Phase-Curvature Effects on Cosmic Time and Chronometer Inference

Cosmic chronometers infer the expansion history from differential age dating of massive passive galaxies and therefore probe cosmological inference through time rather than through integrated distance observables [citation]. Recent work has sharpened both sides of that observational setting: the differential-age signal itself has been clarified as a kinematic observable only under explicit spacetime and tracer assumptions [citation], late-time model-independent ladder calibrations using chronometers together with DESI-era distance data have made the anchor dependence of low-redshift inference increasingly transparent [citation], and covariance-complete chronometer inference has become an explicit methodological issue in its own right [citation]. The analysis below is confined to that late-time chronometer/time-inference layer. Throughout, the imported background history, the untreated distance-dominated anchor, and the clock map remain distinct formally specified objects on the declared window; no mixed late-time background-distance-clock decomposition is constructed here.

Three variables are kept distinct throughout:

\text{metric FRW time } t,
\qquad
\text{raw CHC-coupled clock time } \tau_{\mathrm{CHC}},
\qquad
\text{reconstructed coordinate } t_{\mathrm{eff}}.

The first is the time coordinate of the FRW background metric. The second is the raw output of a CHC-coupled clock. The third is the coordinate reconstructed when that raw output is numerically identified with metric time.

The response branch gives

d\tau_{\mathrm{CHC}}=\sqrt{1-\Xicos(t)}\,dt,
\qquad
0\le \Xicos(t)<1,

with $\Xicos(t_0)\approx 0$TeX\Xicos(t_0)\approx 0 at the present epoch. Equation reference follows from the root-theory probe metric after the normalization $A=1$TeXA=1 and the definition $\Xicos=B(\bar\chi)\dot{\bar\chi}^{\,2}/\Lambda_\Xi^4$TeX\Xicos=B(\bar\chi)\dot{\bar\chi}^{\,2}/\Lambda_\Xi^4. Whether a stellar-population age estimator tracks this universal proper time is an empirical measurement-model question tested below. Distance-dominated late-time reconstructions enter as independent external anchors on a specified comparison interval [citation].

Figure or table content is omitted from the web reader; use the canonical manuscript for the exact object.

Metric time and CHC-coupled clock time

In a spatially flat FRW background,

ds^2=-dt^2+a^2(t)\,d\bm{x}^2,

so $t$TeXt is the metric time and the standard proper time of comoving observers.

On the homogeneous cosmological branch, the root-theory response metric is

\mathrm d\widetilde s^2=A^2(\bar\chi)\left[-\left(1-\frac{B(\bar\chi)\dot{\bar\chi}^{\,2}}{\Lambda_\Xi^4}\right)\mathrm d t^2+a^2(t)\mathrm d\bm x^2\right].

Define $\Xicos=B(\bar\chi)\dot{\bar\chi}^{\,2}/\Lambda_\Xi^4$TeX\Xicos=B(\bar\chi)\dot{\bar\chi}^{\,2}/\Lambda_\Xi^4 and use the local normalization $A=1$TeXA=1 throughout the remainder of this paper.

proposition: Clock map from the covariant response metric. Assume a comoving ideal clock whose matter action is locally Lorentz invariant with respect to $\widetilde g_{\mu\nu}$TeX\widetilde g_{\mu\nu}. If $0\le\Xicos<1$TeX0\le\Xicos<1, its proper time and the coordinate-frequency representation of a constant proper transition energy $\Delta E_0$TeX\Delta E_0 satisfy

\mathrm d\tau_{\mathrm{CHC}}=\sqrt{1-\Xicos(t)}\,\mathrm d t,
\qquad
\Delta E_{\mathrm{CHC}}(t)=\Delta E_0\sqrt{1-\Xicos(t)}.

proof. Along a comoving worldline, $\mathrm d\bm x=0$TeX\mathrm d\bm x=0. The definition $\mathrm d\tau_{\mathrm{CHC}}^2=-\mathrm d\widetilde s^2$TeX\mathrm d\tau_{\mathrm{CHC}}^2=-\mathrm d\widetilde s^2 and reference with $A=1$TeXA=1 give the first relation. An ideal transition accumulates phase $\mathrm d\varphi=\Delta E_0\mathrm d\tau_{\mathrm{CHC}}/\hbar$TeX\mathrm d\varphi=\Delta E_0\mathrm d\tau_{\mathrm{CHC}}/\hbar. Expressing the same phase as $\mathrm d\varphi=\Delta E_{\mathrm{CHC}}\mathrm d t/\hbar$TeX\mathrm d\varphi=\Delta E_{\mathrm{CHC}}\mathrm d t/\hbar gives the second. The inequality $\Xicos<1$TeX\Xicos<1 is precisely the Lorentzian-signature condition on this timelike branch.

Let $\Delta E_0$TeX\Delta E_0 denote the phase-flat reference value. Equation reference gives

\Delta E_{\mathrm{CHC}}(t)=\Delta E_0\sqrt{1-\Xicos(t)},
\qquad
0\le \Xicos(t)<1.

The corresponding clock period is

T_{\mathrm{CHC}}(t)=\frac{2\pi\hbar}{\Delta E_{\mathrm{CHC}}(t)}
=
T_0\,\frac{1}{\sqrt{1-\Xicos(t)}},
\qquad
T_0=\frac{2\pi\hbar}{\Delta E_0},

so the raw clock increment satisfies

d\tau_{\mathrm{CHC}}=\sqrt{1-\Xicos(t)}\,dt.

Equations reference--reference are therefore consequences of the covariant response metric for ideal clocks. Their application to galaxy-age estimators still requires the proxy condition in reference. At the present epoch the local anchoring condition is

\Xicos(t_0)\approx 0.

Raw clock output and reconstructed coordinate

The raw variable $\tau_{\mathrm{CHC}}$TeX\tau_{\mathrm{CHC}} is what a CHC-coupled chronometer records. The reconstructed coordinate $t_{\mathrm{eff}}$TeXt_{\mathrm{eff}} is obtained when that raw clock output is used as if no CHC correction were present. Accordingly,

dt_{\mathrm{eff}}\equiv d\tau_{\mathrm{CHC}}=\sqrt{1-\Xicos(t)}\,dt.

Equation reference is therefore a typing identity: the numerical value of the raw clock increment is promoted to a coordinate increment, but the logical roles of $t$TeXt, $\tau_{\mathrm{CHC}}$TeX\tau_{\mathrm{CHC}}, and $t_{\mathrm{eff}}$TeXt_{\mathrm{eff}} remain distinct.

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Declared late-time window and redshift form

definition: Declared late-time window. A clock-map analysis is stated on a finite redshift window

\Zwin=[0,z_\star],
\qquad
z_\star>0,

for which chronometer pipelines, tracer selections, stellar-population-synthesis (SPS) choices, and any external-anchor assumptions are declared. No statement in the present analysis requires a continuation of $\Xicos(z)$TeX\Xicos(z) beyond $\Zwin$TeX\Zwin. All exact-style chronometric reductions, finite-window age offsets, and conditional chronometer-inference claims below are read only relative to this declared late-time window.

Using

\frac{dz}{dt}=-(1+z)H(z),

one obtains

dt=-\frac{dz}{(1+z)H(z)},

and therefore

dt_{\mathrm{eff}}(z)=-\frac{dz}{(1+z)H(z)}\sqrt{1-\Xicos(z)}.

Equation reference is the basic late-time mapping object used below.

Lookback time

The standard FRW lookback time is

t_L^{\Lambda\mathrm{CDM}}(z)=\int_0^z \frac{dz'}{(1+z')H(z')}.

The present map replaces the inferred time increment by reference, so the reconstructed lookback time becomes

t_L^{\mathrm{clock}}(z)=\int_0^z \frac{dz'}{(1+z')H(z')}\sqrt{1-\Xicos(z')}.

proposition: Clock-mapped lookback reduction on the declared late-time window. This reduction is asserted only on the declared late-time window and not outside it. If $0\le \Xicos(z)<1$TeX0\le \Xicos(z)<1 on the declared interval and the background history $H(z)$TeXH(z) is held fixed, then

t_L^{\mathrm{clock}}(z)\le t_L^{\Lambda\mathrm{CDM}}(z),

with strict inequality whenever $\Xicos(z')>0$TeX\Xicos(z')>0 on a set of nonzero measure.

proof. The integrands in reference and reference differ only by the factor $\sqrt{1-\Xicos(z')}\le 1$TeX\sqrt{1-\Xicos(z')}\le 1.

Finite-window accumulated age offset

For a declared late-time window $\Zwin=[0,z_\star]$TeX\Zwin=[0,z_\star], define the accumulated age offset relative to the standard FRW inference by

\Delta t_{\mathrm{age}}^{\mathrm{clock}}(z_\star)
\equiv
\int_0^{z_\star}\frac{dz}{(1+z)H(z)}\Bigl[1-\sqrt{1-\Xicos(z)}\Bigr].

proposition: Nonnegative finite-window age offset on the declared late-time window. This accumulated offset is asserted only on the declared late-time window and not as a global age relation. If $0\le \Xicos(z)<1$TeX0\le \Xicos(z)<1 on the declared late-time window $\Zwin$TeX\Zwin, then

\Delta t_{\mathrm{age}}^{\mathrm{clock}}(z_\star)\ge 0,

with strict inequality whenever $\Xicos(z)>0$TeX\Xicos(z)>0 on a set of nonzero measure in $[0,z_\star]$TeX[0,z_\star].

proof. The integrand in reference is nonnegative whenever $0\le \Xicos(z)<1$TeX0\le \Xicos(z)<1.

A total-age relation with upper limit $\infty$TeX\infty would require a separate continuation of $\Xicos(z)$TeX\Xicos(z) beyond the declared late-time window together with an early-time admissibility analysis. No such global continuation is part of the present analysis.

Standard chronometer relation

The cosmic-chronometer method estimates the expansion rate through

H(z)=-\frac{1}{1+z}\frac{dz}{dt}.

At the methodological level, the principal systematics are tracer purity, residual young-population contamination, SPS modeling, and the resulting covariance budget [citation].

CHC-coupled chronometer inference

If the physically measured differential time is $d\tau_{\mathrm{CHC}}$TeXd\tau_{\mathrm{CHC}} rather than $dt$TeXdt, then

\frac{dz}{d\tau_{\mathrm{CHC}}}
=
\frac{dz}{dt}\frac{dt}{d\tau_{\mathrm{CHC}}}
=
-(1+z)H(z)\frac{1}{\sqrt{1-\Xicos(z)}}.

If the raw clock increment is identified with metric time, the inferred chronometer expansion rate is

H_{\mathrm{inf}}^{\mathrm{CC}}(z)
=
-\frac{1}{1+z}\frac{dz}{d\tau_{\mathrm{CHC}}}
=
\frac{H(z)}{\sqrt{1-\Xicos(z)}}.

Equation reference is a conditional chronometer-inference relation relative to an imported background rate $H(z)$TeXH(z) on the same declared window. It does not by itself reconstruct $H(z)$TeXH(z) from chronometer data alone; it specifies how an admitted raw-clock pipeline would map a fixed background history into an inferred chronometer rate.

proposition: Conditional chronometer-inference bias on the declared late-time window. This bias relation is asserted only on the declared late-time window and for a fixed imported background history $H(z)$TeXH(z). Within the late-time clock map,

H_{\mathrm{inf}}^{\mathrm{CC}}(z)\ge H(z),

with equality if and only if $\Xicos(z)=0$TeX\Xicos(z)=0.

proof. Equation reference together with $0\le \Xicos(z)<1$TeX0\le \Xicos(z)<1 implies $1/\sqrt{1-\Xicos(z)}\ge 1$TeX1/\sqrt{1-\Xicos(z)}\ge 1.

Equation reference is interpreted only for chronometer pipelines that satisfy the admissibility gates introduced next.

The mapping reference is scientifically admissible only if it can be distinguished from method-specific chronometer systematics and compared to external late-time anchors on a declared window. Admissibility is therefore enforced by the gates G0--G3.

Gate G0: chronometer-proxy admissibility

For an admitted chronometer pipeline $A$TeXA, let $d\hat t_A$TeXd\hat t_A denote the differential age variable returned by the pipeline after nuisance control. Gate G0 requires a reduction of that age variable to the common raw clock increment:

d\hat t_A=\bigl[1+\epsilon_A(z)\bigr]d\tau_{\mathrm{CHC}},
\qquad
|\epsilon_A(z)|\le \sigma_A^{\mathrm{proxy}}(z),

where $\sigma_A^{\mathrm{proxy}}(z)$TeX\sigma_A^{\mathrm{proxy}}(z) is included in the declared pipeline covariance budget.

Pipelines are not presumed admissible by default. Only pipelines passing G0 are admitted into reference. If no independently justified reduction from $d\hat t_A$TeXd\hat t_A to $d\tau_{\mathrm{CHC}}$TeXd\tau_{\mathrm{CHC}} is available on the declared window, the pipeline is excluded from the admitted clock-map set and treated as method-specific rather than as a direct probe of the universal clock factor. This requirement is motivated by the known dependence of chronometer reconstruction on tracer selection, SPS modeling, and spectral-age proxy control [citation].

Gate G1: cross-pipeline consistency

Let $H_{\mathrm{inf}}^{A}(z)$TeXH_{\mathrm{inf}}^{A}(z) and $H_{\mathrm{inf}}^{B}(z)$TeXH_{\mathrm{inf}}^{B}(z) be chronometer inferences obtained from two independent admitted pipelines within the same platform class. Define

R_{AB}(z)\equiv \frac{H_{\mathrm{inf}}^{A}(z)}{H_{\mathrm{inf}}^{B}(z)}.

Gate G1 is passed on the declared window if

|R_{AB}(z_i)-1|\le n\,\sigma_{AB}(z_i)
\qquad
\text{for all declared bins } z_i\in \Zwin,

and if the residuals $R_{AB}(z_i)-1$TeXR_{AB}(z_i)-1 show no coherent monotone redshift trend above the declared significance threshold.

Here $\sigma_{AB}(z_i)$TeX\sigma_{AB}(z_i) is the covariance-propagated uncertainty of the ratio in the declared bin, and $n$TeXn is the declared acceptance multiplier. A coherent trend in $R_{AB}(z)$TeXR_{AB}(z) is evidence for pipeline-specific chronometer systematics rather than a universal multiplicative clock factor.

Gate G2: compatibility with external late-time anchors

Let $H_{\mathrm{dist}}(z)$TeXH_{\mathrm{dist}}(z) denote an untreated external late-time anchor constructed from BAO or SN+BAO(+calibration) pipelines [citation]. In the present analysis, $H_{\mathrm{dist}}$TeXH_{\mathrm{dist}} is never promoted to an internally reconstructed background history; it remains a comparison input only on a declared late-time window. Recent late-time analyses indicate that apparent dark-energy trends can depend on the adopted supernova compilation and on residual systematics, so any use of $H_{\mathrm{dist}}$TeXH_{\mathrm{dist}} must remain explicitly anchor-conditioned rather than being interpreted as an autonomous background reconstruction [citation].

Under the explicit assumption that $H_{\mathrm{dist}}(z)$TeXH_{\mathrm{dist}}(z) approximates the underlying background $H(z)$TeXH(z) on that comparison window, define the diagnostic estimator

\Xiest(z)\equiv 1-\left(\frac{H_{\mathrm{dist}}(z)}{H_{\mathrm{inf}}^{\mathrm{CC}}(z)}\right)^2.

Equation reference is meaningful only under that declared anchor assumption. It is not an autonomous reconstruction of the background history, but an anchor-conditioned diagnostic on a declared late-time comparison window. If no such assumption is adopted, or if the anchor has already absorbed propagation-side corrections, early-time continuation, or a joint clock-map ansatz, then reference is not evaluated as a standalone clock-map diagnostic. Gate G2 is passed if

\Xiest(z_i)\ge -n\,\sigma_{\Xi}(z_i),
\qquad
|\Xiest(0)|\le \epszero,

for all declared bins $z_i$TeXz_i in the comparison window, and if the inferred profile remains mutually compatible across chronometer pipelines that pass G0 and G1.

Here $\sigma_{\Xi}(z_i)$TeX\sigma_{\Xi}(z_i) is the covariance-propagated uncertainty of the estimator in the declared bin and $\epszero$TeX\epszero is the low-redshift anchoring tolerance. Throughout, $H_{\mathrm{dist}}$TeXH_{\mathrm{dist}} denotes an untreated late-time distance-dominated anchor on the declared comparison window. If propagation-side corrections, early-time continuations, or joint clock-map assumptions are absorbed into that anchor, then reference is not used as a standalone clock-map diagnostic and the analysis becomes a joint model rather than a standalone clock-map test.

Gate G3: robustness under chronometer choices

A signal admitted by G0--G2 must remain stable, within the declared covariance budget, under reasonable variations of

- redshift binning, - passive-tracer selection cuts, and - SPS library choices.

A feature that appears only for a narrow or unstable chronometer configuration is not admitted as evidence for the clock map [citation].

For implementation-level diagnostics on a declared late-time window, one may fix a one-parameter witness family together with one untreated anchor convention, one redshift binning, and one covariance prescription. The formulas below define a residual diagnostic under that unchanged comparison convention; they do not elevate the chosen family to a unique physical profile or to an additional calibration layer.

definition: One-parameter witness family. On a declared late-time window $\Zwin=[0,z_\star]$TeX\Zwin=[0,z_\star], fix

\Xicos^{\star}(z;\Xi_0)=\Xi_0\,\frac{z}{1+z},
\qquad
0\le \Xi_0<1.

This family satisfies $\Xicos^{\star}(0)=0$TeX\Xicos^{\star}(0)=0 and $0\le \Xicos^{\star}(z)<1$TeX0\le \Xicos^{\star}(z)<1 on $\Zwin$TeX\Zwin. All witness-family residual comparisons below are read only relative to this fixed one-parameter family on the declared late-time window.

Under the explicit external-anchor assumption of Gate G2, one may form the anchor-conditioned comparison curve

H_{\mathrm{CC|anchor}}^{\star}(z;\Xi_0)
\equiv
\frac{H_{\mathrm{dist}}(z)}{\sqrt{1-\Xicos^{\star}(z;\Xi_0)}}.

Equation reference is an anchor-conditioned comparison curve induced by the declared anchor $H_{\mathrm{dist}}$TeXH_{\mathrm{dist}} and the witness family; it is not a standalone physical prediction of the clock map by itself and not an independent reconstruction of $H(z)$TeXH(z). The corresponding finite-window age-offset corollary is

\Delta t_{\mathrm{age}}^{\star}(z_\star;\Xi_0)
=
\int_0^{z_\star}\frac{dz}{(1+z)H_{\mathrm{dist}}(z)}\Bigl[1-\sqrt{1-\Xicos^{\star}(z;\Xi_0)}\Bigr],

which remains auxiliary and is not promoted to an independent age-fit program.

For an admitted chronometer pipeline $A$TeXA with declared covariance budget $\sigma_A^{\mathrm{CC}}(z_i)$TeX\sigma_A^{\mathrm{CC}}(z_i), define the chronometer residual on that fixed convention

R_A^{\mathrm{CC}}(\Xi_0)
=
\sup_{z_i\in\Zwin}
\frac{\bigl|H_{\mathrm{inf}}^{A}(z_i)-H_{\mathrm{CC|anchor}}^{\star}(z_i;\Xi_0)\bigr|}{\sigma_A^{\mathrm{CC}}(z_i)}.

Likewise define the corresponding anchor-side residual

R_{\Xi}(\Xi_0)
=
\sup_{z_i\in\Zwin}
\frac{\bigl|\Xiest(z_i)-\Xicos^{\star}(z_i;\Xi_0)\bigr|}{\sigma_{\Xi}(z_i)}.

When age-sensitive late-time inferences are available on the same window, one may additionally report

R_{\mathrm{age}}(\Xi_0)
=
\sup_{z_i\in\Zwin}
\frac{\bigl|\Delta t_{\mathrm{age}}^{\mathrm{obs}}(z_i)-\Delta t_{\mathrm{age}}^{\star}(z_i;\Xi_0)\bigr|}{\sigma_{\mathrm{age}}(z_i)},

but this remains an auxiliary corollary rather than a primary chronometer diagnostic.

definition: Fixed-convention residual admissibility. Fix a single one-parameter witness family $\Xicos^{\star}(z;\Xi_0)$TeX\Xicos^{\star}(z;\Xi_0), one untreated external anchor $H_{\mathrm{dist}}$TeXH_{\mathrm{dist}}, one chronometer platform class, one redshift binning, and one covariance prescription on the declared late-time window. The residual diagnostic is admissible only if there exists at least one $\Xi_0$TeX\Xi_0 such that

R_A^{\mathrm{CC}}(\Xi_0)\le n,
\qquad
R_{\Xi}(\Xi_0)\le n,

for all chronometer pipelines $A$TeXA that pass G0 and G1, while remaining stable under G3. Changing the witness family, the anchor convention, the binning, or the covariance prescription defines a different comparison convention. All downstream residual admissibility and benchmark-conditioned residual statements are read only relative to this fixed comparison convention.

The standalone clock map is rejected on a declared window if any of the following occurs:

- the relevant chronometer pipelines fail G0, so no common reduction to the raw clock increment $d\tau_{\mathrm{CHC}}$TeXd\tau_{\mathrm{CHC}} is admitted; - no nonnegative profile $\Xicos(z_i)$TeX\Xicos(z_i) with $\Xicos(0)\approx 0$TeX\Xicos(0)\approx 0 and $\Xicos(z_i)<1$TeX\Xicos(z_i)<1 satisfies reference while remaining compatible with G1 and G3; - the estimator reference violates G2 by becoming significantly negative, losing low-redshift anchoring, or exhibiting unresolved pipeline dependence; - the apparent signal disappears once tracer, binning, SPS, and proxy-admissibility covariances are propagated; - no single fixed witness family $\Xicos^{\star}(z;\Xi_0)$TeX\Xicos^{\star}(z;\Xi_0) can satisfy the chronometer and anchor residual bounds reference on the same window within one unchanged comparison convention; - the claimed effect requires propagation-side corrections or a continuation of $\Xicos(z)$TeX\Xicos(z) beyond the declared late-time window.

A positive result would support only a mapping-level statement: on the declared late-time window, admitted chronometer pipelines are compatible with a nontrivial factor $\sqrt{1-\Xicos(z)}$TeX\sqrt{1-\Xicos(z)} in raw clock inference. It would not by itself identify a microscopic clock Hamiltonian, fix a unique physical family for $\Xicos(z)$TeX\Xicos(z), or redefine the background cosmology.

The compact-target theorem separates two possible chronometer effects. A spatially homogeneous phase history lies in the zero-spatial-winding sector and is governed by the local canonical equations already used here. A nonzero winding configuration necessarily carries gradient energy and cannot be continuously reduced to the homogeneous constant-phase branch. Its contribution to clock comparison must be obtained from a solved inhomogeneous background and an explicit clock coupling; topology alone supplies no universal frequency shift.

If several clock species or time-transfer channels share one phase-dependent metric or coupling, their shifts form a common response map. At fixed background the number of independent first-order responses cannot exceed the rank of the shared parameter sensitivity. Left-null combinations must therefore vanish, providing calibration-independent consistency tests. Allowing a separate phase coefficient for every clock removes these restrictions and converts the construction into unconstrained phenomenology.

The closure test for the cosmic-time and chronometer response is applied to a dimensionless observable vector $y\in\mathbb R^m$TeXy\in\mathbb R^m formed from fixed reference scales and the declared basket of chronometer ages, differential expansion estimates, clock ratios, and distance cross-checks. Let $a$TeXa range over the independent constitutive inputs comprising clock coupling, phase-background profile, formation-time nuisance, and population response.

proposition: Functional saturation, finite closure, and sector admissibility. Suppose the unrestricted prediction map $F:a\mapsto y$TeXF:a\mapsto y is continuously differentiable on a Banach space of constitutive inputs. If $D_aF$TeXD_aF is surjective and has a bounded right inverse at the calibration point, the unrestricted family is locally open in observable space and supplies no nonzero local equality restriction on $y$TeXy. Suppose instead that a single microscopic closure replaces $a$TeXa by finite parameters $\theta\in\mathbb R^p$TeX\theta\in\mathbb R^p, with profiled nuisance coordinates $\eta\in\mathbb R^q$TeX\eta\in\mathbb R^q. If

J=D_\eta F_{\rm cl}D_\theta F_{\rm cl},
 \qquad \operatorname{rank}J=r<m,

then there are $m-r$TeXm-r independent first-order restrictions

w^{\mathsf T}\delta y=0,
 \qquad w\in\ker J^{\mathsf T}.

If the rank is constant locally, these restrictions are tangent to a compatibility manifold of codimension $m-r$TeXm-r. For this sector, the finite closure is admissible only if one microscopic clock coupling and one inhomogeneous phase solution are shared across all clock populations and distance comparisons.

proof. Split surjectivity gives a bounded right inverse $R$TeXR with $D_aF\,R=I_m$TeXD_aF\,R=I_m. The Banach-space submersion theorem then makes $F$TeXF locally onto a neighborhood of the calibrated observable vector. Any smooth equality holding throughout that image must therefore vanish on an open set and contributes no model-specific local restriction. Under finite closure, the attainable first-order variations are exactly the column space of $J$TeXJ. Its orthogonal complement is $\ker J^{\mathsf T}$TeX\ker J^{\mathsf T}, whose dimension is $m-r$TeXm-r by rank--nullity, which proves reference. The constant-rank theorem supplies the stated local manifold. The sector condition is necessary because population-specific phase responses can absorb every chronometer residual without testing a common clock law. Failure of that condition therefore rejects the proposed microscopic closure before parameter estimation can be counted as evidence for it.

This paper studies a late-time clock map in which metric FRW time $t$TeXt, raw CHC-coupled clock time $\tau_{\mathrm{CHC}}$TeX\tau_{\mathrm{CHC}}, and reconstructed coordinate $t_{\mathrm{eff}}$TeXt_{\mathrm{eff}} are kept distinct and related by

d\tau_{\mathrm{CHC}}=dt_{\mathrm{eff}}=\sqrt{1-\Xicos(t)}\,dt,
\qquad
\Xicos(t_0)\approx 0.

On a declared late-time window this yields the reconstructed lookback relation reference, the finite-window accumulated age-offset corollary reference, and the chronometer bias relation

H_{\mathrm{inf}}^{\mathrm{CC}}(z)=\frac{H(z)}{\sqrt{1-\Xicos(z)}}.

The empirical content of the analysis is confined to that chronometer/inference layer. Applicability requires proxy admissibility, cross-pipeline consistency, compatibility with untreated external late-time anchors under an explicit anchor assumption, and robustness under tracer, binning, and SPS choices, with any auxiliary residual diagnostic carried out under one unchanged comparison convention. The paper does not claim that current chronometer data already require $\Xicos(z)\neq 0$TeX\Xicos(z)\neq 0; it states only the admissibility conditions for evaluating that possibility on a declared late-time window. Global age reconstruction, early-time chronology, microscopic clock closure, propagation-side corrections, and any mixed late-time background-distance-clock decomposition are outside the present scope.

The analysis does not require a unique physical form for $\Xicos(z)$TeX\Xicos(z). For descriptive asymptotics one may reuse the witness family of reference in the small-amplitude form

\Xicos(z)=\varepsilon\,\frac{z}{1+z},
\qquad
0<\varepsilon\ll 1,

which obeys $\Xicos(0)=0$TeX\Xicos(0)=0 and grows smoothly across the late-time window.

For small $\varepsilon$TeX\varepsilon,

\sqrt{1-\Xicos(z)}=1-\frac{1}{2}\Xicos(z)+\mathcal O(\varepsilon^2),

so

t_L^{\mathrm{clock}}(z)
=
t_L^{\Lambda\mathrm{CDM}}(z)
-\frac{1}{2}\int_0^z \frac{dz'}{(1+z')H(z')}\Xicos(z')
+\mathcal O(\varepsilon^2),

and

\frac{H_{\mathrm{inf}}^{\mathrm{CC}}(z)}{H(z)}
=1+\frac{1}{2}\Xicos(z)+\mathcal O(\varepsilon^2).

These formulas are illustrative only.

Funding and competing interests..

No external funding was received for this work. The author declares no competing interests.