Worldline Phase-Load and Local Commit Cadence in the CHC Framework

A finite phase-commit system gives a reduced open-system description of phase-linked persistence, local commit, and observable projection on a finite state window [citation]. The present paper addresses a different layer. It asks how a persistent local structure accumulates its internal commits along a spacetime history when that history carries kinetic, gravitational, and possible cohesive phase-load.

The object is not a replacement for metric proper time. The metric member is retained exactly. The object is a factorization of clock accumulation into two parts: a reference internal commit rate and a worldline cadence factor. On the metric member the cadence factor is the usual proper-time lapse relative to a declared comparison convention. In the phase-load representation the same factor is written as an exponential attenuation by a scalar load. This converts the statement ``a moving clock runs slow'' into the statement that the corresponding worldline carries larger load against the comparison convention and therefore admits a lower local commit cadence in that comparison.

The distinction matters because coordinates can always be attached to the travelling clock. Such a coordinate choice places the travelling clock at the spatial origin of its own chart. It does not make the entire history an inertial history. The local frame can be reset at each event; the connection between those resets remains part of the worldline history. The accumulated commit count is an integral over the history, not a property of any single instantaneous frame.

The paper uses standard relativistic notation. Proper time is the invariant time measured by an ideal clock along a timelike worldline [citation]. Atomic clocks, muon lifetimes, aircraft-clock comparisons, optical clock comparisons, and satellite navigation all support the universality of the metric member at current resolution [citation]. The construction below factorizes the metric member by a scalar phase-load and then defines finite residual-cadence extensions on declared comparison windows.

Let $(\Mcal,g)$TeX(\Mcal,g) be a time-oriented Lorentzian spacetime with signature $(-,+,+,+)$TeX(-,+,+,+). Let $c$TeXc denote the invariant local light speed. A timelike worldline is a smooth map

\Gamma:J\to\Mcal,
\qquad
\lambda\mapsto x^\mu(\lambda),

with $g_{\mu\nu}\dot x^\mu\dot x^\nu<0$TeXg_{\mu\nu}\dot x^\mu\dot x^\nu<0. Its metric proper-time element is

\dd\tau_g[\Gamma]
=
\frac1c\sqrt{-g_{\mu\nu}\,\dd x^\mu\dd x^\nu}.

The proper-time functional is

\Tau_g[\Gamma]=\int_\Gamma \dd\tau_g.

This functional depends on the path, not on the coordinates used to describe it.

definition: Comparison-time convention. On a finite comparison window, a comparison convention $C$TeXC is a declared positive one-form $\Theta_C$TeX\Theta_C along the relevant timelike histories. We write

\dd t_C=\Theta_{C,\mu}\dd x^\mu,

and require $\dd t_C>0$TeX\dd t_C>0 on future-directed timelike histories in the comparison. A future-directed unit timelike comparison field $T^\mu$TeXT^\mu gives the local observer convention

\Theta^{(T)}_\mu=-c^{-2}g_{\mu\nu}T^\nu,
\qquad
 g_{\mu\nu}T^\mu T^\nu=-c^2,

so that $\dd t_T=\Theta^{(T)}_\mu\dd x^\mu$TeX\dd t_T=\Theta^{(T)}_\mu\dd x^\mu. It is not an absolute rest frame; it is one declared comparison convention among others.

If $u^\mu=\dd x^\mu/\dd\tau_g$TeXu^\mu=\dd x^\mu/\dd\tau_g is the four-velocity of $\Gamma$TeX\Gamma, then for the local unit-field convention $T$TeXT,

\gamma_T(\Gamma)
=
-\frac1{c^2}g_{\mu\nu}T^\mu u^\nu,
\qquad
\dd t_T=\gamma_T\dd\tau_g.

For any declared comparison convention $C$TeXC, the metric cadence factor is

\chi_g(\Gamma;C)=\frac{\dd\tau_g}{\dd t_C}.

For $C=T$TeXC=T this gives $\chi_g(\Gamma;T)=1/\gamma_T$TeX\chi_g(\Gamma;T)=1/\gamma_T. In a stationary weak-field comparison we use another declared convention, denoted $K$TeXK, with reference-time one-form $\dd t_K$TeX\dd t_K normalized by the chosen stationary clock or asymptotic coordinate time. Gravitational potential factors in the weak-field expansion below are statements about this declared reference-time comparison, not about the local unit-field lapse.

definition: Commit clock. A commit clock of channel type $q$TeXq on $\Gamma$TeX\Gamma is a positive reference rate $\Omega_q(x)$TeX\Omega_q(x) together with a cadence factor $\chi_q(x,u,\mathcal I;C)$TeX\chi_q(x,u,\mathcal I;C), where $\mathcal I$TeX\mathcal I denotes the declared local phase-load data and $C$TeXC is the declared comparison convention. Its accumulated local commit count is

N_q[\Gamma;C]
=
\int_\Gamma \Omega_q(x)\,\chi_q(x,u,\mathcal I;C)\,\dd t_C.

The metric member is obtained by setting

\chi_q=\chi_g
\qquad\text{for all clock channels }q,

so that for every declared comparison convention $C$TeXC,

N_q[\Gamma;C]=\int_\Gamma \Omega_q(x)\,\dd\tau_g.

When $\Omega_q$TeX\Omega_q is constant over the comparison, the accumulated count is proportional to metric proper time.

definition: Worldline phase-load. Given a declared comparison convention $C$TeXC, the metric phase-load density of a worldline is

\Pcal_g(\Gamma;C)=-\log\chi_g(\Gamma;C).

A general CHC cadence factor is written

\chi_q=\exp[-\Pcal_q],
\qquad
\Pcal_q=\Pcal_g+\varepsilon\Rcal_q.

Here $\Rcal_q$TeX\Rcal_q is a scalar residual load density for channel $q$TeXq and $\varepsilon$TeX\varepsilon is a bookkeeping parameter. The standard branch is $\varepsilon=0$TeX\varepsilon=0.

On the standard branch the phase-load is only a rewriting of the metric cadence factor. Larger $\Pcal_g$TeX\Pcal_g means smaller $\chi_g$TeX\chi_g and hence fewer commits per unit comparison time. This is the precise sense in which kinetic motion, and in the weak-field convention below a deeper gravitational potential, appear as positive load contributions that reduce local commit cadence relative to the declared comparison convention; other gravitational comparisons follow their declared sign convention.

The residual term $\Rcal_q$TeX\Rcal_q is not fixed by the metric member. If retained, it must belong to an admitted finite-window residual class.

definition: Admitted residual load. Let $W$TeXW be a finite comparison window and let $q$TeXq be a fixed clock channel. A residual load density $\Rcal_q$TeX\Rcal_q is admitted on $W$TeXW only if it is a dimensionless scalar along the relevant worldline, has its channel assignment fixed before the comparison, is bounded on $W$TeXW, is not retuned between the compared histories, and enters only through the declared cadence factor $\exp[-\varepsilon\Rcal_q]$TeX\exp[-\varepsilon\Rcal_q] with

\sup_W |\varepsilon\Rcal_q|\le \eta <1.

All first-order residual expansions below are taken on finite windows satisfying reference; the $O(\varepsilon^2)$TeXO(\varepsilon^2) remainders are uniform on that window. The metric member is the projection $\varepsilon=0$TeX\varepsilon=0. A residual-free representative sets $\Rcal_q=0$TeX\Rcal_q=0; a retained residual branch may have nonzero $\Rcal_q$TeX\Rcal_q, but it is inactive under the metric projection.

A finite residual expansion may then be written

\Rcal_q
=
\alpha_q\,\frac{a_\mu a^\mu}{a_*^2}
+
\beta_q\,\Ccal_{\rm coh}
+
\delta_q\,\frac{R_{\mu\nu}u^\mu u^\nu}{R_*}
+
\cdots,

where $a^\mu=u^\nu\nabla_\nu u^\mu$TeXa^\mu=u^\nu\nabla_\nu u^\mu is the four-acceleration, $\Ccal_{\rm coh}$TeX\Ccal_{\rm coh} is a declared dimensionless cohesive or internal-structure scalar, and $a_*,R_*$TeXa_*,R_* are scale constants used only to make the coefficients dimensionless. Acceleration and cohesive residuals are not part of the universal ideal-clock member; they are finite-channel residual candidates only after the metric member and ordinary clock-systematic effects have been fixed. Setting all residual coefficients to zero gives the universal metric member.

remark. The term ``load'' does not designate a force acting on a clock. It designates the scalar logarithmic factor controlling local commit cadence relative to a comparison convention: positive load lowers cadence, while negative load raises it. On the metric member this factor is exactly the logarithmic representation of the relativistic lapse $\dd\tau_g/\dd t_C$TeX\dd\tau_g/\dd t_C for the declared comparison convention.

The metric member is the projection $\varepsilon=0$TeX\varepsilon=0. On this branch WPL is only a cadence-level retyping of standard worldline clock accumulation. The residual-free representative further sets $\Rcal_q=0$TeX\Rcal_q=0 for every clock channel. A nonmetric reading is present only when a declared admitted residual load $\Rcal_q$TeX\Rcal_q is retained on a finite comparison window with $\varepsilon\ne0$TeX\varepsilon\ne0. All comparisons below are therefore decomposed into a universal metric part and a residual part.

The residual branch is a finite scalar-cadence extension of the metric member. It has physical content only after an admitted residual scalar, a fixed clock channel, and a comparison window have been specified. Any nonzero residual load must be tested against the standard metric member and ordinary systematic effects before it can be read as a physical separation. This construction does not assert an observed clock anomaly, an acceleration-dependent violation of the metric clock hypothesis, a universal nonmetric time law, a GPS or optical-clock replacement model, or a new empirical constraint on clock physics.

Let $\dd t_K$TeX\dd t_K be the declared stationary reference-time one-form of a weak, slowly varying metric, normalized by a fixed stationary clock or by asymptotic coordinate time. Let $\Phi$TeX\Phi denote the Newtonian potential offset relative to that normalization, so $|\Phi|/c^2\ll1$TeX|\Phi|/c^2\ll1 and $\Phi=0$TeX\Phi=0 at the chosen reference. For coordinate speed $v_K$TeXv_K measured relative to that stationary reference slicing, the standard metric lapse relative to $\dd t_K$TeX\dd t_K has the expansion

\frac{\dd\tau_g}{\dd t_K}
=
1+\frac{\Phi}{c^2}-\frac{v_K^2}{2c^2}+O(c^{-4}).

Consequently, the reference-time phase-load is

\Pcal_g^{(K)}
=-\log\left(\frac{\dd\tau_g}{\dd t_K}\right)
=
-\frac{\Phi}{c^2}+\frac{v_K^2}{2c^2}+O(c^{-4}).

The gravitational term and the kinetic term therefore enter the same reference-time cadence object. A deeper gravitational potential and a larger kinetic speed both increase the load in the sense of reducing cadence relative to the declared stationary reference time, subject to the sign convention for $\Phi$TeX\Phi.

proposition: Metric branch cadence in a weak static comparison. On the metric branch, for constant $\Omega_q$TeX\Omega_q and the declared stationary reference time $t_K$TeXt_K,

\frac{\dd N_q}{\dd t_K}
=
\Omega_q\exp[-\Pcal_g^{(K)}]
=
\Omega_q\frac{\dd\tau_g}{\dd t_K}.

Thus kinetic and gravitational relativistic clock shifts are a single phase-load cadence law relative to the declared weak-field comparison time.

proof. This is the metric-branch identity reference written with $\dd t_K$TeX\dd t_K as the comparison increment. The expansion reference follows by expanding reference.

A coordinate chart can be chosen so that a chosen worldline remains at spatial coordinate origin. For an accelerated worldline this chart is not a single inertial chart on the entire history. The metric components and the comparison convention in that chart carry the acceleration and simultaneity transport information. The local reset of coordinates does not reset the integral reference.

theorem: Diffeomorphism invariance of the commit integral. Let $\Gamma$TeX\Gamma be a timelike worldline and let $\varphi$TeX\varphi be a diffeomorphism from a neighbourhood of $\Gamma$TeX\Gamma to another coordinate representation. If $g$TeXg, the comparison one-form $\Theta_C$TeX\Theta_C, $u$TeXu, and the scalar residual data are transformed tensorially, then on every common segment

N_q[\Gamma;C]=N'_q[\varphi(\Gamma);\varphi_*C],

with $\varphi_*C$TeX\varphi_*C denoting the pushed-forward comparison convention.

proof. The integrand in reference is a scalar times the declared comparison one-form reference. The pullback of that one-form to the worldline is invariant under coordinate change. The proper-time lapse reference and every scalar residual in reference are likewise invariant when their tensor arguments and the comparison convention are transformed. Hence the line integral is unchanged.

corollary: Finite chart-cover transition invariance. If $\Gamma$TeX\Gamma is covered by a smooth family of local charts whose transition maps are regular on overlaps, and if $g$TeXg, $\Theta_C$TeX\Theta_C, $u$TeXu, and the scalar residual data are transformed tensorially across those overlaps, then the same value of $N_q[\Gamma;C]$TeXN_q[\Gamma;C] is obtained by summing the chartwise pullbacks over any finite refinement of the cover.

proof. Apply the theorem on each overlap. Regular transition maps identify the pulled-back one-form integrands on common subsegments, so adjacent chartwise integrals agree on refinements and sum to the same line integral on $\Gamma$TeX\Gamma.

corollary: Moving-origin coordinate reset. Attaching the spatial origin of a regular local chart family to the travelling clock changes the coordinate representation of the other clock but not either clock's accumulated commit count.

proof. A moving-origin description is a particular regular chart-family representation along the worldline. The previous corollary preserves the chartwise line integral, while the transformed metric, comparison one-form, velocity, and residual scalars carry the transport data of the history.

corollary: Local equivalence and path inequivalence. Every local rest frame may recover the same local laws and the same invariant light speed. It does not follow that two histories between the same endpoint events have equal commit count. Equality of local status is not equality of worldline load integral.

proof. Local Lorentz frames put the metric in Minkowski normal form at an event, establishing equality of local ideal laws. The accumulated count is instead a scalar line integral along the full history. Two curves can have identical local normal forms at every point but different proper lengths or different integrals of residual scalars. Local equivalence therefore does not imply equality of path integrals.

This corollary is the CHC resolution of the coordinate-reset objection. The travelling clock may regard itself as locally at rest at each event. The sequence of local frames is nevertheless transported nontrivially through acceleration, reversal, or gravitational displacement. The accumulation is over the transported history.

proposition: Metric twin comparison. Consider flat spacetime and the inertial local-observer convention $C=T$TeXC=T. Let event $A$TeXA be departure and event $B$TeXB be reunion. Let $\Gamma_0$TeX\Gamma_0 be the inertial worldline at rest relative to $T$TeXT between $A$TeXA and $B$TeXB, and let $\Gamma_v$TeX\Gamma_v be a piecewise inertial outbound-return worldline with speed magnitude $v$TeXv on each leg relative to $T$TeXT and negligible turnaround duration. If the comparison coordinate duration is $\Delta t_T$TeX\Delta t_T, then

\Tau_g[\Gamma_0]=\Delta t_T,
\qquad
\Tau_g[\Gamma_v]=\Delta t_T\sqrt{1-\frac{v^2}{c^2}}.

For a constant channel rate $\Omega_q$TeX\Omega_q on the metric branch,

\frac{N_q[\Gamma_v]}{N_q[\Gamma_0]}
=
\sqrt{1-\frac{v^2}{c^2}}.

proof. For the inertial convention $C=T$TeXC=T, reference gives $\dd\tau_g=\sqrt{1-v^2/c^2}\,\dd t_T$TeX\dd\tau_g=\sqrt{1-v^2/c^2}\,\dd t_T on each constant-speed leg. Integrating both equal-duration legs gives reference. Equation reference follows from reference.

Equation reference does not privilege Earth. If a physical situation is built in which the Earth-bound clock follows the outbound-return worldline and the spacecraft clock follows the inertial worldline, the ratio is reversed. What is invariant is the comparison of histories between common events, not the naming of one body as the origin.

In phase-load language, the traveller carries

\Pcal_g=\log\left(\frac1{\sqrt{1-v^2/c^2}}\right)

on the high-speed legs relative to $T$TeXT, while the inertial rest history relative to $T$TeXT carries zero kinetic load. The reduced commit count is the integral of $\Omega_q\ee^{-\Pcal_g}$TeX\Omega_q\ee^{-\Pcal_g} over the comparison duration.

The metric member is universal: all ideal clock channels accumulate in proportion to $\dd\tau_g$TeX\dd\tau_g. A CHC phase-load extension becomes physically distinct only when residual scalar loads are retained. To first order in $\varepsilon$TeX\varepsilon,

N_q[\Gamma;C]
=
\int_\Gamma \Omega_q\ee^{-\Pcal_g}\left(1-\varepsilon\Rcal_q+O(\varepsilon^2)\right)\dd t_C\nonumber

=
\int_\Gamma \Omega_q\dd\tau_g
-
\varepsilon\int_\Gamma \Omega_q\Rcal_q\dd\tau_g
+O(\varepsilon^2).

The first term is the standard metric accumulation. The second term is a channel-dependent displacement of commit count.

proposition: Same-proper-time separation. Let two histories $\Gamma_1,\Gamma_2$TeX\Gamma_1,\Gamma_2 satisfy $\Tau_g[\Gamma_1]=\Tau_g[\Gamma_2]$TeX\Tau_g[\Gamma_1]=\Tau_g[\Gamma_2] and let $\Omega_q$TeX\Omega_q be constant. On the metric branch they give equal accumulated count. With residual loads,

N_q[\Gamma_1]-N_q[\Gamma_2]
=
-\varepsilon\Omega_q
\left(
\int_{\Gamma_1}\Rcal_q\dd\tau_g
-
\int_{\Gamma_2}\Rcal_q\dd\tau_g
\right)
+O(\varepsilon^2).

Thus histories with equal metric proper time can separate if their residual phase-load integrals differ.

proof. Subtract reference for the two histories and use equality of the metric proper-time terms.

corollary: Acceleration-history separation. If $\Rcal_q=\alpha_q a_\mu a^\mu/a_*^2$TeX\Rcal_q=\alpha_q a_\mu a^\mu/a_*^2 on the relevant window, then equal-proper-time histories with different integrated squared four-acceleration have first-order commit displacement

\Delta N_q
=
-\varepsilon\Omega_q\frac{\alpha_q}{a_*^2}
\left(
\int_{\Gamma_1}a_\mu a^\mu\dd\tau_g
-
\int_{\Gamma_2}a_\mu a^\mu\dd\tau_g
\right)
+O(\varepsilon^2).

The metric branch corresponds to $\alpha_q=0$TeX\alpha_q=0 for ideal clocks.

proof. Substitute $\Rcal_q=\alpha_q a_\mu a^\mu/a_*^2$TeX\Rcal_q=\alpha_q a_\mu a^\mu/a_*^2 into reference. Linearity of the integral gives the displayed expression. Setting $\alpha_q=0$TeX\alpha_q=0 removes the residual term.

corollary: Channel separation. For two commit channels $q$TeXq and $r$TeXr carried on the same worldline and normalized by their reference rates, the first-order logarithmic separation is

\Delta_{qr}[\Gamma]
=
\log\frac{N_q/\Omega_q}{N_r/\Omega_r}
=
-\varepsilon
\left\langle \Rcal_q-\Rcal_r\right\rangle_\Gamma
+O(\varepsilon^2),

where $\langle F\rangle_\Gamma$TeX\langle F\rangle_\Gamma denotes the proper-time average of $F$TeXF on the worldline. The metric member gives $\Delta_{qr}=0$TeX\Delta_{qr}=0 for ideal co-moving channels after reference-rate normalization.

proof. Apply reference to channels $q$TeXq and $r$TeXr on the same curve, divide by their constant reference rates, and expand $\log(A-\varepsilon B)=\log A-\varepsilon B/A+O(\varepsilon^2)$TeX\log(A-\varepsilon B)=\log A-\varepsilon B/A+O(\varepsilon^2). The common metric integral cancels, leaving the proper-time average of $\Rcal_q-\Rcal_r$TeX\Rcal_q-\Rcal_r. If both residuals vanish, so does the normalized separation.

These separations are not present in the universal metric member. They are the finite signatures of a nonzero residual load: acceleration history at fixed metric proper time, channel dependence under identical motion, and cohesive-load dependence under identical metric conditions.

The residual scalar $\Ccal_{\rm coh}$TeX\Ccal_{\rm coh} in reference represents the possibility that an internally coherent or strongly bound structure carries additional load not exhausted by the metric proper-time element. A cohesive residual is admitted only after a channel family $q$TeXq, an internal descriptor tuple $Z_q$TeXZ_q, and a dimensionless map $F_q:Z_q\to\mathbb{R}$TeXF_q:Z_q\to\mathbb{R} are fixed on the comparison window. Typical descriptor entries may include dimensionless functions of internal stress, binding-energy fraction, phase rigidity, or coherent mode participation; the map $F_q$TeXF_q is part of the declared channel model, not a quantity retuned after comparison.

For a clock channel $q$TeXq with cohesive coefficient $\beta_q$TeX\beta_q, the cadence factor is

\chi_q
=
\chi_g\exp[-\varepsilon\beta_q\Ccal_{\rm coh}+O(\varepsilon^2)].

A pair of co-located channels with different $\beta$TeX\beta values then has normalized separation

\Delta_{qr}
=
-\varepsilon(\beta_q-\beta_r)
\left\langle \Ccal_{\rm coh}\right\rangle_\Gamma
+O(\varepsilon^2).

If no such separation is present, the finite window constrains the allowed cohesive coefficients. If a separation is present and cannot be absorbed into ordinary systematic shifts, the metric member is insufficient on that window.

A frequency comparison is a comparison of a transported electromagnetic frequency observable expressed in the local commit-clock units at the emission and absorption endpoints. It is therefore an inverse-clock-unit frequency observable associated with endpoint commit cadence, not a direct ratio of the two endpoint clock-cadence factors and not a transported material carrier substance. On the metric branch this reduces to the standard relativistic frequency ratio. If $k_\mu$TeXk_\mu is the null wave covector and $u_e,u_o$TeXu_e,u_o are the emitter and observer four-velocities, let

\omega=-k_\mu u^\mu

be the locally measured positive frequency. The metric ratio is

1+z_g
=
\frac{\omega_e}{\omega_o}.

In homogeneous cosmology this becomes $1+z_g=a(t_o)/a(t_e)$TeX1+z_g=a(t_o)/a(t_e).

Under the phase-load representation, the electromagnetic excitation or frequency observable is not treated as a substance that loses material content along the path. The phrase ``commit carrier,'' when used for such a comparison, is only shorthand for the commit-capable electromagnetic excitation whose emission-local and absorption-local cadence units are compared on the declared global phase-field comparison structure after propagation. A residual clock-channel factor would enter not by altering the null propagation law in reference, but by modifying the local clock-channel cadence used to measure the endpoint frequency. In the convention used here, a positive residual load suppresses the local channel cadence through $\exp[-\varepsilon\Rcal_q]$TeX\exp[-\varepsilon\Rcal_q]; therefore a frequency measured in units of that local channel is represented to first order by

\omega^{\rm meas}_{q,\ell}
=
\omega_\ell\exp\left[\varepsilon\Rcal_{q,\ell}+O(\varepsilon^2)\right],
\qquad \ell\in\{e,o\}.

The channel-measured redshift ratio is then

1+z_q
=
\frac{\omega^{\rm meas}_{q,e}}{\omega^{\rm meas}_{q,o}}
=
(1+z_g)
\exp\left[\varepsilon(\Rcal_{q,e}-\Rcal_{q,o})+O(\varepsilon^2)\right],

whenever the same channel type $q$TeXq is used to define the emitter and observer clock comparison. An inverse convention based on received frequency relative to a local observer reference would reverse the endpoint residual difference. The metric member is recovered when $\varepsilon=0$TeX\varepsilon=0.

Piecewise inertial travel

For constant $v$TeXv on two equal legs in flat spacetime, the phase-load form gives

N_q=\Omega_q\Delta t_T\ee^{-\log\gamma}
=\frac{\Omega_q\Delta t_T}{\gamma},
\qquad
\gamma=\frac1{\sqrt{1-v^2/c^2}}.

This is the metric twin result. Adding an acceleration residual supported only at the turnaround gives a correction depending on the finite acceleration profile, not only on the asymptotic speed.

Circular high-speed channel

For a circular orbit of radius $R$TeXR and speed $v$TeXv in flat spacetime relative to an inertial comparison convention,

\dd\tau_g=\dd t_T\sqrt{1-v^2/c^2}.

If an acceleration residual is retained, $a_\mu a^\mu=\gamma^4v^4/R^2$TeXa_\mu a^\mu=\gamma^4v^4/R^2 for uniform circular motion, and

\frac{\dd N_q}{\dd t_T}
=
\Omega_q\sqrt{1-v^2/c^2}
\left[1-\varepsilon\alpha_q\frac{\gamma^4v^4}{R^2a_*^2}+O(\varepsilon^2)\right].

The first factor is the standard lifetime dilation; the second is an allowed residual phase-load factor only off the metric member.

Weak gravitational comparison

For two clocks at potentials $\Phi_1$TeX\Phi_1 and $\Phi_2$TeX\Phi_2 with negligible relative speed, the metric branch gives

\frac{\nu_2}{\nu_1}
=
1+\frac{\Phi_2-\Phi_1}{c^2}+O(c^{-4}).

The CHC phase-load form reads this as a difference of gravitational load. A cohesive residual would add

\frac{\nu_2}{\nu_1}
=
1+\frac{\Phi_2-\Phi_1}{c^2}
-
\varepsilon\left(\beta_q\Ccal_{{\rm coh},2}-\beta_q\Ccal_{{\rm coh},1}\right)
+O(c^{-4},\varepsilon^2).

Let a finite clock comparison be specified by histories $\Gamma_a$TeX\Gamma_a, channel types $q_a$TeXq_a, reference rates $\Omega_{q_a}$TeX\Omega_{q_a}, and comparison intervals. Choose a positive reference duration $\tau_*>0$TeX\tau_*>0 common to the finite comparison and define the dimensionless normalized log-count observable

Y_a=\log\left(\frac{N_{q_a}[\Gamma_a]}{\Omega_{q_a}\tau_*}
\right).

For two histories or two channels,

\Delta Y_{ab}=Y_a-Y_b.

On the metric member,

\Delta Y_{ab}^{(0)}
=
\log\left(\frac{\Tau_g[\Gamma_a]}{\tau_*}\right)
-
\log\left(\frac{\Tau_g[\Gamma_b]}{\tau_*}\right)
=
\log\left(\frac{\Tau_g[\Gamma_a]}{\Tau_g[\Gamma_b]}\right)

when the reference rates are constant. The first residual displacement is

\delta(\Delta Y_{ab})
=
-\varepsilon
\left(
\langle\Rcal_{q_a}\rangle_{\Gamma_a}
-
\langle\Rcal_{q_b}\rangle_{\Gamma_b}
\right)
+O(\varepsilon^2).

Equation reference is the finite comparison map. It separates three cases: pure metric cadence, same-metric-history channel splitting, and same-proper-time residual history splitting.

A finite-window phase-commit system evolves a reduced density state and produces observable projection maps. The present worldline-load construction supplies the cadence factor by which the internal generator is read along a timelike history. If a reduced generator $\Lcal_{\vartheta,\varepsilon}$TeX\Lcal_{\vartheta,\varepsilon} is parameterized by local commit time $s_q$TeXs_q, with

\dd s_q=\chi_q(\Gamma;C)\,\dd t_C,

then, after choosing the clock channel by which the reduced state is read, its comparison-time representation along $\Gamma$TeX\Gamma is

\frac{\dd\rho}{\dd t_C}
=
\chi_q(\Gamma;C)\,\Lcal_{\vartheta,\varepsilon}(\rho).

On the metric member this is the usual proper-time reparameterization. With residual phase-load the same reduced generator is read through a modified cadence factor. No new sector action is selected by this operation.

A worldline phase load represents action-level exchange only if its pulled-back scalar functional has a formally self-adjoint Fr\'echet derivative. Alignment of the four-force with $\nabla_\mu\mathcal H$TeX\nabla_\mu\mathcal H is necessary for a scalar loading interpretation but is not sufficient for variational realizability. On a contractible segment and star-shaped phase chart, the Helmholtz condition supplies a local action primitive; failure excludes the proposed load rather than merely leaving it unproved.

Local commit cadence requires a separate CPTP completion. A no-event sink $K$TeXK must factor as $\sum_aL_a^\dagger L_a$TeX\sum_aL_a^\dagger L_a, with every lost probability assigned to resolved or explicitly unresolved events. If several clocks or trajectories share one phase coupling, their cadence shifts must also obey the left-null restrictions of the common sensitivity matrix. Trajectory-specific coefficients remove that predictive content.

The closure test for the worldline phase load and commit cadence 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 phase accumulation, tick intervals, commit-time distributions, and clock comparisons. Let $a$TeXa range over the independent constitutive inputs comprising worldline action, clock coupling, environment memory, and event instrument.

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 variational worldline model and one normalized finite-memory instrument generate both loading and cadence.

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 an arbitrary cadence modulation can fail action integrability or probability closure. Failure of that condition therefore rejects the proposed microscopic closure before parameter estimation can be counted as evidence for it.

Worldline phase-load gives a CHC cadence reading of relativistic clock effects. The metric member is exactly the usual proper-time accumulation. Its phase-load representation says that kinetic and gravitational clock shifts are represented as changes of local commit cadence relative to a declared comparison convention; in the weak-field reference convention above, the kinetic term and deeper-potential gravitational term appear as positive load contributions. Coordinate reset is harmless: a travelling clock may be kept at the origin of its own chart, but the transported frame history and the worldline integral remain. The twin comparison is therefore a comparison of two commit histories between common events, not a privilege of Earth as an absolute frame.

The same formalism also exposes finite residual directions. If all residual loads vanish, CHC reduces to universal metric cadence for ideal clocks. If residual loads are retained as an explicitly declared nonmetric residual hypothesis, then acceleration-history, cohesive-channel, or clock-channel separations are admissible comparison-window outputs even where the metric member assigns equal proper time. These separations are not part of the standard branch, do not assert an observed clock anomaly by themselves, and are only the additional content of a nonzero worldline phase-load extension.

A bounded public-clock support summary covers the metric-member consistency side of this statement. It checks Hafele--Keating airborne-clock shifts, Bailey muon storage-ring lifetime dilation, and the Chou/NIST optical-clock height-shift gate against standard metric-member formulas, with local support label WPL-PUBLIC-CLOCK-METRIC-MEMBER-GATES-SATISFIED. This support summary is not a raw clock-data reanalysis and does not claim a clock anomaly; it verifies that the retained metric member is represented on representative public clock surfaces.

In flat spacetime with inertial comparison time $t$TeXt,

\dd\tau_g=\dd t\sqrt{1-v^2/c^2}.

For an outbound-return path with equal coordinate durations $\Delta t/2$TeX\Delta t/2 and constant speed magnitude $v$TeXv on each leg,

\Tau_g
=
2\int_0^{\Delta t/2}\sqrt{1-v^2/c^2}\,\dd t

=\Delta t\sqrt{1-v^2/c^2}.

The turnaround can be replaced by any smooth acceleration segment; the limiting result is recovered as the duration of that segment tends to zero with fixed endpoint velocities. This limiting statement is a metric-member statement; acceleration-residual extensions require a declared smooth profile with finite residual integral.

From $\chi_q=\exp[-\Pcal_g-\varepsilon\Rcal_q]$TeX\chi_q=\exp[-\Pcal_g-\varepsilon\Rcal_q],

\chi_q
=
\chi_g\left(1-\varepsilon\Rcal_q+\frac12\varepsilon^2\Rcal_q^2+O(\varepsilon^3)\right).

On any window satisfying reference, the elementary remainder bound

\left|e^{-\varepsilon R}-(1-\varepsilon R)\right|
\le
\frac12 e^{|\varepsilon R|}|\varepsilon R|^2

shows that the first-order remainder is uniform on the window. Substitution into reference gives reference. If $\Omega_q$TeX\Omega_q varies slowly along the path, the same expansion holds with the weighted average

\langle F\rangle_{\Gamma,q}
=
\frac{\int_\Gamma \Omega_q F\dd\tau_g}{\int_\Gamma \Omega_q\dd\tau_g}.

No new observational data are introduced. All definitions, assumptions, and representative finite comparisons used in the argument are contained in the text. Companion source summaries record public clock metric-member gate calculations for representative literature surfaces; those gates are bounded-support and do not assert a nonmetric clock residual or new clock anomaly. The combined local label PCD-WPL-VP1-FORMAL-AND-PUBLIC-CLOCK-METRIC-COMPATIBILITY-SUPPORT denotes a declared support comparison shared with CHC-PCD, not a clock anomaly claim and not a separate public manuscript claim.

Funding and competing interests..

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