Local and Global Light Speed in the CHC Framework

Precision information on electromagnetic propagation is obtained most stringently from Solar-System ranging, radio science, and laboratory isotropy experiments, whereas cosmological observables are inferred from line-of-sight integrals accumulated over much larger scales. A propagation-sector modification is therefore admissible only if it remains compatible with existing local constraints before it is used in any cosmological inference [citation]. In optical-metric treatments, the passage from a line-of-sight propagation operator to luminosity- and angular-diameter-distance relations depends on the declared export class, because refraction, absorption, and reciprocity assumptions need not be interchangeable [citation]. The present paper therefore fixes the export class explicitly instead of treating distance formulas as automatic consequences of a local propagation law.

On the nondegenerate root branch, the canonical scalar $\chi$TeX\chi satisfies $\nabla_\mu\chi=\Meff\nabla_\mu\HH$TeX\nabla_\mu\chi=\Meff\nabla_\mu\HH. The kinetic hierarchy parameter is therefore

\Xi_{\nabla} \equiv \frac{\mathcal K_\chi}{\Lambda_{\Xi}^{4}},

with the nonnegative norm $\mathcal K_\chi$TeX\mathcal K_\chi selected by the background and the ray direction. The root representation theorem gives the test-probe metric

\widetilde g_{\mu\nu}=A^2(\chi)\left[g_{\mu\nu}+\frac{B(\chi)}{\Lambda_\Xi^4}\nabla_\mu\chi\nabla_\nu\chi\right]

to leading first-derivative order. The present analysis studies its propagation consequences in the negligible-backreaction limit. The quantity of interest is the difference between the gravitational metric-cone speed and the probe-cone speed inferred from finite propagation protocols,

\text{causal speed } c_{\infty}
\qquad\text{and}\qquad
\text{operationally inferred propagation speed } c_{\mathrm{local}}.

The former fixes the tensor characteristic cone; the latter is extracted from finite time-transfer, ranging, or Doppler protocols. When $B\ne0$TeXB\ne0, the two effective cones are physically distinct and their difference is directly constrained; it is not a redefinition of the SI value of $c$TeXc.

The analysis proceeds in three steps. First, an optical-response bridge is used to represent the local propagation correction through a path integral. Second, Solar-System admissibility is imposed on Cassini-class two-way conjunction families at both delay and Doppler levels. Third, a coarse-grained cosmological profile $\Xicos(z)$TeX\Xicos(z) is admitted into distance integrals only if the same underlying realization survives the local contract. In particular, a smooth background profile is not accepted merely because it provides a convenient cosmological fit variable.

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

The present paper remains confined to the propagation branch. No chronometer or cosmic-time complement is developed here, and no late-time mixed-branch observational inference is attempted on the basis of the present propagation export alone.

definition: Propagation variables. The propagation quantities used below are:

- $c_{\infty}=c_g$TeXc_{\infty}=c_g, the local tensor-cone speed of $g_{\mu\nu}$TeXg_{\mu\nu}; - $c_{\mathrm{local}}(x,e)$TeXc_{\mathrm{local}}(x,e), the probe-cone speed inferred along the local spatial unit direction $e^i$TeXe^i; - $\Xi_t=B(u^\mu\nabla_\mu\chi)^2/\Lambda_\Xi^4$TeX\Xi_t=B(u^\mu\nabla_\mu\chi)^2/\Lambda_\Xi^4, the temporal projection measured by an observer $u^\mu$TeXu^\mu; - $\Xi_\parallel=B(e^\mu\nabla_\mu\chi)^2/\Lambda_\Xi^4$TeX\Xi_\parallel=B(e^\mu\nabla_\mu\chi)^2/\Lambda_\Xi^4, the spatial projection along the ray; - $\Xicos(z)$TeX\Xicos(z), the homogeneous value of $B\dot\chi^2/\Lambda_\Xi^4$TeXB\dot\chi^2/\Lambda_\Xi^4 used in the cosmological timelike branch; - $\Xienv(t,\bm{x})$TeX\Xienv(t,\bm{x}), the local environment-dependent remainder around a coarse-grained background realization; and - $z$TeXz, the standard observed spectroscopic redshift imported from the background sector on the declared cosmological export class.

All downstream propositions, corollaries, and quantitative exports in the present paper are read only relative to these declared variables, the declared local path families, the declared redshift interval, and the declared reciprocity-preserving export class fixed below. All cosmological exports below use the imported standard spectroscopic redshift and the imported homogeneous background history on that declared class; no independent clock variable or propagation-renormalized effective Hubble function is introduced in the present propagation sector.

The exact homogeneous timelike and static spacelike specializations of reference are, respectively,

\frac{c_{\mathrm{local}}^{2}}{c_{\infty}^{2}}=1-\Xi_t,
 \qquad
 \frac{c_{\mathrm{local}}^{2}(e)}{c_{\infty}^{2}}=\frac{1}{1+\Xi_\parallel}.

The second relation is the relevant one for a static local spatial profile. Its refractive-index representation is

n_{\HH}(x,e)\equiv \frac{c_{\infty}}{c_{\mathrm{local}}(x,e)}=\sqrt{1+\Xi_\parallel(x,e)}.

Both metrics remain Lorentzian only while $1-B\Xi_{\nabla}>0$TeX1-B\Xi_{\nabla}>0 on a timelike background, as established by the root-theory invertibility condition.

Because the SI metre is defined by fixing the numerical value of the speed of light in vacuum, the experimental content of reference cannot be a direct remeasurement of a free constant. The observable content lies instead in propagation delays, path-integrated refractive response, and frequency-transfer observables [citation].

Local statements below apply to path families for which the background is static or slowly varying on the signal timescale and the derivative expansion is valid. Cosmological statements apply to a specified redshift interval and to the same coefficient function $B(\chi)$TeXB(\chi) constrained by local and multimessenger observations.

theorem: Directional probe-cone speed. Choose a local orthonormal frame of $g_{\mu\nu}$TeXg_{\mu\nu} with observer $u^\mu$TeXu^\mu and a unit spatial ray direction $e^\mu$TeXe^\mu. Put

a=\frac{B(e^\mu\nabla_\mu\chi)^2}{\Lambda_\Xi^4},
\quad
b=\frac{B(u^\mu\nabla_\mu\chi)(e^\nu\nabla_\nu\chi)}{\Lambda_\Xi^4},
\quad
d=\frac{B(u^\mu\nabla_\mu\chi)^2}{\Lambda_\Xi^4}.

If $1+a>0$TeX1+a>0 and the response metric is Lorentzian, the future-directed probe-null speed $\beta=c_{\rm local}/c_\infty$TeX\beta=c_{\rm local}/c_\infty is

\beta=\frac{-b+\sqrt{b^2+(1+a)(1-d)}}{1+a}.

For a purely timelike gradient this reduces to $\beta^2=1-d$TeX\beta^2=1-d; for a static spatial gradient it reduces to $\beta^2=(1+a)^{-1}$TeX\beta^2=(1+a)^{-1}.

proof. A ray tangent can be written $k^\mu=\omega(u^\mu+\beta e^\mu)$TeXk^\mu=\omega(u^\mu+\beta e^\mu). The conformal factor $A^2$TeXA^2 cancels from the null equation. Substitution into $\widetilde g_{\mu\nu}k^\mu k^\nu=0$TeX\widetilde g_{\mu\nu}k^\mu k^\nu=0 gives

(1+a)\beta^2+2b\beta+d-1=0.

The positive, future-directed root is reference. Setting $e\cdot\nabla\chi=0$TeXe\cdot\nabla\chi=0 or $u\cdot\nabla\chi=0$TeXu\cdot\nabla\chi=0 proves the two reductions.

On a static slowly varying background and along a fixed ray direction, the spacelike specialization of reference is conformally equivalent to the optical line element

ds_{\mathrm{opt}}^{2}
 =
 -\frac{c_{\infty}^{2}}{n_{\HH}^{2}(x)}\,dt^{2}+d\ell^{2},

with $n_{\HH}$TeXn_{\HH} given by reference. The optical null condition $ds_{\mathrm{opt}}^{2}=0$TeXds_{\mathrm{opt}}^{2}=0 yields

dt = \frac{n_{\HH}(x)}{c_{\infty}}\,d\ell,

and therefore the time-of-flight relation

t_{\mathrm{tof}}=\frac{1}{c_{\infty}}\int_{\text{path}}n_{\HH}(x)\,d\ell.

For a fixed geometric path length $\ell$TeX\ell, the path-averaged inferred speed is

c_{\mathrm{local}}\equiv \frac{\ell}{t_{\mathrm{tof}}}=\frac{c_{\infty}}{\langle n_{\HH}\rangle_{\text{path}}}.

In the small-$\Xi$TeX\Xi regime,

n_{\HH}(x,e)=\sqrt{1+\Xi_\parallel(x,e)}
 \simeq 1+\frac{1}{2}\Xiloc(x),
 \qquad
 c_{\mathrm{local}}(x)
 \simeq c_{\infty}\left(1-\frac{1}{2}\Xiloc(x)\right).

Here and below $\Xiloc$TeX\Xiloc abbreviates the ray-projected $\Xi_\parallel$TeX\Xi_\parallel on the static Solar-System branch. The square-root form follows from the covariant response metric rather than being imposed as an independent refractive ansatz [citation].

proposition: Operational content. On static slowly varying local backgrounds, the correction induced by reference enters experiments only through the path integral reference and its derivatives on declared ray families.

proof. Equations reference--reference show that the inferred propagation speed is determined by the path integral of $n_{\HH}$TeXn_{\HH}. The measurable correction is therefore an excess time-transfer functional. If $B\ne0$TeXB\ne0, that functional measures a difference between the probe and tensor cones.

Cassini-class path family and excess delay

Let $\Fcass$TeX\Fcass denote a Cassini-class conjunction family of two-way Earth--Sun--spacecraft radio links evaluated on a declared conjunction window $\Wcass$TeX\Wcass. The family is specified by four operational inputs:

- two-way light-time modelling for an Earth station $\to$TeX\to spacecraft $\to$TeX\to Earth station link, - a finite Doppler count time $\Tc$TeX\Tc, - an evolving superior-conjunction geometry with near-grazing impact parameter $b(t)$TeXb(t), and - a fixed plasma-calibration convention for the residual propagation channel.

For one-way propagation, the excess over propagation at $c_{\infty}$TeXc_{\infty} is

\DtCR
 =
 \frac{1}{c_{\infty}}
 \int_{\text{path}}
 \left[\sqrt{1+\Xiloc(x,e)}-1\right]d\ell
 \simeq
 \frac{1}{2c_{\infty}}\int_{\text{path}}\Xiloc(x)\,d\ell.

For a representative superior conjunction, the general-relativistic reference delay is

\DtGR
 =
 \frac{2GM_{\odot}}{c_{\infty}^{3}}
 \ln\left(\frac{4r_{E}r_{R}}{b^{2}}\right),

with standard notation for emitter radius, receiver radius, and impact parameter [citation]. Cassini reported $\gamma-1=(2.1\pm 2.3)\times 10^{-5}$TeX\gamma-1=(2.1\pm 2.3)\times 10^{-5} from solar-conjunction radio links [citation]. Using that result as the representative admissibility scale, the excess delay must satisfy

\abs{\DtCR}
 \lesssim
 \frac{\abs{\gamma-1}}{2}\,\abs{\DtGR},

or equivalently,

\left|\int_{\text{path}}\Xiloc(x)\,d\ell\right|
 \lesssim
 \abs{\gamma-1}\,c_{\infty}\abs{\DtGR}.

Doppler-level admissibility

Cassini constrained the propagation channel through a Doppler observable rather than through a static delay alone [citation]. Let $\rho(t)$TeX\rho(t) denote the modelled two-way light time and define the propagation excess by

\DCR(t)\equiv \rho_{\mathrm{model}}(t)-\rho_{\mathrm{GR}}(t).

A minimal count-time Doppler representation is

y_{\mathrm{CR}}(t;\Tc)
 \equiv
 K_{\mathrm{link}}\frac{\DCR(t+\Tc)-\DCR(t)}{\Tc},

with $K_{\mathrm{link}}$TeXK_{\mathrm{link}} absorbing fixed link-convention factors. The admissibility contract is therefore two-sided:

\sup_{\text{path}(t)\in\Fcass,\; t\in\Wcass}\abs{\DCR(t)}\le \Imax,
 \qquad
 \sup_{\text{path}(t)\in\Fcass,\; t\in\Wcass}\abs{y_{\mathrm{CR}}(t;\Tc)}\le \Dmax.

Here $\Imax$TeX\Imax and $\Dmax$TeX\Dmax denote imported residual budgets from the chosen Cassini-class analysis. They are used as external admissibility bounds and are not rederived in the present work.

remark: Representative order-of-magnitude witness. As a representative order-of-magnitude witness, for a superior-conjunction Earth--Sun--Saturn geometry with $r_{E}\approx 1\,\au$TeXr_{E}\approx 1\,\au, $r_{R}\approx 9.5\,\au$TeXr_{R}\approx 9.5\,\au, and $b\approx R_{\odot}$TeXb\approx R_{\odot}, one has $\DtGR$TeX\DtGR of order a few $10^{-4}\,$TeX10^{-4}\,s and therefore $c_{\infty}\DtGR$TeXc_{\infty}\DtGR of order $10^{5}\,$TeX10^{5}\,m. With Cassini-class $\abs{\gamma-1}\sim 10^{-5}$TeX\abs{\gamma-1}\sim 10^{-5}, reference constrains the path integral of $\Xiloc$TeX\Xiloc to the metre scale and the corresponding path average on a two-way path of order $20\,\au$TeX20\,\au to the $10^{-13}$TeX10^{-13} range. This witness illustrates the scale of the declared local admissibility contract and is not by itself a new theorem-bearing observational bound.

Laboratory optical-cavity experiments constrain anisotropy of light propagation on distinct local path families and therefore provide an independent check on any admissible $\Xiloc$TeX\Xiloc profile [citation]. The local contract is consequently not a weak small-parameter preference; it is the primary admissibility condition.

corollary: Common-parameter consistency. If every solution and coefficient function $B(\chi)$TeXB(\chi) that produces a specified nonzero $\Xicos(z)$TeX\Xicos(z) violates either reference, the response-metric signature condition, or an applicable multimessenger cone bound, then that cosmological response is excluded.

proof. The local, multimessenger, and cosmological observables are evaluations of the same pair $(\chi,B)$TeX(\chi,B). An admissible cosmological response would therefore furnish a counterexample satisfying all three necessary conditions. If no such pair exists, the assumed response cannot belong to the theory's admissible parameter space.

Passing the local contract does not by itself justify a cosmological background profile. Decompose the canonical field and coefficient into a homogeneous background and an environmental perturbation, $\chi=\bar\chi(t)+\delta\chi(t,\bm x)$TeX\chi=\bar\chi(t)+\delta\chi(t,\bm x) and $B=B(\chi)$TeXB=B(\chi). To first order in the perturbation of the projected response, write

\Xi_{\rm obs}(t,\bm{x},e) = \Xicos\bigl(z(t)\bigr)+\Xienv(t,\bm{x},e)+\mathcal O(\delta\chi^2),

where $\Xicos=B(\bar\chi)\dot{\bar\chi}^{\,2}/\Lambda_\Xi^4$TeX\Xicos=B(\bar\chi)\dot{\bar\chi}^{\,2}/\Lambda_\Xi^4 is the homogeneous timelike response and $\Xienv$TeX\Xienv is the direction-dependent linearized environmental contribution. Equation reference is an expansion of an observable response, not an exact additive decomposition of the invariant scalar kinetic norm.

proposition: Existence-level admissible coarse-grained export on the declared propagation class. A background profile $\Xicos(z)$TeX\Xicos(z) on a specified redshift interval is admissible only if there exists at least one solution $\chi(t,\bm{x})$TeX\chi(t,\bm{x}) and one coefficient function $B(\chi)$TeXB(\chi) such that:

- its local remainder $\Xienv(t,\bm{x})$TeX\Xienv(t,\bm{x}) satisfies reference on all declared local path families, and - the same realization yields the claimed $\Xicos(z)$TeX\Xicos(z) as the coarse-grained line-of-sight sector on the declared redshift interval.

No claim is made that every smooth profile $\Xicos(z)$TeX\Xicos(z) admits such a lift.

proof. The local contract and the cosmological profile are both functionals of the same $\chi$TeX\chi and $B$TeXB. If no common pair realizes them, the profile is not in the image of the admissible solution space. Conversely, the proposition asserts only necessity, so existence of one common pair is sufficient for this compatibility test but not for empirical acceptance.

The preceding proposition is an existence-level admissibility statement. It does not claim uniqueness of the underlying realization or of the coarse-graining scale that supports a given admitted $\Xicos(z)$TeX\Xicos(z).

The cosmological specialization is then

c(z)=c_{\infty}\sqrt{1-\Xicos(z)},
 \qquad
 0\le \Xicos(z)<1.

Here $z$TeXz denotes the standard observed spectroscopic redshift imported from the background sector, and $\Hbg(z)$TeX\Hbg(z) denotes the imported homogeneous background expansion history on the declared export class. Equation reference acts only on propagation operators for declared distance exports at fixed imported redshift variable and imported background history; no independent clock variable, no frequency/wavelength redshift split, and no propagation-renormalized effective Hubble function are introduced here [citation]. A nonzero propagation profile on this declared class does not by itself close a distance law, an observational totalization, or a late-time mixed-branch inference. Interpreting $c(z)$TeXc(z) as a modified cosmological clock or as a change in the homogeneous background equations would require additional structure not specified in the present paper.

Distance-bias functional

For $\Xicos\ll 1$TeX\Xicos\ll 1, define

\epsilon(z)\equiv \frac{1}{2}\Xicos(z),
 \qquad
 \Bbias(z)
 \equiv
 \frac{\int_{0}^{z}\epsilon(z')\,\dfrac{c_{\infty}}{\Hbg(z')}\,dz'}{\int_{0}^{z}\dfrac{c_{\infty}}{\Hbg(z')}\,dz'}.

The quantity $\Bbias(z)$TeX\Bbias(z) is the redshift-window-averaged propagation bias relevant to line-of-sight distances. A nontrivial cosmological export requires a realization for which $\Bbias(z)$TeX\Bbias(z) is non-negligible on the declared window while the same realization satisfies the local contract.

Failure conditions

The cosmological export fails if any of the following occurs:

- every realization satisfying the local contract yields a $\Bbias(z)$TeX\Bbias(z) that remains negligible on the declared redshift interval; - every realization that yields a nontrivial $\Bbias(z)$TeX\Bbias(z) necessarily regenerates a forbidden local remainder $\Xienv$TeX\Xienv on Solar-System paths; - a realization satisfies the delay-level bound but violates the Doppler-level part of reference; - the claimed effect requires a redefinition of cosmological time or a modification of homogeneous background dynamics not specified in the present construction; or - the claimed export requires a reciprocity-violating distance map or a separate redshift-variable split not declared in the present construction.

The equations in this section specify only the declared insertion point of the propagation map into standard distance observables. No claim is made that current supernova or BAO data are already fitted by the propagation map alone, no late-time multi-probe parameter estimation is attempted here, and the present propagation insertion is not elevated to a completed distance-closure statement. Throughout this section, $\Hbg(z)$TeX\Hbg(z) denotes the imported homogeneous background expansion history on the declared export class and is not promoted to a propagation-corrected effective Hubble function.

On the declared export class, the load-bearing propagation insertion is the line-of-sight Hubble distance

\DHub(z)\equiv \frac{c(z)}{\Hbg(z)}
 =
 \frac{c_{\infty}}{\Hbg(z)}\sqrt{1-\Xicos(z)}.

The corresponding radial comoving distance is

\DCom(z)=\int_{0}^{z}\DHub(z')\,dz'.

Transverse distances are then written in standard notation as

\DTrans(z)=S_{k}\!\bigl(\DCom(z)\bigr),

with $S_{k}(\chi)=\chi$TeXS_{k}(\chi)=\chi, $R_{0}\sin(\chi/R_{0})$TeXR_{0}\sin(\chi/R_{0}), or $R_{0}\sinh(\chi/R_{0})$TeXR_{0}\sinh(\chi/R_{0}) for the imported flat, closed, or open background branches, respectively. The luminosity- and angular-diameter distances satisfy

\DAng(z)=\frac{\DTrans(z)}{1+z},
 \qquad
 \DLum(z)=(1+z)^{2}\DAng(z)

only on the declared photon-conserving, reciprocity-preserving export class [citation]. These relations are not asserted as universal consequences of the local propagation law alone; they are the declared cosmological exports of the photon-conserving, reciprocity-preserving class fixed here. In the flat-background export used for the illustrative formulas below, $\DTrans=\DCom$TeX\DTrans=\DCom.

The drag-epoch sound horizon $r_{d}$TeXr_{d} entering BAO observables is imported from the chosen background analysis and is not rederived in the present propagation sector. Accordingly, anisotropic BAO constrain $\DHub(z)/r_{d}$TeX\DHub(z)/r_{d} and $\DTrans(z)/r_{d}$TeX\DTrans(z)/r_{d}, with $\DAng(z)=\DTrans(z)/(1+z)$TeX\DAng(z)=\DTrans(z)/(1+z) on the declared export class, while Type-Ia supernovae constrain $\DLum(z)$TeX\DLum(z) [citation]. If a proposed cosmological export requires a breakdown of reciprocity or a separate frequency/wavelength redshift split, that lies outside the present paper and must be declared as a different propagation class.

An early-universe propagation insertion point is correspondingly given by

\frac{d\eta_{\mathrm{CR}}}{dz}=\frac{c(z)}{\Hbg(z)}
 =
 \frac{c_{\infty}}{\Hbg(z)}\sqrt{1-\Xicos(z)}.

Equation reference records only the line-of-sight propagation operator entering conformal-time integrals. It is not by itself a statement about distance duality, clock redefinition, or a completed BAO/CMB implementation.

The construction is ruled out if any of the following is established:

- no nontrivial local profile satisfies the Solar-System contract reference; - any candidate cosmological propagation sector produces measurable distance leverage only by violating the same local contract on some declared local path family; - the delay-level bound is satisfied but the Doppler-level bound is not; - the claimed leverage depends on a clock reinterpretation or a modification of homogeneous background dynamics rather than on the propagation map itself; or - the claimed distance export requires reciprocity violation or a separate redshift-variable split that is not declared in the present propagation class.

The decisive observational structure is therefore not a pointwise determination of a radial $c(r)$TeXc(r) profile. It is a joint test of path-integrated delay, Doppler stability, and the existence of a residual admissible sector that remains capable of producing a nonzero distance bias.

Compactness of the phase target changes global field sectors but does not by itself change a local null cone. Within any smooth winding sector the root kinetic term is locally canonical, so a modified local propagation speed still requires an explicit effective metric, disformal coupling, or higher-derivative principal symbol. The existence of a nonzero winding number alone is therefore not evidence for a local variation of $c$TeXc.

Global travel times may nevertheless depend on a winding background after the coupled field equations and the probe metric have been specified, because the unavoidable gradient energy can alter the solved geometry or enter an admitted matter coupling. Such an effect must be calculated from that declared action and compared with the zero-winding solution; it cannot be read off from target topology. Since disformal probe metrics are established prior art, the distinctive test is a shared, fixed coupling that predicts several timing or propagation observables and survives the resulting compatibility constraints without probe-specific retuning.

The closure test for the local and global light propagation 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 local cone tests, travel times, lensing distances, and clock ratios. Let $a$TeXa range over the independent constitutive inputs comprising conformal and disformal probe couplings, background phase solution, source calibration, and clock 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 probe metric and one solved phase background determine every optical and clock observable on the declared window.

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 compact target topology alone changes neither the local metric cone nor the independent probe couplings. Failure of that condition therefore rejects the proposed microscopic closure before parameter estimation can be counted as evidence for it.

The \ propagation map distinguishes the asymptotic causal speed $c_{\infty}$TeXc_{\infty} from the operationally inferred speed $c_{\mathrm{local}}$TeXc_{\mathrm{local}} in a phase-curved environment. Through the optical-response representation reference, the measurable content is carried by excess delays and their derivatives on declared path families.

Solar-System admissibility is the primary gate. Cassini-class conjunction tracking and laboratory isotropy tests force any admissible local response to remain extremely small on local operational paths, and only the residual sector left open by that contract can be exported to cosmological line-of-sight distance integrals.

The construction therefore fails closed in two stages. If no nontrivial local profile survives the Solar-System contract, the propagation sector terminates locally. If a surviving local sector admits no coarse-grained cosmological lift compatible with the same contract, then no cosmological propagation effect is allowed. Within those bounds, the relevant cosmological outputs are the distance-bias functional reference, the line-of-sight operator $\DHub(z)=c(z)/\Hbg(z)$TeX\DHub(z)=c(z)/\Hbg(z), and the associated standard distance exports on the declared reciprocity-preserving class. A nonzero propagation map does not by itself close observational distance inference or late-time mixed-branch reconstruction. These outputs remain propagation-side exports only and are not by themselves promoted here to chronometer inference, homogeneous-background re-inference, or late-time mixed-branch observational closure.

No chronometer or cosmic-time inference is closed here. No clock-side reinterpretation, no homogeneous-background re-inference, and no late-time mixed-branch observational closure are developed in the present paper. All propagation-side claims in the present paper are read only on the declared local path families and the declared redshift interval, and any appendix-level descriptive scan remains tied to the declared witness family rather than to a benchmark-independent physical profile.

For a spherically symmetric local environment,

\HH(r)=\HH_{0}+\delta\HH_{\odot}(r),
 \qquad
 \Xiloc(r)=\frac{\Meff^{2}}{\Lambda_{\Xi}^{4}}\left|\frac{d\HH}{dr}\right|^{2}.

For a static radial field and a radial ray, the corresponding local propagation law is

c(r)=\frac{c_{\infty}}{\sqrt{1+\Xiloc(r)}}
 \simeq
 c_{\infty}\left(1-\frac{1}{2}\Xiloc(r)\right).

Near Earth, with $\Xiloc(r)=\Xi_{\oplus}+\delta\Xi(r)$TeX\Xiloc(r)=\Xi_{\oplus}+\delta\Xi(r),

c(r)
 \simeq
 c_{\oplus}\left[1-\frac{\delta\Xi(r)}{2\left(1+\Xi_{\oplus}\right)}\right],
 \qquad c_{\oplus}\equiv c(r_{\oplus}).

These expressions provide a convenient local specialization for radial environments; observational admissibility remains governed by the path-family contract of reference.

For descriptive scans on the declared redshift interval one may use the witness family

\Xicos(z)=\Xi_{0}(1+z)^{m},
 \qquad m>0,

with admissible domain

0\le \Xi_{0}< (1+z_{\max})^{-m}

on the declared interval $z\in[0,z_{\max}]$TeXz\in[0,z_{\max}]. The corresponding first-order propagation law is

c(z)
 \simeq
 c_{\infty}\left[1-\frac{1}{2}\Xi_{0}(1+z)^{m}\right].

This family is illustrative only. All descriptive asymptotics in this appendix are read only relative to this declared witness family on the stated interval and are not widened into benchmark-independent claims. Admissibility still depends on the existence statement in Proposition reference, not on the convenience of a particular parametrization.

Funding and competing interests..

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