Galactic Reconstruction Involving No Constants: Heuristic Emergent Structure

Galactic Reconstruction Involving No Constants: Heuristic Emergent Structure Jon Loomis Abstract Gravitational Reconstruction Involving No Constants: Heuristic Emergent Structure (GRINCHES) presents a structural analysis of galactic gravity based solely on mass and distance, without assuming time, physical constants, or spacetime curvature. Starting from the mass–distance field S(R) = ∫dM D,we show that the functional form and environmental variation of the gravitational coupling follow directly from structure through dimensional consistency and invariance arguments. Weak-field phenomena including effective gravitational strength, rotation behavior, time-delay effects, and lensing arise at the correct order of magnitude without invoking dark matter or metric geometry. The results indicate that a significant portion of gravitational phenomenology is encoded in mass–distance structure prior to the introduction of time-normalized constants. 1 Introduction The GRINCHES framework (Galactic Reconstruction Involving No Constants: Heuristic Emergent Structure) isolates the minimal structural content required for gravitational phenomenology by beginning without assumed physical constants, time-dependent quantities, or metric structure. Only mass, distance, and geometry are treated as primitive inputs. Quantities normally regarded as fundamental constants are not deferred or suppressed, but instead introduced only when they arise from structural relationships. The objective is not to propose a competing theory of gravity, but to expose the structural backbone common to any formulation that successfully reproduces weak-field observations. The motivation for this approach follows from the observation that gravitational phenomena are typically modeled using constants whose numerical values depend explicitly on units of time, even though those units are defined by physical processes known to vary with gravitational environment. In particular, the SI definitions of both cand Gembed the second as a primitive unit, despite extensive observational evidence for environment-dependent clock rates and coordinate variability of cin gravitational fields. When such constants are introduced at the outset, standard formulations obscure the extent to which gravitational behavior is determined by mass–distance structure alone. GRINCHES reverses this logic by removing all unit-dependent assumptions and examining which gravitational features emerge prior to the introduction of time or dynamics. At the core of the framework is the mass–distance integral S(R) = ∫dM D, evaluated directly over the full mass distribution of a galaxy without invoking shell-theorem reductions. Unlike the enclosed-mass constructions employed in Newtonian and relativistic dynamics, the integral defining S(R)retains contributions from exterior mass. Such cancellations arise only after equations of motion are specified and are therefore not intrinsic properties of mass–distance structure itself. The resulting quantity defines a global structural field encoding how mass is distributed relative to position, with distance entering directly within the integral rather than appearing only after integration. From S(R), a scale-dependent structural coefficient emerges that reproduces the form of weak-field gravitational couplings once appropriate correlations are identified. When evaluated across galactic radii, this procedure yields an effective gravitational coupling that remains locally consistent while exhibiting systematic departures under assumptions of global invariance. These departures arise from structural constraints rather than from modifications to dynamical laws. Once a method for obtaining Gfrom structure is established, variations in quantities normally attributed to weakfield time dilation begin to manifest as structural scalings. These effects arise without reference to a time variable, metric curvature, or spacetime warping. Instead, they follow directly from dimensional consistency between S(R), which carries units of mass per length, and the empirically inferred gravitational coupling, once a single structural 1 reference point is fixed. This indicates that a substantial portion of gravitational phenomenology is encoded directly in mass–distance structure. The gravitational coupling obtained within the GRINCHES framework is a coordinate quantity derived prior to the introduction of time or dynamical evolution and is therefore not assumed to be universally constant. Constancy is not imposed unless it emerges structurally, and in this case it does not. GRINCHES forbids the assumption of constants a priori; such quantities are introduced only when they are obtained structurally and only in the form in which they arise without invoking time-based assumptions. Proper-time transformations may be applied to reexpress results in invariant form, but these transformations do not alter the underlying structural relations from which observational inferences are drawn. By construction, GRINCHES concerns itself solely with quantities that are observable. Throughout this work, “derived” refers to quantities obtained uniquely from mass–distance structure, up to the unavoidable unit translation required to express results in time-normalized observational coordinates. No dynamical laws or constants are assumed. GRINCHES does not claim completeness, nor does it seek to replace existing dynamical theories. Instead, it isolates the structural environment within which those theories operate. Its purpose is to distinguish which observed gravitational features arise from mass–distance structure alone, which require additional dynamical assumptions, and where conventional formulations implicitly rely on unit conventions rather than invariant physical content. 2. Methods The objective of this analysis is not to introduce new physical laws, but to identify which aspects of galactic gravitational behavior can be recovered strictly from structure. The methods employed here isolate mass–distance relationships while excluding time-based assumptions, unit conventions, and metric prescriptions unless they arise as direct consequences of the structural construction. The following subsections summarize the guiding principles, deliberate exclusions, and structural framework used throughout this work. 2.1 Interpretive Principles The following statements are adopted only as interpretive constraints on the final results. They do not enter directly into the construction of the structural model: 1. Gravitational influence increases with mass and decreases with separation. 2. Time dilation is observed in the presence of gravity and motion. 3. In General Relativity, the rate of time is predicted to decrease without bound near compact objects such as black holes, consistent with strong gravitational time-dilation effects inferred observationally. 4. Mass and energy are related by E=mc2. 5. Comparisons with established constants are allowed, but no constant or measurement beyond mass, distance, or direction may be used unless they are obtained strictly from mass, distance, and direction. These principles serve only to constrain which qualitative behaviors are considered physically admissible when interpreting the structural results. They do not influence the form of the structural fields themselves. 2.2 Deliberate Exclusions To prevent time-based or unit-based assumptions from shaping the structural construction, the following elements are intentionally excluded unless a direct structural correspondence is established: 1. The gravitational constant Gand the speed of light cdo not appear in any structural integral or intermediate quantity. 2. No spacetime metric, field equation, curvature tensor, or interval is assumed. 3. No relativistic corrections, kinematic effects, or time-dilation formulas are used in defining structural quantities. 4. No shell-theorem reductions or enclosed-mass shortcuts are employed; all integrals are evaluated explicitly over the full mass distribution. 2 5. No model fitting or adjustment of mass, radius, or other physical quantities to match expectations is permitted; only transformations implied directly by structurally derived quantities are allowed. Dark matter is excluded from the initial construction of the model, but not from the analysis as a whole. Its omission at the outset is a consequence of methodological restrictions rather than an assertion about its physical existence. In standard gravitational modeling, dark matter is introduced to resolve a mismatch between observed kinematics and predictions derived from time-normalized dynamical laws. Its origin is therefore inherently kinematic: it enters only after velocity, acceleration, and time have been assumed. The present work proceeds in the opposite order. Its aim is to determine what structure arises from mass and distance alone, prior to the introduction of kinematics or any time-based normalization. Including dark matter at this stage would therefore be logically inconsistent, as it would embed a corrective term whose motivation presupposes precisely the quantities this framework intentionally withholds. Importantly, the geometric reconstruction employed here is deliberately sensitive to missing structural components. If an additional, non-baryonic contribution were fundamentally required to reproduce large-scale gravitational behavior, its exclusion would produce severe and unmistakable failure of the model at the structural level. The absence of such failure indicates that the dominant large-scale behavior is already encoded in the mass–distance geometry. Dark matter is reintroduced later for comparison, in order to examine how its inclusion modifies or supplements the resulting structure once kinematic interpretation is permitted. A similar restriction applies to spatial warping. Curvature of space is not assumed at the outset because its conventional formulation requires both time and a specific transformation framework relating coordinates across regions with different temporal rates. These ingredients are not available prior to the construction of the geometric field and would therefore amount to an imposed solution rather than an emergent one. Spatial warping is thus excluded not because it is denied, but because it requires a defined cause and mechanism. Should a structural reason for warping be identified within the mass–distance framework—together with a well-defined means by which geometry responds—it can be incorporated naturally at a later stage. Until such a mechanism is established, introducing curvature would conflate geometric structure with interpretive assumptions derived from time-dependent formalisms. These exclusions ensure that the resulting quantities depend only on spatial structure and mass distribution. 2.3 Structural Length Unit All distances are expressed in a structural length unit numerically equivalent to the meter but not operationally defined through the SI relation. This avoids the implicit introduction of time through spatial units. The structural construction therefore remains purely spatial, with no reference to clocks, durations, or propagation speeds. 2.4 Structural Galaxy Model The galaxy model consists of a Hernquist-like bulge and an exponential disk with surface density Σ(R) = Σ0e−R/Rd, truncated at Rdisk,max.The total baryonic mass is initially set to 1.85 ×1041 kg, with no contribution from dark matter or additional components. This choice serves solely to anchor units within the structural integrals and plays no dynamical role in shaping the resulting fields. The enclosed bulge mass is Mbulge(< R) = Mbulge R2 (R+abulge)2, and the enclosed disk mass is Mdisk(< R) = 2π∫min(R,Rdisk,max ) 0 Σ(R′)R′dR′. Kinematics are not included in the model, but can be derived. 2.4.1 Irrelevance of the Absolute Mass Scale Although the total baryonic mass is normalized to a convenient value, the absolute mass scale is not essential to the structural quantities evaluated here. Disk galaxies with different total masses but similar radial surface– density profiles differ only by a global scaling factor. The structural integrals employed in this work are homogeneous under simultaneous rescalings of mass and length and therefore preserve functional shape across galaxies of different sizes. 3 Consequently, it is the spatial distribution of mass, rather than its absolute magnitude, that governs the structural behavior examined in this analysis. Any inference of absolute mass scales later in this work arises only after structural quantities are conditionally mapped to derived constants. 2.5 Mass–Distance Structural Field The central structural quantity is the mass–distance field S(R) = ∫dM D, evaluated over the full disk. For a point located at radius Rin the midplane, the explicit form is S(R) = ∫Rdisk,max 0∫2π 0 Σ(r′)r′ √R2+r′2−2Rr′cos θ+ϵ2 soft dθ dr′, where Σ(r′)is the surface density, r′is the radial coordinate of the mass element, and D=√R2+r′2−2Rr′cos θ+ϵ2 soft is the Euclidean distance between the evaluation point and the mass element. A softening length of ϵsoft = 100 pc is used solely for numerical stability. The differential mass element is dM = Σ(r′)r′dr′dθ, so S(R)represents the weighted sum of dM/D over the disk. This scalar field encodes the complete mass–distance structure of the galaxy and serves as the foundational quantity from which all subsequent structural scaling relations and weak-field correspondences are derived. 2.6 Structural Identification of the Gravitational Coupling During the structural analysis, no gravitational coupling was assumed. However, once the mass–distance field S(R) had been constructed, it became necessary to determine whether any structural quantity could be associated with the empirically observed gravitational constant. After several unsuccessful attempts to relate S(R)directly to Gthrough algebraic scaling, dimensional analysis, or enclosed-mass constructions, a formal diagnostic expression was examined: ∫GM rdG =G2M 2r. This expression was not introduced as a physical law and carries no immediate dynamical interpretation. Its role was purely diagnostic: to test whether the numerical structure of the observed gravitational constant reflects information about the local mass–distance environment. When evaluated using empirically measured values of Gand known masses and radii, the dimensionless quantity G2M/(2r)was found to closely match the fractional contributions of corresponding masses to the structural field S(R)at the Solar radius. This agreement held across multiple scales, including terrestrial, Solar, and galactic systems. Motivated by this correspondence, the inversion G2Mtot 2R0 = 1 = K2 was adopted as a structural identification condition rather than as a physical constraint or dynamical law. Solving for the gravitational coupling yields Gcoord =K√2 √S(R0). The factor Krepresents the fractional contribution of the mass and distance used in the structural identification of G. Under the unity condition, it specifies the mass–distance ratio required for the observed gravitational coupling. Its numerical value is fixed to unity and carries units of m5/2kg−1/2s−2, supplying the dimensional content required to map the time-free structural field Sinto empirically defined units. No freedom is introduced by K, and no fitting or adjustment is performed at any stage. This step constitutes the first and only point at which time units enter the framework, and they do so solely through dimensional translation into empirically defined units, not through any assumption of temporal dynamics, clock behavior, or time evolution. 4 Once this identification is established, the inversion permits a family of conditional structural inferences. Holding any two of the quantities {G, M, R}fixed determines the third. For example, specifying a reference radius allows the total mass required for a given coordinate value of Gto be inferred, or conversely, fixing the mass determines the structural radius associated with a measured coupling. These inferences are conditional and structural in nature; they do not constitute predictions of global mass content and are not independently verifiable at present. This identification does not assume universality of Gand does not treat it as a fundamental constant. Instead, it establishes a correspondence between the observed gravitational coupling and the local mass–distance environment once a single structural anchor point is fixed. Usage of the standard constant Gstd and the structurally derived quantity Gcoord is explicitly distinguished throughout to avoid ambiguity. 2.7 Structural Treatment of Gravitational Lensing In its original application to galactic interiors, the mass–distance field S(R)was implicitly vectorized by symmetry: the test point is embedded within a surrounding mass distribution, and contributions arrive from a broad range of directions. In this regime, directional information is effectively averaged by the geometry of the system, and the scalar field alone is sufficient to encode the relevant structural influence. Outside such embedded configurations, however, the scalar field Sbecomes insufficient on its own. Because S integrates dM/D without directional discrimination, it cannot distinguish between environments that differ radically in geometry but yield comparable integrated values, such as a compact mass concentration viewed at distance and a large-scale underdensity or supervoid. In these cases, the scalar field retains influence over long distances without encoding whether that influence is coherently directed or geometrically diffuse. For extragalactic lensing configurations, a mechanism is therefore required to prioritize directional structure within the mass–distance environment itself. The angular weighting introduced below serves this role by suppressing isotropic contributions and enhancing geometrically coherent ones, allowing the structural influence to decay appropriately with distance while preserving the purely mass–distance character of the construction. To account for this geometric asymmetry without introducing metric curvature, null geodesics, or relativistic field equations, the structural influence of mass elements is weighted by their angular distribution relative to the propagation direction of the ray. This is implemented by constructing a local directional coherence measure derived solely from the mass–distance environment. At each evaluation point along a ray path, the vector directions from the ray position to each mass element are computed and combined using mass–distance weighting. The resulting weighted mean direction defines a local preferred structural axis. The root-mean-square angular deviation of individual mass directions from this axis provides a scalar measure of angular dispersion. The gradient of this angular dispersion with respect to impact parameter defines a dimensionless weighting function that modulates the contribution of the structural field along the ray path. This weighting suppresses isotropic contributions and enhances regions where the mass distribution produces coherent directional focusing. The resulting deflection arises from the structural gradient alone and does not rely on assumptions of spacetime curvature or light following geodesics. This procedure preserves the purely structural nature of the analysis while allowing scalar mass–distance fields to produce anisotropic focusing when the mass distribution occupies a limited angular extent. The method is applied uniformly across all lensing simulations presented in this work. Formally, let ˆ uidenote the unit vector pointing from the ray position to the ith mass element, and let wi=dMi/Di denote its mass–distance weight. The weighted mean direction is defined as ˆ umean =∑iwiˆ ui ∥∑iwiˆ ui∥. The angular dispersion is then quantified by the root-mean-square deviation θrms =[∑iwiarccos2(ˆ ui·ˆ umean) ∑iwi]1/2 . This quantity provides a scalar measure of the directional coherence of the local mass–distance environment. An intuitive analogy is the difference between standing on the surface of the Earth versus standing on a baseball. In the former case, support is provided coherently from a wide range of directions, while in the latter it is confined to a narrow angular region. Although the integrated support may be similar, the geometric distribution of that support differs substantially. Likewise, identical scalar values of Scan correspond to structurally stable, directionally averaged environments or to highly anisotropic configurations, depending on the angular distribution of surrounding mass. 5 2.8 Coordinate and Proper Time Quantities Within the GRINCHES framework, the distinction between coordinate and proper quantities is not treated as an independent physical input for interpretation. Observations are made exclusively from the observer’s proper time, and any inference about distant regions is necessarily expressed in coordinate terms relative to that frame. Accordingly, all quantities discussed in this work are interpreted as observationally inferred, regardless of whether they are formally labeled as coordinate or proper in other formalisms. As an illustrative example, in the case of Shapiro delay, light propagation does not proceed as it does in the observer’s local proper time. A delay in signal arrival is observed, independent of the interpretive framework used to attribute that delay to coordinate effects, proper-time effects, or metric structure. Within GRINCHES, such distinctions do not alter the observable outcome and therefore do not enter into the structural analysis. This interpretive constraint is crucial within GRINCHES because the foundational structural model does not incorporate time in any capacity. All quantities are evaluated as a static structural snapshot derived from mass and distance alone. While transformations to proper time or alternative temporal parametrizations may be performed for bookkeeping or comparison with existing formalisms, such transformations introduce no new observable content and have no effect on the interpretation of results. Within this framework, time serves only as a representational layer applied after structural relations have been established. 3. Data and Numerical Implementation The structural model is evaluated on a one–dimensional radial grid extending from the galactic center to Rdisk,max. The bulge and disk profiles follow the parameters specified in Section 2, with the total baryonic mass normalized to 1.85 ×1041 kg. No dynamical quantities, time standards, or metric structures enter the computation; only spatial mass–distance relationships are evaluated. 3.1 Radial Sampling and Structural Integrals The enclosed mass M(R)and the mass–distance field S(R)are evaluated on the same uniformly sampled radial grid to ensure internal consistency. The structural field is obtained from the double integral S(R) = ∫Rdisk,max 0∫2π 0 Σ(r′)r′ √R2+r′2−2Rr′cos θ+ϵ2 soft dθ dr′, which incorporates contributions from the full two–dimensional disk. A softening length of ϵsoft = 100 pc is applied uniformly to prevent numerical divergences at small separations. Because the analysis is structural rather than dynamical, these integrals constitute the complete computational procedure required to obtain the foundational structural fields used throughout this work. No gravitational potential, metric coefficient, or time–dependent quantity is computed at this stage. Coordinate–level quantities that resemble effective couplings or weak–field parameters appear only in later sections and are obtained algebraically from S(R) rather than introduced as independent assumptions. 3.2 Reproducibility The numerical implementation is fully specified by: •the radial grid, •the exponential disk parameters Σ0and Rd, •the bulge mass and scale radius, •the softening length ϵsoft, •the structural integrals defined above. Any model constructed with these ingredients reproduces the structural fields to within numerical precision, independent of any dynamical interpretation or choice of physical constants. Table 1 presents representative values of the enclosed mass and structural field across the disk. It is important to emphasize that the initial structural galaxy model used here is not assumed to be final or physically complete. Its purpose is to expose the raw mass–distance structure and identify emergent correlations prior to the introduction of any dynamical interpretation. 6 Tab. 1: Representative structural outputs from the baryonic galaxy model. All quantities are expressed in SI mass units (kg) for reference; the field S(R)carries units of kg/m. R(kpc) Menc(< R)(kg) S(R)(kg/m) 24.4701 ×1040 1.2681 ×1021 49.2481 ×1040 9.6776 ×1020 61.2817 ×1041 7.3294 ×1020 81.5127 ×1041 5.6575 ×1020 10 1.6515 ×1041 4.4923 ×1020 12 1.7314 ×1041 3.6760 ×1020 15 1.7906 ×1041 2.8614 ×1020 20 1.8234 ×1041 2.0835 ×1020 25 1.8322 ×1041 1.6432 ×1020 30 1.8357 ×1041 1.3596 ×1020 Once effective couplings such as Gare structurally identified from S(R), these relations impose admissibility conditions on mass distributions. Galaxy models that fail to satisfy these conditions are structurally excluded, while those that remain consistent are retained for further analysis. This procedure does not involve fitting observational data or adjusting parameters to match expectations; it represents a structural consistency filter applied after the relevant relations have been derived. This approach mirrors standard practice in gravitational modeling, where solutions are constrained by governing relations once those relations are established. In GRINCHES, however, the governing relations themselves are obtained from mass–distance structure rather than postulated a priori. 4. Results The structural model yields a set of radial functions derived entirely from the baryonic mass distribution and Euclidean spatial relationships. No gravitational constants, time-dependent quantities, or dynamical assumptions enter the construction. All results reflect only the intrinsic structural relationships encoded in the mass–distance field and its algebraic consequences. 4.1 Enclosed Mass Profile The enclosed baryonic mass increases steeply within the central kiloparsec due to the bulge and grows more gradually throughout the exponential disk. Beyond several disk scale lengths, the enclosed mass approaches its asymptotic baryonic total. This mass profile defines the structural background against which all subsequent quantities are evaluated. 4.2 Mass–Distance Structural Field The primary structural quantity is the mass–distance field S(R) = ∫Rdisk,max 0∫2π 0 Σ(r′)r′ √R2+r′2−2Rr′cos θ+ϵ2 soft dθ dr′, which sums contributions from all mass elements weighted by inverse distance. The field is largest at small radii, where mass elements lie closer on average, and decreases monotonically with R. Because the disk extends over many scale lengths, the decline in S(R)is gradual: outer regions continue to receive non-negligible contributions from the extended two-dimensional mass distribution. As a result, S(R)falls off much more slowly than any enclosed-mass or point-mass approximation would suggest. This field represents the cumulative mass–distance structure of the galaxy and serves as the foundation for all subsequent quantities. 7 Fig. 1: Three-dimensional visualization of the structural mass–distance field S(R)for the toy galaxy model, defined by S=∫dM/D. The central red marker indicates the position of Sagittarius A*. The field encodes the cumulative structure of the bulge and disk and serves as the basis for all structural scaling relations introduced in later sections. 8 Fig. 2: Structural mass–distance field S(R)computed from the baryonic disk and bulge model. The primary y– axis (solid curve) shows the structural field S(R), while the secondary y–axis (dashed curve) shows the corresponding mass–distance ratio M/D expressed in physical units (kg/m). This plot highlights the direct relationship between the structural field and the ordinary mass–distance distribution, while illustrating the extended contribution of the exponential disk at large radii. 4.3 Multi-Galaxy Structural Consistency Evaluating the same structural integral at the midpoint between the Milky Way and Andromeda yields finite contributions from both galaxies. Although the combined magnitude is significantly lower than within either galaxy individually, it remains structurally well-defined. This demonstrates that the mass–distance formalism generalizes directly to multi-galaxy and group-scale environments without modification. 4.4 Relative Structural Field For relative comparisons within a single galaxy, it is useful to define the dimensionless ratio ˜ S(R) = S(R) S(8.2 kpc), which expresses how strongly the surrounding baryonic mass influences a location at radius Rrelative to the Solar position. Values above unity indicate regions structurally closer to more mass than the Solar radius, while values below unity indicate regions structurally farther. The normalized field highlights two key features of the galaxy’s structural profile: 1. ˜ S(R)declines slowly with radius due to the extended two-dimensional disk; no shell-theorem suppression occurs. 2. S(R)is highest at the center of the galaxy not simply due to central mass concentration, but because the center is structurally positioned to receive contributions from all mass elements in the disk. Shell–theorem constructions eliminate contributions from exterior mass; the structural field does not, and therefore the central value reflects the cumulative influence of the entire system. These properties distinguish ˜ S(R)from the standard gravitational potential −GMenc/R, whose steep radial decline follows from the enclosed-mass approximation and time-normalized formulations. ˜ S(R)therefore represents the underlying structural profile that any time-normalized potential must inherit. 9 Tab. 2: Relative gravitational coupling extracted from the structural mass–distance field. Gext,bar is obtained using baryonic matter only, while Gext,tot includes an extended halo component. Both cases use the same structural extraction formula. R(kpc) Baryonic + Halo Baryonic Only 2.0 0.975 0.637 4.0 0.986 0.750 6.0 0.994 0.868 8.2 1.000 1.000 10.0 1.004 1.106 15.0 1.012 1.377 20.0 1.017 1.606 25.0 1.022 1.805 30.0 1.026 1.982 From the perspective of the present framework, extended mass components act as structural compensators rather than as a consequence of modified dynamics. Their effect is to modify the mass–distance environment such that a time–normalized gravitational coupling appears approximately invariant when interpreted through local measurement conventions. In this sense, within the present framework, extended mass functions as a structural medium rather than as a dynamical agent. No parameters are adjusted to enforce this behavior. The extracted coupling is determined entirely by the mass– distance field. Any resemblance to modified–gravity phenomenology arises automatically from disk geometry and extended mass distributions, not from changes to the gravitational law itself. This result recasts the role of dark matter in purely structural terms. Enforcing a globally invariant gravitational coupling in a universe with non-uniform mass–distance structure requires the introduction of additional mass. Whether this mass corresponds to new matter, effective time normalization, or environment-dependent calibration remains an interpretive question rather than a structural one. A direct structural implication of the GRINCHES framework is that, under the assumption of a globally invariant gravitational coupling, environments with significantly lower mass–distance structure would require the inference of additional effective mass to satisfy the structural identification condition M D=2K2 G2, which evaluates to M D≃4.49 ×1020 kg/m in SI units using the locally measured value of G. This value arises directly from the structural relationship and does not depend on any particular galactic radius. In standard gravitational interpretations, satisfying this mass–distance requirement in under-dense environments — for example, within large voids in the cosmic web — would correspond to inferring a higher effective concentration of mass than is present in baryons alone. From the structural perspective of GRINCHES, this is not a dynamical claim about new matter, but a prediction about how assumptions of invariant coupling translate into inferred mass distributions when mass–distance structure is low. The same structural requirement applies in the opposite regime. In regions of high baryonic density, maintaining an invariant gravitational coupling would instead require a suppression of effective mass contributions relative to the local structural field. Thus, enforcing global invariance produces a symmetric structural prediction: extended mass must be inferred in low-density environments, while effective mass must be reduced or suppressed in high-density environments. In the specific case of the Milky Way toy model examined here, the transition between these two regimes occurs near the Solar radius (R0≃8.2 kpc), where the baryonic mass–distance field happens to satisfy the structural requirement M/D = 2K2/G2. This radius is not fundamental; it reflects the particular mass distribution of the model galaxy and would shift in systems with different structural profiles. This behavior is not pathological within the standard framework; it is an inevitable consequence of enforcing global invariance on a structurally heterogeneous system. Importantly, this implication does not rely on the failure of any existing observations. It arises solely from the mass–distance structure itself. If future observations instead support a systematic dependence of the gravitational coupling on structural environment, the need for compensatory extended mass would diminish accordingly. 16 6A.3 Universality of Gas a Consistency Constraint The gravitational constant Gis conventionally treated as universal because no statistically significant variation has been detected in local experiments, with typical constraints at the level of ∼10−5. It is therefore natural to regard universality as an empirically supported assumption. At the same time, the manner in which this universality is maintained is not made explicit, particularly given that Genters gravitational observables through time-based dimensions while other coordinate-dependent effects, such as Shapiro delay, are directly observed despite the speed of light also being treated as universally invariant. One may further note that if Gdepended weakly on structural environment through a time-related mechanism, existing experimental precision would be insufficient to resolve such variation in terrestrial or near-Earth measurements. For example, if such dependence scaled with weak-field gravitational time dilation, the associated effect on coordinate light speed would need to reach differences of order ∼103m/sto become detectable given the current experimental sensitivity to variations in Gat the level of ∼10−5. This suggests that the most favorable environments for direct tests would be strongly time-dilated regions, such as the Solar surface or the vicinity of compact objects. No gravitational environment accessible in the immediate vicinity of Earth produces time-dilation effects of this magnitude, and existing measurements are therefore confined to a very narrow structural regime. This limitation can be quantified directly. Near the Earth’s surface, the fractional gravitational time dilation between two clocks separated by a height hsatisfies ∆s s≃gh c2≈1.1×10−16 h(per meter), where gis the local gravitational acceleration. If, hypothetically, the gravitational coupling scaled with time as G∝s−2, the corresponding fractional change in Gwould be ∆G G≃ −2∆s s≈2.2×10−16 h(per meter), This expression represents a multiplicative scaling of Gwith relative time rate, not an additive offset; the absolute change would be ∆G≃(2.2×10−16 h)G. Reaching the current experimental sensitivity threshold ∆G/G ∼10−5would therefore require a vertical separation of order h∼10−5 2.2×10−16 ≈4.5×1010 m, corresponding to ∼4.5×107km, far exceeding the radius of the Earth and comparable to a substantial fraction of an astronomical unit. Accordingly, the absence of detected variation in Gin terrestrial or near-Earth experiments does not, by itself, place strong constraints on time-related dependence at the level implied by weak-field gravitational dilation. Beyond experimental sensitivity, however, the assumption of universality also entails a nontrivial consistency requirement once time-based assumptions are removed. The numerical value of Gmust remain invariant not only in local proper-time measurements, but also under comparison between observers situated in environments with substantially different time rates. For instance, consider an observer on Earth and an observer located just outside the Schwarzschild radius of a compact object. Each must recover the same value of Gin local experiments, and each must describe gravitational interactions in the other environment in a manner consistent with the same coupling. This consistency is not automatically guaranteed by coordinate-versus-proper time bookkeeping. Observable gravitational acceleration satisfies g∼GM r2, so any relative scaling of acceleration by a time-rate factor is algebraically equivalent to scaling GM itself. Propertime transformations alone do not uniquely fix how such scaling should be attributed: compensating a time-rate difference by a factor of γcan be interpreted as an effective rescaling of G, an altered gravitational participation of mass M, or a combination of both. Treating Gas invariant therefore requires that these effects cancel in a specific and coordinated manner across environments. Within the GRINCHES framework, no mechanism is identified that enforces such cancellation once constants and time-based assumptions are removed. This should not be interpreted as a claim that universality is incorrect. Rather, the framework highlights a structural tension: the empirical universality of Gmust either emerge as a consequence of the mass–distance environment or be imposed as an external consistency condition beyond the structural model itself. 17 Crucially, this requirement is not merely formal. Universality implies a physical explanation for how gravitational interactions are directly observed to agree across environments with vastly different time rates. In particular, an observer situated in a strongly time-dilated region—such as near the Schwarzschild radius of a compact object—would be required to observe gravitational interactions at Earth as having the same effective coupling that we measure locally, without reliance on post hoc coordinate transformations, proper-time rescalings, or inferred corrections. The need for such agreement at the level of observation, rather than reinterpretation, constitutes a nontrivial consistency condition. At present, no empirically established mechanism is known that explicitly enforces this agreement independent of time-based assumptions. 6B. Structural Variation as an Indicator of Time-Related Behavior Structural Derivation of Time, Propagation Speed, and Energy This section derives time scaling, propagation speed, and energy directly from the mass–distance field S, without assuming clocks, velocities, or energetic quantities as primitives. All quantities with time dimensions enter only through a single observer-level calibration and are thereafter used exclusively in relative form. 6B.1 Structural Time Scaling The structural field S(R) = ∫dM D has units [S] = kg m−1,[1 S]= m kg−1. A time-free structural coupling. Because Sis built from mass and distance alone, any structural coupling that maps Sinto a dimensionless response must also be time-free. The unique mass–length unit carried by 1/S therefore identifies a natural structural coupling scale G c2∝1 S,[G c2] = m kg−1. This step does not assume any dynamical law or metric structure; it is a unit identity forced by the form of S. Recovering time-normalized constants requires a time-bearing bridge. Empirical constants such as Gand ccarry time dimensions, [G] = m3kg−1s−2,[c] = m s−1. Since Scontains no time, introducing either Gor cin SI units requires a single observer-level conversion (a “timenormalization bridge”) that supplies the missing s-dimensions. Once such a bridge is fixed at one reference location, all other values must be obtained as relative structural scalings. Uniqueness of the quarter-power. Let ccoord(R)denote the time-normalized propagation speed associated with the structural environment at R. The only structural dependence available is through S(R), so the most general power-law form is ccoord(R)∝S(R)−p, for some exponent p. A time-free coupling G0can be formed from the time-normalized constants by the ratio G/c2, whose units are [G c2]=m3kg−1s−2 m2s−2= m kg−1. This matches the structural unit [1/S], so the only consistent way to relate time-normalized observables back to structure is Gcoord(R) ccoord(R)2∝1 S(R). 18 Using ccoord(R)∝S(R)−p, this implies Gcoord(R)∝ccoord(R)2 S(R)∝S(R)−2p S(R)=S(R)−(2p+1). But the structural identification already obtained from the G2M/(2R)diagnostic yields Gcoord(R)∝S(R)−1/2. Equating the structural exponents gives −(2p+ 1) = −1 2=⇒p=1 4. Therefore, ccoord(R)∝S(R)−1/4. Since a characteristic time scale is the inverse of a propagation rate, the corresponding structural time scaling is τ(R)∝1 ccoord(R)∝S(R)1/4. Relative form (no clocks assumed). For two locations Rand R0, τ(R) τ(R0)=(S(R) S(R0))1/4 . This ratio is the only time-related content required by the structural model. Absolute time in SI units enters only through a single observer-level normalization. 6B.2 Structural Derivation of Propagation Speed Propagation speed is defined as a length per unit time. In the present framework, the only spatially varying input is the structural field S(R). Once the structural time scaling has been established as τ(R)∝S(R)1/4, any propagation speed that is time-normalized in the observer’s unit system must scale inversely with this quantity. Therefore the structural scaling of propagation speed is ccoord(R)∝τ(R)−1∝S(R)−1/4. This fixes both the exponent and the sign. Larger mass–distance structure (larger S) corresponds to a larger structural time factor and therefore a smaller coordinate propagation speed when expressed in a fixed observer time unit. Observer-level calibration. The structural field determines only relative variation. To express ccoord(R)in SI units, a single observer-level calibration constant is required: ccoord(R) = KcS(R)−1/4. Dimensional consistency fixes the units of Kc. Since [S−1/4] = kg−1/4m1/4and [c] = m s−1, one must have [Kc] = [c] [S]1/4= m s−1·kg1/4m−1/4= kg1/4m3/4s−1. The constant Kcis fixed once at the observer’s reference location R0: Kc≡cstd S(R0)1/4, where cstd is the locally measured speed of light at R0in the observer’s unit system. With this single anchoring, all other coordinate values follow purely from the structural ratios: ccoord(R) cstd =(S(R0) S(R))1/4 . No additional time assumptions enter beyond this one-point calibration. The structural model fixes the spatial dependence of ccoord, while the observer-level anchor supplies the SI normalization. 19 Interpretation of the Calibration Constant The calibration constant Kcintroduces no new structural content. Its role is limited to converting the dimensionless structural ratios fixed by S(R)into the observer’s time-normalized units. Once anchored at a single reference location, Kcpackages the observer’s local time convention together with the characteristic structural scale of the surrounding environment. All spatial variation in ccoord(R)is thereafter determined solely by ratios of S1/4. For intuition only, one may note that near the Solar radius the implied time scale associated with this calibration is of order the light-crossing time of the local structural environment. This correspondence is not an assumption and is not used elsewhere in the analysis; it merely reflects how observer-level normalization encodes the background scale against which structural variation is measured. 6B.3 Structural Energy Scaling Energy does not appear anywhere in the structural construction. Once a propagation speed has been expressed in time-normalized units, energy emerges automatically through the standard bookkeeping relation E(R) = m c(R)2. Substituting the structural scaling for c(R)yields E(R)∝m S(R)−1/2. For two locations, E(R) E(R0)=(S(R0) S(R))1/2 . No independent energy postulate is required. Energy differences reflect nothing more than how structural variation is expressed once time units are introduced. It is noteworthy that this scaling matches the inverse dependence found for the structurally extracted gravitational coupling, Gcoord(R)∝√2 √S(R). Both quantities depend on the same structural factor S−1/2once expressed in time-normalized form. This correspondence is not imposed and does not rely on any dynamical assumption; it follows directly from dimensional consistency and the shared role of time normalization in both c2and G. 6B.4 The Invariant Structural Coupling G0=G/c2 A central outcome of the GRINCHES program is that while Gand care not introduced as primitives, a particular combination of them acts as a genuine structural invariant once they are conditionally recovered from the mass– distance field. That combination is G0≡G c2. Complete unit breakdown. In SI units, [G] = m3kg−1s−2,[c2] = m2s−2. Therefore [G0] = m3kg−1s−2 m2s−2= m kg−1. This is the key dimensional fact: G0carries no residual time units. It is therefore immune to any reparameterization of time that rescales Gand cin the usual way. 20 G0is invariant under time reparameterization. If the observer’s time unit is rescaled by a factor λ(i.e. one adopts a different clock rate as the operational second), then dimensional consistency alone implies G∝s−2, c ∝s−1, c2∝s−2. Hence the ratio satisfies G c2∝s−2 s−2=constant. Thus, even if Gand cappear to vary as coordinate quantities when viewed from a fixed observer frame, the combination G0is structurally invariant. G0matches the structural units of the mass–distance field. The GRINCHES structural field is S(R) = ∫dM D,[S] = kg m−1. Therefore [1 S]= m kg−1= [G0]. This unit identity is not rhetorical; it is the reason G0functions as the bridge between a time-free structural environment and time-normalized observables. G0is the unique coupling that maps a mass–distance ratio into a dimensionless structural response: ξ(R)≡G0S(R),[ξ] = 1. In other words, the dimensionless structural “strength” parameter ξcan be constructed from Swithout introducing any explicit time unit. Once ξis known, coordinate quantities that embed time normalization (such as G and cseparately) may be reconstructed consistently, but G0itself remains the time-free structural constant of the framework. Invariance within the GRINCHES extraction rules. In the present work, the extracted coordinate coupling is written Gcoord(R) = √2K1 √S(R), ccoord(R) = KcS(R)−1/4, where Kand Kcare observer-level conversion factors that package the chosen time normalization into SI units. Forming the ratio gives Gcoord(R) ccoord(R)2=√2K S(R)−1/2 K2 cS(R)−1/2=√2K K2 c≡G0, which is explicitly independent of R. The S(R)dependence cancels identically. This cancellation is the operational signature that G0is the structural invariant underlying the coordinate variability of Gand c. Interpretive significance. GRINCHES therefore treats G0as the true structural coupling: it connects mass–distance structure to dimensionless response without importing time as a primitive. The familiar constants Gand care then understood as time-normalized expressions of that structural coupling within a chosen observer clock standard. Whether one chooses to enforce local invariance of Gand cby proper-time transformations does not alter the underlying structural role of G0. Coordinate invariance of G0and its interpretation as a reference frame. A central hinge quantity in the GRINCHES analysis is G0≡G c2, whose units are [G0] = m/kg and therefore contain no time dimension. Because Gcarries units of t−2while ccarries units of t−1, the ratio G/c2is invariant under any reparameterization of the time unit. This invariance holds in two distinct senses: in proper-time descriptions (where local transformations may be applied to enforce constant measured values) and in coordinate descriptions (where nonlocal inferences are made from a fixed observer frame). In both cases, coordinate variance of Gand cleaves G0unchanged. This dual invariance admits a useful interpretation: G0can be treated as a reference frame (or reference layer) against which coordinate quantities are compared. If coordinate values are transformed while G0is held fixed, 21 structural outcomes are preserved even as the numerical values of Gand cchange. Under this interpretation the coordinate speed of light in the G0-reference description may take an arbitrarily small numerical value, e.g. ccoord = 1 m/s, without implying a physical pathology. Coordinate light speeds are known (even within standard relativistic treatments) to approach zero continuously at an event horizon; a small coordinate value therefore represents a highly time-dilated description rather than a discontinuity. This viewpoint also clarifies why GRINCHES avoids using the conventional potential ratio ϕ/c2as a primary measure of relative time. The potential ϕis defined in a time-normalized system and therefore depends on the adopted time unit, whereas the structural ratios used in GRINCHES are time-free. One may note that ϕ/c2is unitless and therefore free of explicit time dimensions. In practical terms, however, using ϕ/c2as a time indicator requires foreknowledge of the coordinate values entering ϕ(or equivalently, foreknowledge of how time normalization has already been imposed). Otherwise, the computation yields a proper-time result, which is not directly observable from an external frame. GRINCHES instead privileges time-free structural expressions (e.g. M/D and S) and uses G0only as an invariant bridge when time-normalized observables are introduced. 6B.4 The Invariant Structural Coupling G0=G/c2 A central outcome of the GRINCHES program is that while Gand care not introduced as primitives, a particular combination of them acts as a genuine structural invariant once they are conditionally recovered from the mass– distance field. That combination is G0≡G c2. Complete unit breakdown. In SI units, [G] = m3kg−1s−2,[c2] = m2s−2. Therefore [G0] = m3kg−1s−2 m2s−2= m kg−1. This is the key dimensional fact: G0carries no residual time units. It is therefore immune to any reparameterization of time that rescales Gand cin the usual way. G0is invariant under time reparameterization. If the observer’s time unit is rescaled by a factor λ(i.e. one adopts a different clock rate as the operational second), then dimensional consistency alone implies G∝s−2, c ∝s−1, c2∝s−2. Hence the ratio satisfies G c2∝s−2 s−2=constant. Thus, even if Gand cappear to vary as coordinate quantities when viewed from a fixed observer frame, the combination G0is structurally invariant. G0matches the structural units of the mass–distance field. The GRINCHES structural field is S(R) = ∫dM D,[S] = kg m−1. Therefore [1 S]= m kg−1= [G0]. This unit identity is not rhetorical; it is the reason G0functions as the bridge between a time-free structural environment and time-normalized observables. G0is the unique coupling that maps a mass–distance ratio into a dimensionless structural response: ξ(R)≡G0S(R),[ξ] = 1. In other words, the dimensionless structural “strength” parameter ξcan be constructed from Swithout introducing any explicit time unit. Once ξis known, coordinate quantities that embed time normalization (such as G and cseparately) may be reconstructed consistently, but G0itself remains the time-free structural constant of the framework. 22 Invariance within the GRINCHES extraction rules. In the present work, the extracted coordinate coupling is written Gcoord(R) = √2K1 √S(R), ccoord(R) = KcS(R)−1/4, where Kand Kcare observer-level conversion factors that package the chosen time normalization into SI units. Forming the ratio gives Gcoord(R) ccoord(R)2=√2K S(R)−1/2 K2 cS(R)−1/2=√2K K2 c≡G0, which is explicitly independent of R. The S(R)dependence cancels identically. This cancellation is the operational signature that G0is the structural invariant underlying the coordinate variability of Gand c. Interpretive significance. GRINCHES therefore treats G0as the true structural coupling: it connects mass–distance structure to dimensionless response without importing time as a primitive. The familiar constants Gand care then understood as time-normalized expressions of that structural coupling within a chosen observer clock standard. Whether one chooses to enforce local invariance of Gand cby proper-time transformations does not alter the underlying structural role of G0. Coordinate invariance of G0and its interpretation as a reference frame. A central hinge quantity in the GRINCHES analysis is G0≡G c2, whose units are [G0] = m/kg and therefore contain no time dimension. Because Gcarries units of t−2while ccarries units of t−1, the ratio G/c2is invariant under any reparameterization of the time unit. This invariance holds in two distinct senses: in proper-time descriptions (where local transformations may be applied to enforce constant measured values) and in coordinate descriptions (where nonlocal inferences are made from a fixed observer frame). In both cases, coordinate variance of Gand cleaves G0unchanged. This dual invariance admits a useful interpretation: G0can be treated as a reference frame (or reference layer) against which coordinate quantities are compared. If coordinate values are transformed while G0is held fixed, structural outcomes are preserved even as the numerical values of Gand cchange. Under this interpretation the coordinate speed of light in the G0-reference description may take an arbitrarily small numerical value, e.g. ccoord = 1 m/s, without implying a physical pathology. Coordinate light speeds are known (even within standard relativistic treatments) to approach zero continuously at an event horizon; a small coordinate value therefore represents a highly time-dilated description rather than a discontinuity. This viewpoint also clarifies why GRINCHES avoids using the conventional potential ratio ϕ/c2as a primary measure of relative time. The potential ϕis defined in a time-normalized system and therefore depends on the adopted time unit, whereas the structural ratios used in GRINCHES are time-free. One may note that ϕ/c2is unitless and therefore free of explicit time dimensions. In practical terms, however, using ϕ/c2as a time indicator requires foreknowledge of the coordinate values entering ϕ(or equivalently, foreknowledge of how time normalization has already been imposed). Otherwise, the computation yields a proper-time result, which is not directly observable from an external frame. GRINCHES instead privileges time-free structural expressions (e.g. M/D and S) and uses G0only as an invariant bridge when time-normalized observables are introduced. Invariant combinations and structural interpretation. Additional insight follows from examining alternative factorizations of the invariant combination G0≡G c2. Because G0is invariant under both proper–time normalization and coordinate reparameterization, it may be combined with mass or energy without reintroducing time dependence. For example, the combination G mc2 has units [G mc2]= m kg−2. 23 Fixing the reference mass to m= 1 kg defines the invariant structural quantity ℓ0≡G (1 kg) c2, which remains invariant under both proper–time normalization and coordinate reparameterization. When this invariant quantity is multiplied by the time–normalized energy per unit mass, E m=c2, the gravitational coupling is reconstructed identically: ℓ0×E m=G. While algebraically redundant, this decomposition is structurally suggestive: a specific invariant quantity multiplied by an energy–per–mass scale yields the gravitational coupling. This indicates that Gmay be viewed not as a primitive constant, but as a composite quantity encoding how mass and energy are related through an invariant structural bridge. Similarly, the combination GM c2 has units of length. Although the physical interpretation of this length is not fixed within the present framework, its appearance alongside a quantity that is invariant in both proper and coordinate forms suggests structural significance. Notably, this length arises without invoking dynamics, curvature, or time evolution, and therefore cannot be dismissed as a coordinate artifact. These observations do not assert new physical laws. Rather, they indicate that invariant combinations involving G0admit multiple structurally meaningful factorizations. The recurrence of mass, length, and energy scales derived from the same invariant quantity suggests that what are conventionally treated as independent constants may instead reflect different projections of a single underlying structural relationship once time normalization is imposed. 6B.5 Additive vs. Nested Interpretations of Time Structure In General Relativity, gravitational environments are commonly treated as additive at the level of the potential, Φtot = Φgal + Φ⊙+ Φ⊕, with relative time dilation inferred from the resulting metric. This construction implicitly assumes that all gravitational contributions act on a single, shared time coordinate and may therefore be summed directly. Within the GRINCHES framework, this assumption is no longer justified. The mass–distance field S=∫dM D is constructed without time and therefore does not encode how time must compose across nested gravitational environments. While individual mass–distance contributions may be formally summed, Ssum =Sgal +S⊙+S⊕, such a sum represents only a geometric aggregation of structure and carries no prescription for how relative clock rates should combine. A fundamental inconsistency appears when additive reasoning is pushed to its logical limit. Using weak–field expressions, the combined terrestrial, Solar, and galactic contributions already exceed the total relative time normalization required to reconcile locally measured proper time with the invariant structural reference defined by G0≡G c2. Relative to this reference, the total normalization implied by the observer’s environment corresponds to a factor of 1/c. When gravitational contributions are treated additively, this factor is exceeded rapidly once multiple structural layers 24 are included. Direct summation therefore overshoots the required normalization well before all relevant environments are accounted for. This failure indicates that additive composition cannot be the correct rule for time behavior across nested gravitational environments. GRINCHES therefore adopts a hierarchical interpretation. Each gravitational environment is experienced relative to a background defined by the level above it: terrestrial structure is embedded within the Solar environment, the Solar environment is embedded within the galactic background, and the galaxy itself is embedded within a broader cosmological background. Any time behavior inferred from structure must therefore compose multiplicatively rather than additively. A natural representation of this nesting is a composite structural form, Seff =√√Sgal S1 AU S⊕, where S1 AU denotes the Solar mass–distance field evaluated at 1 AU and S⊕the terrestrial contribution at Earth’s surface. This construction reflects the fact that structure itself does not transform; only the observer’s relation to successively embedded structural layers changes. The role of the invariant quantity G0=G c2 becomes particularly transparent in this context. Because G0carries no time dimension, it defines a reference description against which coordinate normalizations may be compared. In particular, the G0–reference admits a coordinate description in which the propagation speed takes the value ccoord = 1 m/s. This description does not correspond to a physical frame, but serves as a normalization reference. Comparing the locally measured coordinate speed cto this reference fixes the relative time normalization uniquely: tlocal tG0 =cG0 clocal =1 c. This factor does not represent an absolute time dilation, nor does it imply a global time field. It expresses only the normalization required to relate coordinate descriptions that share the same invariant structural coupling. When the nested structural composition is evaluated using terrestrial, Solar, and galactic contributions, the resulting normalization remains insufficient to reach the full factor implied by 1/c. This shortfall identifies a missing structural contribution, naturally associated with mass beyond the galaxy. Importantly, this inference does not require modification of galactic-scale relations or redefinition of G. Structural quantities such as Gare determined locally within their respective layers and vary only weakly; within the Solar System any such variation would be far below current experimental sensitivity. Hierarchical composition therefore emerges as a structural necessity. Additive reasoning fails to satisfy the invariant reference condition, while nested composition succeeds without invoking new dynamics, global synchronization, or assumed metric structure. 6B.6 Composite Structural Environment and the Emergence of c/G Using the hierarchical (nested) structural composition proposed in the previous subsection, the effective structural environment at Earth is not given by a simple sum of mass–distance fields, but by a geometric composition of nested backgrounds. Explicitly, the total structural field at Earth is defined as Stot,⊕=√√√Sbg Sgal S1AU S⊕. Here, •Sgal is the Milky Way mass–distance field evaluated at the Solar radius (R0= 8.2 kpc), •S1AU is the Solar contribution at Earth’s orbital radius, •S⊕is the terrestrial surface contribution, •and Sbg represents a residual large–scale structural background implied by hierarchical closure rather than introduced as a fitted parameter. 25 Fig. 3: Rotation curves obtained by applying Newtonian orbital relations to a structure-dependent gravitational coupling derived from the mass–distance field S(R). The curves illustrate how a slowly varying effective coupling can qualitatively reproduce flat rotation behavior using purely baryonic matter. These results are not intended as precision fits to Milky Way data and serve only to demonstrate the structural mechanism. declines more gradually than in models that enforce a universal coupling. This produces extended regions of approximately flat rotational velocity using only baryonic matter, without introducing dark matter, interpolation functions, or adjustable parameters. Several limitations must be emphasized. The structural model isolates only the galactic contribution to the gravitational environment. It does not include kinematic time dilation, stellar substructure, hierarchical time composition, or external structural backgrounds. Moreover, because the mass normalization is derived directly from the locally measured value of G, discrepancies in velocity amplitude are not interpreted as missing mass. Any remaining mismatch with observed rotation curves may therefore arise from either incomplete baryonic modeling or from differences between the time rate governing orbital motion and the locally measured proper time. Both effects lie outside the scope of the present structural analysis. It is nevertheless noteworthy that the characteristic offset between the model velocities and conventional expectations is of order ∼√2. This factor recurs elsewhere in the framework and corresponds numerically to the ratio S1/4between the Solar radius and the outermost galactic radii evaluated here. No fitting or adjustment is performed on the basis of this observation, and no structural derivation of this factor is asserted in the present work. Accordingly, these results should be interpreted strictly as a proof of principle. They demonstrate that flat rotation behavior can emerge naturally from mass–distance structure alone once the assumption of a universal gravitational coupling in both coordinate and proper time is relaxed. GRINCHES does not assert the nonexistence of dark matter; rather, it shows that the necessity of dark matter within standard GR arises from enforcing coupling universality in environments where the underlying structural field is strongly non-uniform. 6D. Possibility of Gravitational Attraction as a Coordinate Energy Gradient Within the GRINCHES framework, gravitational attraction may be interpreted as a coordinate energy gradient rather than as a fundamental force or as a consequence of spatial curvature. This interpretation follows directly from the dependence of coordinate quantities on the mass–distance field and requires no assumptions beyond those already introduced. At any location x, the coordinate energy associated with a mass element mmay be written as Ecoord(x) = m c2 coord(x), where ccoord denotes the coordinate value of the speed of light associated with the local structural environment. Because c2 coord varies spatially with structure, this quantity generally varies in coordinate descriptions, even though the locally measured energy of the mass remains invariant in its own proper frame. Energy ordering as the origin of motion. A spatial variation in coordinate energy establishes an energetic ordering between neighboring regions. No explicit force law is required to respond to such an ordering: systems evolve toward configurations that are energetically lower relative to the external description. In this sense, gravitational motion appears as a relational consequence of coordinate energy differences rather than as the action of a force acting at a distance. 32 For infinitesimally separated locations xand x+ dx, the difference dEcoord =md(c2 coord) represents the local coordinate energy imbalance between adjacent regions. Dividing by the separation dxand normalizing by mass yields an acceleration scale, a∼1 m dEcoord dx=dc2 coord dx, which points in the direction of decreasing c2 coord and therefore decreasing coordinate energy. This relation is not asserted as an equation of motion. It expresses only the magnitude and direction of the coordinate energy gradient. Motion follows as the only relational response available once such an ordering exists. Connection to structural gravity. Using the invariant structural relation G0=G/c2, coordinate quantities satisfy c2 coord G0=Gcoord, and the structural field obeys Gcoord S∼Φ, where Φplays the role of a coordinate gravitational potential. Accordingly, the coordinate energy gradient reproduces the standard gravitational acceleration numerically, while arising entirely from structural variation in time–dependent quantities rather than from a force postulate or spatial curvature. Illustrative Earth-surface scale. Near the Earth’s surface, the fractional change in relative time rate per meter of elevation is of order 1 t dt dx ∼1.09 ×10−16 m−1. Multiplying this structural time gradient by c2yields a coordinate energy gradient per unit mass of the correct order of magnitude to reproduce the observed gravitational acceleration, 1 m dEcoord dx ∼9.8 m/s2, at the surface. This correspondence is illustrative only. It introduces no new constants or calibration and serves solely to demonstrate how familiar gravitational behavior emerges from extremely small spatial variations in relative time rate when expressed through coordinate energy. Scale independence of the gradient. The coordinate energy gradient is not defined per meter or per any preferred length scale. It exists simultaneously at all scales. Whether a body spans meters, nanometers, or astronomical units, each constituent experiences the same relational imbalance with its immediate surroundings. The gradient is embedded within matter itself through the structural dependence of coordinate quantities, not imposed externally. Macroscopic smoothness versus microscopic variability. On macroscopic scales, the structural field varies smoothly across space. As a result, the associated coordinate energy gradient appears uniform and stable, producing the familiar steady acceleration attributed to gravity. This smoothness, however, should not be assumed to persist at all scales. At smaller or subatomic scales, the structural environment is continuously perturbed by the motion of the Earth, the Solar System, the galaxy, and larger structures. At these scales the gradient may fluctuate rapidly in both magnitude and direction. The apparent calm of the macroscopic field may therefore conceal substantial microscopic variability, even though such effects average out at observable scales. Interpretational scope. This description does not assert that gravity is energy loss, nor does it introduce a new dynamical law. It provides a structural interpretation of why acceleration arises when coordinate time rates vary with mass–distance environment. The gradient is entirely coordinate in nature: it governs how motion appears when described relative to an external frame, while leaving all local conservation laws intact. 33 Proper-time invariance and coordinate appearance. No physical energy is gained or lost in this process. In GRINCHES, all mass–energy remains invariant in its own proper time. The quantity Ecoord =m c2 coord represents only how that same invariant energy is described when expressed in an external coordinate frame whose time rate differs from the local proper time. Equivalence is therefore maintained at all levels. Each mass element evolves according to its own proper time and experiences no internal force or energy imbalance. Apparent energy gradients arise only when neighboring regions are compared across differing time rates, in direct analogy with other coordinate effects in relativistic physics. Matter and energy do not exist in a global time. They exist only in their own proper frames and respond to external structure through relational comparison. Whether those comparisons are described as coordinate, observational, or gravitational does not alter their physical content. Conclusion This work began from a deliberately minimal premise: mass and distance were taken as the only primitive quantities. No assumptions were made regarding time, spacetime geometry, dynamical laws, or the numerical values of physical constants. Time entered the analysis only at the stage required to express structurally derived quantities in observational units. Outside of this translation, time and the constants ordinarily associated with it were not imported into the framework but instead emerged as consequences of mass–distance structure. The central result is the structural extraction of the gravitational coupling from the mass–distance field. The quantity identified with Gis neither assumed nor fitted; it arises as a normalization of the structural field S(R) = ∫dM/D. Unless additional assumptions are imposed, the extracted coupling varies with environment. This result is purely structural and does not rely on relativistic field equations, metric assumptions, or prescribed dynamics. It is this outcome that underlies all subsequent results in the analysis. Once the gravitational coupling is treated as a structural quantity, several phenomena commonly attributed to additional physics follow directly. In extended mass distributions, the effective gravitational response varies only slowly with radius, producing approximately flat rotation behavior using baryonic matter alone. This occurs without modifying Newtonian orbital relations and without invoking new forces, new dynamical laws, or additional particle species. Within this interpretation, dark matter does not appear as a structurally required component of the mass–distance field. Instead, its conventional role may be reinterpreted as a structural renormalization mechanism. Introducing additional, non-luminous mass at galactic scales increases the effective mass–distance ratio so that the gravitational coupling extracted from structure coincides with the locally measured standard value of G. From this perspective, dark matter functions to enforce universality of the gravitational coupling rather than to modify orbital dynamics directly. The present analysis shows that if universality of the coordinate gravitational coupling is not imposed a priori, then observed galactic-scale behavior can be reproduced using baryonic matter alone. Dark matter is therefore not required to explain rotation curves or lensing within the structural framework developed here. Nothing in the framework forbids the existence of additional mass components; however, their inferred necessity arises specifically from the assumption that Gmust take the same coordinate value in all environments. When that assumption is relaxed, environmental variation in gravitational response follows naturally from mass–distance structure. Gravitational lensing likewise emerges from structure alone. Photon trajectories are redirected by transverse gradients in the mass–distance field, modulated by geometric angular coherence. Numerical simulations reproduce lensing magnitudes and spatial features of the correct order without invoking spacetime curvature, new interaction terms, or non-baryonic matter. The appearance of a preferred angular scale associated with peak lensing strength arises from geometry rather than enclosed mass, indicating a structural origin for lensing behavior. Throughout the analysis, equivalence is preserved in all proper frames. Local physics remains unchanged: energy conservation, atomic processes, and locally measured quantities are unaffected. All variations discussed are coordinate in nature and arise from relative time structure implied by mass–distance environments. Observed gravitational effects therefore reflect relational comparisons between structurally distinct regions rather than intrinsic changes to matter or energy. Crucially, the framework introduces no new forces, no additional laws of motion, and no new particle content. All results follow from requiring that time, gravitational coupling, and propagation speed be obtained from mass– distance structure alone, rather than being imported as independent primitives. Constants and time enter only as observer-level normalizations required for measurement, not as foundational inputs. In this sense, the framework is conservative in content even where it is nonstandard in interpretation. Equally important are the elements not found in the structural construction. No structural basis is identified for universally invariant values of Gin both proper time and observed coordinate descriptions, for spatial warping as 34 a physical mechanism, or for the preservation of constant propagation speed across disparate environments. These features are widely employed in standard formulations but are introduced as mutually supporting assumptions rather than derived consequences. While these assumptions often appear together in practice, each requires its own physical justification. In particular, enforcing a universal gravitational coupling constrains the form of spacetime structure, and that structure is then used to preserve invariant local propagation speed by construction, with observable effects such as Shapiro delay arising through coordinate rather than local variation. The preservation of constant propagation speed follows naturally if spatial distance itself varies with relative time rate; no analogous structural mechanism is evident for the invariance of Gor for spatial warping as a physical process. If such properties are fundamental, a physical explanation for how and why each arises individually remains to be established. This requirement is not merely philosophical: the SI system relies critically on time-based units across much of its structure, and unresolved assumptions about invariance propagate directly into derived constants, including those involving ¯h, which, while not addressed in this structural analysis, does contain explicit dependence on units of time. Establishing cause, rather than relying on mutual consistency, is therefore a necessary step for any foundational account of gravitational and temporal structure. The conclusions reached here are unavoidably nonstandard relative to conventional gravitational formulations. This was not a design goal. It followed from the decision to restrict the analysis to mass and distance alone and to pursue the resulting structure without importing additional primitives. The framework presented does not constitute a complete dynamical theory. Rather, it establishes a structural foundation from which gravitational coupling, effective light propagation, lensing behavior, and gravitational attraction emerge prior to the introduction of metrics, curvature, or fixed constants. Whether this structural description corresponds to physical reality is ultimately an empirical and mathematical question. Further work will be required to extend the analysis to fully realistic mass distributions, to incorporate cosmological structure self-consistently, and to develop a complete dynamical formulation. The purpose of the present study is more limited: to make explicit what follows from mass and distance alone, and to clarify which gravitational assumptions are structural consequences and which are imposed by convention. 35 References 1. Einstein, Albert. The Foundation of the General Theory of Relativity.Annalen der Physik, vol. 49, no. 7, 1916, pp. 769–822. English translation reprinted in The Principle of Relativity, Dover Publications, 1952. 2. Misner, Charles W., Kip S. Thorne, and John A. Wheeler. Gravitation. W. H. Freeman, 1973. 3. Will, Clifford M. “The Confrontation between General Relativity and Experiment.” Living Reviews in Relativity, vol. 17, no. 4, 2014, doi:10.12942/lrr-2014-4. 4. Shapiro, Irwin I. “Fourth Test of General Relativity.” Physical Review Letters, vol. 13, no. 26, 1964, pp. 789–791. 5. Uzan, Jean-Philippe. “The Fundamental Constants and Their Variation: Observational and Theoretical Status.” Reviews of Modern Physics, vol. 75, no. 2, 2003, pp. 403–455. 6. Planck, Max. The Theory of Heat Radiation. Translated by Morton Masius, Dover Publications, 1959. 7. Clowe, Douglas, et al. “A Direct Empirical Proof of the Existence of Dark Matter.” The Astrophysical Journal Letters, vol. 648, no. 2, 2006, pp. L109–L113. 36 Appendix A: Coordinate Time Limits and Entropy Considerations Black holes are not modeled within the present structural framework. Nevertheless, their existence highlights a conceptual issue that is shared by both General Relativity and the structural approach developed here. In General Relativity, the coordinate speed of light approaches zero at the Schwarzschild radius, and clocks defined relative to a distant observer asymptotically freeze. Within the GRINCHES framework, an analogous conclusion follows from structural considerations: as the mass–distance field S(R)grows without bound, coordinate values of both cand Gdecrease toward zero, reflecting extreme relative time dilation with respect to the observer. Taken at face value, this behavior suggests a paradox. If coordinate time truly vanishes relative to a distant observer, then all coordinate phenomena should cease: no motion, no energy exchange, no evolution. In such a limit, black holes would appear not merely dark, but dynamically inert. This paradox arises only in descriptions tied to an external time parameter and does not imply that local physics halts within the strongly dilated region. Observationally, however, black holes are not inert. They accrete matter, launch relativistic jets, merge, emit gravitational waves, and interact dynamically with their environments. These processes are inferred through time– dependent signals originating outside the putative horizon and are not consistent with a literal interpretation of complete coordinate time cessation. This tension is not resolved by appealing to proper time alone. While General Relativity correctly predicts that infalling observers experience finite proper time, the observables accessible to distant observers are necessarily coordinate-based. Both GR and GRINCHES therefore encounter the same unresolved question: how can systems exhibit ongoing, energetic, and time–dependent behavior when the formal coordinate description suggests an asymptotic freezing of all change? Within the present work, this issue is not taken as evidence against either framework. Rather, it indicates that the standard language of “time stopping” near black holes is an incomplete description. Coordinate limits may signal the breakdown of a particular parametrization rather than the cessation of physical processes. The structural framework developed here reproduces the same asymptotic behavior as General Relativity without invoking a metric or a horizon surface. It therefore supports the interpretation that black holes represent a regime in which time comparisons between vastly different structural environments lose operational meaning. What replaces this description remains an open problem and lies beyond the scope of the present analysis. Entropy and the zero–time limit. A related tension arises when black holes are considered from the standpoint of entropy. In standard treatments, black holes are assigned a maximum entropy, proportional to the area of the event horizon. Entropy, however, is not an instantaneous quantity: it is defined through counting of accessible microstates and is operationally inferred through time–dependent processes such as equilibration, coarse–graining, or information loss. In both General Relativity and the structural framework presented here, the coordinate time rate relative to a distant observer approaches zero in the black–hole limit. If coordinate time truly vanishes, the operational basis for entropy measurement also vanishes. No transitions occur, no coarse–graining can proceed, and no entropy increase can be observed. In this limit, the assignment of maximum entropy becomes ambiguous, as entropy is ordinarily defined through the accumulation of inaccessible microstates over time. This is not merely a semantic concern. If entropy is maximal while coordinate time is zero, then entropy would have to exist independently of any temporal process. Such a notion departs from the statistical foundations of entropy and suggests that the standard black–hole entropy assignment is not a direct operational measurement in the zero–time limit, but an extrapolation beyond the regime in which time–based quantities remain well defined. More generally, this highlights a broader issue with entropy in relativistic contexts: entropy comparisons implicitly assume a shared, or at least comparable, time structure. When relative time rates differ drastically between observers or frames, entropy ceases to be an invariant quantity in any operational sense. Black holes represent the extreme case of this problem, where the time required to define entropy formally diverges. Within the GRINCHES framework, this observation reinforces the view that entropy should be treated as a derived, observer–dependent construct rather than a fundamental quantity. The apparent coexistence of maximal entropy with vanishing coordinate time is therefore interpreted not as a failure of thermodynamics, but as an indication that entropy is being applied outside its domain of empirical meaning. Appendix B: Experimental Verification of Spatial Warping Spatial warping is difficult to falsify directly. However, it admits a clear experimental test through measurement of the speed of light as a velocity over a fixed physical distance. 37 Why as a velocity rather than the conventional frequency–wavelength approach? It is well established experimentally that atomic clocks are affected by relative time rate. For example, two clocks operating at different altitudes, each locally controlling light emission and detection, will both observe themselves as producing the same frequency. Relative to one another, however, those frequencies differ. Because wavelength is not measured independently in such experiments, both the wavelength and the speed of light are inferred from frequency alone. Under a global change in relative time rate—such as that produced by the Earth’s changing distance from the Sun over its elliptical orbit—what is primarily tested is the stability of the atomic clock under time-rate variation, not the behavior of light propagation itself, unless spatial warping occurs. This is why measuring cexplicitly as a velocity is essential. If the experimental baseline itself adjusts with relative time rate, no variation in cwill be detected. If spatial warping does not occur, then no such compensating mechanism exists, and variation in measured propagation speed becomes locally observable. This test does not require GRINCHES methodology. Standard weak-field gravitational time-dilation expressions suffice. The expected variation in cin the absence of spatial warping can be computed directly from the change in Solar gravitational potential along the Earth’s orbit. Interpretational significance. A null result is not merely consistent with spatial warping; it constitutes direct observational evidence for it. If no variation in measured propagation velocity is observed despite a known change in gravitational time rate, then spatial distances must adjust in proportion to time in order to preserve the observed invariance of c. In this case, spatial warping is not a coordinate artifact but a physical mechanism required to maintain local constancy. Conversely, a positive detection of propagation-speed variation would carry equally strong implications. If a fixed experimental baseline does not rescale with gravitational time rate, then the measured velocity of light must change. Such a result would demonstrate that spatial warping does not occur and would directly falsify the notion that the locally measured speed of light is universally invariant under changes in gravitational environment. Importantly, this outcome could not be dismissed as a coordinate effect. The measurement compares elapsed time against a physically fixed distance, both realized locally. A detected variation would therefore demonstrate that local propagation speed depends on environment even in proper-time measurements, and cannot be eliminated by coordinate transformations or clock redefinitions. In that case, the empirical constancy of cwould be shown to be contingent rather than universal. A positive detection would indicate that no physical mechanism operates to preserve invariant local propagation speed under changes in gravitational environment. Local invariance could therefore no longer be treated as a fundamental property, but only as an approximate condition valid within restricted regimes. Accordingly, this experiment does more than test spatial warping. It directly discriminates between two mutually exclusive possibilities: •local invariance of propagation speed enforced by physical geometry, or •environment-dependent propagation speed reflecting time-rate variation without spatial compensation. Either outcome resolves a foundational ambiguity. A null result establishes spatial warping as a real physical process. A positive result rules out both spatial warping and universal local constancy of cas a fundamental principle. 38