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.

In , the relevant control variable is the phase-gradient hierarchy parameter

\Xi \equiv \frac{\Meff^{2}(\nabla \HH)^{2}}{\Lambda_{\Xi}^{4}},

with domain-specific evaluations used below. The present analysis isolates the propagation response associated with reference. The Einstein--Hilbert causal structure is not replaced, and the homogeneous background Einstein equations are not modified. The quantity of interest is the difference between the asymptotic causal speed and the 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 underlying relativistic null structure; the latter is extracted from finite time-transfer, ranging, or Doppler protocols in a phase-curved environment.

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.

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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:

- $c_{\infty}$TeXc_{\infty}, the asymptotic causal speed associated with the underlying relativistic null structure; - $c_{\mathrm{local}}(x)$TeXc_{\mathrm{local}}(x), the speed inferred from a finite local propagation protocol; - $\Xiloc(x)$TeX\Xiloc(x), the local phase-curvature measure governing the propagation response on inhomogeneous backgrounds; - $\Xicos(z)$TeX\Xicos(z), a coarse-grained cosmological profile admitted only after the local contract is satisfied; - $\Xienv(t,\bm{x})$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 working propagation law is taken to be

c_{\mathrm{local}}^{2}(x)=c_{\infty}^{2}\bigl(1-\Xiloc(x)\bigr),
 \qquad 0\le \Xiloc(x)\ll 1,

on declared local path families, with corresponding refractive-index representation

n_{\HH}(x)\equiv \frac{c_{\infty}}{c_{\mathrm{local}}(x)}=\frac{1}{\sqrt{1-\Xiloc(x)}}.

Throughout, the phrase ``varying $c$TeXc'' refers only to the operationally inferred propagation speed in reference. It does not introduce a second causal cone.

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].

The admissibility window is fixed as follows. Local statements are claimed only on declared path families for which the background is static or slowly varying on the signal timescale and the small-$\Xi$TeX\Xi expansion is valid. Cosmological export is claimed only on a declared redshift interval after the same realization passes the Solar-System contract. No closure claim is made outside those windows. All downstream propagation claims are read only on that declared local/cosmological admissibility window.

On a static slowly varying background, reference is 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)=\frac{1}{\sqrt{1-\Xiloc(x)}}
 \simeq 1+\frac{1}{2}\Xiloc(x),
 \qquad
 c_{\mathrm{local}}(x)
 \simeq c_{\infty}\left(1-\frac{1}{2}\Xiloc(x)\right).

The square-root form is therefore a direct consequence of the optical-response bridge [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 causal structure remains encoded in $c_{\infty}$TeXc_{\infty}, so the measurable correction is an excess time-transfer functional rather than a deformation of the relativistic null cone.

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[\frac{1}{\sqrt{1-\Xiloc(x)}}-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: Local-first consequence. If no nontrivial local profile satisfies reference, then no cosmological propagation sector is admissible.

Passing the local contract does not by itself justify a cosmological background profile. A further existence statement is required: an underlying realization must admit a coarse-grained cosmological component without regenerating forbidden local structure. Write

\Xi(t,\bm{x}) = \Xicos\bigl(z(t)\bigr)+\Xienv(t,\bm{x}),

where $\Xicos$TeX\Xicos is a coarse-grained line-of-sight background and $\Xienv$TeX\Xienv is the local remainder constrained directly by reference.

proposition: Existence-level admissible coarse-grained export on the declared propagation class. A background profile $\Xicos(z)$TeX\Xicos(z) on a declared redshift interval is admissible only if there exists at least one realization of $\Xi(t,\bm{x})$TeX\Xi(t,\bm{x}) 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 constrains the full realization of $\Xi$TeX\Xi on Solar-System paths, not only its coarse-grained average. A cosmological profile is therefore admissible only if it can be embedded in at least one realization whose local remainder remains within the same contract. Otherwise the cosmological profile is not compatible with the local sector of the theory.

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.

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}.

The corresponding local propagation law is

c(r)=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.