A Chronometric Closure Program for Proper-Time Differences in an Inhomogeneous Expanding Universe

1. Introduction

Modern precision cosmology is organized around the FLRW metric, which idealizes the Universe as homogeneous and isotropic on sufficiently large scales. For a spatially flat background,

$$d{s}^2=-c^2\mathrm{d}{t}^2+a^2\left(t\right){\delta }_{ij}\mathrm{d}{x}^i\mathrm{d}{x}^j,$$

(1)

where $a\left(t\right)$ is the scale factor. In the exact FLRW spacetime, t is not merely an arbitrary label: it equals the proper time of the idealized comoving fundamental observers whose worldlines are orthogonal to the homogeneous spatial hypersurfaces [2,3].

The question considered here is narrower. Once nonlinear structure formation produces physically distinct environments, must the proper time accumulated along every late-time matter worldline remain equal to the background FLRW time between physically corresponding states? General Relativity does not impose such an identity. Proper time is a local invariant attached to a timelike worldline,

$$\tau \left[\gamma \right]={\displaystyle \frac{1}{c}}{\int }_{\gamma }\sqrt{-g_{\mu \nu }\mathrm{d}{x}^{\mu }\mathrm{d}{x}^{\nu }},$$

(2)

and any comparison between two clocks requires specified worldlines, endpoint events, and a common physical comparison prescription [2].

The RTD-EU program therefore asks an operational question: under what conditions is FLRW cosmic time an adequate proxy for the proper time accumulated by matter in a specified environment? The present paper does not assume that the answer is a percent-level difference. Instead, it defines the invariant quantity to be calculated, identifies the gauge-dependent intermediate quantities that must not be mistaken for observables, and specifies the minimum closure conditions required before cosmological implications can be claimed.

An earlier RTD-EU preprint proposed broad possible implications for cosmic age, dark-energy inference, and the Hubble tension [33]. The present work deliberately narrows and strengthens that proposal: phenomenological clock modifications are not treated as predictions unless they are recovered from a self-consistent metric, worldline, source, and null-geodesic calculation.

This more conservative formulation is important because several neighboring approaches already study inhomogeneous expansion, averaging, regional clocks, and relativistic structure formation. Buchert averaging quantifies kinematical backreaction for irrotational dust [9]; Wiltshire’s timescape program explicitly considers regional clock-rate differences [11]; and Green and Wald have derived strong restrictions on large backreaction effects within a different approximation framework [12,13]. A viable RTD-EU implementation must be mathematically distinguishable from these approaches and confront the same observational constraints.

2. Cosmic Time, Killing Symmetry, and What Does Not Follow

A Killing vector field ${\xi }^{\mu }$ satisfies

$${\mathcal{L}}_{\xi }{g}_{\mu \nu }={\nabla }_{\mu }{\xi }_{\nu }+{\nabla }_{\nu }{\xi }_{\mu }=0.$$

(3)

If ${\xi }^{\mu }$ is timelike in a stationary region, the scalar

$$E=-p_{\mu }{\xi }^{\mu }$$

(4)

is conserved along geodesic motion. This is the standard relation between a spacetime isometry and a particle energy associated with time translations [1,2].

For the standard FLRW comoving time-flow vector ${\xi }^{\mu }={\left({\partial }_t\right)}^{\mu }$,

$${\mathcal{L}}_{{\partial }_t}{g}_{ij}=2H\left(t\right)g_{ij},H\equiv {\displaystyle \frac{\dot{a}}{a}},$$

(5)

so ${\partial }_t$ is not a Killing flow when $a\left(t\right)$ evolves. For the ordinary matter- and radiation-filled cosmologies of interest here, the spacetime is nonstationary and a global stationarity-based conserved energy cannot be invoked.

Two qualifications are essential. First, expansion written in a particular coordinate system is not, by itself, a proof that every possible timelike Killing field is absent; special coordinate representations, such as the Milne slicing of Minkowski spacetime, provide a familiar warning against that inference. Second, the absence of a timelike Killing field does not make FLRW cosmic time unphysical. It establishes only that global stationarity does not force every physical worldline to share a single time-translation symmetry. Local covariant conservation, ${\nabla }_{\mu }{T}^{\mu \nu }=0$, remains part of the Einstein equations independently of this issue [2].

Thus Killing symmetry motivates the question but does not answer it. A macroscopic environmental clock differential must come from the metric and the worldlines themselves.

3. Invariant Worldline Comparison

Let ${\gamma }_A$ and ${\gamma }_B$ denote two physically specified timelike worldlines, or representative worldlines of two specified timelike congruences $u_A^{\mu }$ and $u_B^{\mu }$. Their endpoints should be defined by physical conditions rather than arbitrary coordinate values. Let

$${\Sigma }_i:S\left(x\right)=S_i,{\Sigma }_f:S\left(x\right)=S_f,$$

(6)

where $S\left(x\right)$ is a scalar constructed from measurable matter or geometric fields. On the domain of interest the level sets should be spacelike,

$${\nabla }_{\mu }S{\nabla }^{\mu }S<0,$$

(7)

and each selected worldline should cross each boundary once. These conditions prevent a coordinate label from being mistaken for a physical synchronization convention.

The proper time accumulated by worldline $K\in \{A,B\}$ is

$${\tau }_K={\displaystyle \frac{1}{c}}{\int }_{{\gamma }_K\cap {\Sigma }_i}^{{\gamma }_K\cap {\Sigma }_f}\sqrt{-g_{\mu \nu }\mathrm{d}{x}^{\mu }\mathrm{d}{x}^{\nu }}.$$

(8)

The proper-time misclosure is

$$\Delta {\tau }_{AB}\equiv {\tau }_A-{\tau }_B.$$

(9)

Once the spacetime, the worldlines, and the boundary states are physically specified, Eq. (9) is a scalar comparison and cannot be eliminated by a coordinate transformation.

The choice of S is part of the physical model. Possible examples include a matter density measured in a specified frame, a radiation temperature in a specified local rest frame, a recombination or ionization criterion, or another scalar state variable appropriate to the process being studied. Different choices need not define identical experiments.

4. Foliation, Lapse, and the Metric-Level Closure Problem

A $3+1$ decomposition is useful for exposing how a coordinate clock is related to proper time. We adopt the convention

$$d{s}^2=-N^2{c}^2\mathrm{d}{t}^2+h_{ij}(\mathrm{d}{x}^i+{\beta }^i\mathrm{d}t)(\mathrm{d}{x}^j+{\beta }^j\mathrm{d}t),$$

(10)

where N is the lapse, $h_{ij}$ the induced spatial metric, and ${\beta }^i$ has dimensions of coordinate velocity [4,5]. Along a timelike worldline with $v_K^i\equiv \mathrm{d}{x}_K^i/\mathrm{d}t$,

$${\Gamma }_K^2\left(t\right)\equiv {\left({\displaystyle \frac{\mathrm{d}{\tau }_K}{\mathrm{d}t}}\right)}^2=N^2-{\displaystyle \frac{h_{ij}}{c^2}}(v_K^i+{\beta }^i)(v_K^j+{\beta }^j).$$

(11)

Thus

$${\tau }_K={\int }_{t_i^{\left(K\right)}}^{t_f^{\left(K\right)}}{\Gamma }_K\left(t\right)\mathrm{d}t.$$

(12)

The general closure relation is therefore

$$\Delta {\tau }_{AB}={\int }_{t_i^{\left(A\right)}}^{t_f^{\left(A\right)}}{\Gamma }_A\mathrm{d}t-{\int }_{t_i^{\left(B\right)}}^{t_f^{\left(B\right)}}{\Gamma }_B\mathrm{d}t.$$

(13)

Only when the chosen endpoint surfaces coincide with common $t=\mathrm{const}$ slices do the coordinate integration limits become identical and allow the abbreviated form $\int ({\Gamma }_A-{\Gamma }_B)\mathrm{d}t$.

The lapse N, shift ${\beta }^i$, and therefore ${\Gamma }_K$ for a chosen coordinate description are foliation dependent. They are useful bookkeeping devices, not standalone observables. In coordinates adapted to one selected congruence, one may arrange $v_K^i=0$ and, locally or under suitable conditions, choose vanishing shift, in which case ${\Gamma }_K=N$. Such an adaptation cannot generally be imposed simultaneously for multiple distinct congruences.

Equation (13) is consequently not a prediction. It is a chronometric closure relation: it identifies the intermediate quantity that must be calculated from a specified spacetime solution while leaving the invariant result in Eq. (9) unchanged.

5. Raychaudhuri Evolution Is Not Clock Closure

For a timelike congruence with four-velocity $u^{\mu }$, the expansion scalar is

$$\theta ={\nabla }_{\mu }{u}^{\mu }.$$

(14)

Its evolution is governed by the Raychaudhuri equation [3,6],

$${\displaystyle \frac{\mathrm{d}\theta }{\mathrm{d}\tau }}=-{\displaystyle \frac{1}{3}}{\theta }^2-{\sigma }_{\mu \nu }{\sigma }^{\mu \nu }+{\omega }_{\mu \nu }{\omega }^{\mu \nu }-R_{\mu \nu }{u}^{\mu }{u}^{\nu }+{\nabla }_{\mu }{a}^{\mu },$$

(15)

where ${\sigma }_{\mu \nu }$ is the shear, ${\omega }_{\mu \nu }$ the vorticity, and $a^{\mu }=u^{\nu }{\nabla }_{\nu }{u}^{\mu }$ the four-acceleration.

Equation (15) rigorously permits different density, shear, curvature, vorticity, and acceleration histories to produce different environmental expansion histories. A virialized matter congruence can have negligible mean expansion while an underdense region continues to expand. These are physically meaningful differences in the deformation of neighboring worldlines.

They are not, however, equivalent to a derived difference in $\mathrm{d}\tau /\mathrm{d}t$. Raychaudhuri evolution does not by itself fix the lapse, the slicing, or the complete metric. The relevant logical chain is

$$\begin{array}{|c|}\hline \{T_{\mu \nu },\mathrm{ICs}\}⟹\{g_{\mu \nu },u_K^{\mu }\}⟹{\Gamma }_K\left[\mathcal{F}\right]⟹{\tau }_K⟹\Delta {\tau }_{AB}\\ \hline\end{array}$$

(16)

where $\mathcal{F}$ denotes the chosen foliation used to represent the intermediate clock rate. There is one physical spacetime metric $g_{\mu \nu }$ in a given model; the environmental label K belongs to the congruence or worldline, not to a different spacetime metric.

6. Relation to Averaging, Regional-Clock, and Relativistic Structure Methods

The RTD-EU closure question overlaps with, but is not identical to, several established research programs.

6.1. Buchert Averaging

Buchert’s standard irrotational-dust formalism averages scalar Einstein equations on hypersurfaces orthogonal to the dust flow and introduces a kinematical backreaction term containing expansion variance and shear [9]. It is directly relevant to the question of how nonlinear structure affects average expansion. In its standard synchronous comoving form, however, the time parameter is the dust proper time. The formalism therefore does not, by itself, provide an environmental lapse difference between independently specified congruences. Any RTD-EU use of Buchert averaging must add an explicit, physically justified clock-comparison prescription rather than identifying backreaction with time dilation by definition.

6.2. Timescape and Regional Clocks

Wiltshire’s timescape cosmology is the closest conceptual precedent for large regional clock differences: it attributes apparent acceleration in part to differences between clocks associated with bound structures and volume-average regions [11]. The present closure program takes a deliberately different methodological stance. It does not assume that a percent-level regional clock effect exists. It requires the metric/worldline calculation to produce the effect and requires the same model to predict redshift, distance, lensing, and structure observables.

6.3. Small-Scale Backreaction Bounds

Green and Wald developed a framework in which small-scale inhomogeneities, under specified assumptions, cannot mimic a dark-energy stress tensor in the background dynamics [12,13]. These results provide an important counterpoint. Any RTD-EU realization that claims a large cosmological effect must state clearly whether it satisfies those assumptions, lies outside them, or predicts an observational effect distinct from background dynamical backreaction.

6.4. Exact and Numerical Inhomogeneous Spacetimes

Exact Lemaître–Tolman–Bondi-type solutions and Szekeres models provide controlled non-FLRW geometries in which worldline and null-geodesic observables can be calculated consistently [14,15,16]. Relativistic large-scale-structure codes such as gevolution provide a complementary weak-field numerical route in which metric degrees of freedom and particle geodesics are evolved together [17]. These approaches are natural test beds for the closure program because they permit the same geometry to determine both clocks and photon propagation.

7. Weak-Field Benchmark and Gauge Discipline

In a weak-field scalar perturbation, a slowly moving observer has, schematically,

$$\mathrm{d}\tau \simeq \left(1+{\displaystyle \frac{\Phi }{c^2}}-{\displaystyle \frac{v^2}{2c^2}}\right)\mathrm{d}t,$$

(17)

when written in a suitable weak-field gauge. Cosmological perturbations and gravitational potentials are gauge-sensitive objects unless combined into gauge-invariant variables or tied to a fully specified coordinate prescription [7,8]. Equation (17) is therefore a benchmark, not a substitute for a covariant observable calculation.

For a representative potential contrast $\left|\Delta \Phi \right|/c^2\sim {10}^{-5}$ sustained over a time of order ${10}^{10}$ yr, the fractional clock difference is of order ${10}^{-5}$, corresponding to roughly ${10}^5$ yr. Peculiar-velocity terms are typically of similar or smaller weak-field order for nonrelativistic large-scale structure. Ordinary static halo potentials therefore do not automatically produce a percent-level chronometric effect.

Potential offsets must also be handled operationally. A constant redefinition of a Newtonian potential is not a measurable clock signal. Only differences and integrals that survive the complete observable construction have physical meaning. A larger RTD-EU effect would have to arise from a demonstrable feature of the relativistic inhomogeneous solution, with any averaging scheme shown to preserve the relevant physical comparison.

8. Minimum Viable RTD-EU Model

Before RTD-EU can be used to explain a cosmological anomaly, one concrete implementation should obtain the following quantities from the same self-consistent model:

• Geometry $g_{\mu \nu }$: solve or approximate the spacetime from specified matter, initial data, and boundary conditions.
• Congruences $u_K^{\mu }$ and worldlines ${\gamma }_K$: define the physical matter and observer histories being compared.
• Boundary states ${\Sigma }_i,{\Sigma }_f$: specify invariant endpoint states and unique worldline crossings.
• Proper time ${\tau }_K$ and misclosure $\Delta {\tau }_{AB}$: integrate the selected worldlines and form the invariant clock comparison.
• Photon propagation $k^{\mu }$ and redshift z: propagate null geodesics in the same geometry and compute the observed redshift from $k_{\mu }{u}^{\mu }$.
• Distances $D_L,D_A$: obtain luminosity and angular-diameter distances from the same null congruence and optical map.
• Source physics: predict the astrophysical clock, transient, or population quantity actually measured.
• Joint likelihood: compare the relevant probes without reusing a fitted anomaly as an input or inserting an unmodified FLRW light cone by hand.

This requirement prevents a common failure mode: using one geometry to motivate a proper-time correction while retaining an unmodified FLRW redshift–distance relation for the observational fit. In General Relativity the same metric controls proper time, null geodesics, lensing, redshift, and the growth of structure. A viable model must close all of these channels simultaneously.

9. Inferred Expansion Rates and the Hubble-Tension Target

Suppose a reference scale factor $a_{\mathrm{ref}}\left(t\right)$ is reparameterized by the proper time of congruence K. Then

$$H_K^{\left(\mathrm{ref}\right)}\equiv {\displaystyle \frac{1}{a_{\mathrm{ref}}}}{\displaystyle \frac{\mathrm{d}{a}_{\mathrm{ref}}}{\mathrm{d}{\tau }_K}}={\displaystyle \frac{H_{\mathrm{ref}}}{{\Gamma }_K}}.$$

(18)

Here $H_K^{\left(\mathrm{ref}\right)}$ is only the reference-background expansion rate expressed in a different time parameter. It must not be identified automatically with the local kinematical expansion ${\theta }_K/3$; that identification is valid only if it follows from the specified spacetime and regional symmetry assumptions.

The standard early- and late-Universe determinations of the Hubble constant use very different observable pipelines. CMB-based inferences depend on the acoustic scale and an assumed cosmological model [18]; BAO measurements constrain combinations of distances and expansion rates relative to the sound horizon [19]; and local distance-ladder measurements use calibrated standard candles [22]. A valid RTD-EU explanation cannot therefore consist only of the algebraic replacement $H\to H/\Gamma$.

The several-percent discrepancy motivating the Hubble-tension discussion provides a target scale rather than a derived result. Schematically,

$$\left|{\displaystyle \frac{\Delta {\tau }_{AB}}{\tau }}\right|\sim {10}^{-2}$$

(19)

would be potentially relevant only if it appears in the complete observable mapping at a comparable level. If every observationally viable implementation yields

$$\left|{\displaystyle \frac{\Delta {\tau }_{AB}}{\tau }}\right|\ll {10}^{-2},$$

(20)

then the proposed macroscopic RTD-EU mechanism for cosmological parameter tensions is ruled out, even though ordinary GR proper-time differences remain.

10. High-Redshift Galaxy Ages Without Assuming an FLRW $t\left(z\right)$ Map

JWST has spectroscopically confirmed luminous galaxies at very high redshift, including JADES-GS-z14-0 at $z\simeq 14.3$ [32]. Early JWST samples have motivated extensive discussion of rapid galaxy assembly and the efficiency required by standard formation models [31]. These observations do not, by themselves, establish a failure of $\Lambda \mathrm{CDM}$; inferred stellar masses, star-formation histories, dust, metallicity, feedback, and selection effects all matter.

A chronometric calculation must avoid assuming the very redshift–time mapping it is intended to test. Let the emission event e be defined observationally by

$$1+z_{\mathrm{obs}}={\displaystyle \frac{{\left(k_{\mu }{u}^{\mu }\right)}_e}{{\left(k_{\mu }{u}^{\mu }\right)}_o}}.$$

(21)

The source-worldline age available between an invariant initial state ${\Sigma }_i$ and that emission event is

$$\Delta {\tau }_K\left(z_{\mathrm{obs}}\right)={\tau }_K\left[e\left(z_{\mathrm{obs}}\right)\right]-{\tau }_K\left[{\Sigma }_i\right].$$

(22)

The corresponding FLRW quantity may be calculated separately as a reference using $t_{\mathrm{FLRW}}\left(z\right)$, but that mapping should not be inserted into the non-FLRW model before photon propagation has been solved.

A positive difference relative to the FLRW reference could, in principle, provide additional source-frame time for stellar evolution, chemical enrichment, and assembly. It would not automatically produce a mature galaxy. The same calculation must model or interface with baryonic evolution, including gas accretion, cooling, star-formation efficiency, feedback, metal production, and halo growth. Until both the chronometric and baryonic requirements are quantified, RTD-EU should be described as a possible geometric contribution to early-galaxy timescale questions rather than an established resolution.

11. Invariant Redshift and Transient-Time Tests

For a photon with wavevector $k^{\mu }$, emitted by a source with four-velocity $u_e^{\mu }$ and received by an observer with four-velocity $u_o^{\mu }$, the measured frequency is proportional to $-k_{\mu }{u}^{\mu }$. Hence

$$1+z={\displaystyle \frac{{\left(k_{\mu }{u}^{\mu }\right)}_e}{{\left(k_{\mu }{u}^{\mu }\right)}_o}}.$$

(23)

This expression is coordinate invariant and is the appropriate starting point for a high-redshift timing test [2,3].

For successive photons emitted by the same local process over a sufficiently short source-frame proper-time interval, the observed interval inherits the corresponding redshift stretching. Type Ia supernova observations have directly detected the expected approximately $1+z$ time dilation [25,26], and the Dark Energy Survey has measured the power-law exponent in $\Delta t_{\mathrm{obs}}=\Delta t_{\mathrm{em}}{(1+z)}^b$ as $b=1.003\pm 0.005\left(\mathrm{stat}\right)\pm 0.010\left(\mathrm{sys}\right)$ over $0.1\lesssim z\lesssim 1.2$ [27].

Cumulative source-worldline aging therefore does not, by itself, justify replacing $1+z$ by a nonlinear phenomenological dilation law. Let $T_{\mathrm{src}}(z,\mathcal{E})$ be a standardized source-frame timescale in environment $\mathcal{E}$, and let $T_{\mathrm{ref}}$ be its calibrated reference value. Then

$$T_{\mathrm{obs}}(z,\mathcal{E})=(1+z)T_{\mathrm{src}}(z,\mathcal{E}),$$

(24)

and define

$$\mathcal{R}(z,\mathcal{E})\equiv {\displaystyle \frac{T_{\mathrm{obs}}(z,\mathcal{E})}{(1+z)T_{\mathrm{ref}}}}={\displaystyle \frac{T_{\mathrm{src}}(z,\mathcal{E})}{T_{\mathrm{ref}}}}.$$

(25)

After accounting for population evolution, progenitor physics, metallicity, selection, and measurement bias, the standard expectation is approximately $\mathcal{R}\simeq 1$.

RTD-EU may predict $\mathcal{R}\ne 1$ only if the derived spacetime changes the source-frame physical evolution or the operational relation between measured redshift and the source’s cosmological history. A previously proposed phenomenological high-redshift dilation curve should therefore be regarded as an exploratory ansatz until it is recovered from a specified metric, source model, and null-geodesic calculation.

12. Independent Chronometric and Relativistic Constraints

A percent-level environmental clock effect cannot be tested in isolation. Relevant cross-checks include:

• Cosmic chronometers. Differential ages of passively evolving galaxies constrain $\mathrm{d}z/\mathrm{d}t$ under population assumptions [23,24]. RTD-EU must predict what time variable those ages operationally sample rather than assuming they equal a metric lapse.
• Atomic-clock and gravitational-redshift tests. Modern optical clocks verify relativistic gravitational and kinematic clock shifts with extremely high precision in controlled local environments [28,29]. These tests validate the local GR clock framework that any cosmological extension must respect.
• CMB and BAO. The acoustic angular scale and standard-ruler distances constrain the integrated expansion and light-cone geometry [18,19].
• Lensing and distances. The same metric perturbations that alter time and redshift also alter null focusing and magnification; they cannot be adjusted independently [30].
• Structure growth and peculiar velocities. A metric large enough to create a percent-level clock effect may also produce observable signatures in velocities, clustering, and relativistic light-cone effects. Relativistic simulations provide a direct way to test this consistency [17].

These constraints are not merely external checks. They are part of the definition of closure: one spacetime must account for all channels at once.

13. Dark Energy: What Would Have to Be Demonstrated

RTD-EU does not, at its present stage, establish that dark energy is unnecessary. The original Type Ia supernova evidence for late-time acceleration [20,21] is only one part of the evidence entering the standard cosmological model. To argue that a chronometric effect replaces a cosmological constant or another dark-energy component, a model with the proposed clock mapping would have to fit, at minimum, supernova luminosity distances, BAO, the CMB acoustic scale and lensing, structure growth, and local distance calibration without inserting the required acceleration through another free function.

This standard is intentionally stronger than reproducing a single Hubble diagram. It is also where existing backreaction and regional-clock literature becomes essential: some approaches have argued that inhomogeneity can produce important apparent effects [10,11], whereas other analyses find strong limits on the ability of small-scale structure to mimic dark energy [12,13]. RTD-EU can contribute to this question only through an explicit model that is testable against both sets of constraints.

14. Numerical Implementation and Companion Chronometric Tool

The closure program is naturally suited to numerical implementation. A companion Cosmological Proper-Time Geodesy (CPG) tool has been developed within the broader RTD-EU research program as a diagnostic architecture for integrating clock histories and comparing independent chronometric channels. The role of such software is diagnostic: numerical agreement between integration pathways checks the implementation, but it does not create a physical lapse or validate a nonstandard cosmology by itself.

For a formal software release, reproducibility requires at least: a versioned source archive, dependency and parameter files, test cases with known FLRW/weak-field limits, convergence tolerances, a machine-readable output definition, and a permanent repository identifier. The software DOI or archival identifier should be inserted into the final public metadata when available. The numerical code should also retain a strict separation between (i) invariant worldline proper time, (ii) gauge-dependent intermediate lapse variables, and (iii) observable residuals inferred from actual data.

A practical sequence is:

1.

verify exact FLRW closure, $\Delta {\tau }_{AB}=0$, for identical comoving histories;

2.

reproduce the weak-field estimate in Eq. (17);

3.

test an exact inhomogeneous spacetime such as an LTB or Szekeres model;

4.

compare with a relativistic structure simulation;

5.

propagate null geodesics through the same model; and

6.

confront the derived residuals with chronometer, transient, distance, and lensing data.

15. Falsification Criteria

The macroscopic RTD-EU proposal should be considered falsified as an explanation of cosmological tensions if observationally viable implementations consistently satisfy all of the following:

$$\left|\Delta {\tau }_{AB}\right|/\tau \ll {10}^{-2},$$

(26)

no derived source-frame transient residual distinguishable from $\mathcal{R}=1$, and no distance/redshift correction large enough to modify the relevant parameter inference.

Conversely, a potentially significant result would require a self-consistent model that produces a percent-level invariant $\Delta {\tau }_{AB}$ without fitting that value by construction, while simultaneously remaining compatible with the redshift relation, supernova time dilation, BAO, CMB, lensing, local clock tests, and structure observables.

The distinction is important: ordinary gravitational time dilation is already part of General Relativity and is not at stake. What is being tested is specifically the claim that realistic cosmological inhomogeneity can generate a macroscopic integrated proper-time effect capable of altering cosmological inference.

16. Conclusion

RTD-EU is most defensible when formulated as a problem of relativistic chronometric closure rather than as a rejection of FLRW cosmic time. FLRW cosmic time is the proper time of the homogeneous comoving congruence. Other physically specified worldlines can accumulate different proper times, but the magnitude of that difference must be calculated from the geometry rather than inferred directly from the existence of inhomogeneous expansion.

The central program can be summarized as

$$\begin{array}{cc}\hfill \mathrm{matter}+\mathrm{initial}\mathrm{data}& ⟶\{g_{\mu \nu },u_K^{\mu }\},\hfill \\ \hfill & ⟶{\tau }_K\left[{\gamma }_K\right]⟶\Delta {\tau }_{AB},\hfill \\ \hfill & ⟶\{z,D_L,D_A,\mathrm{clocks}\}.\hfill \end{array}$$

(27)

with any lapse factor ${\Gamma }_K$ understood as a foliation-dependent intermediate representation rather than an observable in its own right.

The weak-field benchmark indicates that ordinary halo-scale potentials naturally produce effects near ${10}^{-5}$, not at the percent level. Therefore the burden of proof for a macroscopic RTD-EU mechanism is clear: a relativistic inhomogeneous model must generate the larger effect and must do so while preserving the full set of cosmological and local relativistic constraints. If it succeeds, the result could provide a geometric contribution to cosmological parameter tensions or early structure formation. If all viable implementations remain near the weak-field scale, the proposed macroscopic mechanism is ruled out. Either outcome is scientifically useful.