A universal-lapse null theorem and an STF propagation template
Version: 1.0
Version 1.0 — 24 August 2026
Companion code:
stf_cross_messenger_clock_closure_checks.py
We introduce a cross-messenger clock-closure observable that compares the redshift drift inferred from a long gravitational-wave chirp with the redshift drift measured electromagnetically for the same cosmological source population. For an FLRW spacetime written with an arbitrary universal lapse, and for electromagnetic and gravitational signals sharing the same null cone, the two measurements obey an exact kinematic identity. Defining
\[ X_{\rm EM}(z)\equiv \frac{\dot z_{\rm EM}}{2(1+z)}, \qquad \mathcal R_{\rm 2C}(z)\equiv X_{\rm GW}(z)-X_{\rm EM}(z), \]
we prove that
\[ \boxed{\mathcal R_{\rm 2C}(z)=0} \]
for every common-lapse realization, independently of the functional form of the lapse. A nonzero residual therefore cannot be attributed to a mere universal reparametrization of cosmological time. It requires differential propagation, nonuniversal clock coupling, or an unmodelled systematic.
For a weak time-dependent tensor speed, \(c_T=1+\delta c_T\), the leading local residual is
\[ \boxed{ \mathcal R_{\rm 2C} =\frac12\left[ \frac{\dot{\delta c}_{T,e}}{1+z} -\dot{\delta c}_{T,o} \right] } \]
to first order in \(\delta c_T\). We also derive the corresponding finite-baseline endpoint phase. Applied conditionally to the selective transient field (STF) scalar–Gauss–Bonnet propagation branch, the result is an oscillatory source–observer template governed by the internal scalar phase, while the cosmological acceleration term is governed by the universal clock. The endpoint sinusoid itself is shared with recent ultralight-scalar propagation work; the distinctive proposal here is the cross-messenger closure relation and its universal-lapse invisibility theorem.
Using the response-matched coefficient and scalar benchmark recorded in STF First-Principles V8.1, the instantaneous tensor-speed envelope is \(3.6\times10^{-30}\), the internal period is \(3.326\) yr, and the maximum closure residual at \(z=1\) is approximately \(2.60\times10^{-37}\,{\rm s}^{-1}\), about \(2.27\times10^{-18}\) of the standard \(\Lambda\)CDM acceleration signal. This benchmark is not observationally competitive. The main result is instead a clean null test that separates universal-clock effects from genuinely messenger-dependent propagation and supplies a falsifiable interface for future two-clock completions.
The selective transient field framework distinguishes two time structures:
The question addressed here is deliberately narrower than the construction of a complete STF action:
Can the two-clock distinction be turned into an operational, cross-messenger null test that remains meaningful even before the full nonlinear closure sector is completed?
The answer is yes. The test rests on a kinematic theorem and therefore does not require the unresolved nonlinear parent completion. Its mapping to the specific STF scalar–Gauss–Bonnet branch is conditional on that branch being part of the final healthy parent theory.
This paper makes four claims, with different logical status:
We do not claim in this paper that the full STF parent action has been completed, that the cosmological split-leg closure has passed every scalar constraint test, or that a detectable tensor-speed signal follows from the dark-matter capacity coefficient. In particular, the bare support coefficient \(\kappa\sim10^{70}\) is not substituted into the metric propagation leg. That substitution was an invalid identification in an earlier exploratory calculation and is excluded here.
Three time variables appear in the measurement, although only two are fundamental STF clock structures.
| Symbol | Operational role | STF interpretation |
|---|---|---|
| \(\tau_o\) | detector proper time and electromagnetic time standard | realization of the universal clock at the observer |
| \(\tau_e\) | source proper time | realization of the same universal clock at emission |
| \(\Theta_I\) | phase of the oscillating scalar | internal clock |
The distinction is not between two coordinate labels. It is between a shared causal/proper-time structure and a dynamical phase that can vary relative to it. A lapse redefinition changes how a coordinate \(T\) parametrizes the universal clock, but cannot by itself create a difference between two messengers that couple to that same clock.
We write a spatially flat homogeneous metric as
\[ ds^2=-L_U^2(T)dT^2+a^2(T)\left(d\chi^2+\chi^2d\Omega^2\right), \qquad d\tau=L_U(T)dT. \]
The physical Hubble rate is
\[ H\equiv \frac1a\frac{da}{d\tau} =\frac{1}{L_U}\frac1a\frac{da}{dT}. \]
The function \(L_U(T)\) may be nontrivial. No choice of time coordinate or gauge is made in the theorem below.
For a radial electromagnetic null ray,
\[ 0=-L_U^2dT^2+a^2d\chi^2, \]
so a comoving source at fixed \(\chi\) satisfies
\[ \chi=\int_{T_e}^{T_o}\frac{L_U(T)}{a(T)}dT. \]
Varying the endpoints for two neighboring wavefronts gives
\[ \frac{L_o}{a_o}dT_o=\frac{L_e}{a_e}dT_e. \]
Since \(d\tau=L_UdT\), this becomes
\[ \frac{d\tau_o}{a_o}=\frac{d\tau_e}{a_e}, \qquad 1+z=\frac{d\tau_o}{d\tau_e}=\frac{a_o}{a_e}. \]
Differentiate with respect to observer proper time:
\[ \frac{d}{d\tau_o}\ln(1+z) =H_o-\frac{H_e}{1+z}. \]
Therefore
\[ \dot z_{\rm EM} =(1+z)H_o-H_e, \]
and
\[ \boxed{ X_{\rm EM}(z) \equiv\frac{\dot z_{\rm EM}}{2(1+z)} =\frac12\left(H_o-\frac{H_e}{1+z}\right). } \]
Every explicit lapse factor has disappeared. The physical expansion rate retains whatever dynamics the theory assigns to it, but an arbitrary coordinate lapse does not become a separately measurable redshift-drift signal.
Assume:
Then, for an arbitrary lapse \(L_U(T)\),
\[ X_{\rm GW}(z)=X_{\rm EM}(z), \]
and hence
\[ \boxed{\mathcal R_{\rm 2C}(z)=0.} \]
Under the assumptions, the gravitational-wave eikonal obeys the same null equation as the electromagnetic eikonal. The endpoint variation is therefore
\[ \frac{L_o}{a_o}dT_o=\frac{L_e}{a_e}dT_e \]
for both messengers. Converting both endpoints to the same physical proper times gives
\[ 1+z_{\rm GW}=1+z_{\rm EM}=\frac{a_o}{a_e}. \]
Differentiation with respect to detector proper time yields the same drift,
\[ \frac{\dot z_{\rm GW}}{1+z} =\frac{\dot z_{\rm EM}}{1+z} =H_o-\frac{H_e}{1+z}. \]
The difference vanishes. \(\square\)
A universal modification of the time parametrization, including a nontrivial common lapse, may change the background solution \(a(\tau)\), but it cannot generate a messenger difference at fixed physical expansion history. The closure residual is blind to universal time reparametrization.
After controlled systematics are removed, \(\mathcal R_{\rm 2C}\ne0\) implies that at least one assumption of the theorem fails. Possibilities include a distinct tensor cone, nonuniversal matter coupling, frequency-dependent propagation, phase-dependent endpoint physics, or a breakdown of the assumed homogeneous propagation model.
This is the main reason the test is useful. It is a discriminator, not merely another measurement of \(H(z)\).
For a quasi-circular inspiral, let \(\mathcal M_z=(1+z)\mathcal M\) be the redshifted chirp mass and define the leading time remaining to coalescence at observed frequency \(f\):
\[ \Delta T(f) =\frac{5}{256}\mathcal M_z^{-5/3}(\pi f)^{-8/3}, \]
in units \(G=c=1\). A slowly varying redshift produces the Fourier-phase correction
\[ \boxed{ \Psi_{\rm acc}(f) =+2\pi fX_{\rm GW}\Delta T^2(f) =+\frac{25}{32768} X_{\rm GW}\mathcal M_z^{-10/3}(\pi f)^{-13/3}. } \]
The positive sign is important. Liu, Feng, and Guo rederived this term by matched asymptotic expansion in 2025 and corrected the opposite sign used in part of the earlier literature. We adopt their convention throughout. Changing the Fourier-transform convention flips all Fourier phases together but does not alter the closure identity.
The EM-side quantity is measured independently:
\[ X_{\rm EM}(z)=\frac{1}{2(1+z)}\frac{dz}{d\tau_o}. \]
In practice, the two measurements need not come from the same individual object. One may compare a hierarchical GW inference of \(X_{\rm GW}(z)\) with a redshift-drift reconstruction in matched redshift bins. A genuinely simultaneous standard-siren counterpart is nevertheless valuable because it constrains the redshift, peculiar acceleration, host environment, and line-of-sight nuisance model.
The observable is then
\[ \boxed{ \mathcal R_{\rm 2C}(z) =X_{\rm GW}(z)-\frac{\dot z_{\rm EM}(z)}{2(1+z)}. } \]
Now let gravitational waves propagate with a weak time-dependent speed
\[ c_T(t)=1+\delta c_T(t), \qquad |\delta c_T|\ll1, \]
while photons remain luminal in the matter frame. For neighboring tensor wavefronts,
\[ \frac{c_{T,o}}{a_o}dt_o =\frac{c_{T,e}}{a_e}dt_e. \]
The gravitational-wave redshift is therefore
\[ 1+z_{\rm GW} =\frac{dt_o}{dt_e} =\frac{a_o}{a_e}\frac{c_{T,e}}{c_{T,o}}. \]
Taking an observer-time derivative gives the exact geometric-optics relation
\[ \frac{d}{dt_o}\ln(1+z_{\rm GW}) =H_o-\frac{H_e}{1+z_{\rm GW}} +\frac{1}{1+z_{\rm GW}} \frac{\dot c_{T,e}}{c_{T,e}} -\frac{\dot c_{T,o}}{c_{T,o}}. \]
Expanding to first order in \(\delta c_T\) and comparing with the EM drift yields
\[ \boxed{ \mathcal R_{\rm 2C}(z) =\frac12\left[ \frac{\dot{\delta c}_{T,e}}{1+z} -\dot{\delta c}_{T,o} \right] +\mathcal O(\delta c_T^2). } \]
The ordering is emitter minus observer. This corrects the sign ordering in the exploratory conversation that motivated the present paper. The magnitude estimates made there are unchanged, but the phase labels are not.
The local drift formula is the short-baseline expansion of a more general endpoint effect. For fixed comoving distance,
\[ \chi=\int_{t_e}^{t_o}\frac{1+\delta c_T(t)}{a(t)}dt. \]
Relative to luminal propagation, the arrival-time perturbation is
\[ \delta t_{\rm prop} =-\int_{t_e}^{t_o}\frac{\delta c_T(t)}{a(t)}dt +\mathcal O(\delta c_T^2). \]
For an oscillatory speed
\[ \delta c_T(t)=\epsilon(t)\cos\Theta_I(t), \qquad \dot\Theta_I=m_s, \]
and amplitudes varying slowly compared with \(m_s^{-1}\), integration by parts gives
\[ \delta t_{\rm prop} \simeq \frac{1}{m_s}\left[ -\epsilon_o\sin\Theta_o +(1+z)\epsilon_e\sin\Theta_e \right]. \]
With the Fourier convention used above,
\[ \boxed{ \delta\Psi_T(f) =\frac{2\pi f}{m_s} \left[ -\epsilon_o\sin\Theta_o(f) +(1+z)\epsilon_e\sin\Theta_e(f) \right]. } \]
The frequency-dependent endpoint phases are
\[ \Theta_o(f)=\Theta_{c,o}-m_s\Delta T(f), \]
\[ \Theta_e(f)=\Theta_{c,e}-\frac{m_s\Delta T(f)}{1+z}. \]
This exact leading endpoint template is preferable whenever the observation spans an appreciable fraction of the internal period. Expanding both sines through second order in \(m_s\Delta T\) produces:
Explicitly,
\[ \delta\Psi_T^{(2)} =2\pi f\Delta T^2 \frac12\left[ \frac{\dot{\delta c}_{T,e}}{1+z} -\dot{\delta c}_{T,o} \right]. \]
Thus the endpoint calculation and the redshift-drift calculation agree.
Oscillating ultralight fields coupled through parity-even Gauss–Bonnet terms are already known to generate endpoint-dependent gravitational-wave speed, redshift, amplitude, and phase effects. Jenks and Kamionkowski developed this physics explicitly for the axiverse in 2026. The sinusoidal endpoint structure should therefore not be presented as uniquely STF.
The new object proposed here is the closure construction:
\[ \text{GW chirp drift} -\text{EM redshift drift}, \]
together with the proof that an arbitrary universal lapse cancels exactly. The STF contribution is a specific interpretation and coefficient map for one conditional propagation branch.
Consider the parity-even tensor propagation sector
\[ S\supset\int d^4x\sqrt{-g}\,F(\phi)\,\mathcal G, \]
where \(\mathcal G\) is the Gauss–Bonnet invariant. To keep sign and action-normalization conventions explicit, write the quadratic tensor coefficients as
\[ Q_T=M_{\rm Pl}^2+8\sigma_{\rm GB}H\dot F, \qquad F_T=M_{\rm Pl}^2+8\sigma_{\rm GB}\ddot F, \]
so
\[ c_T^2=\frac{F_T}{Q_T}, \qquad \delta c_T \simeq\frac{4\sigma_{\rm GB}}{M_{\rm Pl}^2} (\ddot F-H\dot F). \]
Here \(\sigma_{\rm GB}=\pm1\) records the curvature and coupling-sign convention. It can equivalently be absorbed into \(F\). The no-ghost and no-gradient conditions are
\[ Q_T>0, \qquad F_T>0. \]
For a linear coupling and a coherently oscillating scalar,
\[ F(\phi)=F_{,\phi}\phi, \qquad \phi(t)=\Phi(t)\cos\Theta_I(t), \qquad \dot\Theta_I=m_s, \]
with \(m_s\gg H\),
\[ \delta c_T \simeq -\frac{4\sigma_{\rm GB}F_{,\phi}m_s^2\Phi}{M_{\rm Pl}^2} \cos\Theta_I. \]
Therefore
\[ \dot{\delta c}_T \simeq \frac{4\sigma_{\rm GB}F_{,\phi}m_s^3\Phi}{M_{\rm Pl}^2} \sin\Theta_I, \]
up to relative corrections of order \(H/m_s\). The two-clock residual becomes
\[ \boxed{ \mathcal R_{\rm 2C}(z) \simeq \frac{2\sigma_{\rm GB}F_{,\phi}m_s^3}{M_{\rm Pl}^2} \left[ \frac{\Phi_e\sin\Theta_e}{1+z} -\Phi_o\sin\Theta_o \right]. } \]
For a coherently oscillating field that supplies a matter-like mean density,
\[ \Phi_e=\Phi_o(1+z)^{3/2}, \]
so
\[ \boxed{ \mathcal R_{\rm 2C}(z) \simeq \frac{2\sigma_{\rm GB}F_{,\phi}m_s^3\Phi_o}{M_{\rm Pl}^2} \left[ \sqrt{1+z}\sin\Theta_e -\sin\Theta_o \right]. } \]
The endpoint phase is
\[ \delta\Psi_T(f) \simeq \frac{8\pi\sigma_{\rm GB}F_{,\phi}m_s f}{M_{\rm Pl}^2} \left[ \Phi_o\sin\Theta_o(f) -(1+z)\Phi_e\sin\Theta_e(f) \right]. \]
These equations exhibit the two clocks directly:
A common lapse alters neither their closure theorem nor the messenger difference. A differential propagation coefficient allows the internal phase to appear in the GW channel without appearing in the EM redshift drift.
The STF source record contains several coefficients with different physical roles. Keeping them separate is essential.
The parent-level closure analysis defines
\[ Q=P_A\mathcal C^A=M_*^2q_N, \]
with first-order memory
\[ (D_U+\omega_c)y=\omega_cQ, \qquad Z=Q-y, \]
and high-pass response
\[ K^R_{\rm sel}(\omega) =\frac{-i\omega}{\omega_c-i\omega}. \]
The scalar support normalization is controlled by
\[ \kappa=\gamma\tau_{\rm eff}M_*^2 =\frac{\zeta/\Lambda}{L_*^2}. \]
This coefficient belongs to the saturated closure readout. It is not automatically the coefficient of the metric tensor-propagation leg.
The V8.1 tensor-speed estimate uses a separately response-matched coupling
\[ F_{,\phi} =c_{\rm GB}\left(\frac{\zeta}{\Lambda}\right)M_{\rm Pl}, \qquad c_{\rm GB}=\mathcal O(1). \]
The symbol \(c_{\rm GB}\) is used here to avoid collision with the large support coefficient \(\kappa\). This distinction removes the erroneous \(10^{-80}\) tensor-speed estimate and the associated \(10^{68}\) coefficient-gap claim from the earlier exploratory branch.
If the metric propagation leg is itself filtered by the closure memory, its Fourier-domain amplitude acquires an additional factor
\[ K^R_{\rm sel}(\omega) =\frac{-i\omega}{\omega_c-i\omega}. \]
At the STF benchmark \(\omega\sim\omega_c\), the derivative expansion is not controlled and the exact kernel must be retained. Whether this factor multiplies the cosmological scalar–Gauss–Bonnet leg is a parent-theory question; it is not assumed in the numerical benchmark below.
We use the parameters recorded in STF First-Principles V8.1:
\[ m_s=3.94\times10^{-23}\ {\rm eV}, \]
\[ \omega_s=\frac{m_s}{\hbar} =5.9859\times10^{-8}\ {\rm s}^{-1}, \]
\[ T_s=\frac{2\pi}{\omega_s}=3.32618\ {\rm yr}, \]
and the instantaneous tensor-speed envelope
\[ |\delta c_T|_{\rm env}=3.6\times10^{-30}. \]
For a flat \(\Lambda\)CDM reference cosmology with
\[ H_0=67.4\ {\rm km\,s^{-1}\,Mpc^{-1}}, \quad \Omega_m=0.315, \quad \Omega_\Lambda=0.685, \]
the electromagnetic acceleration variable at \(z=1\) is
\[ X_{\rm EM}(1) =1.14538\times10^{-19}\ {\rm s}^{-1}. \]
If \(\Phi_e=\Phi_o(1+z)^{3/2}\), the phase-independent maximum of the closure residual is
\[ |\mathcal R_{\rm 2C}|_{\max} =\frac{\omega_s}{2}|\delta c_T|_{o,\rm env} \left(1+\sqrt{1+z}\right). \]
At \(z=1\),
\[ \boxed{ |\mathcal R_{\rm 2C}|_{\max} =2.60123\times10^{-37}\ {\rm s}^{-1} } \]
and
\[ \boxed{ \frac{|\mathcal R_{\rm 2C}|_{\max}}{X_{\rm EM}} =2.27107\times10^{-18}. } \]
This is not a plausible detection target with the present response-matched coefficient.
For a source-frame binary-neutron-star chirp mass \(\mathcal M=1.22M_\odot\) at \(z=1\), \(\mathcal M_z=2.44M_\odot\). The reproducibility script gives:
| \(f\) (Hz) | \(\Delta T\) (yr) | \(\Psi_{\rm acc}\) (rad) | \(\lvert\delta\Psi_T\rvert\) endpoint bound (rad) |
|---|---|---|---|
| 0.05 | 13.6639 | \(6.6904\times10^{-3}\) | \(1.2577\times10^{-22}\) |
| 0.10 | 2.15193 | \(3.3189\times10^{-4}\) | \(2.5155\times10^{-22}\) |
| 0.20 | 0.338907 | \(1.6464\times10^{-5}\) | \(5.0310\times10^{-22}\) |
The acceleration phase is small but conceptually measurable through a population of long-lived decihertz sources. The STF propagation phase at this benchmark is about nineteen orders of magnitude smaller.
The tiny benchmark has two implications:
The observable remains useful because its common-lapse null prediction is exact. A measurement can be reported independently of whether the STF benchmark is large enough to explain it.
A model-independent analysis can fit
\[ \Psi(f)=\Psi_{\rm src}(f) +2\pi f\left[X_{\rm EM}(z)+\mathcal R_{\rm 2C}(z)\right] \Delta T^2(f), \]
with \(X_{\rm EM}\) constrained externally and \(\mathcal R_{\rm 2C}\) treated as a free parameter in each redshift bin.
This is appropriate when the differential propagation changes slowly over the observed chirp. It directly tests the universal-lapse null theorem.
For an internal clock with a period comparable to the mission duration, the local expansion is insufficient. The fit should instead include
\[ \delta\Psi_T(f) =\frac{2\pi f}{m_s} \left[ -\epsilon_o\sin(\Theta_{c,o}-m_s\Delta T) +(1+z)\epsilon_e \sin\left(\Theta_{c,e}-\frac{m_s\Delta T}{1+z}\right) \right]. \]
The source and observer phases should initially be nuisance parameters. A coherent Galactic or cosmological scalar model may correlate them, but that correlation must be justified using the scalar coherence length and propagation history rather than assumed.
A practical hierarchical model separates:
\[ X_{\rm GW}(z,\hat n,t_o) =X_{\rm cos}(z) +X_{\rm pec} +X_{\rm env} +X_{\rm lens} +X_{\rm prop}, \]
where:
The EM analysis must use the corresponding source and line-of-sight model. A clean cosmological residual is expected to be coherent in redshift and, for the oscillatory STF template, periodic in observer time with the same \(m_s\) across sources.
The following patterns help separate a two-clock propagation signal from nuisance effects:
No single event is expected to establish the effect. The test is naturally hierarchical and cross-messenger.
The clock-closure theorem is kinematic. The STF interpretation is not. It inherits the current conditional status of the split-leg completion.
The archived parent-level audits establish the following limited results:
The same audit chain also found real obstructions:
The Floquet obstruction has since been closed at the effective-theory level: the activation-fork resolution shows that on the conservative total-rate branch the minimal admissible low-rate activation susceptibility is quadratic, \(g_2(d)=d^2/(d^2+\mathcal D_X^2)\), pointwise stable through scalar-oscillation onset in the deterministic weak-coupling baseline, with a minimum margin above \(4\times10^3\) while retaining order-one response at the late-inspiral thresholds. The action-level Dirac/Ward variation of that activated map remains outstanding.
The surviving construction is therefore not the minimal reduced local action. It is a parent-level split-leg open-system completion with:
\[ Q=P_A\mathcal C^A \]
defined before reduction, a first-order closure channel, a preserved lapse–shift–clock constraint rank, an explicit Ward-completing environment, and a nontrivial physical readout.
The null theorem is valid whether the final STF completion succeeds or fails. If a healthy parent theory produces universal coupling only, it predicts closure. If it produces a differential tensor leg, the residual template measures that leg. If no healthy parent channel exists, the STF mapping is discarded but the observable remains a useful generic test of differential propagation.
This makes clock closure a good interface between theory construction and data analysis: the observable can be defined now, while coefficient priors can be updated as the parent theory matures.
| Observation | Universal-lapse sector | Differential STF propagation branch | Interpretation |
|---|---|---|---|
| \(\mathcal R_{\rm 2C}=0\), no periodicity | passes | unconstrained if below sensitivity | common clock remains viable |
| \(\mathcal R_{\rm 2C}\ne0\), no common period | fails as sole explanation | not the minimal coherent-scalar template | investigate systematics or other propagation physics |
| common-period residual with endpoint/redshift scaling | fails as sole explanation | qualitatively supported | fit full STF/ultralight-scalar endpoint model |
| same periodic drift in EM and GW channels | closure may still hold | differential branch disfavored | likely common background or timing systematic |
| birefringent periodic residual | fails as sole explanation | parity-even branch disfavored | parity-odd propagation candidate |
| detectable residual at the V8.1 period but amplitude \(\gg10^{-37}\,{\rm s}^{-1}\) | fails | V8.1 coefficient benchmark fails | requires enhanced/activated branch or different theory |
A nonzero residual is not automatically evidence for STF. The distinctive evidence would be the joint pattern: nonclosure, common internal period, endpoint structure, correct redshift scaling, and compatibility with independent tensor-sector constraints.
The source and observer scalar phases need not be coherent across cosmological distances. The phase variables should be marginalized independently unless a concrete scalar halo or cosmological coherence model is supplied. The phase-independent envelope remains valid without coherence.
A scalar–Gauss–Bonnet coupling can also affect binary generation. The present paper isolates propagation. A full waveform analysis must include or bound generation-side corrections so that they are not absorbed into \(\mathcal R_{\rm 2C}\).
Time-dependent tensor kinetic normalization also modifies the amplitude. We use the phase/redshift channel because it maps directly onto clock closure. A joint amplitude–phase analysis may improve discrimination but requires a careful luminosity-distance and selection model.
Long-lived binaries may experience acceleration in host galaxies, clusters, triples, or nuclear environments. These effects can generate the same \(f^{-13/3}\) scaling as cosmological redshift drift. Cross-messenger host characterization and population marginalization are mandatory.
The null test compares clocks. Detector timing calibration, barycentric corrections, and long-term ephemeris errors therefore require exceptional control. A periodic observer endpoint term can be especially vulnerable to a common timing systematic.
The metric coefficient \(F_{,\phi}\), the closure support coefficient \(\kappa\), and any cosmological activation factor must be derived from one explicit healthy parent completion. The current response matching is a conditional coefficient map, not yet such a derivation.
The paper is sufficient to define the test, prove its null theorem, and provide a conditional STF template. It is not the end of the theory program. The next calculations should be performed in this order:
Only after steps 1–3 can the tiny V8.1 benchmark be promoted from a conditional estimate to a prediction of a complete theory.
The two-clock idea admits a clean operational test.
For any cosmology in which photons, gravitational waves, and matter clocks share one physical metric, an arbitrary universal lapse cancels from the comparison of GW-inferred and EM-measured redshift drift. The exact null prediction is
\[ \mathcal R_{\rm 2C}=0. \]
A weak differential tensor speed violates closure according to
\[ \mathcal R_{\rm 2C} =\frac12\left[ \frac{\dot{\delta c}_{T,e}}{1+z} -\dot{\delta c}_{T,o} \right]. \]
An oscillating scalar–Gauss–Bonnet branch converts the STF internal clock into a periodic endpoint phase while the standard acceleration phase continues to track the universal clock. This is the precise sense in which the two clocks become separately observable.
The present STF V8.1 coefficient benchmark is far too small for foreseeable detection: at \(z=1\), its maximum closure residual is about \(2.60\times10^{-37}\,{\rm s}^{-1}\). That numerical result does not weaken the theorem. It clarifies the scientific role of the proposal. Clock closure is a model-independent null test; STF supplies one conditional, highly suppressed template for violating it.
The conceptual result is therefore stronger than a claimed signal:
A universal clock can alter cosmological dynamics, but it cannot by itself make two messengers disagree. Cross-messenger nonclosure isolates genuinely differential propagation.
Let
\[ \delta c_T=\epsilon\cos\Theta, \qquad \dot\Theta=m_s. \]
The endpoint phase is
\[ \delta\Psi_T =\frac{2\pi f}{m_s} \left[-\epsilon_o\sin(\Theta_{c,o}-m_s\Delta T) +(1+z)\epsilon_e \sin\left(\Theta_{c,e}-\frac{m_s\Delta T}{1+z}\right) \right]. \]
Using
\[ \sin(\Theta-x) =\sin\Theta-x\cos\Theta-\frac{x^2}{2}\sin\Theta+\mathcal O(x^3), \]
the quadratic term is
\[ \delta\Psi_T^{(2)} =2\pi f\Delta T^2 \frac{m_s}{2} \left[ \epsilon_o\sin\Theta_{c,o} -\frac{\epsilon_e\sin\Theta_{c,e}}{1+z} \right]. \]
But
\[ \dot{\delta c}_T=-m_s\epsilon\sin\Theta, \]
so
\[ \delta\Psi_T^{(2)} =2\pi f\Delta T^2 \frac12\left[ \frac{\dot{\delta c}_{T,e}}{1+z} -\dot{\delta c}_{T,o} \right]. \]
The endpoint and redshift derivations therefore agree exactly at this order.
Starting from
\[ \Psi_{\rm acc}=2\pi fX\Delta T^2, \qquad \Delta T=\frac5{256}\mathcal M_z^{-5/3}(\pi f)^{-8/3}, \]
we obtain
\[ \Psi_{\rm acc} =2\pi fX\frac{25}{65536} \mathcal M_z^{-10/3}(\pi f)^{-16/3}. \]
Since \(f=(\pi f)/\pi\),
\[ \Psi_{\rm acc} =\frac{25}{32768} X\mathcal M_z^{-10/3}(\pi f)^{-13/3}. \]
The companion code verifies the equality numerically to machine precision.
| Item | Convention used here |
|---|---|
| Metric signature | \((-+++ )\) |
| Proper time | \(d\tau=L_UdT\) |
| Hubble rate | \(H=a^{-1}da/d\tau\) |
| GW speed | \(c_T=1+\delta c_T\) |
| Closure residual | \(X_{\rm GW}-X_{\rm EM}\) |
| Fourier acceleration phase | \(+2\pi fX\Delta T^2\) |
| sGB sign | carried by \(\sigma_{\rm GB}\) |
| Scalar frequency | \(\dot\Theta_I=m_s\), with \(m_s\) converted from eV using \(\hbar\) |
Run:
python stf_cross_messenger_clock_closure_checks.pyThe script checks:
No nonstandard Python packages are required.
Established in this paper
Not established in this paper