapping. At this level, no metric, connection, curvature tensor, or stress–energy tensor is introduced. The primitive content consists only of the discrete configuration domain, adjacency-defined transport, conservation of count, and admissibility constraints on the mapping, including normalization, repeatable realization, and holonomy-free closure.

From this structure, the objects treated as fundamental in GR are derived. The metric

arises as the coarse-grained realization of the separation functional induced by the mapping, with its scaling fixed by normalization of conserved count rather than coordinate choice. The associated connection and curvature tensors follow from the continuum limit of the induced second-difference structure encoded in adjacency relations, yielding the standard geometric form without additional postulates.

The stress–energy tensor is likewise not introduced phenomenologically. It is defined variationally from the matter contribution to the MQ Action through variation with respect to

, with the matter sector expressed through encoded availability and its normalization. Geometry and source therefore arise from the same functional structure.

The gravitational coupling is reduced to the dimensional structure of the fundamental measures (l_f, m_f, t_f, θ_si), with its value fixed by mapping-consistent scaling relations. The cosmological contribution is determined by the MQ expansion sector through Ω_Λ, H₀, and the associated MQ cosmological contribution, rather than introduced as an independent phenomenological parameter.

Taken together, these results imply that the field equation is not fundamental but a derived consistency condition on the realization of conserved count under the admissible Frames mapping. Each term in the macroscopic equation is fixed by Internal Frame count structure and its realization in the System Frame.

What remains primitive is correspondingly minimal: the configuration domain, adjacency-defined transport, conservation of count, and the admissibility conditions defining the Frames mapping. These are not derived within the present framework.

By contrast, the continuum structures of GR – the metric, connection, curvature, stress–energy tensor, coupling constant, and cosmological contribution – are secondary. They arise as continuum representations of conserved count realized in the System Frame, consistent with the Frames Principle.

The reduction achieved is therefore structural. The continuum description of gravitation is retained in standard form, but its constituent elements are reinterpreted as emergent from a single discrete kinematic structure governed by the MQ Action.

4.2 Relation to GR

The relationship between MQ and GR is one of structural equivalence at the macroscopic level combined with a difference in foundational origin. The MQ framework does not modify the Einstein field equation within its established domain of empirical validity but reproduces its standard form as the continuum realization of a kinematic structure defined by the Frames mapping.

At the level of observable physics in the System Frame, gravitational dynamics are governed by the same field equation as in GR. The distinction lies in the interpretation of its constituent terms. In GR, the metric is postulated as a fundamental geometric field defined on a differentiable manifold, with curvature constructed intrinsically from that structure. In MQ, the metric

is a derived representation of the realized separation functional induced by the mapping. Curvature correspondingly arises from spatial variation of this separation functional.

The stress–energy tensor retains its operational role but differs in origin. In MQ it arises from variation of the matter contribution to the MQ Action, with matter content expressed through encoded availability and normalization under the Frames mapping. Geometry and source therefore originate from a common variational structure.

The gravitational coupling is likewise reinterpreted. In GR it is introduced empirically, whereas in MQ it follows from the dimensional structure of the fundamental measures and mapping-consistent scaling relations. The cosmological contribution is similarly derived from the MQ expansion structure rather than treated as a free phenomenological parameter.

These distinctions place MQ within the class of approaches in which gravitational dynamics emerge from a deeper substrate while preserving the empirical content of GR. In MQ, this substrate is explicitly kinematic and discrete, governed by conservation of count under the admissible Frames mapping.

An MQ description of expansion admits a homogeneous realization in which the separation functional defines a global background relation between System Frame and Internal Frame measures. This background fixes the normalization of expansion but does not determine local curvature, which remains governed by spatial variation of the realized separation functional.

In physical terms, this mapping-induced residual does not imply departure from spatial flatness of the homogeneous background realization. Rather, it introduces a normalization offset in the realized expansion rate while preserving the continuum geometric structure of the field equation. The MQ construction therefore predicts that a spatially flat background may coexist with a systematic mapping-induced offset in the realized expansion rate, providing a mapping-based interpretation of the Hubble tension [18].

Operationally, the homogeneous realization serves as a reference configuration fixing the relation between Internal Frame count evolution and System Frame expansion. Local gravitational dynamics arise from deviations from this background through spatial variation of the realized separation functional, ensuring that cosmological expansion and local curvature remain unified within a single separation functional while remaining dynamically distinct.

The significance of this formulation is twofold. First, the expansion scale follows from the same count structure that generates metric and curvature. Second, expansion normalization and local gravitational dynamics occupy distinct roles: the former fixes global normalization, while the latter governs local geometry.

4.3 Relation to prior unification efforts

The present construction is best understood in the context of prior unification programs seeking a common origin for gravitational and gauge structure. MQ shares with these approaches the objective of deriving gravitational dynamics from a deeper substrate while differing in both the nature of that substrate and the mechanism by which observable structure emerges through the Frames mapping.

Early unification efforts such as Kaluza–Klein theory [3] extend the dimensionality of spacetime, embedding gauge structure into a higher dimensional metric. In contrast, MQ introduces no additional dimensions. Instead, geometric and gauge related behavior arise from mapping-induced realization of count relations within a single kinematic domain.

Weyl’s unification program [4] similarly sought to incorporate electromagnetic structure into geometry through scale connection. In MQ, gauge behavior arises from normalization under the Frames mapping rather than from an independently postulated geometric connection.

Closer parallels exist with induced and emergent gravity approaches, in which the Einstein equation is treated as derived rather than fundamental. Sakharov’s induced gravity [5], Jacobson’s thermodynamic derivation [6], and related formulations [7,8] all treat gravitational dynamics as emergent. MQ aligns with this perspective in treating the field equation as derived but differs in that the underlying structure is neither statistical nor thermodynamic. It is kinematic and discrete, governed by the Frames Principle and conservation of count.

The distinction becomes explicit at the level of the action. In emergent approaches, one typically begins with an effective continuum action and interprets its origin through coarse-graining arguments. In MQ, the action is defined directly on the Internal Frame count domain, and continuum expressions appear only after normalization under the Frames mapping and coarse-graining. The Einstein–Hilbert and matter contributions are therefore interpreted as continuum representations of mapping-induced structure.

A further distinction lies in the treatment of scale. In higher dimensional and gauge-theoretic models, scale separation and coupling parameters are typically introduced independently. In MQ, scale dependence is governed by count separation within the Internal Frame, with distinct physical regimes arising from this dependence without introducing independent adjustable couplings.

The resulting position of MQ relative to prior work can therefore be stated succinctly: like geometric unification programs, it seeks a common origin for gravitational and gauge structure; like emergent-gravity approaches, it treats the Einstein equation as derived. It differs from both by locating this structure in a single discrete kinematic mapping between count and continuum representations.

4.4 Restricted uniqueness of admissible Frames mappings

The remaining question is whether the Frames mapping introduces hidden freedom. Within MQ, an admissible Frames mapping is not an arbitrary map from the Internal Frame to the System Frame. It must preserve conserved count, induce a normalized System Frame realization, support repeatable macroscopic records, and satisfy holonomy-free closure on all contractible reference cycles. A valid realization exists only on a referenceable universe domain U, where the mapping establishes the consistent reference relations required for observables. An admissible Frames mapping is therefore defined only on domains U satisfying the referenceability conditions given in Appx. A and developed through the realization structure of Appx. K.

Let M₁ and M₂ be two admissible Frames mappings on the same referenceable universe domain U. For every relational region ω^'⊂U, each mapping realizes the same conserved Internal Frame count structure into its corresponding System Frame relational image,

M_i:ω^'→Ω_i(ω^'), i = 1,2. (66)

For each admissible mapping M_i, the realized relational image is Ω_i=M_i(ω^')⊆R, where R is the System Frame realization structure. The images Ω₁ and Ω₂ do not denote physically distinct System Frames. They represent candidate realizations obtained from admissible Frames mappings prior to equivalence identification within the same realization structure R.

The realized separation functional is the mapping-normalized realization of conserved count,

, i = 1,2, (67)

where C(ω^') is the count functional fixed by conserved Internal Frame count and N[ω^'] is the mapping normalization functional defined in Appx. A.3. Both are determined by count conservation, invariance under admissible relabelings, and the global normalization and holonomy constraints imposed by the Frames mapping. Admissibility excludes dependence on System Frame coordinates or non-conserved Internal Frame structure.

For any relational region ω^', admissibility restricts normalization to a scalar action on the conserved count functional,

where λ(M_i) is a mapping-dependent normalization factor fixed globally by the requirement of consistent boundary-flux realization and holonomy-free closure on all contractible cycles, and the scalar form is restricted to admissible configurations satisfying these constraints. This scalar form follows from the requirement that N preserve additivity over disjoint relational regions and invariance under admissible relabelings, as required by the definition of the count functional. No region-dependent or configuration-dependent modification is admissible, as such dependence would violate conservation or induce path-dependent realizations.

Since ω^' is the same relational region in the same referenceable domain, conservation fixes C(ω^'). Admissibility then requires equality of realized separations,

Substituting Eq. (68) into Eq. (69) yields λ(M₁)=λ(M₂). Thus, the normalization factor is uniquely fixed across all admissible Frames mappings on U. The normalization is therefore not free but determined by admissibility constraints on the domain.

The closure constraint follows from the requirement that realized observables be independent of reference path. Because the Frames mapping defines System Frame relations through coarse-grained realization of Internal Frame count, consistency requires that all contractible reference cycles produce no residual mismatch. For every contractible reference cycle γ⊂U, admissibility requires

, i = 1,2, ∀γ⊂U. (70)

This condition enforces path-independence of realized observables on contractible cycles and follows directly from the definition of a referenceable universe domain.

Repeatable realization requires that physically admissible Frames mappings produce stable observables under repeated evaluation of the same relational region. Let

denote the ordered realized data associated with ω^', consisting of realized separation, contractible-cycle closure, and stable record state. For admissible Frames mappings on the same domain U,

This equation is the substantive uniqueness constraint. It states that admissible Frames mappings on the same referenceable domain produce identical realized observables for every relational region. Any remaining difference between M₁ and M₂ is therefore representational.

Define the realized equivalence relation induced by M_i as

Using Eq. (72), the two equivalence relations coincide,

. Therefore M₁ and M₂ induce the same quotient of relational regions into realized System Frame observables.

The induced map between admissible realizations is

This map is well-defined on equivalence classes induced by realized observables. Because Φ₂₁ preserves normalization, repeatable realization, and holonomy-free closure by Eqs. (68–70), it defines a frames automorphism. Hence M₂=Φ₂₁∘M₁, with

Equations (72–74) establish restricted uniqueness. On a fixed referenceable universe domain, admissible Frames mappings form a single physical equivalence class,

Every representative of [M]_U produces the same realized separations, record states, metric, curvature tensors, stress–energy tensor, gravitational coupling, and scalar contribution governing expansion. These quantities therefore depend only on the equivalence class [M]_U, not on the representative mapping.

This result removes the immediate concern that the field equation derivation depends on arbitrary mapping choice. The derivation depends only on the equivalence class [M]_U, not on a particular representative M. The local gravitational construction is therefore unique up to frames automorphism.

The remaining question is whether locally equivalent representatives can assemble into globally inequivalent realizations due to non-contractible structure. Let {U_α} be an admissible cover of a referenceable universe domain U, with each U_α satisfying normalization, repeatable realization, and holonomy-free closure on contractible cycles. From the restricted uniqueness result above, any two admissible Frames mappings on each U_α differ only by a frames automorphism,

On overlaps U_α∩U_β, admissibility requires the realized observables to agree. Therefore,

Φ_α∘M₁=Φ_β∘M₁ on U_α∩U_β, (77)

so the transition automorphism

acts trivially on all realized observables in the overlap. This trivial action holds only at the level of realized observables and does not imply identity on the Internal Frame configuration domain.

Thus, for every contractible chain of overlaps,

This cocycle condition ensures consistency of gluing across overlaps within the referenceable domain. It states that locally unique admissible Frames mappings assemble into a unique global equivalence class whenever no residual automorphism accumulates around contractible overlap cycles.

Nontrivial global freedom can arise only from non-contractible cycles or mapping defects. Let D denote the mapping defect class and let γ_* denote a non-contractible reference cycle in U∖D. The only possible global invariant is defect holonomy,

Two globally admissible mappings are therefore physically equivalent if and only if they have the same realized local observables and the same defect-holonomy data,

The equivalence is defined over all relational regions and all non-contractible reference cycles in U∖D. This yields the global classification result in restricted form: admissible Frames mappings on a referenceable universe domain are classified by their local automorphism class together with their non-contractible defect holonomy data,

Since the local class [M]_U is uniquely determined by observable and normalization constraints through Eqs. (71–75), any remaining inequivalence is entirely global and is exhausted by defect data. No additional invariant is introduced unless it corresponds to a distinct mapping defect class or non-contractible holonomy.

The mapping used in the field equation derivation is therefore not arbitrary. On each referenceable universe domain, admissible Frames mappings are locally unique up to frames automorphism and globally classified by non-contractible defect holonomy data. Accordingly, any remaining non-uniqueness belongs strictly to global topological structure and not to the construction of the Einstein field equation within MQ.

4.5 Falsifiability and scope

The derivation presented here is constrained by internal consistency conditions that follow directly from the Frames Principle and the MQ variational principle. These conditions define explicit operational failure modes at the level of realized observables and therefore establish the falsifiability of the construction.

A necessary requirement is that the realized separation functional admit a well-defined continuum limit under admissible coarse-graining. This limit must remain stable under admissible refinement and independent of coarse-graining choice within a fixed referenceable universe domain. The emergence of the differentiable metric structure

depends on the convergence and stability of this limit. Failure of convergence, or dependence on admissible coarse-graining choice, would invalidate the emergence of the metric and falsify the geometric construction.

A second requirement is that the mapping-induced variational structure produce a unique correspondence between deformation and gravitational coupling. Predictive determinacy requires that all admissible realizations consistent with the MQ variational principle yield the same gravitational coupling in the continuum limit on a fixed referenceable universe domain. Any residual freedom would imply that the coupling is not uniquely fixed by the underlying kinematic structure.

A third requirement is that conservation of count in the Internal Frame map to a conserved current in the System Frame without additional dynamical structure. The resulting continuity relation must arise solely from normalization under the Frames mapping and preserved count flux. Failure of conservation under admissible mapping would invalidate the derivation of realized conservation laws.

A fourth requirement is that the macroscopic field equation governing

be recovered without independent postulates beyond the MQ variational principle and admissible Frames mapping. The Einstein field equation derived in Sections 2–3 must emerge solely from the MQ variational principle and mapping structure in the continuum limit of admissible realizations. Dependence on externally imposed geometric identities or independently specified matter structure would invalidate the derivation.

These conditions define the internal falsifiability of the framework. Each condition constitutes an independent necessary requirement, and failure of any single condition produces a breakdown of the derivation. Their satisfaction establishes derivational closure.

In addition to these internal criteria, the present formulation has two scope limitations that bound its present completeness without affecting the field equation derivation.

First, admissible Frames mappings have been shown to be locally unique up to frames automorphism on any fixed referenceable universe domain, while global realizations remain classified by the mapping defect class and associated non-contractible holonomy data established in Section 4.4. The remaining open problem is therefore not local uniqueness but the exhaustive classification of globally inequivalent defect-bearing domains and their associated non-contractible holonomy classes. These global classes correspond to physically distinguishable realizations only insofar as their defect-holonomy structure induces observable differences in large-scale realized relational structure. They therefore provide a falsifiable aspect of the framework through possible observational signatures in global topology, large-scale anisotropy, or expansion structure. Such signatures, if present, would appear as departures from mapping-consistent predictions in large-scale angular correlations or distance–redshift structure.

Second, the present work derives the classical gravitational field equation in the continuum limit but does not provide a complete dynamical unification of relativistic and quantum field degrees of freedom within the MQ framework. In this context, “classical” refers to the continuum field equation governing the emergent metric

. A complete unification requires extension of the same mapping-consistent action structure to relativistic quantum dynamical degrees of freedom. Such extensions are necessary to establish full dynamical unification but lie beyond the scope of the present work.

APPENDICES

APPENDIX A: DEFINITIONS OF KEYWORDS, TERMS AND SYMBOLS

Notes

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

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

Keywords

Internal Frame

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

System Frame

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

Frames mapping

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

Frames Principle

The Frames Principle asserts that all physically meaningful observables arise from the Frames mapping between the Internal Frame and the System Frame. No observable quantity is intrinsic to the Internal Frame alone. All measurement is relational and defined through the mapping. This principle replaces the assumption of a pre–existing spacetime manifold with a kinematic origin of geometry.

MQ Action

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

MQ variational principle

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

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

Informativity differential

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

Metric differential

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

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

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

Electromagnetic demarcation

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

Count separation

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

Realization manifold

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

Mapping–induced deformation

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

Fixed interaction scale

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

Curvature–resolved regime

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

Terms

A.1 Primitive definitions

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

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

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

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

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

A.2 Discrete Internal Frame primitives

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

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

= 0. The mapping between the Internal Frame update index A and System Frame time t is established only through the Frames mapping M, and is generally nontrivial. Consequently, A should be interpreted as a kinematic sequencing parameter intrinsic to the discrete count substrate, rather than as a temporal observable.

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

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

, or equivalently,

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

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