Temporal Equivalence Principle: Synchronization Holonomy in Pulsar Scintillation

1. Introduction: Temporal Equivalence Principle and Observable Holonomy

General Relativity (GR) is built upon the Einstein Equivalence Principle (EEP), which postulates that at any spacetime point, a local inertial frame can be chosen such that the laws of physics take their special relativistic form. In this framework, spacetime is represented as a single pseudo-Riemannian manifold $(M, g_{\mu\nu})$, where proper time $d\tau$ is a purely geometric quantity defined by the line element $d\tau^2 = g_{\mu\nu} dx^\mu dx^\nu$. The standard scalar-factorizable thin-screen scintillation model assigns each scattered propagation path an additive scalar delay, making differential delays simple differences and the closure residual identically zero by arithmetic.

The Temporal Equivalence Principle (TEP) retains the EEP's local validity—freely falling frames still recover special relativity—but relaxes the global assumption that a single metric governs both gravitational dynamics and matter coupling. Where Einstein's derivation assumes $g_{\mu\nu}$ universally defines proper time for all physical processes, TEP introduces a matter metric $\tilde{g}_{\mu\nu}$ that coincides with $g_{\mu\nu}$ in dense, screened environments while permitting deviations in unscreened regions.

The divergence between TEP and standard theory occurs at the level of the path-delay model. Whereas the additive-delay null hypothesis assumes each scattered path carries a scalar delay—making the closure residual vanish by simple arithmetic—TEP allows non-exact structure through disformal coupling, yielding path-dependent proper time accumulation that varies spatially. In this framework, proper time is no longer a passive coordinate but a dynamical entity affected by the local gravitational potential through a two-metric scalar-tensor structure.

This dynamical structure yields a specific, falsifiable prediction: synchronization holonomy, a geometric phase effect where proper time transport around closed loops fails to close by measurable amounts. A robust non-zero closure signal rejects the tested scalar-factorizable additive-delay null; if conventional multi-path propagation explanations are excluded by independent replication and matched-screen controls, it demonstrates non-integrable synchronization transport. This work presents observational evidence consistent with this phenomenon, providing candidate evidence for synchronization holonomy as a feature of macroscopic clock transport in the interstellar medium.

The TEP framework is a two-metric scalar-tensor theory defined on a single spacetime manifold endowed with two metrics: a gravitational metric \(g_{\mu\nu}\) (governing gravitational interactions) and a canonical causal matter metric \(\tilde{g}_{\mu\nu} = -A_{\rm loc}^2(u_{\rm grav}(\mathbf x))U_\mu U_\nu + A_{\rm sp}^2(u_{\rm drift}(t,\mathbf x))\Pi_{\mu\nu} + B(\phi)\bigl(\Pi_\mu^\alpha\Pi_\nu^\beta\nabla_\alpha\phi\nabla_\beta\phi - 2(U\cdot\nabla\phi)U_{(\mu}\Pi_{\nu)}{}^\alpha\nabla_\alpha\phi\bigr)\) (\(U_\mu\equiv-\nabla_\mu\bar\phi/\sqrt{2\bar X}\), \(\Pi_{\mu\nu}=g_{\mu\nu}+U_\mu U_\nu\); Paper 0, §2.2, Step 80) to which all non-gravitational fields and clocks universally couple. In conformal–disformal shorthand, the metrics are related by the map:

where \(\phi\) is the time field, \(A(\phi) = \exp(\beta_A \phi / M_{\text{Pl}})\) is a universal conformal factor, \(B(\phi)\) encodes tiny, direction-dependent deformations of the light cone, and \(\nabla_{\mu}\) denotes the covariant derivative with respect to the gravitational metric \(g_{\mu\nu}\). The constants \(\beta_A\) and \(M_{\text{Pl}}\) represent the scalar coupling strength and Planck mass, respectively. General Relativity is recovered exactly when \(A(\phi) = 1\) and \(B(\phi) = 0\). The TEP framework is founded on four axioms:

The four axioms that define the TEP framework are: (A1) a two-metric structure where gravity is governed by \(g_{\mu\nu}\) while matter couples to a causal matter metric \(\tilde{g}_{\mu\nu}\); (A2) the Temporal Equivalence Principle, requiring that all non-gravitational processes evolve according to proper time defined by \(\tilde{g}_{\mu\nu}\); (A3) causal safety, ensuring that deviations from GR remain within current constraints on photon-graviton speed differences; and (A4) environmental suppression of observable Temporal Shear through \(\Sigma_\mu^{\rm obs}=\mathcal{S}_\Sigma(\mathcal{E})\,\Sigma_\mu\), which suppresses Temporal Shear in screened environments while leaving dilute, unscreened regions such as the ISM accessible to TEP dynamics.

The critical consequence for this work is that the ISM—being a dilute environment where Temporal Shear remains unsuppressed—is precisely where TEP effects should manifest maximally. The disformal coupling \(B(\phi) \neq 0\) introduces non-exact structure into the time-transport 1-form, yielding path-dependent proper time accumulation around closed scattering loops.

The breakdown of global simultaneity is formalized through a synchronization-transport law, deriving the convention-independent synchronization holonomy \(H \equiv \oint_C d\tilde{\tau}\) as an invariant measure of non-integrability of time transport around closed loops. In purely conformal theories this holonomy vanishes once general-relativistic contributions are removed; a nonzero holonomy at leading order requires non-exact structure provided by disformal coupling \(B(\phi) \neq 0\).

This study applies the TEP framework to pulsar scintillation across 15 pulsars, providing a comprehensive multi-pulsar test in the interstellar medium—a dilute environment where Temporal Shear remains unsuppressed and TEP effects manifest maximally. The phase-domain falsification of the \(\psi=0\) null is driven by PSR J0437-4715; the remaining objects verify environmental scaling predictions and provide complementary geometric structure.

The dataset combines measurements from Parkes observations of PSR J0437-4715 as the primary target and PSR J1603-7202 for velocity geometry testing with a 73.8° velocity separation. The velocity vector geometry provides a critical geometric test: TEP predicts opposite-sign holonomy for pulsars with different velocity projections onto scattering geometry.

Additionally, the study incorporates Jiamusi telescope observations of ten bright, distant pulsars (distances 1–3.7 kpc; see Table 3.1 for the full catalog), extending the distance baseline by an order of magnitude to enable rigorous tests of TEP scaling with scattering strength. These distant pulsars provide noise-limited bounds because they contain only 1–2 closure-capable epochs; they provide bounds but do not independently diagnose environmental suppression.

1.1 Key Terminology

Temporal Equivalence Principle (TEP): A theoretical framework where proper time is elevated to a dynamical field that couples to the local gravitational environment, creating path-dependent time accumulation that varies spatially.

Synchronization holonomy (H): The non-closure of proper time transport around closed loops, representing path-dependent time accumulation. This effect is analogous to a geometric phase for clock transport. The Phase Closure \(\psi\) is the primary detection statistic.

Closure delay: The sum of differential delays around a triplet of scattered propagation paths through the interstellar medium, constituting a direct measurement of synchronization holonomy.

Stokes alignment: Geometric rectification of closure delays using Stokes' theorem, which aligns delays according to the orientation of the scattering geometry and the effective velocity through the scattering screen.

Arc curvature (\(\eta\)): A measure of scattering strength in the interstellar medium, which TEP predicts should scale with the holonomy magnitude.

Temporal Topology: The spatial and covariance structure of \(\ln A(\phi)\).

Temporal Shear: The canonical synchronization gradient \(\Sigma_\mu \equiv \nabla_\mu \ln A(\phi) = \alpha(\phi)\,\nabla_\mu \phi\). Its observable response is \(\Sigma_\mu^{\rm obs} = \mathcal{S}_\Sigma(\mathcal{E})\,\Sigma_\mu\), where \(\mathcal{E}\) denotes the environmental state (density, compactness, gradients, source structure, coherence scale, boundary conditions).

Signed mean vs. Phase Closure \(\psi\): The signed mean (average of signed delays) serves as a physical metric and is expected to be near zero due to bipolar partial cancellation. The Phase Closure \(\psi\) is the primary zero-centered circular detection statistic (measuring the loop holonomy phase within the TEP measurement model).

Methodology Status: The prediction of non-zero synchronization holonomy follows directly from the disformal metric structure defined above; a self-contained derivation is given in Appendix A. The pulsar scintillation closure delay methodology presented here was developed independently to test that prediction. The empirical tests reported below serve to test the predictions of this framework.

1.2 Summary of Findings

The analysis reports the following core results, which together form a multiscale consistency check on the Temporal Equivalence Principle:

• Primary Phase Falsification: PSR J0437-4715 rejects the additive scalar path-delay null through a non-zero Phase Closure \(\psi\) (\(\bar{\psi}=0.984 \pm 0.046\) rad, 95% circular-bootstrap CI \([0.737,1.235]\) rad excluding zero; unweighted \(\psi=1.120\) rad, 95% CI \([0.990,1.253]\) rad excluding zero; circular concentration confirmed by Rayleigh \(p=1.39\times10^{-13}\) weighted and \(1.34\times10^{-44}\) unweighted, V-test \(p=1.02\times10^{-5}\)).
• Geometric Corroboration: PSR J1603-7202 exhibits a frame-independent bipolar decomposition and elevated circular variance directionally consistent with the high-dispersion pattern; its Phase Closure is noise-limited, so the decomposition is a consistency check rather than demonstrated parity structure.
• Environmental Bounds: Jiamusi and MeerKAT pulsars are noise-limited, consistent with TEP's predicted environmental suppression on distant, noise-limited sightlines where Temporal Shear is suppressed.
• Orbital Kinematics: While the naive signed-delay orbital mean is attenuated by within-epoch bipolar cancellation (amplitude \(1.11 \pm 0.79\) ns, LR \(p = 0.372\)), the identical REML mixed-effects model resolves significant orbital modulation on the unsigned magnitude \(M_{\rm cl}\) (\(A = 5.71 \pm 1.10\) ns, LR \(p = 1.70 \times 10^{-6}\); \(6.42 \pm 1.19\) ns, \(p = 5.86 \times 10^{-7}\) on the 1,093 independent epochs) and on the paired noise-subtracted excess \(M_{\rm cl} - E[M_{\rm cl}]\) (\(A = 1.14 \pm 0.45\) ns, \(\sigma_\alpha = 10.60\) ns, LR \(p = 0.0425\), Step 046b), corroborated in the phase domain by the Thiele–Innes binary reflex velocity projection onto the \(128^\circ\) scattering axis (\(2\Delta\ln L = 5.70\), 1 df \(p = 0.0170\), \(\Delta\mathrm{AIC} = -3.70\), Step 051).

Together, these findings exclude the \(\psi = 0\) null hypothesis across multiple standard interstellar scattering models, supporting the existence of synchronization holonomy.

2. Synchronization Holonomy in Pulsar Scintillation

2.1 The Closure Delay Observable

When pulsar signals traverse the interstellar medium, scattering from plasma inhomogeneities creates multiple propagation paths. These paths interfere in the observer's dynamic spectrum, producing characteristic patterns that encode information about the scattering geometry. In standard scintillation theory, each scattered propagation path is assigned a scalar delay \(\tau_i\). Differential delays are differences of these scalars, so the closure sum around any triplet vanishes identically by arithmetic—a property termed synchronization consistency.

The closure residual \(C_{ijk}\) tests this additive representation. Each cross-term \(\hat{\tau}_{ab}\) is independently measured from the secondary-spectrum peak between scattered paths \(a\) and \(b\) via sub-pixel interpolation. For a triplet of scintillation arclets indexed by \(i, j, k\) forming a closed loop, the closure residual is:

Under the additive-delay null hypothesis, there exists a scalar assignment \(\tau_i\) such that \(\hat{\tau}_{ab} = \tau_b - \tau_a\) within measurement noise, implying \(C_{ijk} \equiv 0\). The closure construction cancels common-mode contributions—including standard relativistic effects that apply equally to all scattered paths—by design, isolating only path-dependent contributions that violate the additive representation. TEP predicts a non-zero synchronization holonomy \(H_{\rm resid}^{(\tau)}\), defined as the GR-subtracted loop integral of the synchronization connection \(\Delta\sigma^{(\tau)} = \tilde{\sigma}^{(\tau)} - \sigma_{\rm GR}^{(\tau)}\) around the closed path (Jakarta, Paper 0). This holonomy—a geometric phase for clocks—arises because the disformal sector \(B(\phi) \neq 0\) of the matter metric \(\tilde{g}_{\mu\nu}\) generates a non-exact contribution to the synchronization connection when \(u \cdot \nabla\phi \neq 0\). The observed closure delay relates to the holonomy magnitude as:

where \(\langle \cdot \rangle_{\text{aligned}}\) denotes the Stokes-aligned average accounting for the relative orientation of the path triplet and the observer's velocity (see Section 4.7 for the alignment algorithm specification). The alignment ensures that delays correlate with geometric orientation as predicted by TEP.

The primary detection statistic is the Phase Closure \(\psi\), constructed directly from the geometric phase of the complex secondary-spectrum cross-term triplet. Unlike unsigned delay-domain amplitude estimators (\(M_{\rm cl}\), Appendix B), which are strictly positive under folded-normal Rice noise, the Phase Closure \(\psi\) is a zero-centered, odd-parity circular observable that cancels identically in the absence of path-dependent synchronization transport:

where \(\phi_{ab}\) is the phase of the cross-term between scattered paths \(a\) and \(b\), measured directly from the secondary spectrum. The minus sign on \(\phi_{ik}\) reflects the actual geometry: the three independently measured cross-terms are \((i,j)\), \((j,k)\) and the direct difference \((i,k)\); their closure is \(\phi_{ij} + \phi_{jk} - \phi_{ik}\). For a sample of \(N\) triplet measurements, the circular mean is:

TEP predicts \(\bar{\psi} \neq 0\) while the additive-delay null hypothesis predicts \(\bar{\psi} = 0\).

Why ordinary ray phases close but TEP synchronization transport need not. If every measured cross-term is factorizable as \(Q_{ab} = X_a X_b^*\), then \(\arg(Q_{ij}Q_{jk}Q_{ki}) = 0\) identically. This is precisely the scalar-delay null tested here: each ray carries a scalar accumulated phase \(\Theta_a\), and the closure \((\Theta_i - \Theta_j) + (\Theta_j - \Theta_k) + (\Theta_k - \Theta_i) = 0\) vanishes regardless of path dependence, because endpoint phases are state functions. TEP modifies the operational comparison itself: the pairwise observable is the transport of synchronization between the path-defined events associated with rays \(a\) and \(b\),

where \(\Gamma_{ab}\) is the comparison path in spacetime induced by the scattering geometry, \(\Delta\sigma^{(\tau)}\) is the time-valued residual synchronization transport (units of seconds), and \(\omega_{\rm eff}(x)\) is the effective angular frequency that converts synchronization transport to phase. The time-valued residual holonomy is:

and the phase closure is:

Only when \(\omega_{\rm eff}\) is common to all three legs may this be reduced to \(\psi_{ijk} = \operatorname{wrap}\!\left[\omega_{\rm eff}\,H_{\rm resid}^{(\tau)}(C_{ijk})\right]\). In geometric-optics (eikonal) electrodynamics on \(\tilde{g}_{\mu\nu}\), the electromagnetic wave covector \(k_\mu = \partial_\mu \Phi_{\rm EM}\) transports phase at the local carrier angular frequency \(\omega_{\rm eff} = 2\pi\nu_0\) calibrated against the observer's \(\tilde{g}\)-proper time congruence, while the secondary-spectrum extraction kernel introduces a subband-overlap transfer weight evaluated explicitly in Appendix A.4 and Step 061. A direct empirical test of the phase–time mapping is available through the subband and wideband frequency scaling:

valid for an unwrapped phase and a frequency-independent residual over the fitted subband. If frequency dependence is allowed, \(d\psi_{\rm unwrap}/d\omega = H + \omega\,dH/d\omega\). The derivative connects the phase and delay channels and is testable across observing subbands. The null \(\psi = 0\) is recovered whenever \(\Delta\sigma^{(\tau)}\) is exact, including the conformal-only sector, whereas the disformal non-exact sector permits \(\psi \neq 0\). The TEP measurement operator modifies each cross-term as:

Within the TEP measurement model, synchronization transport contributes an edge-dependent phase, so the three pairwise terms need not factorize into endpoint phases and their closure can be non-zero. Appendix A.4 defines the associated events and paths; electromagnetic transfer determines the channel coefficient. The closure product is then \(Q_{ij}^{\rm TEP} Q_{jk}^{\rm TEP} Q_{ki}^{\rm TEP} = |X_i X_j X_k|^2 \exp(i \oint_C \omega_{\rm eff}\,\Delta\sigma^{(\tau)})\), so \(\psi_{ijk} = \operatorname{Arg}\exp(i \oint_C \omega_{\rm eff}\,\Delta\sigma^{(\tau)})\). The measured cross-term is non-factorizable under TEP because the synchronization transport along each edge is an independent path integral, not a difference of state functions. The full identification of the three measured cross-terms with the edges of a genuine closed spacetime loop, including the event and path definitions, is given in Appendix A.4.

2.1.1 Statistical Motivation for Phase Closure

The choice of \(\psi\) as the primary detection statistic follows from the symmetry properties of the cross-term closure phase, independent of any specific theoretical framework. Define the complex closure product:

Under scalar factorization (\(Q_{ab} = X_a X_b^*\)), \(\mathcal{B}_{ijk} = |X_i X_j X_k|^2\), and therefore \(\psi_{ijk} = 0\). Measurement and extraction errors broaden the observed distribution around the scalar-null direction. TEP synchronization curvature adds a nonfactorizable link phase, shifting the circular mean away from zero. The folded closure magnitude \(M_{\rm cl} = \langle |C_{ijk}| \rangle\), by contrast, is strictly positive under any noise model—it measures the mean absolute deviation, not a physical holonomy. This is the Rice-distribution bias: \(M_{\rm cl} > 0\) even when no signal is present.

The Phase Closure \(\psi\) is therefore the appropriate zero-centered test statistic: it cancels to zero for rank-one scalar-factorizable delays, while any physical mechanism producing non-factorizable delay-Doppler cross-term phase structure yields \(\langle \psi \rangle \neq 0\). Because the Rayleigh statistic \(Z=2n_{\rm eff}\bar{R}^2\) measures circular concentration (rejecting circular uniformity) and is invariant under rotating all angles by a constant offset, rejection of \(\langle\psi\rangle = 0\) is established by the circular mean and its bootstrap confidence interval (\(\bar{\psi}=0.984\pm0.046\) rad, 95% CI \([0.737,1.235]\) rad; unweighted \(\psi=1.120\) rad, 95% CI \([0.990,1.253]\) rad, both excluding zero), while Rayleigh \(Z=59.21\) (\(p=1.39\times10^{-13}\) weighted; \(p=1.34\times10^{-44}\) unweighted) establishes that the phase distribution is sharply concentrated rather than uniform circular noise. The interpretation of this non-zero mean offset as a synchronization-holonomy candidate follows from the TEP measurement model (Appendix A.4–A.5), while the statistical exclusion of \(\langle\psi\rangle=0\) under the tested estimator is model-independent.

2.2 TEP Predictions for Closure Delays

The TEP framework—a two-metric scalar-tensor theory introduced in Paper 0—makes a central prediction that directly contrasts with the standard additive-delay model: the synchronization holonomy must be non-zero. The theory posits a disformal coupling between a scalar field $\phi$ and the matter metric $\tilde{g}_{\mu\nu}$ (Bekenstein 1993; Koivisto & Zumalacárregui 2012):

where $g_{\mu\nu}$ is the Einstein metric governing gravitational dynamics, $A(\phi)$ is the conformal coupling function, and $B(\phi) \geq 0$ is the disformal coupling parameter. The observational observable is not a raw one-way speed or the integral of a scalar proper-time differential. Following the Jakarta construction (Paper 0, Section 3.3 and Appendix A3), let $\tilde{\sigma}^{(\tau)}$ denote the matter-frame synchronization transport one-form induced by $\tilde{g}_{\mu\nu}$ and the chosen clock congruence, and let $\sigma_{\rm GR}^{(\tau)}$ denote the corresponding GR one-form including Sagnac, gravitomagnetic/Lense–Thirring, Shapiro delay, gravitational redshift, station motion, clock-scale realization, and reference-frame corrections. The residual synchronization connection is:

and the time-valued residual holonomy around a closed loop $C = \partial\Sigma$ is:

This quantity is invariant under admissible synchronization re-gaugings because both $\tilde{\sigma}^{(\tau)}$ and $\sigma_{\rm GR}^{(\tau)}$ shift by the same exact one-form, leaving $\Delta\sigma^{(\tau)}$ unchanged. In the purely conformal subclass ($B = 0$), the scalar contribution to clock-rate transport is generated by the exact one-form $\omega^{(A)} = d\ln A$, so $\oint_C d\ln A = 0$ for any closed loop in a simply connected region. A leading-order nonzero $H_{\rm resid}^{(\tau)}$ requires residual synchronization curvature $d(\Delta\sigma^{(\tau)}) \neq 0$, supplied in the minimal TEP model by the disformal sector $B(\phi) \neq 0$.

2.2.1 Projected Disformal Contribution

To state the disformal contribution in a measurement-defined way, let $u^\mu$ denote the four-velocity field of the physical clock network or observer congruence, and let $P_\mu{}^\nu = \delta_\mu{}^\nu + u_\mu u^\nu$ be the spatial projector into the local rest space of that congruence. To leading order in the disformal correction, the projected contribution to the synchronization connection has the representative form (Jakarta, Appendix A3):

up to the sign convention used for the synchronization one-form and higher-order disformal corrections. In a hypersurface-orthogonal $3+1$ slicing, where $u^\mu = n^\mu$, this reduces to:

where $N$ is the lapse and $n^\mu$ is the unit normal to the spatial slices. Unlike $d\ln A$, the disformal contribution is not generically exact. Its curvature satisfies $d(\delta\tilde{\sigma}^{(\tau)}) \neq 0$ whenever the prefactor multiplying the projected scalar gradient varies independently around the loop.

2.2.2 Physical Condition for Non-Vanishing Holonomy

A static scalar gradient in a hypersurface-orthogonal frame does not generate the leading holonomy. The pulsar observable requires relative motion through the field, time dependence, mixed temporal-spatial structure, or independently varying lapse/shift and disformal response around the effective loop. The key factor is $u \cdot \nabla\phi \neq 0$: the four-velocity of the clock congruence must have a component along the scalar field gradient.

For the J0437−4715 scintillation geometry, $u \cdot \nabla\phi \neq 0$ is generated by three independent contributions:

• Pulsar proper motion: J0437−4715 has a large transverse proper motion ($\mu_\alpha \simeq 121$ mas/yr, $\mu_\delta \simeq -71$ mas/yr; Deller et al. 2008), corresponding to a tangential velocity $v_\perp \simeq 104$ km/s at $d \simeq 156$ pc. The pulsar's worldline sweeps through the interstellar scalar field gradient, producing a time-varying $u \cdot \nabla\phi$ along the line of sight.
• Earth's orbital motion: The Earth's orbital velocity ($v_{\rm orb} \simeq 30$ km/s) modulates the observer's four-velocity projection onto $\nabla\phi$ over annual timescales, producing a periodically varying contribution to $u \cdot \nabla\phi$.
• Screen motion: The scattering screens at $\sim 89.8$ pc and $\sim 124$ pc have their own peculiar velocities relative to the local standard of rest, contributing additional temporal structure to the effective loop geometry.

These kinematic ingredients enter the theoretical prediction through the $u \cdot \nabla\phi$ factor in equations $\eqref{eq:disformal_sigma}$ and $\eqref{eq:disformal_3plus1}$, not merely through the later Stokes-alignment algorithm. The predicted holonomy is therefore a function of the pulsar's proper motion, the Earth's orbital phase, and the screen kinematics — all of which are independently measured. The Stokes-aligned analysis (Section 4.7) is the observational implementation of this kinematic projection; the theoretical prediction precedes it.

Where the single-metric null hypothesis requires the closure residual to vanish, TEP predicts path-dependent residual synchronization holonomy as a consequence of disformal coupling in dilute environments where Temporal Shear remains unsuppressed. The Phase Closure $\psi$ provides a zero-centered test: $\psi = 0$ under the additive-delay null hypothesis, $\psi \neq 0$ for TEP.

This non-zero holonomy manifests with specific geometric properties. In a Stokes-aligned distribution, delays converge to a consistently negative mean ($-H$) when aligned with the orientation kinematics vector, while the magnitude $M_{\rm cl}$ remains invariant regardless of specific triplet geometry. The sign of the holonomy correlates with geometric orientation, defined by clockwise versus counter-clockwise configurations.

A critical consequence of this sign-geometry correlation is that simple aggregate averaging over all orientations yields $\langle \tau \rangle \approx 0$. This bipolar cancellation is expected by construction, not evidence against TEP. Analysis by orientation is therefore required to reveal the underlying signal, with the effect manifesting maximally in dilute environments where Temporal Shear is unsuppressed, such as the interstellar medium.

Key distinction: TEP predicts a bipolar signal where the holonomy sign flips with velocity projection. The primary detection statistic is the Phase Closure $\psi$, which measures the loop closure phase (interpreted as the holonomy phase within the TEP measurement model) and is immune to folded-magnitude noise-floor bias (Appendix B), corroborated in the delay domain by the paired noise-subtracted closure excess ($1.290 \pm 0.11$ ns, $t \approx 12.2\sigma$; $\approx 7.6$–$10.8\sigma$ under 15–240-epoch contiguous blocks). The signed mean serves as a parity metric expected near zero due to bipolar cancellation across orientations.

2.3 The ISM as an Unscreened Laboratory

Temporal Topology is the spatial and covariance structure of the conformal clock field \(\ln A(\phi)\), characterized locally by its gradients and statistically by \(C_A(x,x')\). Temporal Shear is the canonical synchronization gradient:

The observable Temporal Shear is screened by the environmental operator \(\mathcal{S}_\Sigma(\mathcal{E})\):

where \(\mathcal{E}\) includes density, compactness, gradients, source structure, coherence scale and boundary conditions. Density is one environmental projection, not a universal switch. The disformal response is correlated with the same field configuration but is not fixed by \(\mathcal{S}_\Sigma\) alone. It depends additionally on \(B(\phi)\), the field solution, the observer congruence and the measurement channel.

The interstellar medium toward PSR J0437-4715 provides a suitable testbed for TEP effects. With typical densities $\rho \sim 10^{-24} \text{ g/cm}^3$, the ISM is a dilute environment. The scattering screens at $\sim 89.8$ pc and $\sim 124$ pc (Gwinn et al. 2006) create multiple path configurations that enable closure delay measurements around closed loops through the Galactic plasma. The diffuse ISM provides a long, low-compactness multipath baseline and is therefore a motivated candidate environment for an active J0437 response. However, $\rho \ll \rho_T$ alone does not determine $\mathcal{S}_\Sigma(\mathcal{E})$, nor does conformal unscreening by itself fix the disformal transfer. The J0437 analysis therefore constrains the channel-integrated non-exact response rather than inferring it from ambient density alone.

The anisotropic nature of scattering toward J0437-4715, documented by previous studies (Walker et al. 2004; Coles et al. 2011), naturally introduces preferred orientations that may affect the distribution of clockwise versus counter-clockwise triplet configurations. This ISM anisotropy does not invalidate the TEP test; rather, it provides a physical explanation for orientation asymmetries while preserving the core prediction of magnitude equality between orientations.

It is important to distinguish the unscreened ISM scattering screens from the pulsar's immediate binary environment. The white dwarf companion to J0437-4715 generates a deep gravitational potential that locally suppresses observable Temporal Shear through the environmental screening operator $\mathcal{S}_\Sigma(\mathcal{E})$. Within the companion's screening radius, the scalar field gradient flattens toward its minimum, attenuating the locally observable Temporal Shear. Consequently, orbital-scale kinematics are expected to be partially shielded, while proper-motion-scale kinematics—sampling the unscreened ISM at much larger distances—remain the dominant TEP channel.

3. Data and Methodology

3.1 Multi-Pulsar Dataset

This study analyzes scintillation data from 15 pulsars. The primary detection is obtained from PSR J0437-4715; PSR J1603-7202 provides complementary geometric evidence through bipolar structure and variance decomposition; Jiamusi and MeerKAT provide noise-limited bounding data. The dataset combines Parkes observations of nearby millisecond pulsars with Jiamusi telescope observations of more distant pulsars and MeerKAT observations of three additional pulsars (J0908-1739, J0922-0638, J1731-4744), spanning a range of distances (~156 pc to ~3700 pc), proper motions, and scattering environments.

Data Provenance

Note: Jiamusi epoch and triplet counts reflect raw scintillation data availability. Four Jiamusi pulsars (B0355+54, B0540+23, B1508+55, B2154+40) produced viable closure triplets in the current pipeline configuration; the remaining six had raw data but did not yield closure-capable measurements. Of the three MeerKAT pulsars, only J1731-4744 produced a viable closure measurement (1 epoch, 20 triplets); J0908-1739 and J0922-0638 did not yield closure-capable data. Unknown proper motions limit MeerKAT geometric utility.

3.1.1 Parkes Pulsars (J0437-4715, J1603-7202)

The Parkes Pulsar Timing Array (PPTA) Data Release 2 provides multiband ($732\text{--}3104 \text{ MHz}$) observations spanning 2002–2018. The J0437-4715 dataset includes 10,201 epochs spanning L-band ($\sim 1400 \text{ MHz}$) and UHF ($\sim 3000 \text{ MHz}$) frequencies; J1603-7202 contributes 761 epochs.

The current analysis processes all cataloged epochs through steps 002–003 (secondary spectra and closure analysis). Of 1,391 epochs with measurable closure triplets, 1,093 have ≥5 triplets and form the independent sample; the remaining 298 are excluded from primary inference but retained for downstream analyses. The analysis includes both L-band and UHF observations, with the L-band subset providing the primary detection.

J0437-4715 serves as the primary target with 1,391 closure-capable epochs. J1603-7202 provides a complementary velocity geometry test with 73.8° velocity vector separation from J0437-4715, yielding 248 viable epochs from 761 total processed.

3.1.2 Jiamusi Pulsars (B0329+54, B0355+54, B0540+23, B0740-28, B1508+55, B1933+16, B2154+40, B2310+42, B2324+60, B2351+61)

The Jiamusi 66m telescope (China) provides observations of ten bright, distant pulsars with strong scintillation. These pulsars extend the distance baseline by \(\sim 10\times\), enabling critical tests of TEP scaling with scattering strength.

3.1.3 MeerKAT Pulsars (J0908-1739, J0922-0638, J1731-4744)

The MeerKAT radio telescope (South Africa) provides observations of three pulsars via the Thousand-Pulsar-Array programme. Proper motions for these pulsars are not available in the current catalog; velocities are treated as unknown. These pulsars provide noise-limited bounding checks but do not contribute to geometric sign tests due to the missing velocity data.

Dataset rationale: The 15-pulsar sample spans diverse distances, velocities, and ISM environments. The robust Phase Closure detection is confined to J0437-4715; J1603-7202 contributes complementary bipolar geometric validations from the same Parkes/PPTA DR2 data family, supplying an independent line-of-sight geometric test of the Stokes-aligned holonomy pattern rather than a separate telescope systematics check. The Jiamusi and MeerKAT data provide noise-limited bounding checks against simple universal baseline offsets; their null status is consistent with TEP's predicted environmental suppression in distant, noise-limited sightlines, though this interpretation remains noise-limited and not independently diagnostic. The velocity vector geometry (particularly the 73.8° proper motion vector separation between J0437 and J1603) provides a geometric test of TEP's velocity-dependent predictions.

3.2 Data Processing Pipeline

Dynamic spectra are processed through a standardized, automated pipeline comprising 60 sequential steps (Steps 000–061) organized into validation stages:

3.3 Physical Metrology of Synchronization

The closure delay measurement rests on precise differential time metrology across scattered signal paths. The step-by-step physical procedure is:

Step 1: Signal Acquisition. Pulsar signals are recorded at the telescope with timestamps derived from hydrogen maser frequency standards (typical stability $10^{-15}$ at 100 s integration). Raw voltage samples are digitized at MHz sampling rates with GPS-disciplined clock synchronization.

Step 2: Dynamic Spectrum Formation. Voltage data undergo coherent dedispersion to remove interstellar dispersion, then channelization via polyphase filter banks yields intensity as a function of frequency and time $I(f, t)$ with microsecond temporal resolution.

Step 3: Secondary Spectrum Computation. The conjugate Fourier transform converts $I(f, t)$ to the secondary spectrum $S(\tau, f_D)$, where differential delay $\tau$ and differential Doppler $f_D$ conjugate variables encode scattering geometry. The 2D power distribution reveals parabolic scintillation arcs—signatures of scattered paths through anisotropic plasma screens.

Step 4: Arclet Identification. Local maxima in $S(\tau, f_D)$ above SNR $\geq 5$ threshold identify discrete scintillation arclets at coordinates $(\tau_i, f_{D,i})$. Cross-correlation of each arclet with parabolic templates refines arc positions with sub-pixel precision (0.1 ns interpolation).

Step 5: Triplet Formation and Closure Construction. For each epoch with $\geq 3$ arclets, all possible triplets $(i, j, k)$ are formed. The closure residual $C_{ijk} = \hat{\tau}_{ij} + \hat{\tau}_{jk} + \hat{\tau}_{ki}$, with each $\hat{\tau}_{ab}$ independently measured from the corresponding cross-term peak in the secondary spectrum, geometrically closes the path loop. By construction, this quantity is independent of absolute pulse arrival time and cancels all common-mode contributions—including standard relativistic effects and instrumental timing offsets—leaving only the path-dependent TEP holonomy signal.

Step 6: Stokes Alignment. Each triplet's oriented area in the $(\tau, f_D)$ plane determines its geometric sign. The effective velocity vector (pulsar proper motion plus Earth's orbital velocity, including annual aberration and Sagnac corrections) projects onto this area to yield the kinematic weighting factor. This alignment ensures that the signed delay correlates with velocity geometry as predicted by TEP. Earth's orbital velocity is obtained from the solar-system barycentric Earth state vector via astropy.coordinates.get_body_barycentric_posvel (astropy 6.1, built-in SOFA/ERFA ephemeris epv00), yielding agreement with JPL DE440 at the centimetre-per-second level.

Step 7: Statistical Aggregation. The Phase Closure \(\psi\) across all triplets provides the primary TEP detection statistic. Under GR, \(\psi = 0\); under TEP, \(\psi \neq 0\) with magnitude scaling with scattering strength and path geometry. The bipolar prediction—opposite signs for opposite velocity projections—is tested via signed mean validations. The magnitude $M_{\rm cl} = \langle |C_{ijk}| \rangle$ is analyzed as a secondary validation, noting its susceptibility to noise floor bias.

This metrology chain transforms raw telescope voltages into a physically interpretable measure of path-dependent time transport, with each step traceable to atomic clock standards and geometric observables.

3.4 Error Propagation and Uncertainty Quantification

The uncertainty in the Stokes-aligned closure delay is derived from the standard error of the mean across all measurements. For PSR J0437-4715, this comprises 19,167 triplet measurements from 1,391 closure-capable epochs. Individual triplet delay uncertainties (\(\sigma_{\text{triplet}} \approx 0.5\text{--}2 \text{ ns}\), measured via cross-correlation peak fitting) propagate through the analysis as follows:

Individual measurement uncertainty: Each closure residual \(C_{ijk} = \hat{\tau}_{ij} + \hat{\tau}_{jk} + \hat{\tau}_{ki}\) (for triplet paths \(i, j, k\)) has an associated uncertainty \(\sigma_{\tau}\). In the pipeline implementation, this is approximated by the characteristic grid resolution of the secondary spectrum:

where \(d_{\tau}\) is the median grid spacing in the delay domain and the \(\sqrt{3}\) factor accounts for the three cross-terms in the closure sum. This provides a characteristic scale for triplet uncertainties; the final reported uncertainties use the empirical standard error of the mean across all measurements.

Epoch-level independence: The conservative framework treats the observational epoch as the fundamental unit of independence. Epochs with fewer than 5 triplets are excluded from the primary statistical analysis because low triplet counts produce unreliable epoch-level means. For PSR J0437-4715, of 1,391 closure-capable epochs, 298 have fewer than 5 triplets and are excluded from epoch-level aggregation, leaving \(N_{\text{epochs}} = 1{,}093\) independent epoch means for primary inference. The uncertainty in the global mean \(\sigma_{\text{global}}\) is given by the standard error across these independent samples:

Because the triplets formed within a single epoch share arclets and propagation paths, triplet-level measurements are not mutually independent; the operative SEM for the inverse-variance-weighted global \(M_{\rm cl}\) is therefore obtained by resampling epochs rather than triplets. An epoch-level bootstrap of the identical weighted estimator (10,000-resample audit, step_060) yields \(\sigma_{\rm global} = 0.26\) ns under independent-epoch resampling, rising to \(0.39\text{--}0.55\) ns under contiguous 15–240-epoch moving blocks (≈77–1200 d median span) that absorb residual campaign/screen-state clustering; the epoch-mean autocorrelation persists to ~200-epoch lags, so the blocked value is a lower envelope rather than a converged plateau. The internal inverse-variance SEM \(1/\sqrt{\sum_i w_i} = 0.102\) ns is retained in the pipeline output as the measurement-noise diagnostic but is not the quoted aggregate uncertainty.

3.4.1 Sample Accounting

The following table reconciles every reported J0437-4715 statistic to a single pipeline stage. All counts are extracted directly from step_003_closure_final_summary_j0437.json and the per-epoch results file step_003_closure_final_per_epoch_j0437.json.

Every reported primary statistic refers to the 1,093-epoch independent sample. The 19,167 triplet count describes the full closure-capable dataset and is reported in the abstract and data tables for sample-size transparency. The 298 epochs excluded by the ≥5-triplet filter are retained in the raw results and used by downstream analyses (e.g., probabilistic weighting, bootstrap) that do not require epoch-level aggregation.

Global Stokes-aligned Phase Closure: The primary TEP detection statistic is the Phase Closure \(\psi\), computed with circular statistics on the 1,093-epoch sample. The circular mean is \(\bar{\psi} = 0.984 \pm 0.046 \text{ rad}\) (circular SE, \(\bar{R}\) = 0.308, Rayleigh p = 1.39 × 10⁻¹³, V-test p = 1.02 × 10⁻⁵). An epoch-level circular bootstrap (10,000 iterations) yields a 95% CI of [+0.737, +1.235] rad, which excludes zero. The raw absolute magnitude is \(M_{\rm cl} = 8.100 \pm 0.26 \text{ ns}\) (epoch-level bootstrap SEM; the internal inverse-variance SEM \(1/\sqrt{\sum_i w_i} = 0.102\) ns counts triplet measurement noise only and understates the epoch-to-epoch scatter), but a per-epoch folded-normal noise floor \(E[M_{\rm cl}]\) = 6.810 ns quantitatively saturates this value, leaving a noise-subtracted excess of \(1.290 \pm 0.11\) ns (paired epoch-level SEM of the mean-minus-floor difference, \(t \approx 12.2\sigma\); \(\approx 7.6\)–\(10.8\sigma\) under 15–240-epoch contiguous-block resampling, step_060 audit). The Phase Closure \(\psi\) is therefore the primary TEP validation; \(M_{\rm cl}\) remains a noise-floor-dominated validation in the current dataset.

The signed mean ($-0.184 \pm 0.157 \text{ ns}$ epoch-level, $-1.17\sigma$)—computed on Stokes-aligned closure delays—shows minimal bipolar bias, consistent with nearly isotropic sampling of the scattering geometry. This near-zero value indicates successful alignment of the bipolar signal (where positive and negative contributions cancel), confirming the expected parity behavior. The Phase Closure \(\psi\) is the primary zero-centered circular detection statistic (measuring the loop holonomy phase within the TEP measurement model).

The total standard deviation (\(\sigma_{\text{total}} \approx 16 \text{ ns}\) for raw \(M_{\rm cl}\)) reflects measurement uncertainty, ISM variability, and the heavy-tailed noise distribution. The frequentist MAD-based folded-normal subtraction reported above (\(1.290 \pm 0.11\) ns epoch-level, \(t \approx 12.2\sigma\)) and the hierarchical Bayesian posterior for triplet-level holonomy amplitude \(H\) (\(0.000 \pm 0.459\) ns; 95% CI [0.000, 0.899] ns) answer different questions and must not be conflated: the former subtracts a per-epoch Rice floor from the inverse-variance-weighted epoch mean, whereas the latter marginalizes triplet delays under a generative \(\text{Normal}(s_i H, \sigma_i)\) model with inverse-variance weights and a HalfNormal prior, testing whether latent \(H\) exceeds zero after probabilistic weighting. Because the orientation sign is included in the model, \(H\) estimates the orientation-aligned amplitude. The posterior near zero shows that this particular signed Gaussian model does not recover a coherent delay-domain amplitude. The folded trimmed excess measures a different, unsigned feature of the distribution and is not by itself a signed-holonomy estimate. Phase Closure \(\psi\) remains the primary detection statistic.

3.5 Statistical Analysis Framework

The statistical validation employs multiple cross-validated methods:

Multiple Comparison Correction: Bonferroni correction is applied for the 4 primary validation tests (aggregate mean, \(M_{\rm cl}\) magnitude, negative delays, positive delays). Uncorrected \(\alpha = 0.05\), corrected \(\alpha = 0.0125\). All reported significance values account for this correction.

The closure delay distribution for J0437-4715 exhibits the expected behavior for interstellar scattering data. The Bayesian analysis provides robust estimates of the mean and uncertainty despite the presence of outliers, which are handled through probabilistic weighting.

The standard mean and Bayesian hierarchical model are reported as robust estimators, with the latter naturally weighting measurements by their individual uncertainties to reduce sensitivity to heavy tails while preserving the sign and central tendency of the signal.

Fixed random seeds ensure full reproducibility. All exclusion decisions are logged with explicit reasons to prevent cherry-picking.

3.6 Data Sources and Acknowledgments

The primary scintillation data used in this study are sourced from the Parkes Pulsar Timing Array Data Release 2. Specifically, dynamic spectra for PSR J0437-4715 were obtained from the CSIRO Data Access Portal (Reardon et al. 2020, DOI: 10.25919/5f3cd2bc1c213), while data for PSR J1603-7202 were retrieved from the same portal following the work of Walker et al. (2022, DOI: 10.25919/82f5-mh79). Additionally, 5 archival epochs were sourced from the ATNF Scintools repository. For the more distant Jiamusi pulsars B0329+54, B0355+54, B0540+23, B0740-28, B1508+55, B1933+16, B2154+40, B2310+42, B2324+60, and B2351+61, dynamic spectra were obtained from the Jiamusi 66m telescope Pulsar Scintillation Archive, collected during monitoring campaigns and processed using standard radio astronomy pipelines.

Both CSIRO datasets are provided under the Creative Commons Attribution 4.0 International Licence.

4. Results: Detection of Synchronization Holonomy

4.1 Primary Detection and validations

Analysis of 19,167 scintillation triplets from the primary PSR J0437-4715 dataset (2002–2018) reports evidence consistent with Stokes-aligned proper-time holonomy through the Phase Closure ψ metric. Of 1,391 epochs with closure triplets, 298 had fewer than 5 triplets and were excluded from the independent sample, leaving 1,093 epochs for primary inference.

Observable map (aggregation level). Triplet-level quantities include per-triplet aligned closure delay \(H_i\) (geometric delta_us), per-triplet phase closure \(\psi_i\) on \([-\pi,\pi)\), and triplet-level \(|H_i|\) magnitudes used in some null sweeps. Epoch-level summaries form one mean signed delay and one mean \(M_{\rm cl}\) per epoch (before global weighting), plus the SNR²-weighted circular mean of \(\psi_i\) within each epoch for the primary phase stack. Global Step 003 statistics inverse-variance-weight those epoch delay means for \(M_{\rm cl}\) and delay-SEM-weight the epoch circular \(\psi\) means for the headline Rayleigh/V/bootstrap pipeline. Step 007’s epoch-mean \(M_{\rm cl}\) cross-validation averages \(|\text{epoch mean } H|\) over holdout epochs; Step 007’s phase_closure_epoch_cv applies the same epoch \(\psi\) construction with a shuffle split (dedicated RNG seed in the JSON). These channels must not be numerically cross-compared without noting the aggregation.

The Phase Closure ψ is the primary detection statistic—a zero-centered, odd-parity metric theoretically immune to the Rice bias that affects folded delay magnitudes. All phase statistics are computed with circular statistics (circular mean, circular standard deviation, and branch-cut-safe bootstrap confidence intervals) because ψ lives on the circle [−π, π). Because the Rayleigh test (\(Z=2n_{\rm eff}\bar{R}^2\)) and V-test test circular uniformity against concentration (and a distribution tightly concentrated at zero would also yield a small Rayleigh \(p\)-value), concentration and non-zero mean direction are tested by distinct statistics: (i) circular concentration is established by Rayleigh \(Z = 59.21\) (\(p = 1.39 \times 10^{-13}\) weighted; \(Z_{\rm uw}=202.05\), \(p=1.34\times 10^{-44}\) unweighted) and V-test \(V = +4.26\) (\(p = 1.02 \times 10^{-5}\) relative to \(\mu_0 = 0\)), and (ii) rejection of zero mean closure (\(\langle\psi\rangle = 0\)) is established by the measured circular mean \(\bar{\psi} = 0.984 \pm 0.046\) rad (analytical circular SE, \(21.2\sigma\), \(\bar{R} = 0.308\); unweighted \(\psi = 1.120\) rad) and its 10,000-iteration epoch-level circular bootstrap 95% CI of \([+0.737, +1.235]\) rad (\(\text{SE}_{\rm boot} = 0.125\) rad at effective sample size \(n_{\rm eff} = 313\), excluding zero at \(7.9\sigma\) i.i.d. and \(5.2\sigma\)–\(6.6\sigma\) across 15–240-epoch contiguous MJD blocks; Step 059) alongside the unweighted 95% CI \([+0.990, +1.253]\) rad (\(\text{SE}_{\rm boot} = 0.067\) rad, excluding zero at \(16.8\sigma\), i.e. \(>15\sigma\), and \(5.7\sigma\)–\(12.7\sigma\) across 15–240-epoch MJD blocks).

Weighted Rayleigh convention. The reported \(Z\) uses the inverse-variance-weighted epoch sample: \(Z = 2 n_{\rm eff}\bar{R}^2\) with effective sample size \(n_{\rm eff} = (\sum_i w_i)^2/\sum_i w_i^2 \approx 313\), and \(p = e^{-Z/2} = e^{-n_{\rm eff}\bar{R}^2}\). With \(\bar{R} = 0.3076\) this gives \(Z = 59.21\), \(p = 1.39\times10^{-13}\). The unweighted construction on the same 1,093 independent epochs gives \(Z_{\rm uw} = 202.05\) (\(p = 1.34\times10^{-44}\), \(\bar{R}_{\rm uw} = 0.304\)); the unweighted statistic is the frame-invariant primary. Both conventions reject circular uniformity.

In the delay domain, subtracting the per-epoch folded-normal noise floor ($E[M_{\rm cl}] = 6.810$ ns) from the epoch-level inverse-variance weighted closure magnitude ($M_{\rm cl} = 8.100 \pm 0.26$ ns) yields a paired noise-subtracted excess of $1.290 \pm 0.11$ ns (paired epoch-level bootstrap SEM; $t \approx 12.2\sigma$; $\approx 7.6$–$10.8\sigma$ under 15–240-epoch contiguous blocks, Steps 052 and 060), corroborating the primary phase-domain detection. The statistical properties of raw unsigned delay estimators ($M_{\rm cl}$) and trimmed diagnostics ($M_{\rm cl,sym10}$) are detailed in Appendix B.

4.1.1 Hierarchical Variance Decomposition

A hierarchical Bayesian uncertainty model accounts for variance contributions at multiple levels: triplet, epoch, screen state, pulsar, and telescope. This variance decomposition yields the effective sample size accounting for correlations:

The primary statistical analysis uses the 1,093-epoch independent sample (epochs with ≥5 triplets). The epoch-level variance contribution (1.88%) indicates moderate correlation within epochs, as expected for scintillation data. The 298 epochs with <5 triplets are retained in the raw results for downstream analyses that do not require epoch-level aggregation.

4.1.2 Systematic Error Budget and Phase-Closure Robustness

The primary detection statistic ψ is estimated via circular statistics on 1,093 independent epochs. Its precision is quantified by the circular standard error, which already encodes all sources of phase-angle variance (measurement noise, pixel discretization, thermal variation, and any residual calibration structure) through the empirical dispersion of the angles themselves. The measured circular mean is ψ = 0.984 ± 0.046 rad (circular SE, \(\bar{R}\) = 0.308). The significance of this detection is self-contained: Rayleigh Z = 59.21 (p = 1.39 × 10⁻¹³) rejects uniformity, V-test V = +4.26 (p = 1.02 × 10⁻⁵) rejects circular uniformity toward the reference direction μ₀ = 0, and an epoch-level circular bootstrap (10,000 iterations) yields a 95% CI of [+0.737, +1.235] rad that excludes zero. The bootstrap CI establishes confidence-interval resolution: no ordinary bootstrap replicate crossed zero, so the directly resolved empirical tail is p < 10⁻⁴. A formal null p-value would require a null-centred block bootstrap or permutation distribution. No linearized delay-to-phase conversion is required or used for these inference statements.

Below the primary detection, a separate delay-domain metrology budget is maintained for the \(M_{\rm cl}\) magnitude validations. Because closure delays cancel common-mode contributions—including standard relativistic effects and instrumental calibration offsets—by construction, the systematic terms relevant to \(M_{\rm cl}\) are bounded and qualitatively different from those that would affect a single-path delay measurement. The budget quantifies the residual uncertainty in delay-domain amplitude metrology; it does not propagate linearly into the phase-domain circular error because the circular SE already captures the operative phase variance.

The core metrology systematic of 0.272 ns is dominated by the pixel discretization bound (0.197 ns). This bound is a conservative theoretical upper limit for parabolic sub-pixel interpolation. The actual contribution to phase-angle variance is already subsumed in the empirical circular SE of 0.045 rad: if pixel discretization were a dominant phase error, the circular distribution would be broadened and \(\bar{R}\) would be reduced; the observed \(\bar{R}\) = 0.308 is consistent with the measured phase scatter.

Threshold robustness (step_041): The SNR and arclet threshold sweeps in step_041 test the stability of the signed mean across selection cuts. The signed mean is a physical metric expected to be near zero due to bipolar cancellation (Section 2.2); its variation across thresholds (span ±0.57 ns across the tested SNR grid) reflects the inherent noise of a near-zero bipolar estimator, not a systematic bias in ψ. The unsigned \(M_{\rm cl}\) magnitude is robust across all threshold cuts (t-statistic ≥ 35.3σ minimum). The Phase Closure ψ itself was not threshold-dependent: the circular mean and \(\bar{R}\) are stable because ψ is extracted from complex cross-term phases prior to any SNR-based weighting. Threshold-robustness concerns therefore apply to the signed-mean validation, not to the primary phase detection.

4.1.3 Validation Summary

The Phase Closure distribution across the primary dataset shows a non-zero central value with concentrated circular scatter (\(\bar{R}\) = 0.308, circular σ = 1.50 rad), consistent with a physical signal superimposed on measurement noise. The following table summarizes the primary and secondary validation metrics:

4.2 ISM Null Model Simulations

Standard ISM models were tested to determine whether they can produce the observed non-zero Phase Closure ψ. Seven scalar-factorizable ISM null implementations were simulated in step_008. All returned zero mean Phase Closure and none reproduced the observed coherent signal — a loop-ordered monopole whose orientation parity is verified algebraic and whose placement classes share one mean direction (Step 056). The observed Phase Closure reported in Section 4.1 is inconsistent with all tested scalar-factorizable implementations.

None of the tested scalar-factorizable ISM implementations reproduces the observed coherent non-zero Phase Closure. All simulated null distributions from step_008 are centered at zero, while the observed circular mean has a 95% bootstrap CI that excludes zero. The instrumental-noise simulation produces \(M_{\rm cl}\) ~ 8 ns through unsigned noise bias, but its Phase Closure distribution remains centred at zero and does not reproduce the observed coherent circular mean.

4.3 Velocity-Sign Consistency Check

An informal velocity-sign consistency check was performed. Pulsar names and proper motion vectors were temporarily withheld while ψ was computed, then matched to velocity vectors to test whether the observed sign pattern matched the TEP prediction.

Note on B2324+60: This pulsar has no measured proper motion in the literature (Wang et al. 2018, Table 1), so a zero proper motion reference value of (0.0, 0.0) mas/yr was used.

Out of 15 pulsars analyzed, one shows a robust circular Phase Closure detection (J0437-4715, Section 4.1). J1603-7202 shows a strong bipolar Stokes decomposition (ψ₀ = −0.821, ψ₁ = −1.965) with high circular dispersion at its lower transverse speed (Rayleigh p = 0.936, Section 4.1), matching the TEP expectation that the monopole is diluted while bipolar structure remains. The table lists inverse-variance–weighted circular ψ for both Parkes targets; those weights are delay-variance dominated, so the weighted circular row is not the primary J1603 statistic. The informal velocity-sign stress test instead uses the signed epoch-mean closure delays, which are negative for J0437 and positive for J1603 (Section 4.9), matching the opposite bias expected when the two proper-motion vectors are separated by ~74°. This pattern is consistent with TEP across two independent sightlines with complementary observables.

The non-zero Phase Closure signal reported in Section 4.1 provides robust, zero-centered evidence for TEP holonomy. The Phase Closure distribution across the primary dataset shows a non-zero circular mean with concentrated scatter (\(\bar{R}\) = 0.308), consistent with a physical signal superimposed on measurement noise.

4.4 Velocity-Direction Controls

A critical review concern is whether the Stokes/velocity-projection alignment procedure encodes the expected sign. Five independent controls are implemented to demonstrate that the observed sign structure is physical, not an artifact of the alignment algorithm.

Control 1 (Velocity-Label Permutation): With 2 pulsars, there are 2! = 2 possible velocity-label assignments. In the current Step 047 implementation, both permutations tie the best match score, which is the expected score degeneracy at N=2; the row documents that outcome rather than supplying extra degrees of freedom for label swapping.

Control 2 (Pre-Alignment validation): The epoch-level phase-closure distribution is already coherent before any velocity-domain weighting, demonstrating that the primary phase observable is not created by the alignment step.

Control 3 (Freeze Record): The Step 047 freeze record stores a hash of the Step 003 phase-closure values and file timestamps at control runtime. This records the internal chronology of artifacts for reproducibility; external blinding protocols are outside the scope of the present manuscript, so the freeze record is classified as a documentation control.

Control 4 (Wrong-Velocity): Not applicable to the primary phase closure observable. Individual triplet phases (and therefore epoch-level phase closure) are extracted from the complex secondary spectrum prior to any velocity weighting, so modifying the velocity vector cannot change phase_closure_rad.

Phase-Scramble Specificity (Step 047): The phase-scramble control is a uniform-null Monte Carlo: it draws 1,000 sets of \(N\) independent uniform phases on \([-\pi, \pi)\) and compares the resulting circular concentration (\(\bar{R}\)) to the observed value. This generates fresh random angles from the uniform distribution; it does not permute the observed angles (permuting a fixed set of angles would leave \(|\sum_i e^{i\psi_i}|\) exactly unchanged). In 1,000 Monte Carlo samples of independent uniform phases, none reached the observed concentration; the finite-simulation estimate is \(p = (0+1)/(1000+1) = 9.99\times10^{-4}\).

Leg-Permutation Null and Three-Leg Loop Amplification (Steps 055 and 059): The deferred bispectrum control—permuting one cross-term leg within each epoch before reconstructing \(\mathcal{B}_{ijk}\)—is now implemented. The per-triplet leg phases \(\phi_{01}, \phi_{12}, \phi_{02}\) were recomputed through the unchanged Step 002–003 chain on all 1,391 viable epochs (the recomputed aggregation reproduces the headline values exactly: \(\psi = 0.984\) rad, \(\bar{R} = 0.308\), \(n = 1{,}093\)). Permuting one leg's phases across the triplets of each epoch destroys loop-level association while preserving every leg's marginal distribution, the estimator chain, and the epoch weights. Over 400 replicates the concentration collapses from \(\bar{R} = 0.308\) to \(\bar{R}_{\rm perm} = 0.097\text{--}0.129\) depending on which leg is permuted (and to \(0.066\) when the leg is drawn from a different epoch entirely), with no replicate reaching the observed value (\(p \le 2.5\times10^{-3}\)). An extended 5{,}000-replicate audit of the strictest cross-epoch mode (step_059) resolves the floor: no replicate attains \(\bar{R} = 0.308\) (null maximum \(0.190\), empirical \(p < 2\times10^{-4}\)); the observed concentration sits \(7.6\) null standard deviations above the cross-epoch null mean (Gaussian-tail \(p \approx 2\times10^{-14}\)). Crucially, the stored leg-phase cache directly rules out any hypothesis in which \(\psi\) arises from static or deterministic FFT-axis leg phases (such as the \(f_D = 0\) zero-Doppler row reached when two nested parabolic screens \(\tau = \eta_1 f_D^2\) and \(\tau = \eta_2 f_D^2\) interfere at matched Doppler velocity \(\Delta f_D \approx 0\) with \(\Delta\tau > 0\), where Hermitian symmetry \(F(-f_D,-\tau)=F^*(+f_D,+\tau)\) leaves \(\arg F(0,+\tau)\) unconstrained at \(\tau > 0\)): across all 19,167 triplets, the two-leg sum \(\phi_{01}+\phi_{12}\) is nearly uniform (\(\bar{R}_{01+12}=0.0364\), mean \(-0.030\) rad) and the third leg \(\phi_{02}\) is uniform (\(\bar{R}_{02}=0.0145\)), so uncoupled legs would yield \(\bar{R}_{\rm indep}=\bar{R}_{01+12}\bar{R}_{02}=5.28\times10^{-4}\), whereas subtracting \(\phi_{02}\) on the same triangle cancels the random screen-image phase \(\theta_2-\theta_0\) and amplifies the resultant length by \(231\times\) to \(\bar{R}(\psi)=0.1220\) (\(324\times\) on the \(13{,}849\) triplets containing an \(f_D=0\) leg, and \(49.5\times\) on the \(5{,}318\) strictly off-axis triplets with no \(f_D=0\) leg, which independently yield \(\psi=+0.801\) rad, Rayleigh \(Z=47.93\), \(p=3.92\times10^{-11}\); excising all \(2{,}580\) shared-pixel triplets likewise leaves \(16{,}587\) distinct-pixel triplets with \(\psi=+1.071\) rad, \(\bar{R}=0.0640\), \(103\times\) amplification, Rayleigh \(Z=135.82\), \(p=3.21\times10^{-30}\); Step 059). The measured \(\psi\) is therefore a genuine per-loop phase association—a real three-leg closure property of the data—not an artifact of the leg marginals or the estimator.

Non-Factorizable Screen Null (Step 055): The second control synthesizes dynamic spectra \(I(\nu,t) = |E(\nu,t)|^2\) where \(E\) is a complex Gaussian field whose conjugate spectrum concentrates on the measured J0437 two-arc geometry (\(\eta_1 = 0.0072\), \(\eta_2 = 0.0076\) \(\mu\)s/mHz\(^2\), arclet \(|f_D|_{\max} = 6.5\) mHz modelled as a \(\sigma = 3.25\) mHz Gaussian extent, flux statistics matched to the data). For this \(\chi^2\) field the secondary-spectrum cross-terms are convolutions, not factorizable products, so the closure phase is generically non-zero under standard scintillation physics—the channel the scalar-factorizable step_008 suite cannot reach. Pushed through the identical chain, 71 of 3,000 synthetic epochs met the cross-screen viability cuts; their concentration at matched sample size (\(\bar{R} = 0.26\), vs the real distribution resampled to the same \(n\): \(0.35 \pm 0.13\)) is statistically comparable to the data—closure-phase concentration of this magnitude is not anomalous per se. What the screen null does not reproduce is the direction persistence: synthetic epoch-mean directions are consistent with uniform (Rayleigh \(p = 0.55\); unweighted direction \(\bar{R} = 0.09\) overall, \(0.14\) symmetric and \(0.04\) for the asymmetric-arm variant), while the 1,093 real epoch means persist at \(+0.984\) rad over fifteen years. The anomaly thus contracts to the stable direction of the closure phase, not its existence.

Forward Scintillation-Null Battery (Step 058): To test whether the circular-uniformity null is mis-specified—i.e., whether standard thin-screen scintillation can concentrate closure phases near the observed direction through mundane physics—a suite of additive-image thin-screen forward models was propagated through the unchanged Step 002–003 chain (Hough arc detection, arclet selection, the cross-screen triplet requirement, sub-pixel cross-term measurement, and complex-patch phase extraction), with epoch dimensions, sampling intervals, and observing bands drawn from the empirical distribution of the real J0437 catalogue. Ten models spanning single-screen, two-screen, non-quadratic curved-arc, discrete arclet-family, interior-fill, and three-screen propagation, together with a 30%-persistent-arclet variant, a fully persistent image field, pure noise, and a phase-scrambled control, contribute \(\sim\)14,500 synthetic epochs and \(\sim\)85,000 simulated triplets. Every model in which the image population decorrelates between epochs—the appropriate ISS null, since arc curvatures persist while image content decorrelates on the scintillation timescale—returns a pooled \(\psi\) distribution consistent with circular uniformity (\(\bar{R} \le 0.05\); epoch-direction \(\bar{R} \le 0.12\)), against the observed \(\bar{R} = 0.304\) and direction persistence \(0.311\); the phase-scrambled control—identical amplitude spectrum, identical arc detection and triplet selection, with only the measured phases randomized—yields \(\bar{R} = 0.003\) and epoch-direction \(\bar{R} = 0.035\) over 668 viable epochs, the deepest null in the battery at full yield, confirming that the clustering resides entirely in the phase content rather than the selection chain. Concentration appears only in the frozen-field limit in which every image's \((\tau, f_D)\) position, amplitude, and phase is held fixed across all epochs; each such screen realization concentrates \(\psi\) at a realization-specific direction scattered over the circle (per-realization \(\bar{R} \sim 0.03\)–\(0.39\), directions spanning \([-2.8, +1.5]\) rad), with no systematic convergence near the observed \(+1.1\) rad. The measured-position sign-pattern channel is reproduced structurally by the simulators—origin-straddling peak captures occur at comparable rates—but the flipped-sign triplets remain circularly uniform in every fresh-population model (per-pattern \(\bar{R} \lesssim 0.15\)), in contrast to the real sample where those patterns concentrate at \(+1.3\)–\(+1.4\) rad. The closure-phase concentration is therefore not a generic property of multi-screen propagation, non-quadratic curvature, discrete arclet families, or of the measurement channel itself; it is specific to the data.

Together, these controls demonstrate that the primary phase observable exists before velocity weighting, that scramble controls reject a trivial circular-null interpretation, and that the closure is a genuine loop-level association under leg permutation. With only two Parkes pulsars, velocity-label permutation is necessarily score-degenerate; the discriminating power is carried by the phase-scramble, pre-alignment, leg-permutation, and direction-persistence checks together with the J0437 circular statistics.

4.5 Independent Statistical Validation

The primary detection statistic is the Phase Closure \(\psi\), whose 95% circular-bootstrap interval \([0.737,1.235]\) rad excludes zero and whose directional V-test rejects uniformity toward \(\mu_0 = 0\) at \(p = 1.02 \times 10^{-5}\) (Rayleigh \(Z = 59.21\), \(p = 1.39 \times 10^{-13}\) weighted; \(Z_{\rm uw} = 202.05\), \(p = 1.34 \times 10^{-44}\) unweighted). Five independent validation suites test out-of-sample generalization, parity symmetry, seasonal stability, hierarchical weighting, and synthetic injection recovery; legacy delay-domain amplitude estimators (\(M_{\rm cl}\)) and their folded-normal Rice noise-floor diagnostics are documented in Appendix B.

4.5.1 Phase Closure \(\psi\) — Epoch Holdouts (Circular Cross-Validation)

Step 007 implements phase_closure_epoch_cv: a 5-fold shuffle split on the same 1,093-epoch stack used for primary circular inference (epochs with \(\geq 5\) triplets; within-epoch SNR²-weighted circular mean of triplet \(\psi\); across-epoch inverse-delay-variance weights, matching the Step 003 construction). A dedicated RNG seed (\(42 + 70007\)) generates the fold assignment. Each holdout fold recomputes circular mean \(\psi\), Rayleigh uniformity, and a directed V-test of uniformity toward \(\mu_0 = 0\) using only held-out epochs. Pass rule in the ledger: ≥4/5 folds have Rayleigh \(p < 0.05\) and the fold circular mean lies in the same open half-plane as the full-sample mean \(\cos(\bar\psi_{\text{fold}} - \bar\psi_{\text{full}}) > 0\). Frozen ledger: 5/5 folds meet the Rayleigh threshold.

Full-sample (recomputed from arrays, Step 003-consistent): \(\bar\psi = +0.984\) rad, Rayleigh \(Z = 59.21\) (\(p = 1.39e-13\)), V-test \(p = 1.02e-05\); the recomputed full-sample \(\bar\psi\) matches the Step 003 summary weighted mean at machine precision (\(|\Delta\psi| = 0\) rad in the frozen ledger).

5/5 folds reject circular uniformity at \(\alpha = 0.05\) (Rayleigh) and retain the same half-plane as the global mean; the step flag test_passed is true in the frozen JSON. V-test power drops in some folds because the holdout subsample is weaker than the full 1,093-epoch stack—expected behaviour, not a contradiction with Section 4.1’s full-sample V-test.

4.5.2 Signed-Delay Bipolar Cancellation and Noise-Subtracted Excess

The signed mean across all triplets and epochs (\(-0.184 \pm 0.157\) ns epoch-level, \(-1.17\sigma\), \(d \approx -10^{-3}\)) confirms minimal parity leakage, consistent with nearly isotropic sampling of a bipolar holonomy signal whose sign flips with orientation. In the unsigned delay domain, the observed independent-epoch mean (\(12.008\) ns in Step 007) lies significantly below the uncancelled iid Gaussian expectation (\(19.141\) ns, \(z = -18.40\sigma\), \(p < 10^{-16}\)), directly reflecting within-epoch bipolar cancellation from Stokes parity, while the inverse-variance-weighted epoch excess above the empirical per-epoch folded-normal floor (\(E[M_{\rm cl}] = 6.810\) ns) remains positive at \(1.290 \pm 0.11\) ns (\(t \approx 12.2\sigma\) iid, \(\approx 7.6\)–\(10.8\sigma\) under 15–240-epoch contiguous blocks, Step 060; Appendix B).

4.5.3 Seasonal Stability and Annual Modulation

The Phase Closure \(\psi\) remains independently significant across both half-year partitions (Jan–Jun: \(5,176\) triplets; Jul–Dec: \(5,714\) triplets), and the paired noise-subtracted delay excess is likewise positive in both halves (\(1.586 \pm 0.18\) ns in Jan–Jun, \(t \approx 8.9\); \(1.141 \pm 0.13\) ns in Jul–Dec, \(t \approx 8.7\), Step 060), tracking the global \(1.290\) ns excess rather than a single season.

A dedicated sinusoidal annual-modulation fit was performed on the per-epoch \(\psi\) and on the orientation-separated signed delays (step_054). Each series was regressed on \(\{1, \sin 2\pi t, \cos 2\pi t\}\) with \(t\) in years and compared against the constant model by a nested F-test. For J0437-4715 (\(n = 1093\) epochs over 16.6 yr) the annual \(\psi\) amplitude is \(0.037 \pm 0.090\) rad (F = 0.16, p = 0.853) and the signed-delay amplitude is \(0.63\) ns (p = 0.715). For J1603-7202 (\(n = 215\) epochs over 13.9 yr) the annual \(\psi\) amplitude is \(0.138 \pm 0.247\) rad (p = 0.753) and the signed-delay amplitude is \(1.01\) ns (p = 0.482). No statistically significant annual sinusoidal term is detected: the 95% upper bound on the J0437 \(\psi\) annual amplitude is \(\approx 0.21\) rad. Under the kinematic model in which Earth's ~30 km/s orbital velocity modulates \(u \cdot \nabla\phi\) against the dominant ~104 km/s pulsar transverse motion, a fully projected orbital contribution could reach \(\sim 0.3\) rad; the measured bound is compatible with partial projection and does not yet isolate the annual term. The annual channel is therefore a bounded kinematic diagnostic rather than a detected modulation.

4.5.4 Probabilistic Weighting Analysis

To address concerns about threshold sensitivity and selection bias, a fully Bayesian hierarchical model is implemented with probabilistic weighting.

This approach replaces hard SNR cuts with smooth uncertainty-based weighting, preserving the full geometric distribution of the 1,391 observational epochs from PSR J0437-4715.

The hierarchical model structure:

where \(\delta_i\) are the Stokes-aligned closure delays, \(s_i\) is the known orientation sign for triplet \(i\), and \(H\) is the orientation-aligned holonomy amplitude. The HalfNormal prior ensures \(H > 0\). Multiplying by the known sign gives \(s_i \delta_i \sim \text{Normal}(H, \sigma_i)\), so balanced positive and negative orientations do not make \(H\) cancel; \(H\) estimates the orientation-aligned amplitude. Each triplet contributes with inverse-variance weight \(w_i \propto 1/\sigma_i^2\), ensuring high-precision measurements contribute more while low-SNR triplets are retained with appropriately reduced influence.

The Phase Closure ψ detection validates the TEP holonomy signal. The raw inverse-variance-weighted \(M_{\rm cl}\) lies near the MAD-based folded-normal floor, with a measurable post-floor excess (\(1.290 \pm 0.11\) ns) in the frequentist row of the table above; the signed-Gaussian Bayesian posterior row (\(0.000 \pm 0.459\) ns, 95% CI \([0.000, 0.899]\) ns) is the orientation-aligned latent-\(H\) estimate from the signed model in Eq. $\eqref{eq:hierarchical_model}$, which cancels near zero because of within-epoch bipolar sign balance and provides the conservative upper bound used in Step 057, whereas the sign-marginalized bipolar mixture model of Step 044 (\(\delta_i \sim \tfrac{1}{2}\mathcal{N}(+H,\sigma_{{\rm tot},i}) + \tfrac{1}{2}\mathcal{N}(-H,\sigma_{{\rm tot},i})\), with intrinsic scatter \(\tau_{\rm int} = 40.85\) ns) recovers \(H = 14.159 \pm 0.441\) ns (95% CI \([13.225, 15.071]\) ns, \(32.1\sigma\)). Phase Closure ψ remains the primary odd-parity statistic.

SNR stratification by percentiles reveals a weak negative correlation (\(r = -0.107, p = 9.4 \times 10^{-12}\)) between SNR and \(M_{\rm cl}\), consistent with selection bias: low-SNR triplets are preferentially selected when the underlying signal is larger. Probabilistic weighting corrects this bias by allowing all triplets to contribute smoothly according to their measurement precision.

4.5.5 Synthetic Signal Injection Validation

To validate the threshold degradation model, synthetic signal injection is performed into pure noise. Known TEP signals (\(H = 0\text{--}25 \text{ ns}\)) are injected into 200-triplet noise realizations, and recovery is tested under both hard threshold cuts and probabilistic weighting.

Note: Recovery fraction is computed as (recovered excess) / (expected excess), where expected excess = \(0.4 \times\) injected \(H\) and recovered excess = (recovered \(H\) \(-\) noise floor). Values from probabilistic weighting (50 repetitions, 200 triplets/rep). The noise floor of ~2.4 ns arises from the unsigned mean of Gaussian noise (Rice bias), not from residual TEP signal.

Null hypothesis testing (100 trials, $H = 0$) yields 0/100 false positives, corresponding to a 95% binomial upper bound of approximately 3.6% on the false-positive probability in this finite synthetic experiment.

The synthetic model demonstrates monotonically increasing recovery with injected signal amplitude, with the recovered \(M_{\rm cl}\) rising from the ~2.4 ns noise floor toward higher values as injection strength increases. Recovery fraction improves from 0.53 at \(H = 5\) ns to 0.90 at \(H = 25\) ns, reflecting the expected geometric averaging over random arclet orientations. Threshold variation across SNR cuts (0, 3, 5, 7, 10) is minimal (\(\text{CV} = 0.2\%\)), indicating that in this controlled model, probabilistic weighting and hard thresholds yield consistent results when the SNR-quality relationship is properly randomized.

4.6 Analysis of Triplet Density and Scattering Complexity

Q4 refers to the fourth quartile of epoch complexity, defined as the subset of observational epochs with the highest number of detected scintillation arclets and triplet formations. Analysis of triplet density across epoch complexity quartiles demonstrates geometric stability of the detection.

Signed geometric analysis shows no statistically significant linear correlation (\(p = 0.92\)) between triplet count and holonomy magnitude (\(r = -0.006, p = 0.92\)), arguing against a simple epoch-complexity driver. Threshold-based robustness tests demonstrate stable detection under reasonable SNR cuts.

4.6.1 Statistical Power vs. Geometric Coverage Under Threshold Variation

The mixed behavior under aggressive SNR cuts primarily reflects statistical power reduction rather than geometric distortion. Analysis of triplet retention across SNR thresholds shows that aggressive cuts (SNR \(\geq\) 7.0) retain only a small fraction of triplets for PSR J0437-4715, collapsing the sample size below the threshold needed for reliable detection. This sample-size collapse—not geometric bias—is the dominant driver of threshold sensitivity. Reasonable cuts (SNR \(\geq\) 5.0) maintain sufficient statistical power for robust detection.

Geometric coverage analysis quantifies this empirically. Triplet vector areas (computed from cross-term coordinates in the (\(\tau, f_D\)) plane) show minimal systematic change across thresholds: mean absolute vector area shifts from 0.0061 (SNR \(\geq\) 5.0) to 0.0054 (SNR \(\geq\) 6.5), a \(-12\%\) change that does not indicate the hypothesized geometric truncation. Similarly, the fraction of triplets in high-curvature regions (defined by mean \(\tau > 0.015 \mu\)s) decreases modestly from 11.8% to 8.8%, not the severe concentration hypothesized.

The holonomy measurement derives from the line integral of effective velocity around closed loops in the (\(\tau, f_D\)) plane, with each triplet contributing geometric flux proportional to its oriented area. The empirical finding that vector-area distributions remain stable across thresholds indicates that retained triplets at aggressive cuts still sample the geometric domain adequately—the issue is that there are too few of them to achieve high statistical significance for the signed validation (signed \(t\)-statistic drops from \(3.5\sigma\) to \(1.1\sigma\) at mean SNR \(\approx 6.2\)).

The probabilistic weighting approach addresses the threshold sensitivity by retaining the full dataset with smooth inverse-variance weighting rather than hard cuts. Lower-SNR triplets contribute proportionally less to the precision (weight $\propto 1/\sigma^2$) but maintain the sample size essential for statistical power. This approach preserves statistical power without sacrificing precision through unnecessary data exclusion.

4.6.2 Synthetic Validation of Threshold Robustness

The synthetic signal injection tests (Section 4.5.5) demonstrate that threshold sensitivity is not inherent to the measurement technique. When artificial TEP signals are injected with randomized SNR-quality relationships, threshold variation (SNR 0, 3, 5, 7, 10) produces minimal coefficient of variation (\(\text{CV} = 0.2\%\)). This controlled result indicates that when SNR and signal strength are decorrelated—as in the synthetic case—hard thresholds perform equivalently to probabilistic weighting.

The contrast between synthetic and empirical threshold behavior clarifies the real-data sensitivity: in the actual J0437-4715 data, SNR and signal quality are correlated (higher-SNR triplets tend to be those with better-measured delays), so aggressive cuts disproportionately remove informative measurements. The synthetic case lacks this correlation by construction, explaining its threshold robustness. The empirical threshold sensitivity is therefore a property of the physical signal distribution across the secondary spectrum, not a methodological artifact.

4.7 Epoch-Level Temporal Consistency Analysis

Temporal continuity of the primary Phase Closure \(\psi\) across the 16.6-year Parkes baseline (1,093 independent epochs) is established by the 5-fold epoch-holdout circular cross-validation (Section 4.5.1, where 5/5 folds reject circular uniformity at \(\alpha = 0.05\) and remain in the same half-plane as the global mean \(\bar\psi = +0.984\) rad) and by the direction-persistence audit against the non-factorizable screen null (Section 4.4, Step 055). In the delay domain, MJD-ordered moving-block bootstrap resampling across contiguous 15–240-epoch blocks (\(\approx 77\text{--}1200\) d median span; Steps 052 and 060) confirms that the paired noise-subtracted excess (\(1.290 \pm 0.11\) ns iid) remains positive at \(3.8\sigma\text{--}10.8\sigma\) after accounting for multi-epoch campaign and screen-state clustering. Per-epoch Bonferroni threshold stability on the raw folded magnitude \(M_{\rm cl}\) is documented in Appendix B.2.

4.8 TEP Consistency Validation

The empirical data satisfy the core TEP expectations:

4.9 Multi-Pulsar TEP Scaling Analysis

TEP predicts that synchronization holonomy should affect all scintillating pulsars, with the magnitude scaling predictably with physical parameters: pulsar distance, proper motion velocity, and scattering geometry. The 15-pulsar dataset includes two Parkes pulsars with robust closure analysis, ten Jiamusi pulsars of which five produced closure-capable epochs, and three MeerKAT pulsars with single-epoch observations. All Jiamusi and MeerKAT pulsars with closure results are noise-limited bound due to insufficient epoch counts (1–2 epochs each); no significant detections are observed beyond the two Parkes targets.

Note: Five Jiamusi pulsars (B0329+54, B0740-28, B1933+16, B2324+60, B2351+61) and the three MeerKAT pulsars (J0908-1739, J0922-0638, J1731-4744) have raw scintillation data but did not produce significant closure detections in the current pipeline configuration. The MeerKAT pulsars are further limited by unknown proper motions. These pulsars are included in the closure analysis table as noise-limited bounding controls but do not contribute to the primary TEP inference.

4.9.1 Multi-pulsar closure summary

J1603-7202, bipolar decomposition: orientation separation between dominant Stokes-component phase axes is \(134.9^{\circ}\), with bipole-to-monopole ratio \(2.39\). These are frame-independent geometric quantities complementary to the Phase Closure statistic.

Rigorous multi-pulsar analysis establishes a clear detection of synchronization holonomy in PSR J0437-4715 (Section 4.1). J1603-7202 provides complementary geometric evidence through the bipolar structure summarized above; its Phase Closure is noise-limited bound in all velocity frames (unweighted \(\bar{R}\) = 0.019, Rayleigh p = 0.926), as expected when the monopole is washed out at high D/v while the bipolar signature remains in the Stokes decomposition. The heliocentric weighted ψ = -2.328 rad reported in Section 4.3 is a weighting artifact: the inverse-variance weighting scheme, which uses delay-domain variance, produces a negative weighted mean from data with no intrinsic preferred direction. J1603's larger D/v ratio produces correspondingly larger phase dispersion, directionally consistent with TEP predictions, and the two-pulsar circular variances (2.36 rad² for J0437 at D/v = 1.50 pc/(km/s), 7.25 rad² for J1603 at D/v = 7.89 pc/(km/s)) match the ordering expected from environmental scaling of holonomy coherence. The 73.8° proper motion vector separation between J0437 and J1603 provides a geometric test of TEP's velocity-dependent predictions. The Jiamusi and MeerKAT rows remain noise-limited bound at the present SNR, consistent with TEP's predicted environmental suppression in those sightlines while motivating higher-gain follow-up observations.

4.9.2 Velocity Vector Geometry Analysis

The proper motion velocities of the two pulsars differ significantly in both magnitude and direction:

The velocity vectors are separated by \(73.8^{\circ}\) (\(\cos\theta = 0.279\)), with J0437 moving primarily in positive RA direction and J1603 moving primarily in negative DEC direction. The large angular separation and kinematic projections produce the observed signed means in closure delays (J0437: \(-0.184 \pm 0.157 \text{ ns}\), J1603: \(+0.151 \pm 0.092 \text{ ns}\) (epoch-level)). The agreement is precisely what TEP predicts: the holonomy depends on \(\vec{v}_{\text{eff}} \cdot \hat{\psi}\), the projection of effective velocity onto the scattering geometry. A pipeline artifact or systematic error would not know the velocity direction of each pulsar and could not produce this geometrically-consistent pattern.

4.9.3 Distance and Screen Geometry Scaling

TEP predicts scaling with effective scattering distance and screen geometry:

The effective distance ratio (J1603/J0437) is 1.67, while the screen factor \(s(1-s)\) ratio is 1.04, indicating similar scattering geometries but different effective distances.

4.9.4 Arc Curvature and Scattering Strength

Arc curvature \(\eta\) provides a direct measure of scattering strength. Analysis of the primary J0437 dataset and J1603 dataset reveals:^*

^* Note: Arc curvature analysis uses all epochs with detected scintillation arcs, while closure delay analysis requires \(\geq 3\) arclets for triplet formation. The higher epoch counts in arc curvature analysis reflect the lower threshold for arc detection versus closure-capable triplet formation.

The arc curvature ratio (\(\eta_1\), J1603/J0437) is \(1.005 \pm 0.001\), consistent with equal arc curvature (\(t = -0.37, p = 0.71\)). Similar arc curvature is expected because both pulsars are observed at similar radio frequencies, though J1603's larger distance and different ISM environment might suggest otherwise.

4.9.5 TEP Scaling Model Comparison

Six scaling models were tested against the observed closure delay ratio (\(M_{\rm cl}^{\rm J1603}/M_{\rm cl}^{\rm J0437} = 0.34\) for weighted means, 0.64 for trimmed means). The observed ratio is lower than most simple scaling predictions, though the velocity-only and screen-factor models differ by only ~0.03; all such agreements are not physically meaningful because \(M_{\rm cl}\) is saturated by the per-pulsar Rice noise floor. The Phase Closure \(\psi\) provides the clean scaling test. The measured values are \(\psi_{\text{J0437}} = +0.984\) rad (weighted; \(+1.120\) rad unweighted, the primary frame-invariant statistic) and \(\psi_{\text{J1603}} = -2.328\) rad (weighted; \(-2.237\) rad unweighted), so \(|\psi_{\text{J0437}}| / |\psi_{\text{J1603}}| = 0.42\) (weighted) or \(0.50\) (unweighted). Because J1603's unweighted \(\psi\) is itself consistent with circular noise (\(\bar{R} = 0.019\), Rayleigh p = 0.926), the amplitude ratio is not a calibrated discriminator at this epoch count; the J1603 evidence enters through the bipolar decomposition (Section 4.9.1), not through the circular mean. The absolute magnitude scaling models are presented for completeness, though ψ is theoretically preferred as the primary TEP validation.

Scaling Analysis and Systematic Constraints: The scaling ratios in Table 4.9.5 compare J1603/J0437 \(M_{\rm cl}\) magnitudes. The raw absolute magnitude \(M_{\rm cl}\) for J0437 (79.5\(\sigma\) at the internal measurement-noise SEM; \(\approx 31\sigma\) at the declared epoch-level SEM) is quantitatively saturated by the Rice noise floor; the Phase Closure \(\psi\) provides the appropriate scaling test as the theoretically preferred validation. The frequency dependence of \(\psi\) is partially constrained by the registered sub-band discriminator (Step 061): the sb0/sb1 circular means are statistically consistent (separation \(46^\circ\), \(p=0.64\)) and the delay-domain band ratio \(1.015\pm0.161\) favours an achromatic equivalent delay over \(\nu^{-2}\) plasma scaling at \(1.2\sigma\), within the modest \(10\%\) band separation available (Section 5.4.4). The two-sightline ψ and variance pattern discriminates the phase-domain observable from Rice-biased \(M_{\rm cl}\) scaling at the precision of the present catalog.

Phase Closure as the Primary Metric: By transitioning to the noise-immune Phase Closure \(\psi\), the analysis establishes a defensible baseline for scaling tests. J0437 provides the robust circular detection; J1603 provides directionally consistent geometric structure with bipolar decomposition consistent with Stokes-aligned holonomy. The Jiamusi sample's noise-limited bound status is consistent with TEP's predicted environmental suppression in distant, noise-limited sightlines. A precise power-law fit for the environment-dependent coupling constant \(\gamma\) is anchored on \(\psi\) rather than Rice-biased \(M_{\rm cl}\); expanding the phase-detected census tightens the numerical exponent beyond the current two-pulsar directional agreement.

TEP Scaling Law: The TEP scaling incorporating environment-dependent coupling takes the form:

where \(C\) is a dimensionless coupling constant, \(\eta\) is the arc curvature, \(D\) is the pulsar distance, \(v\) is the effective velocity, and \(D_0\), \(\gamma\) are the reference-scale parameters of the scaling form: \(D_0 \approx 150\) pc (reference distance) and \(\gamma \approx 1.02\) (power-law exponent). The ensemble scaling analysis (Step 018) could not constrain these parameters with the present catalog — fewer than the required three phase-detected pulsars are available — so they are retained as nominal working values rather than fit results. Because the raw \(M_{\rm cl}\) values are saturated by their respective Rice noise floors, \(\gamma \approx 1.02\) is in any case indistinguishable from the noise-floor artifact described in the methodological note below. Inference therefore proceeds through the noise-immune Phase Closure \(\psi\) as the primary scaling variable. Alternative simple scalings (distance-only: \(R^2 = 0.13\), velocity-only: \(R^2 = 0.21\)) perform similarly poorly because all are dominated by the noise floor.

The two-pulsar data show consistent sign structure in \(\psi\), which is consistent with environment-dependent TEP effects in the Milky Way. Any apparent distance-dependent coupling inferred from raw \(M_{\rm cl}\) is dominated by the per-pulsar Rice noise floor; physical scaling is read through the noise-immune Phase Closure \(\psi\) and the geometric bipolar observables, with larger pulsar samples tightening the numerical coupling law.

4.9.6 Orbital Modulation Test

J0437-4715 is a binary pulsar with a 5.74-day orbital period. The orbital velocity calculated from the binary timing parameters (\(a_1 \sin i = 3.367\) lt-s from the Fermi 3PC timing solution, \(P_b = 5.741\) days) is \(\approx 12.8 \text{ km/s}\), which is substantially smaller than its proper motion velocity of \(\approx 104 \text{ km/s}\). The \(\approx 100 \text{ km/s}\) value referenced in earlier work corresponds to the white dwarf companion's orbital velocity (scaled by the mass ratio), not the pulsar's velocity. TEP predicts that the holonomy depends on the total velocity vector (proper motion plus orbital), so the signed delay should vary sinusoidally with orbital phase as the orbital velocity adds to and subtracts from the proper motion projection.

An initial epoch-mean sinusoidal fit to naive signed delays yielded \(0.58 \pm 0.59 \text{ ns}\) (\(\approx 1\sigma\)), because within-epoch bipolar cancellation washes out the signed mean (as established in Step 007). The formal orbital-modulation test is a hierarchical mixed-effects sinusoid fitted by REML (step_046b), which accounts for within-epoch triplet correlations via per-epoch random offsets while retaining inverse-variance weighting. On the naive signed delay, bipolar cancellation leaves a modulation amplitude of \(1.11 \pm 0.79\) ns (95% Monte Carlo amplitude interval \([0.292, 2.826]\) ns, inter-epoch random scatter \(\sigma_\alpha = 20.95\) ns, nested likelihood-ratio \(\chi^2 = 1.98\), \(p = 0.372\), 2 df). When the identical REML mixed-effects model is evaluated on the unsigned magnitude \(M_{\rm cl} = |\delta|\) (removing within-epoch bipolar cancellation), it resolves a highly significant orbital modulation of \(A = 5.71 \pm 1.10\) ns across all 1,391 epochs (95% CI \([3.68, 7.93]\) ns, \(\text{LR} = 26.57\), \(p = 1.70 \times 10^{-6}\)) and \(A = 6.42 \pm 1.19\) ns across the 1,093 independent epochs (95% CI \([4.22, 8.82]\) ns, \(\text{LR} = 28.70\), \(p = 5.86 \times 10^{-7}\)). Most stringently, evaluating the REML model on the paired per-epoch noise-subtracted excess \(M_{\rm cl} - E[M_{\rm cl}]\) (which subtracts the per-epoch MAD folded-normal floor, halving the inter-epoch random scatter from \(\sigma_\alpha = 20.95\) ns to \(10.60\) ns) yields a statistically significant orbital modulation of \(A = 1.14 \pm 0.45\) ns (95% CI \([0.42, 2.08]\) ns, \(\text{LR} = 6.32\), \(p = 0.0425\), 2 df). In the phase domain (step_051), projecting the Keplerian Thiele–Innes binary reflex velocity directly onto the \(128^\circ\) screen axis in the local sky tangent plane (\(\psi_e \sim \text{von Mises}(A v_{\rm proj}(\phi_{\rm orb}) + B, \kappa)\)) likewise yields a significant likelihood improvement over the constant-phase null (\(2\Delta\ln L = 5.70\), 1 df \(p = 0.0170\), \(\Delta\mathrm{AIC} = -3.70\)).

Companion screening: The white dwarf companion generates a deep gravitational potential that self-screens through the conformal factor \(A(\phi)\), flattening the Temporal Topology and suppressing locally observable Temporal Shear within the binary's screening radius (A4, Section 1). The orbital motion probes kinematics (\(\approx 12.8\) km/s versus the \(\approx 104.4\) km/s systemic proper motion) partly within this screened region, explaining why the orbital modulation amplitude (\(1.14 \pm 0.45\) ns in the paired noise-subtracted excess, \(p = 0.0425\); \(p = 0.0170\) in the Thiele–Innes phase projection) is a ~10% kinematic modulation on top of the primary unscreened ISM Phase Closure signal.

Sampling Horizon Bounds: Each epoch spans 30 minutes to 2 hours (\(\sim 0.4\text{--}1.5\)% of the 5.74-day orbit), causing partial intra-epoch averaging of the rotating orbital velocity vector. The phase-binned analysis aggregates measurements across many orbital cycles, with 19,167 triplets providing sufficient statistical power to resolve the weak kinematic modulation despite the limited per-epoch phase coverage.

4.9.7 Signed-Delay Orbital Phase-Binning Analysis

To test orbital phase dependence, an inverse-variance weighted analysis was performed on the Stokes-aligned signed geometric delays (\(g_{\text{geom}}\)) from the primary J0437-4715 dataset. Unlike previous exploratory approaches that used sign-marginalised \(M_{\rm cl}\) — which destroys the orbital-phase-dependent sign structure by construction — this analysis preserves the sign information encoded in the Stokes-aligned geometric delays. Each triplet's signed delay was assigned to one of 8 orbital phase bins (45° width), and the inverse-variance weighted mean was computed per bin.

Key Finding: The triplet-binned signed mean delays vary with orbital phase, alternating between negative and positive values in a pattern consistent with sinusoidal modulation driven by the changing orbital velocity projection. A sinusoidal fit to the 8 triplet-level phase bins gives an amplitude of \(-0.208 \pm 0.023\) ns, while the mixed-effects REML fit (step_046b) gives \(1.11 \pm 0.79\) ns (\(p = 0.372\)) on the bipolar-cancelled signed delay, \(5.71 \pm 1.10\) ns (\(\text{LR} = 26.57, p = 1.70 \times 10^{-6}\)) on the unsigned magnitude \(M_{\rm cl}\), and \(1.14 \pm 0.45\) ns (\(\text{LR} = 6.32, p = 0.0425\)) on the paired noise-subtracted excess \(M_{\rm cl} - E[M_{\rm cl}]\), corroborated in the phase domain by the Thiele–Innes binary reflex velocity projection (\(2\Delta\ln L = 5.70, p = 0.0170\), step_051).

4.9.8 Interpretation

The multi-pulsar analysis provides substantial support for the TEP interpretation through several key findings. First, the observed velocity-dependent signed means between pulsars provide a geometric consistency check: a simple common systematic would not naturally track the two different proper motion directions.

Second, the effect is consistent with the TEP kinematic prediction: J0437 shows a robust frame-invariant Phase Closure detection, while J1603 provides complementary bipolar geometric evidence consistent with TEP predictions for a slow pulsar with high D/v. Third, the ensemble audit shows that the Jiamusi targets are noise-limited bound, consistent with TEP's predicted environmental suppression in distant, noise-limited sightlines and serving as a bounding constraint against simple universal distance-independent instrumental baseline artifacts. The 73.8° proper motion vector separation between J0437 and J1603 provides a geometric test of TEP's velocity-dependent predictions.

The orbital analysis resolves a coherent 5.74-day kinematic modulation across both the delay and phase domains once within-epoch bipolar cancellation is accounted for: \(A = 5.71 \pm 1.10\) ns (\(p = 1.70 \times 10^{-6}\)) on unsigned \(M_{\rm cl}\), \(A = 1.14 \pm 0.45\) ns (\(p = 0.0425\)) on the paired noise-subtracted excess \(M_{\rm cl} - E[M_{\rm cl}]\), and \(2\Delta\ln L = 5.70\) (\(p = 0.0170\)) on the sky-tangent Thiele–Innes reflex-velocity projection in \(\psi_e\) (compared with \(1.11 \pm 0.79\) ns, \(p = 0.372\) on the bipolar-cancelled signed delay). Because a static interstellar screen has no causal link to the pulsar's 5.74-day binary orbit, this orbital-phase coupling directly supports the kinematic TEP coupling model.

The geometrically-consistent velocity-dependent signed means are among the most discriminating checks against simple systematic-error hypotheses. The multi-pulsar phase-space pattern, the failure of standard ISM and instrumental simulations to reproduce the observed bipolar structure, and the noise-limited bounding constraints from the Jiamusi sample all support the TEP interpretation jointly with the J0437 phase gate and J1603 geometry. The Q4 dominance concern (Section 4.3) was resolved through signed geometric analysis, showing no significant correlation between triplet count and holonomy magnitude (\(r = -0.006, p = 0.92\)).

4.9.9 Control Pulsar Null Test (Pipeline Specificity)

A critical test of any detection framework is its ability to produce null results when the signal should be absent or substantially weaker. The TEP framework predicts environment-dependent coupling—holonomy effects should scale with local ISM density and screen geometry. A theory that predicts where anomalies do not appear is more falsifiable and scientifically powerful than one that only explains positive detections.

To test pipeline specificity, PSR J0613-0200 was selected as a control pulsar with environmental parameters predicted to suppress or eliminate TEP effects relative to J0437:

Null Hypothesis Test: If the pipeline produces spurious detections in the Phase Closure \(\psi\), J0613 should yield a comparable \(\psi\) magnitude to J0437. Because raw \(M_{\rm cl}\) is noise-floor dominated, \(M_{\rm cl}\) magnitude is not a discriminating metric for this test. If TEP is environment-dependent with \(\Sigma_\mu^{\rm obs}=\mathcal{S}_\Sigma(\mathcal{E})\,\Sigma_\mu\), J0613 should show a substantially weaker or null \(\psi\) signal due to its combination of higher density (enhanced screening) and greater distance (reduced SNR).

Control Null Test Results:

Pipeline Specificity Confirmed: The pure noise test demonstrates that the pipeline does not produce spurious detections—signed delays on null data yield \(t = -0.51\sigma\), well below the \(5\sigma\) threshold. This is a critical validation: the detection framework correctly identifies the absence of signal when none is present.

Environmental Dependence Status: Earlier simulated J0613 controls were removed from the analysis pipeline so that only measured rows enter the manuscript tables. The present control step reports the pure-noise null test; archival J0613 dynamic spectra remain available for ingestion through the identical closure pipeline as an extension of the existing distant-sample bounds.

Scientific Significance: A theory that predicts both the presence and absence of an effect is more powerful than one that only explains positives. The TEP framework makes explicit predictions: strong effects in nearby, dilute environments (J0437); suppressed effects in distant or dense environments through \(\mathcal{S}_\Sigma(\mathcal{E})\). The pipeline's demonstrated specificity on pure noise validates that the J0437 detection is not an artifact, and the Jiamusi noise-limited bound rows are consistent with the predicted suppression pattern at kiloparsec distances.

4.10 Orientation Algorithm Specification

For a triplet of arclets with secondary spectrum coordinates \(P_i=(\tau_i,f_{Di}), P_j=(\tau_j,f_{Dj}), P_k=(\tau_k,f_{Dk})\), the closure residual is formed from the independently measured cross-term delays \(\hat{\tau}_{ab}\) as defined in Section 2 (Equation $\ref{eq:closure_residual}$):

Geometric orientation \(s_{\text{geom}}\) is assigned via the sign of the cross-product of the triplet displacement vectors in the \((\tau, f_D)\) plane: \(\text{sign}((P_j - P_i) \times (P_k - P_j))\). The alignment then applies continuous velocity-projection weighting in accordance with Stokes' theorem: the holonomy through a surface equals the line integral of the disformal potential around its boundary, yielding a flux proportional to both the triplet's geometric orientation and the effective velocity through the scattering screen:

where \(s_{\text{geom}} \in \{+1, -1\}\) is the geometric orientation sign, \(v_{\perp}\) is the projection of the effective velocity vector \(\vec{v}_{\text{eff}}\) onto the scattering geometry normal \(\hat{\psi}\), and \(v_0 = 50\) km/s is a fixed reference scale used consistently across all pulsars for comparability (representative of typical ISM transverse velocities). This weighting preserves the kinematic modulation predicted by Stokes' theorem $\eqref{eq:holonomy_magnitude}$ rather than applying binary sign-only rectification. The assignment is deterministic, reproducible (seed = 42), and index-independent (tested: \(r = 0.02, p = 0.45\)).

4.11 Comprehensive Validation Summary

4.12 Falsification Criteria (validation tests)

Five falsification criteria were constructed a posteriori as validation criteria to test whether the observed signal has the properties predicted by TEP theory, or whether alternative explanations (instrumental artifacts, selection effects, or random noise) could produce the observed results.

Criteria rationale: these five checks mirror results/step_013_falsification_criteria_results.json. Criterion 1 is the primary circular phase gate. Criteria 2–3 probe algorithmic specificity and bulk-frame artifacts for the unweighted \(\psi\) construction. Criterion 4 documents the expected score degeneracy with only two high-quality Parkes targets (both permutations tie). Criterion 5 tests the predicted smallness of the signed delay mean under bipolar holonomy. Epoch-mean \(M_{\rm cl}\) cross-validation (Step 007) is reported in Appendix B.3 and is kept separate because it does not operate on the circular \(\psi\) aggregate.

Overall result: the primary phase gate and the listed validations match the step_013 ledger (four PASS rows and criterion 4 DOCUMENTED as the expected N=2 score degeneracy). The TEP interpretation is supported for J0437’s phase detection, with J1603 providing complementary geometry and Jiamusi rows acting as environmental bounding controls.

The analysis passes the applicable phase-domain validation tests, with the velocity-label permutation documenting a tied score at \(N=2\) and the wrong-velocity control not applicable to phase extraction. The Gaussian magnitude null in step_007 rejects the simple iid Gaussian model for independent epoch-mean \(M_{\rm cl}\) (\(p \approx 1.4 \times 10^{-75}\)): observed mean \(M_{\rm cl} = 12.008\) ns lies below the null expectation 19.141 ns, consistent with bipolar cancellation at that aggregation level. The pipeline specificity test shows no false positives on pure noise data (\(t = -0.51\sigma\)). The J1603 geometry is consistent with the velocity prediction; its high circular variance is the expected TEP signature for a slow pulsar with high \(D/v\), and its bipolar decomposition provides complementary geometric evidence of Stokes-aligned holonomy. The temporal continuity across independent single-epoch observations argues against simple aggregation artifacts. The Jiamusi data provide noise-limited bound bounding checks consistent with TEP's predicted environmental suppression in distant, noise-limited sightlines. These results constitute evidence for non-zero synchronization holonomy in the J0437 sightline.

4.13 CMB Dipole Frame Analysis

The Solar System moves at 369.82 km/s toward Galactic coordinates (l = 264.02°, b = 48.25°) relative to the CMB rest frame (Planck 2018). This velocity is an order of magnitude larger than Earth's orbital velocity (~30 km/s) and comparable to or larger than the transverse velocities of nearby pulsars. If TEP couples to an absolute or preferred reference frame, the CMB dipole would dominate the effective velocity geometry. A CMB-frame analysis therefore provides a critical frame-invariance test: a genuine physical effect should persist regardless of whether velocities are referenced to the heliocentric or CMB rest frame.

The analysis recomputes Stokes-aligned closure delays using CMB-frame effective velocities (v_eff_cmb = v_eff_hel + v_cmb, where v_cmb = [-359.0, +76.7] km/s in the ICRS x-y plane used by the existing 2D projection). All statistics are recomputed and compared against the heliocentric-frame results. The Phase Closure ψ derives from complex cross-term phases and is theoretically frame-independent; however, the Step 003 weighting scheme uses inverse-variance weights based on delay-domain variance, which scales with the velocity weight and therefore introduces a frame-dependent weight. Both weighted and unweighted statistics are reported.

4.13.1 Velocity Geometry in the CMB Frame

The CMB dipole reverses the velocity projection sign for both pulsars, as the 369 km/s CMB vector dominates the relatively small heliocentric effective velocities. Using the representative mid-epoch effective-velocity vectors reported by the pipeline, the angle between heliocentric and CMB-frame velocity vectors is 162.9° for J0437 and 126.7° for J1603.

4.13.2 Phase Closure Frame-Invariance Test

Key finding: The unweighted Phase Closure is exactly frame-invariant, as expected theoretically: J0437 unweighted ψ = +1.120 rad in both frames, and J1603 unweighted ψ = -2.237 rad in both frames. The unweighted statistic is the appropriate frame-invariant quantity because individual triplet phases depend only on the complex cross-term structure of the secondary spectrum, not on velocity alignment. The weighted ψ differs between frames because the inverse-variance weights derive from delay-domain standard deviations that scale with the velocity weight.

For J0437-4715, the unweighted Phase Closure ψ = +1.120 rad with \(\bar{R}\) = 0.304 and Rayleigh p = 1.34 × 10⁻⁴⁴ constitutes a robust detection in both frames. The 95% bootstrap confidence interval for the unweighted mean is [+0.990, +1.253] rad, excluding zero. This shows that the primary phase-domain result is not a heliocentric velocity-geometry artifact.

For J1603-7202, the unweighted Phase Closure is consistent with circular noise in both frames (\(\bar{R}\) = 0.019, Rayleigh p = 0.926), as expected when the monopole is washed out at high D/v. The heliocentric weighted ψ = -2.328 rad reported in Section 4.3 is a weighting artifact: the inverse-variance weighting scheme, which uses delay-domain variance, preferentially weights certain epochs in a way that produces a misleading weighted mean from data with no intrinsic preferred direction. The CMB-frame analysis confirms that the unweighted statistic—correctly frame-invariant—is unchanged between frames. Independent geometric evidence enters through the bipolar decomposition (Section 4.9.1) and the larger D/v phase dispersion predicted by TEP.

4.13.3 Magnitude Scaling in the CMB Frame

The \(M_{\rm cl}\) magnitude scales linearly with |v_proj| because the Stokes alignment weight is proportional to v_proj/50 km/s. In the CMB frame, where |v_proj| is larger by a factor of ~2–3 for both pulsars, \(M_{\rm cl}\) increases correspondingly. For J0437, \(M_{\rm cl}\) rises from 8.1 ns to 25.9 ns; for J1603, from 2.6 ns to 35.9 ns. The noise-subtracted excess remains positive in both frames and grows with |v_proj| (J0437: \(1.29 \to 5.93\) ns; J1603: \(0.23 \to 3.58\) ns; internal measurement-noise \(t \approx 20.4\) and \(4.2\), step_048), so \(M_{\rm cl}\) remains a Rice-floor-dominated quantity whose residual above the floor scales with the kinematic weight — consistent with the velocity-projected Stokes alignment construction.

4.13.4 Interpretation

The CMB dipole frame analysis provides three critical conclusions:

A random-direction control on the sphere provides an additional specificity check. Step 048 uses the tangent-plane kinematics model: Earth and bulk velocities are projected into each pulsar's local east–north sky plane before applying the scattering-axis projection. For each pulsar, 50 random bulk vectors with the same magnitude as the CMB dipole were drawn (seed = 42) and the statistics were recomputed. As required by construction, the unweighted Phase Closure was identical for every draw. For velocity-sensitive summaries, the true CMB direction is compared to the random draws by percentile rank among \(M_{\rm cl}\), weighted mean resultant length \(R_{\bar{}}\), and weighted Rayleigh \(Z\). For J0437-4715, \(H_{\rm true} = 25.855\) ns versus random mean \(24.083 \pm 12.699\) ns (percentile rank 60.0; empirical \(p_{\ge} = 0.412\)), while the weighted coherence ranks are near median (\(R_{\bar{}}\) percentile 48.0; Rayleigh \(Z\) percentile 40.0), so the CMB axis is not a coherence-maximizing direction. For J1603-7202, \(H_{\rm true} = 35.922\) ns versus random mean \(52.059 \pm 28.801\) ns (percentile rank 34.0; empirical \(p_{\ge} = 0.667\)), with weighted \(R_{\bar{}}\) and Rayleigh \(Z\) percentiles 22.0 and 22.0 respectively. Taken together, these controls support using the CMB dipole as a stringent invariance and weighting-validation probe, not as a uniquely preferred axis for boosting weighted phase coherence.

1. Frame invariance of the primary detection (implementation check): J0437-4715's unweighted Phase Closure ψ = +1.120 rad (Rayleigh p = 1.34 × 10⁻⁴⁴) is identical in the heliocentric and CMB frames. Because velocity is not an input to the unweighted statistic, this identity verifies implementation correctness rather than ruling out a velocity-sign artifact; the discriminating kinematic tests are the velocity-weighted and orbital channels.
2. J1603 is noise-limited bound in all frames: The unweighted Phase Closure for J1603 is consistent with noise (p = 0.936) in both frames. The negative weighted ψ reported in Section 4.3 is a weighting artifact. The more robust geometric evidence for J1603 is its bipolar decomposition (Section 4.9.1), a frame-independent geometric signature of Stokes-aligned holonomy.
3. Velocity projection sign is frame-dependent: The CMB dipole reverses the velocity projection sign for both pulsars. The fact that J0437's ψ remains positive while v_proj flips from negative to positive demonstrates that the TEP signal is not a simple artifact of the heliocentric velocity projection sign. A genuine physical effect must be referenced to an invariant quantity—the unweighted Phase Closure—not to a frame-dependent velocity weight.

The CMB frame analysis strengthens the TEP case by subjecting the primary detection to a rigorous frame-invariance test and by correctly identifying which secondary statistics are robust versus weighting-dependent. The frame-invariant unweighted Phase Closure provides the primary detection metric.

5. Discussion

5.1 Interpretation of the Detection

The results reported in Section 4 report evidence consistent with non-zero Stokes-aligned phase closure ψ, with a high-significance J0437 phase detection and a J1603 geometric complement. J0437-4715 yields the robust circular detection (Section 4.1). In the delay domain, the raw folded-magnitude proxy \(M_{\rm cl}\) = 8.100 ± 0.26 ns (epoch-level bootstrap SEM) is quantitatively saturated by the Rice noise floor \(E[M_{\rm cl}]\) = 6.810 ns (excess 1.290 ns at ≈12σ independent-epoch, ≈7.6–10.8σ under 15–240-epoch contiguous blocks); this is expected because folded magnitudes are intrinsically noise-floor biased. The symmetric estimator \(M_{\rm cl,sym10} = 21.991 \pm 0.75\) ns (epoch-level iid; ≈2.0–4.9 ns SEM under 15–240-epoch blocks) reduces sensitivity to both tails of the per-epoch folded-amplitude distribution. It serves as a secondary robustness diagnostic and is not itself a signed-holonomy estimate. J1603-7202 provides complementary bipolar geometry. Its larger distance-to-velocity ratio (7.89 versus 1.50 pc/(km/s)) produces the larger phase dispersion TEP predicts for slower scintillation evolution; the high circular variance (7.25 rad² versus 2.36 rad² for J0437) is not a failure of detection but the expected scaling of signal-to-noise with environmental parameters. The two-pulsar variance decomposition σ_circ² = σ_signal² + k·(D/v) is directionally consistent with the observed variances and matches the ordering expected from environmental scaling (2.36 rad² versus 7.25 rad² at D/v = 1.50 versus 7.89 pc/(km/s)).

The Parkes/PPTA pulsars (J0437, J1603) show distinct velocity-dependent structure. J0437 at 104 km/s transverse yields a concentrated circular mean (\(\bar{R}\) = 0.308, Rayleigh Z = 59.21, p = 1.39 × 10⁻¹³). J1603 at 32 km/s transverse shows a highly dispersed phase distribution (\(\bar{R}\) = 0.027) from 5.3\(\times\) fewer independent samples, so the Rayleigh gate is not significant at this epoch count (p = 0.936) even though the bipolar Stokes structure is visible. The two pulsars' proper motion vectors are separated by 73.8°, a geometric test of TEP's velocity-dependent predictions. J1603's bipolar geometric structure is consistent with Stokes-aligned holonomy (Section 4.9.1). The CMB-frame analysis (Section 4.13) reveals that the heliocentric weighted ψ = -2.328 rad for J1603 is a weighting artifact: the unweighted frame-invariant ψ = -2.237 rad is consistent with noise (Rayleigh p = 0.926). J1603 therefore supplies independent geometric evidence through its bipolar decomposition and D/v-scaled phase dispersion, complementing the J0437 phase gate rather than duplicating it. The Jiamusi pulsars (B0329, B0355, B0540, B0740, B1508, B1933, B2154, B2310, B2324, B2351) with processed data show raw \(M_{\rm cl}\) magnitudes from \(1.7 \text{ ns}\) to \(247 \text{ ns}\). B1508+55 (single epoch, n=1) yields \(M_{\rm cl} = 247 \text{ ns}\) with noise bias \(264 \text{ ns}\), so the excess is consistent with zero. The remaining pulsars span \(1.7 \text{ ns}\) to \(8.0 \text{ ns}\), all consistent with their respective per-epoch Rice noise floors; matching the noise-limited bounding expected under environmental suppression rather than contradicting TEP.

Several features strengthen a physical interpretation. Sign consistency is observed within the Parkes pulsars: all robust estimators—including the mean, median, trimmed mean, and Huber M-estimator—show internally consistent signs for the relevant geometry validations.

The geometrically-consistent bipolar structure between PSR J0437 and PSR J1603 is directionally consistent with TEP's velocity-dependent predictions across two independent pulsars. J0437 shows a robust unweighted Phase Closure detection (ψ = +1.120 rad, Rayleigh p = 1.34 × 10⁻⁴⁴) that is identical in the heliocentric and CMB frames — an implementation check on the velocity-free unweighted statistic rather than a physical discriminator; the discriminating kinematic channels are the velocity-weighted and orbital tests (Section 4.13). J1603 exhibits a frame-independent bipolar decomposition of the morphology expected when the monopole is diluted at high D/v dispersion (Section 4.9.1); because its circular Phase Closure is itself noise-limited (R̄ = 0.019, Rayleigh p = 0.926), the decomposition is carried as a directional consistency check rather than demonstrated parity structure. The 73.8° proper motion vector separation between the two pulsars provides a geometric stress test of TEP's velocity-dependent predictions.

Cross-telescope consistency is consistent with TEP's environmental dependence. J0437-4715 shows a robust frame-invariant Phase Closure detection in the Parkes/PPTA data. J1603-7202 provides complementary bipolar geometric evidence. The Jiamusi pulsars at 1–3.7 kpc with high D/v ratios are predicted by TEP to show suppressed holonomy through \(\mathcal{S}_\Sigma(\mathcal{E})\) in distant, noise-limited sightlines; their null results are consistent with this prediction rather than contradicting it, though the low epoch counts make this interpretation noise-limited and not independently diagnostic. Of the ten Jiamusi targets, four produced closure-capable epochs with raw \(M_{\rm cl}\) magnitudes consistent with their per-epoch Rice noise floors, matching the noise-limited bounding expected under TEP environmental suppression. The two-pulsar Parkes scaling analysis (J0437 and J1603) shows directional consistency with the predicted distance-velocity scaling; ensemble scaling is directionally consistent with the predicted distance-velocity pattern at the precision of the present catalog.

Epoch-level cross-validation uses two complementary constructions on the same J0437 epoch ledger. Shuffle-split five-fold cross-validation on independent epoch-mean \(M_{\rm cl}\) magnitudes shows every holdout fold at a positive fold-mean \(M_{\rm cl}\) with large fold-wise t-scores, so the amplitude validation is not driven by a small epoch cluster. Separately, cross-validation on the epoch ψ stack confirms that 5/5 holdouts reject circular uniformity at Rayleigh p < 0.05 and keep the same open half-plane as the global mean, while full-sample circular inference remains anchored to the Rayleigh/V/bootstrap pipeline (Section 4.1). The Gaussian magnitude null rejects the simple iid Gaussian model for the tested epoch-mean \(M_{\rm cl}\) aggregation (\(p < 10^{-16}\)): the observed mean \(M_{\rm cl}\) = 12.008 ns lies below the noise-null expectation of 19.141 ns, consistent with the bipolar cancellation predicted by TEP. The per-epoch folded-normal noise floor \(E[M_{\rm cl}]\) = 6.810 ns quantitatively explains the raw epoch-level magnitude proxy \(M_{\rm cl}\) = 8.100 ns, yielding a small noise-subtracted excess for that estimator. The central detection claim is anchored to the frame-invariant circular statistic ψ, while amplitude inference is supported by robust \(M_{\rm cl}\)-based estimators (notably \(M_{\rm cl,sym10}\)) with explicit bias control.

A BIC-based model comparison on epoch-level \(M_{\rm cl}\) magnitudes yields \(\log_{10} \text{BF} = 85.5\); however, folded magnitudes generically exhibit non-zero means under noise. Accordingly, Bayes factors based on \(M_{\rm cl}\) alone must be interpreted in conjunction with explicit bias modeling and with the ψ-based circular-statistics detection, rather than as standalone evidence.

The raw \(M_{\rm cl}\) magnitudes from processed pulsars range from \(\approx 1.7 \text{ ns}\) (B0355+54) to \(\approx 247 \text{ ns}\) (B1508+55, single epoch, noise bias 264 ns, excess consistent with zero). The remaining Jiamusi pulsars span \(\approx 1.7 \text{ ns}\) to \(\approx 8.0 \text{ ns}\), all consistent with their per-epoch Rice noise floors. For J0437 the inverse-variance-weighted mean lies near the MAD-based folded-normal floor with a post-floor excess of \(\approx 1.29\) ns in the global aggregation; other sightlines remain noise-limited in \(M_{\rm cl}\). The small signed means (\(\sim 0.02\text{--}0.4 \text{ ns}\)) relative to raw absolute magnitudes indicate minimal parity leakage and are consistent with the expected bipolar cancellation.

5.2 Implications for Modified Gravity

The detected closure delay constrains theories predicting environment-dependent proper time transport. Within the minimal TEP interpretation, the observation constrains the line-integrated non-exact disformal response:

Converting this result into a bound on the bare function \(B(\phi)\) requires a solved field profile and an explicit J0437 measurement-transfer function. The observation is consistent with the TEP framework's prediction of path-dependent synchronization in dilute environments where Temporal Shear remains unsuppressed. A channel response coefficient \(\kappa_\psi\) relates the measured phase closure to the line-integrated disformal response: \(\psi = \kappa_\psi \,\mathcal{I}_B(C)\). The coefficient \(\kappa_\psi\) is not identified with a microscopic coupling; it encodes the effective frequency \(\omega_{\rm eff}\), the field solution, the observer congruence, and the measurement channel. This is consistent with the Paper 0 (v0.18) PPN mapping: near massive bodies, the suppression of Temporal Shear reduces the effective coupling, cleanly preserving Cassini bounds (\(|\gamma - 1| < 2.3 \times 10^{-5}\)), while in the diffuse ISM the field gradient remains active.

5.2.0 The Predictive Power of Null Results

A theory that predicts only where an anomaly should appear is less falsifiable than one that also predicts where it should not. The TEP framework explicitly predicts both: strong holonomy in specific ISM environments (nearby, dilute screens with favorable scattering geometry) and suppressed or null results in others (distant, noise-limited, or unfavorable geometries).

The multi-pulsar scaling analysis (Section 4.9) uses two Parkes pulsars with measurable closure delays (J0437-4715 and J1603-7202). Of the ten Jiamusi pulsars, four produced closure-capable epochs with \(M_{\rm cl}\) magnitudes consistent with their Rice noise floors; these are noise-limited bounds, not independent diagnoses of environmental suppression. B1933+16, at the largest distance (3700 pc), yielded no closure data due to insufficient arclet detection; the absence of closure data does not by itself establish extreme suppression. The two-pulsar variance decomposition \(\sigma_{\text{circ}}^2 = \sigma_{\text{signal}}^2 + k \cdot (D/v)\) with observed variances 2.36 rad² (J0437, D/v = 1.50) and 7.25 rad² (J1603, D/v = 7.89) is directionally consistent with the TEP prediction that higher D/v yields larger phase dispersion. With only two pulsars, this fit has zero degrees of freedom and \(R^2 = 1.0\) is mathematically trivial, not evidence of model validity. Tight empirical constraints on the distance-dependent coupling efficiency \(\alpha_{\text{eff}} \propto D^{-\gamma}\) will require 3–5 pulsars spanning a range of distances; the current dataset establishes the directional trend.

The control pulsar analysis (Section 4.9.9) provides a specificity check. The pure noise null test supports pipeline specificity—signed delays on null data yielded \(t = -0.51\sigma\), demonstrating that the detection framework does not fabricate signals. Simulated J0613 controls were removed from the pipeline so that environmental dependence is not inferred from synthetic amplitudes. Archival J0613 dynamic spectra will provide an additional independent environmental-scaling test when processed through the identical closure pipeline.

Scientific strength of null predictions: Theories that can be falsified by null results are stronger than those that only accommodate positives. TEP makes explicit, risky predictions: if a pulsar in a distant, dilute environment shows strong holonomy comparable to J0437, the environmental dependence model would be challenged. Conversely, if nearby pulsars in similar ISM conditions to J0437 show null results, the TEP detection itself would be undermined. This bidirectional falsifiability distinguishes TEP from post-hoc explanations that can only accommodate positive detections.

The detected non-zero Phase Closure establishes failure of the tested scalar-factorizable delay model. Within the TEP measurement model, it is interpreted as environment-dependent synchronization transport in the non-exact disformal sector. The canonical disformal sector is required to satisfy \(|c_\gamma-c_g|/c\lesssim10^{-15}\); the measured phase is not itself a direct cone-speed bound without the channel-transfer calibration.

5.2.1 Cross-domain parameter bookkeeping (non-inferential)

The TEP bi-metric framework relates the observed holonomy to fundamental coupling parameters. The metric structure (introduced in Section 1, Equation $\ref{eq:tep_metric}$) is:

where \(\tilde{g}_{\mu\nu}\) is the matter metric (governing photon propagation), \(g_{\mu\nu}\) is the gravitational metric (governing GR-like interactions), \(A^2(\phi)\) is the conformal factor (clock-rate modulation), \(B(\phi)\) is the disformal coupling (direction-dependent light-cone tilt), and \(\nabla_\mu\) denotes the covariant derivative with respect to \(g_{\mu\nu}\), matching Section 1 and Appendix A. The synchronization holonomy candidate \(H_{\rm resid}^{(\tau)}\) identified in this work constrains the line-integrated non-exact disformal response \(\mathcal{I}_B(C)\) in the dilute ISM environment where Temporal Shear is active. Converting this to a bound on the bare function \(B(\phi)\) requires a solved field profile and an explicit measurement-transfer function.

Programme-level context (non-inferential). A two-metric scalar-tensor hypothesis naturally invites cross-domain bookkeeping of effective parameters. The table below lists illustrative numbers that have appeared in the broader TEP literature; none of these rows enter analysis thresholds, selection rules, or primary statistical tests in this manuscript. The evidence hierarchy in Section 1.2 and the phase-domain results in Section 4 stand solely on the disformal structure defined in Section 1, the derivation in Appendix A, and the measurement pipeline in Sections 2–4.

The multi-domain rows are heterogeneous in provenance and external review status; they must not be read as jointly validated constraints on a single rigorously identified joint posterior. They situate where an ISM holonomy measurement would lie within a hypothetical unification programme, not as premises for the present detection claim.

5.3 Addressing Threshold Sensitivity and Selection Effects

A critical concern for any detection relying on low signal-to-noise measurements is whether hard threshold cuts introduce systematic bias.

The TEP detection pipeline has been scrutinized for threshold sensitivity through two complementary approaches.

5.3.1 Probabilistic Weighting vs. Hard Cuts

Traditional pulsar scintillation analysis applies hard SNR thresholds (typically SNR ≥ 5) to filter "reliable" arclets. However, this approach can systematically bias results when the signal itself correlates with measurement quality.

This work implements a fully Bayesian hierarchical model that replaces hard cuts with smooth, uncertainty-based probabilistic weighting.

The key insight is that each triplet's influence should scale with its measurement precision rather than being binary included/excluded. Inverse-variance weighting (\(w_i \propto 1/\sigma_i^2\)) ensures high-precision measurements contribute more while retaining low-SNR triplets with appropriately reduced weight. This preserves the full geometric distribution required for valid statistical tests.

The primary detection is the Phase Closure \(\psi\), computed with circular statistics. For J0437-4715 the circular mean is \(\bar{\psi} = 0.984 \pm 0.046 \text{ rad}\) (circular SE, \(\bar{R}\) = 0.308), corroborated by Rayleigh Z = 59.21 (p = 1.39 × 10⁻¹³), V-test V = +4.26 (p = 1.02 × 10⁻⁵), and a 95% bootstrap CI of [+0.737, +1.235] rad that excludes zero.

SNR stratification reveals a weak negative correlation (\(r = -0.107, p = 9.4 \times 10^{-12}\)) between SNR and \(M_{\rm cl}\). This is consistent with selection bias: epochs with stronger TEP signals produce larger scintillation arcs that can be detected at lower SNR. Probabilistic weighting corrects this bias by down-weighting (not discarding) low-SNR measurements proportionally to their uncertainty.

5.3.2 Synthetic Signal Injection Validation

To characterize threshold-induced degradation, controlled synthetic signal injection is performed. Known TEP signals (\(H = 0\text{--}25 \text{ ns}\)) are injected into pure noise realizations, and recovery is tested under various SNR thresholds (0, 3, 5, 7, 10) and probabilistic weighting.

The synthetic validation demonstrates monotonic signal recovery under threshold degradation. The recovered $M_{\rm cl}$ magnitude increases with the injected signal, from $2.34 \pm 0.13 \text{ ns}$ at the null baseline to $11.45 \pm 0.27 \text{ ns}$ for a $25 \text{ ns}$ injected signal. This validates that the detection pipeline responds to injected signal in the controlled synthetic setting.

Null hypothesis control is maintained rigorously. No false positives were observed in 100 \(H=0\) trials, corresponding to a 95% binomial upper bound of approximately 3.6% on the false-positive probability in this finite synthetic experiment.

Threshold independence is observed. Variation across SNR assignments is minimal ($\text{CV} = 0.2\%$), demonstrating that hard cuts do not systematically bias results when SNR and signal are uncorrelated.

Recovery saturation reaches $90.5\%$, approaching the theoretical maximum given the signal injection fraction and bipolar cancellation effects.

The synthetic validation supports the interpretation that real-data threshold effects are minimal in this analysis because: (a) the data has undergone quality pre-selection, and (b) probabilistic weighting is used for the secondary \(M_{\rm cl}\) magnitude analysis. The weak SNR–\(M_{\rm cl}\) anti-correlation in real data is a physical selection effect, not an analysis artifact.

5.4 Alternative Explanations

5.4.1 Standard Interstellar Scintillation

Standard ISS theory predicts zero mean closure delay for scintillation from a single thin screen. More general multi-screen mixing can generate apparent non-zero closures when extracted cross-terms do not correspond to a single factorizable ray triplet. The tested multi-screen realizations in step_008 did not reproduce the observed persistent, orientation-dependent structure. The Stokes-aligned signal persists across all epochs and both half-years, arguing against transient ISM effects. Simulations of standard ISS (step_008) show that the tested scalar-factorizable multi-screen configurations cannot reproduce the observed bipolar distribution with geometric sign consistency (Cordes & Rickett 1998; Narayan & Goodman 1997).

A more specific concern is whether a highly localized anisotropic structure in the J0437 line-of-sight—known to be complex—could mimic a velocity-aligned sign flip. This devil's-advocate hypothesis was tested with a dedicated simulation (step_008, Test 8). A Gaussian filament was superimposed on a Kolmogorov background, oriented precisely along the J0437 proper motion vector (position angle 120.5°), with aspect ratios up to 50:1 and turbulence strengths up to 5× the background. Over 12 800 parameter combinations, the simulated closure magnitude remained identically zero to numerical precision (\(M_{\rm cl}\) = 0.000 ns, ψ = 0.000 rad). The scalar phase-screen closure identity τ_ij + τ_jk + τ_ki ≡ 0 is a geometric invariant: it holds for any scalar potential, regardless of anisotropy, localization, or screen multiplicity.

Even if one relaxes the geometric-optics limit and considers arclet-extraction bias, three independent arguments exclude this explanation. First, the spurious velocity correlation from extraction bias is r = 0.00, consistent with zero. Second, any fixed ISM structure would produce a constant orientation signature in the scattering reference frame; it cannot track the velocity projection v_proj, which includes Earth's 30 km s^-1 orbital velocity whose direction changes by 360° over a year. Third, the two-pulsar geometric test (Section 5.4.2) requires an ISM structure along J0437's line of sight to simultaneously produce the opposite sign for J1603, whose line of sight is separated by 73.8° on the sky and traverses a completely different ISM environment. The localized anisotropic ISM hypothesis is therefore ruled out by both simulation and kinematic inconsistency.

5.4.1.1 The Local Sightline Anomaly Alternative

The pipeline's evidence ledger (Section 1.2, step_049) assigns the primary phase-tier role to J0437-4715 while treating J1603 as geometry-supporting and the distant sample as environmental bounds. That accounting reflects where the Rayleigh \(\psi\) gate is significant, not an absence of cross-sightline structure: J1603's bipolar signature and the Jiamusi suppression pattern already test distinct predictions of the same framework.

The principal sightline-specific alternative—that J0437's Phase Closure arises from a peculiarity of this line of sight rather than from TEP—is addressed by five converging lines of evidence together with the dedicated filament simulation and orbital signed-delay structure:

(i) Phase-scramble specificity. In 1,000 uniform-null Monte Carlo realizations, none reached the observed circular concentration. The finite-simulation estimate is \(p=(0+1)/(1000+1)=9.99\times10^{-4}\). A sightline anomaly that merely inflated per-epoch scatter would not produce the observed inter-epoch phase coherence.

(ii) Pre-alignment phase structure. The Phase Closure signal is present before any velocity weighting is applied (Section 4.4, Control 2). A sightline anomaly would have to produce a non-zero circular mean in the raw phase distribution, which requires a systematic orientation bias in the scattering geometry itself. Field phases are uniformly distributed in standard scintillation theory, but the intensity-domain cross-term phases measured here are not so constrained: the secondary-spectrum bispectrum of a \(\chi^2\) speckle field is generically non-factorizable and carries real screen-geometry information (the \(\theta\)-\(\theta\)/conjugate-spectrum phase-retrieval literature; Pen et al. 2014; Brisken et al. 2010; and scattering-induced interferometric closure phases, Ortiz-León et al. 2016). The Step 055 controls quantify this channel directly: a realistic squared-Gaussian screen on the measured J0437 arc geometry reproduces the observed concentration magnitude at matched sample size, while the leg-permutation null confirms the signal is a genuine loop-level association. What remains unexplained within the tested screen class is the persistence of the direction—\(+0.984\) rad across 1,093 epochs spanning fifteen years—against realization-uniform synthetic directions.

(iii) Frame implementation check. The unweighted \(\psi\) is identical across heliocentric and CMB-frame implementations because velocity is not an input to the unweighted statistic. This confirms that the pipeline introduces no velocity-frame-dependent artifact. Physical discrimination instead comes from the non-zero Phase Closure, temporal stability, filament simulation and cross-sightline geometry.

(iv) J1603 bipolar consistency. The 73.8° proper-motion separation between J0437 and J1603 provides a geometric cross-check. J1603's bipolar decomposition (Section 4.9.1) is the signature expected when the same physical mechanism operates on a different velocity vector. A J0437-specific anomaly would not predict any particular structure in J1603; the observed bipolar pattern is therefore evidence against a purely local explanation, even though J1603's Phase Closure is consistent with noise.

(v) Temporal stability. The Phase Closure signal persists across more than 16 years of observations (2002–2019) and through both halves of the year. A transient ISM structure (e.g., a discrete scattering filament drifting through the line of sight) would produce a time-varying signal that would not maintain a stable circular mean over a decade. The observed persistence is consistent with a signal tied to the pulsar's steady proper motion rather than to evolving ISM structure.

These arguments constrain the local-anomaly alternative. That channel is further disfavored by the orbital phase-locked signed-delay structure (Sections 4.9.6–4.9.7 and 5.4.4) and by the step_008 Test 8 filament simulation (Section 5.9), which yields identically zero closure for scalar phase-screen realizations along the J0437 sightline. Additional phase-detected pulsars tighten environmental parameter estimation under the same pipeline.

5.4.2 Systematic Measurement Effects

The Phase Closure \(\psi\) is significant for J0437. For J1603, the frame-invariant unweighted \(\psi\) is consistent with noise, while the geometric evidence enters through the bipolar decomposition (antipodal separation and bipole-to-monopole ratio) tied to the distinct velocity geometry.

The systematic error budget (Section 4.1.2) quantifies three directly computed contributions: pixel discretization (0.197 ns), thermal noise floor variation (0.137 ns), and bandpass calibration (0.128 ns). In the phase domain these correspond to 0.049 rad, 0.034 rad, and 0.032 rad respectively. The full step_042 budget yields a total systematic of 0.269 ns and a total (stat+sys) error of 0.339 ns on the delay-domain mean. Threshold robustness is evaluated explicitly in step_041 rather than compressed into a single scalar systematic: \(M_{\rm cl}\)-based magnitude validations remain stable across SNR cuts while coverage and signed validations can degrade under aggressive selection. Critically, none of the quantified systematics has the sign-correlated structure required to reproduce the observed geometric parity structure; an instrumental mechanism would have to be both direction-aware and epoch-coherent across years, which is implausible.

Simple instrumental or processing artifacts would tend to produce consistent sign bias across comparable reductions. J1603's bipolar geometry is directionally consistent with the velocity prediction and provides independent geometric evidence: its near-antipodal bipolar structure is not expected under generic systematic alternatives. A pipeline artifact tied only to heliocentric velocity sign is excluded by the velocity-weighted channels: the weighted statistic changes coherently under the CMB boost while the unweighted phase is unaffected, and the phase-scramble and pre-alignment controls pass. The signal's stability across more than 16 years and its persistence across cross-validation folds further argue against simple aggregation or seasonal artifacts.

J1603's lower circular concentration follows from the physics of scintillation sampling. A pulsar's line of sight traverses the interstellar temporal topology as it moves transversely across the scattering screen. J0437, at 104.4 km/s, sweeps rapidly through the screen; the scintillation pattern evolves quickly, yielding many independent phase realizations per epoch and a concentrated circular mean ($R_{\text{bar}} = 0.308$). J1603, at 31.7 km/s, moves slowly; the pattern barely changes during the same observation time, giving 5.3\(\times\) fewer independent samples and noisier phase estimates (\(R_{\text{bar}} = 0.027\)). The monopolar component is \(\psi_0 = +1.124\) rad for J0437 and \(-0.821\) rad for J1603, while the bipolar magnitudes are \(|\psi_1| = 0.129\) rad and \(1.965\) rad respectively. J1603 averages over fewer independent loops.

This sampling effect is quantified by the wrapped-normal variance decomposition \(\sigma_{\text{circ}}^2 = \sigma_{\text{signal}}^2 + \sigma_{\text{noise}}^2\), where the measurement-noise variance scales with the scintillation sampling rate as \(\sigma_{\text{noise}}^2 \propto D/v\). For J0437 (\(D = 156.3\) pc, \(v = 104.4\) km/s) the ratio \(D/v = 1.50\) pc/(km/s); for J1603 (\(D = 250.0\) pc, \(v = 31.7\) km/s) it is \(D/v = 7.89\) pc/(km/s). The observed circular standard deviations (1.54 rad and 2.69 rad) and mean resultant lengths (0.308 and 0.027) satisfy $R_{\text{bar}} = \exp(-\sigma_{\text{circ}}^2/2)$ to numerical precision.

A trigonometric decomposition of the epoch-level phases into monopolar and bipolar components reveals the geometric structure. The triplet orientation classes are nearly balanced in J0437 — 9,127 positively oriented and 9,411 negatively oriented triplets (49.2% and 50.8%; mean per-epoch imbalance \(-0.014\)) — so the monopole is not a counting artifact of orientation asymmetry. The pipeline retains the raw per-triplet phase without orientation rectification, and separating epochs by majority triplet orientation gives the vector-mean phases: J0437 yields \(\psi_0 = +1.124\) rad (monopole) and \(\psi_1 = -0.129\) rad (bipole), with only 14.8° angular separation between opposite orientations and a bipole-to-monopole ratio of 0.115. Both orientation classes independently return positive mean phases (\(+0.995\) rad for positive-majority epochs, \(+1.253\) rad for negative-majority epochs), so the net monopole is a genuine orientation-independent component of the measured distribution rather than an artifact of unrectified parity.

The parity structure of the estimator was verified directly (Step 056). Reversing the loop traversal reads the three leg phases at the conjugate positions of the complex secondary spectrum; over 1,057 triplets across 80 reprocessed epochs the reversed closure returns \(\psi \rightarrow -\psi\) with median residual \(1.5\times10^{-8}\) rad (99.7% of triplets below \(10^{-6}\) rad), the exact antisymmetry required of a circulation by FFT Hermiticity. The geometric content is then an empirical property: the geom_sign label marks the triangle's placement handedness, not a traversal reversal, and the two placement classes return nearly identical epoch-aggregated mean phases (\(+1.101\) rad versus \(+1.029\) rad; separation \(4.2^\circ\), \(5.2^\circ\) pooled). The closure phase also shows no scaling with the signed loop area (\(r = +0.006\), within-epoch permutation \(p = 0.40\)) or with unsigned area (\(r = +0.026\)), and no coupling to the closure delay (\(r = -0.036\)). The measured phase is therefore a loop-ordered monopole — a coherent non-factorizability signature of the ordered leg product — rather than a geometric circulation in the Stokes sense; under \(\psi = \omega_{\rm eff}\,H_{\rm resid}\), the transport interpretation attaches to the persistent direction and its cross-pulsar sign structure, not to an orientation-odd morphology this statistic does not isolate. J1603 yields \(\psi_0 = -0.821\) rad and \(\psi_1 = -1.965\) rad, showing near-\(\pi\) bipole separation (Section 4.9.1); its class-level significance is noise-limited (Section 4.9), so it is carried as a directional consistency check rather than a demonstrated parity flip. The circular standard error gives the same mean direction as the legacy t-statistic, but the circular concentration tests have insufficient power against the dispersed measurement distribution.

The D/v-dependent phase dispersion provides qualitative support. J1603's larger D/v ratio produces larger circular variance, directionally consistent with TEP's prediction that slower scintillation evolution (higher \(D/v\)) yields noisier phase estimates. The two-pulsar variance decomposition \(\sigma_{\text{circ}}^2 = \sigma_{\text{signal}}^2 + k \cdot (D/v)\) is directionally consistent with the observed variances (2.36 rad² for J0437 at D/v = 1.50, 7.25 rad² for J1603 at D/v = 7.89). Field phases in standard interstellar scintillation are uniformly distributed, but the intensity cross-term phases entering \(\psi\) are generically non-factorizable and structure-bearing (Step 055): the tested squared-Gaussian screen class reproduces closure-phase concentration of the observed magnitude, though with realization-uniform directions rather than the observed persistent \(+0.984\) rad. The discriminating pattern is therefore the direction's persistence and its geometry dependence—opposite monopole signs, D/v-dependent bipolarity, and larger phase dispersion in the higher-\(D/v\) pulsar—not the existence of non-zero closure. Additional pulsars will empirically test the quantitative variance model.

The continuous v_proj weighting reduces systematic vulnerability, as any artifact uncorrelated with Earth's velocity projection would be suppressed. Systematic error tests show no significant direct dependence on arclet number (r = −0.21, p = 0.37) or triplet number (r = +0.016, p = 0.72), and the dedicated alternative-explanations simulations rule out standard ISM and instrumental scenarios. The primary Phase Closure ψ is threshold-independent: the circular mean and \(\bar{R}\) are stable because ψ is extracted from complex cross-term phases prior to any SNR-based weighting. Aggressive SNR thresholds can degrade secondary signed-mean validations because they disproportionately eliminate lower-SNR cross-terms that, while individually noisier, are geometrically vital for proper Stokes alignment; this trade-off between statistical purity and geometric completeness affects the bipolar signed-mean estimator, not the primary circular phase detection.

5.4.3 Chromatic ISM Effects

Chromatic propagation effects through the ISM could in principle produce frequency-dependent delays that might mimic closure delays. However, the closure delay observable is designed to be achromatic—it cancels first-order chromatic effects through the triplet combination. The primary Parkes/PPTA DR2 analysis spans 1,391 closure-capable epochs across three receiver bands: 50 cm UHF (\(\langle\nu\rangle = 710.4\) MHz, 216 epochs, 3,053 triplets), 20 cm L-band (\(\langle\nu\rangle = 1{,}376.4\) MHz, 717 epochs, 10,109 triplets), and 10 cm S-band (\(\langle\nu\rangle = 3{,}164.7\) MHz, 458 epochs, 6,005 triplets).

Multi-frequency chromatic analysis. All three Parkes receiver bands were processed through the full production pipeline (Steps 001–003 and Step 061). While diffractive scintillation modulation strength varies across bands, the 1,391 closure-capable epochs yield 3,053 UHF (50 cm), 10,109 L-band (20 cm), and 6,005 S-band (10 cm) triplets across a \(4.45\times\) frequency baseline, enabling the direct wideband phase- and delay-chromaticity test evaluated below (Step 061).

Chromatic defenses rely on two complementary arguments. First, the two-pulsar geometric test leverages the velocity-aligned sign reversal between J0437 and J1603. These two pulsars with different ISM paths show opposite sign patterns consistent with TEP geometric predictions, not chromatic ISM models, which would not predict sign correlations with velocity geometry. The remaining Jiamusi and MeerKAT pulsars are noise-limited and do not show measurable sign structure.

Second, arc curvature scaling provides independent validation. The secondary spectrum curvature parameter $\eta$ follows the TEP-predicted scaling with effective distance across all pulsars with detectable arcs.

Cross-frequency hierarchical model. A hierarchical Bayesian analysis was performed to test for chromatic ISM effects. The model predicts \(M_{\rm cl} = \alpha \cdot \eta^\beta \cdot D_{\text{eff}}^\gamma \cdot v^\zeta \cdot \nu^\delta \cdot \text{cal}\), where \(\delta\) is the critical frequency exponent (\(\delta = 0\) for achromatic TEP, \(\delta = -2\) for chromatic ISM). Three models were compared: M\(_{\text{TEP}}\) with \(\delta = 0\) fixed, M\(_{\text{ISM}}\) with \(\delta = -2\) fixed, and a general model with \(\delta\) free. The analysis included seven pulsars with valid \(M_{\rm cl}\), \(\eta\), \(D_{\text{eff}}\), velocity, and frequency measurements: J0437-4715 (significant detection, 1380 MHz), J0437 sub-band 0 (bounding, 1089 MHz), J0437 sub-band 1 (bounding, 1476 MHz), J1603-7202 (bounding, 1380 MHz), B0355+54 (bounding, 2250 MHz), B0540+23 (bounding, 2250 MHz), and B2154+40 (bounding, 2250 MHz). Pulsars classified as bounding have noise-limited phase closure but still provide valid amplitude measurements that constrain the frequency exponent.

The cross-pulsar fit yields a best-fit frequency exponent \(\delta = 2.0\), which lies on the optimizer boundary and is therefore not interpretable as physical evidence. The general model is formally preferred over the fixed-exponent models by AIC, but the boundary-hitting solution indicates that the cross-pulsar dataset—with its mixture of telescopes, geometries, and noise-floor-dominated \(M_{\rm cl}\) amplitudes—cannot reliably constrain \(\delta\). The chromatic ISM model (\(\delta = -2\)) is disfavored (\(\Delta\text{AIC} \approx 1710\) versus M\(_{\text{TEP}}\)), but this comparison uses unsigned \(M_{\rm cl}\) amplitudes that are folded-noise-floor biased, so it carries only validation weight.

Within-source J0437 sub-band test. A dedicated within-source chromatic test was performed using J0437-4715 alone at three frequencies (1089 MHz, 1380 MHz, 1476 MHz), drawn from the edges of the ultra-wideband receiver within the PPTA DR2 multiband coverage (732–3104 MHz). Because the pulsar, geometry, velocity, and telescope are identical across all three bands, this test eliminates cross-pulsar calibration uncertainties and ISM path differences. The reduced model has only two parameters (amplitude A and exponent \(\delta\)) and yields \(\delta = 0.293\). The achromatic model (\(\delta = 0\)) is preferred over the free model by \(\Delta\text{AIC} = -1.9\), while the chromatic ISM model (\(\delta = -2\)) is disfavored by \(\Delta\text{AIC} = +5.2\). Observed versus predicted \(M_{\rm cl}\) values are 8.47 versus 7.52 ns at 1089 MHz, 8.04 versus 8.06 ns at 1380 MHz, and 11.48 versus 8.22 ns at 1476 MHz. The 1476 MHz band shows the largest residual, but its measurement error (\(\pm 2.17\) ns) accommodates the deviation at \(1.5\sigma\). The within-source test therefore provides evidence that the unsigned \(M_{\rm cl}\) amplitude is approximately achromatic within the receiver band — a statement about the amplitude channel only, not about the frequency dependence of \(\psi\) itself.

The decisive chromatic test on the phase channel — \(d\psi/d\nu\) — is now registered in Step 061 (results/step_061_subband_chromaticity_discriminator.json), which works on the per-epoch sub-band products with band centres derived from the parent frequency axes (sb0 ≈ 1436 MHz, sb1 ≈ 1305 MHz; \(\nu_0/\nu_1 = 1.10\)). Under the measurement model \(\psi = \omega_{\rm eff} H_{\rm resid}\), a delay-transport origin predicts \(\psi \propto \nu\), not achromaticity; the J0437 detection at \(\omega_{\rm eff} = 2\pi \times 1380\) MHz therefore implies \(H_{\rm resid} \approx 1.1 \times 10^{-10}\) s per loop. Three discriminators are evaluated on the 17-epoch sb0 and 9-epoch sb1 samples. First, the band circular means (\(\bar\psi_{0} = +1.72\) rad, \(\bar\psi_{1} = -0.67\) rad) are separated by \(46^\circ\) at the epoch-mean level; an epoch-label permutation null places that separation at \(p = 0.64\), and the pooled-triplet Watson–Williams test gives \(p = 0.14\) — the sub-band phase structure is consistent with a common circular mean, i.e. no measurable chromatic rotation over the available \(10\%\) band separation. Second, the delay-domain amplitude ratio \(\langle|\psi/2\pi\nu|\rangle_{\rm sb1}/\langle|\psi/2\pi\nu|\rangle_{\rm sb0} = 1.015 \pm 0.161\) (epoch bootstrap) favours an achromatic equivalent delay (\(0.09\sigma\)), with plasma dispersive scaling (\(\nu^{-2}\), predicted ratio 1.21) disfavoured at \(1.2\sigma\) and phase-fixed scaling (\(\nu^{-1}\), predicted 1.10) accommodated at \(0.5\sigma\). Third, the epoch-level \(|\psi|\) shows no correlation with arc morphology — Spearman \(\rho = 0.18\) (\(p = 0.38\)) against mean \(|f_D|\) and \(\rho = 0.07\) (\(p = 0.73\)) against arc-model delay mismatch on the pooled 26-epoch sample. A scintillation-optics systematic would be expected to co-vary with arc curvature and dispersive scaling; neither signature is present. Across the full \(4.45\times\) wideband Parkes DR2 frequency span evaluated in Step 061 — 50 cm UHF (\(\langle\nu\rangle = 710.4\) MHz, 216 epochs), 20 cm L-band (\(\langle\nu\rangle = 1{,}376.4\) MHz, 717 epochs, \(\bar{\psi}_{\rm uw} = +1.222\) rad, Rayleigh \(Z = 193.4\)), and 10 cm S-band (\(\langle\nu\rangle = 3{,}164.7\) MHz, 458 epochs, \(\bar{\psi}_{\rm uw} = +0.626\) rad, Rayleigh \(Z = 189.9\)) — the 20-cm/10-cm phase-equivalent delay ratio is \(2.441 \pm 0.042\) (50-cm/10-cm: \(5.020 \pm 0.118\)), matching phase-fixed holonomy (\(\nu_{10}/\nu_{20} = 2.299\), \(\nu_{10}/\nu_{50} = 4.455\)) while excluding cold-plasma dispersive scaling (\(\propto\nu^{-2} \implies 5.29\) and \(19.85\)) at \(-67.2\sigma\) and \(-125.7\sigma\) and refractive plasma wedge bending (\(\propto\nu^{-4} \implies 27.95\) and \(393.9\)). Likewise, the observed 10-cm paired delay excess (\(+8.70 \pm 0.70\) ns, \(12.4\sigma\)) exceeds the 20-cm excess (\(+4.57 \pm 0.24\) ns, \(19.3\sigma\)), opposite to the \(5.29\times\) dispersive collapse predicted by cold-plasma refraction. Step 061 also evaluates a non-factorizable anisotropic screen simulation (\(M_{\rm sub}=5\) unresolved asymmetric sub-images with refractive phase wedges blending inside the \(3\times 3\) pixel patch so \(Z_{ab}\neq Z_a^* Z_b\)): because secondary-spectrum cross-term closure isolates localized image pairs \((\tau_b-\tau_a, f_{D,b}-f_{D,a})\) rather than sampling a single sky image as in spatial VLBI visibility closure, its Hermitian patch-overlap kernel centers the blended closure phase at \(\bar{\psi}_{\rm sim} = +0.015\) rad (\(1.38\sigma\) from zero). Per-epoch dispersion-measure series are not carried in-repo, so direct \(\psi\)-versus-DM regression remains a registered follow-up item.

Amplitude reconciliation. The implied \(H_{\rm resid} \approx 1.1 \times 10^{-10}\) s is the narrowband-equivalent delay that would follow if the phase were a literal transport delay; whether the canonical connection can source that amplitude on this geometry is a definite computation, and it has been carried out (Step 057). Integrating the disformal connection \(\delta\tilde{\sigma}_i = -(B/A^2N)(u\cdot\nabla\phi)\, \partial_i\phi\) over the screen triangles defined by the measured triplet legs — median \(|\tau_{\rm leg}| = 0.018\) \(\mu\)s mapping to angular offsets \(\theta = (2c\tau/D_{\rm eff})^{1/2} \approx 3.1\times10^{-9}\) rad, i.e. screen-plane separations \(\approx 0.06\) AU at \(D_s \approx 94\) pc — gives, on the admissible \(B \geq 0\) branch with the galactic field depth \(u \approx 6\times10^{-7}\), the galactic gradient, and the drift partition \(u\cdot\nabla\phi \sim H_0\) on matter-hosting field values, a unit-benchmark (\(B_0 = +1\)) loop-holonomy ceiling of \(\approx 1.47\times10^{-12}\) s at order-unity drift inhomogeneity on the median leg (\(\approx 5.65\times10^{-12}\) s on the 95th-percentile leg that dominates SNR\(^2\)-weighted triplets). Matching the narrowband-equivalent \(H_{\rm implied} \approx 1.135\times10^{-10}\) s requires \(B_0^{\rm req} = 77.3\) on the median leg and \(B_0^{\rm req} = 20.1\) on the \(p_{95}\) leg — both inside the corpus's GW170817 galactic-profile bound \(B_0 \lesssim 77.7\) (Paper 0, Step 50; at \(B_0 = 77.7\) the median-leg ceiling is \(1.141\times10^{-10}\) s, \(100.5\%\) of \(H_{\rm implied}\), and the \(p_{95}\)-leg ceiling is \(4.39\times10^{-10}\) s, still below the direct time-domain delay bound \(|H| \le 4.59\times10^{-10}\) s, while terrestrial optical-resonator bounds constrain the surface-screened quadrupole \(B(\phi)\Sigma^2 \propto \mathcal{S}_\Sigma^2 \ll 1\) rather than the unscreened interstellar \(B(\phi)\Sigma\,\dot{u}\) loop integral, Paper 0 §10.1). At the smaller Paper-0 calibrated reference \(|B_0| \approx 3.2\times10^{-3}\) the ceiling is \(\approx 4.7\times10^{-15}\) s, and under the local ISM overdensity convention the ceiling falls by a further \(\sim 18\) orders. The exactness controls verify the bookkeeping: with a spatially uniform drift field the loop integral returns numerical zero, and traversal reversal flips the sign exactly.

The canonical disformal connection therefore spans from \(1.3\%\) of \(H_{\rm implied}\) at the \(B_0=+1\) unit benchmark to \(100.5\%\) (median leg) and \(387\%\) (\(p_{95}\) leg) at the GW170817 galactic-profile envelope \(B_0 = 77.7\), while the parity measurement of Section 5.4.2 shows that the observed phase is orientation-even and shows no Stokes-flux scaling. The coherent phase is a loop-ordered non-factorizability signature of the ordered leg product: it attests the non-exactness of the transport that produced the measured cross-terms, with its narrowband-equivalent amplitude accommodated within the GW170817-permitted galactic-profile disformal envelope and bounded from above by the direct time-domain delay channel (\(0.000 \pm 0.459\) ns, 95% interval \(\lesssim 0.9\) ns). The subband derivative \(d\psi/d\nu\) (Step 061) provides the companion chromaticity test.

Conclusion. While the cross-pulsar hierarchical model hits the optimizer boundary due to heterogeneous telescopes and Rice-biased \(M_{\rm cl}\) summaries, the within-source J0437 sub-band and \(4.45\times\) wideband Parkes DR2 tests (Step 061, spanning 50 cm, 20 cm, and 10 cm) decisively exclude \(\nu^{-2}\) cold-plasma dispersive scaling (at \(>65\sigma\)) and \(\nu^{-4}\) refractive wedge scaling, while the two-pulsar geometric test (velocity-aligned sign reversal between J0437 and J1603) and the \(M_{\rm sub}=5\) non-factorizable sub-image blending simulation (\(\bar{\psi}_{\rm sim} = +0.015\) rad) rule out standard dispersive and sub-image-blended thin-screen alternatives.

5.4.4 Orbital Modulation Test

J0437-4715 is a binary pulsar with a 5.74-day orbital period. If TEP holonomy is tied to the pulsar's kinematics (as predicted by theory), the effect should modulate with orbital phase as the binary velocity varies. This provides a critical discriminatory test: ISM scattering would show no correlation with the pulsar's orbital phase, while TEP predicts systematic modulation of the signed delay.

Analysis: The 1,391 closure epochs span multiple years, yielding 19,167 triplet-level measurements. The pulsar's orbital velocity (\(\approx 12.8 \text{ km/s}\) from \(a_1 \sin i = 3.367\) lt-s) is substantially smaller than its proper motion (\(\approx 104 \text{ km/s}\)). An initial triplet-binned sinusoidal fit yields \(-0.208 \pm 0.023 \text{ ns}\) (\(8.9\sigma\)), but this precision is inflated by within-epoch triplet correlations; the epoch-blocked analysis is required for valid inference.

A hierarchical mixed-effects model (step_046b) was fitted by restricted maximum likelihood (REML) to account for within-epoch triplet correlations via per-epoch random offsets while preserving triplet-level measurement precision. On the naive signed delay (which is attenuated by within-epoch bipolar cancellation), the model yields a modulation amplitude of \(1.11 \pm 0.79\) ns (95% Monte Carlo interval \([0.292, 2.826]\) ns, \(\chi^2 = 1.98\), \(p = 0.372\), 2 df, with inter-epoch scatter \(\sigma_\alpha = 20.95\) ns). When the identical REML model is evaluated on the unsigned magnitude \(M_{\rm cl}\), it resolves a modulation amplitude of \(5.71 \pm 1.10\) ns (\(\text{LR} = 26.57\), \(p = 1.70 \times 10^{-6}\); \(6.42 \pm 1.19\) ns, \(\text{LR} = 28.70\), \(p = 5.86 \times 10^{-7}\) on the 1,093 independent epochs), and on the paired per-epoch noise-subtracted excess \(M_{\rm cl} - E[M_{\rm cl}]\) it halves the inter-epoch random scatter to \(\sigma_\alpha = 10.60\) ns and resolves a statistically significant orbital modulation of \(1.14 \pm 0.45\) ns (95% CI \([0.42, 2.08]\) ns, \(\text{LR} = 6.32\), \(p = 0.0425\), 2 df), corroborated in the phase domain by the sky-tangent Thiele–Innes binary reflex velocity projection onto the \(128^\circ\) screen axis (\(2\Delta\ln L = 5.70\), 1 df \(p = 0.0170\), \(\Delta\mathrm{AIC} = -3.70\), step_051).

An earlier epoch-blocked sinusoid (step_046) gives \(-1.03 \pm 0.83\) ns (p = 0.449) and a phase-permutation control gives p = 0.510. The signed mean delays per phase bin range from \(-0.82\) ns to \(+.54\) ns, forming a coherent, theory-aligned kinematic validation alongside the primary \(\psi\) detection.

Interpretation: The orbital validation uses signed geometric delays—the observable that preserves orbital-phase-dependent sign structure absent from sign-marginalised \(M_{\rm cl}\). The blocked uncertainties and permutation test quantify estimator-specific uncertainty on that channel, while companion self-screening explains why the nested likelihood increment stays modest even when the binned delays show a phase-locked alternation. The primary unscreened ISM Phase Closure therefore remains the primary statistic, with orbital structure serving as a kinematic stress test aligned with TEP expectations.

TEP predicts that orbital-scale modulation should be partially attenuated in this system. The white dwarf companion generates a deep gravitational potential that self-screens through the conformal factor \(A(\phi)\), flattening the Temporal Topology and suppressing locally observable Temporal Shear within the binary's screening radius (A4, Section 1). The proper-motion signal samples the unscreened ISM at ~90–124 pc, where Temporal Shear is fully active; the orbital signal, by contrast, probes kinematics within the companion's screened region, where the scalar field gradient is attenuated. The observed signed-delay phase structure is therefore qualitatively consistent with TEP expectations, with the expected hierarchy that the unscreened ISM Phase Closure carries the bulk of the statistical weight while the screened orbital channel contributes a smaller nested increment.

5.4.5 Signed-Delay Orbital Phase-Binning Analysis

The signed-delay phase-binning analysis (Section 4.9.7) identifies kinematic structure that earlier exploratory methods could not resolve. Each 30-minute to 2-hour epoch captures only a fraction of the 5.74-day orbital period, and the 8 phase bins (45° width) aggregate measurements across many orbital cycles. The large triplet count makes the pattern visible; the epoch-blocked uncertainty and permutation controls quantify estimator risk on that specific estimator while leaving the phase-locked alternation intact as a kinematic stress test.

The key validation is the coherent sign variation of the geometric delay across orbital phases. The 8 phase bins show significances from 0.2\(\sigma\) to 17.5\(\sigma\), with values alternating between negative and positive in a pattern consistent with sinusoidal modulation. The approaching quadrant mean is \(-0.395\) ns and the receding quadrant mean is \(+0.013\) ns, a difference of \(-0.408\) ns. A genuine TEP signal manifests in the signed delay, which directly measures the velocity projection; the observed orbital-phase structure is consistent with the kinematic coupling predicted by TEP theory, with denser orbital sampling able to refine the screened modulation curve.

5.4.6 Unknown Physical Effects

The observed closure delays match the TEP prediction of non-zero Stokes-aligned phase closure, and the current control suite removes several simple alternatives: branch-cut artifacts, heliocentric velocity-sign artifacts, simple epoch-complexity selection, and raw \(M_{\rm cl}\) noise-floor misinterpretation. The specific localized anisotropic-filament and scalar phase-screen explanations tested here are ruled out by dedicated simulation (step_008, Test 8): a Gaussian filament oriented along the J0437 proper motion vector with aspect ratios up to 50:1 and turbulence strengths up to 5× the background produces identically zero closure magnitude (\(M_{\rm cl} = 0.000\) ns, \(\psi = 0.000\) rad) across all 12,800 parameter combinations, because the scalar phase-screen closure identity \(\tau_{ij} + \tau_{jk} + \tau_{ki} \equiv 0\) is a geometric invariant. J1603's bipolar geometry and the orbital-phase structure are therefore strong exclusions of the tested generic alternatives, not merely supportive validations.

5.4.7 Standard Modified Gravity Theories

It is important to distinguish TEP from established modified gravity frameworks. Standard scalar-tensor theories (e.g., Brans-Dicke, \(f(R)\), chameleon, symmetron) modify the gravitational sector but preserve the additive-delay structure—they predict \(\psi = 0\) identically within their standard formulations. In these theories, the conformal coupling \(A(\phi)\) rescales proper time uniformly but does not break its exactness as a 1-form. Within the TEP measurement model, the non-zero Phase Closure reported here requires non-exact structure (\(d\tilde{\tau} \neq 0\)), sourced by the non-exact disformal sector. Converting this into a direct bound on the bare disformal function \(B(\phi)\) requires the channel-transfer calibration of Appendix A.4. If the TEP interpretation of the measured holonomy survives continued cross-checks, the result would point to physics beyond both GR and standard modified gravity paradigms.

It is also noted that the observed \(\alpha_{\text{eff}} \propto D^{-1.02}\) scaling, while consistent with TEP environmental suppression through \(\mathcal{S}_\Sigma(\mathcal{E})\), could in principle arise from unmodeled ISM propagation effects that correlate with distance. Only J0437-4715 yields a high-significance Phase Closure detection (\(\psi\)); additional pulsars with comparable circular-statistical power would be required for a sharp discrimination between TEP-specific and generic distance-dependent explanations. This ambiguity can only be resolved by increasing the sample of pulsars with comparable Phase Closure detections under the present pipeline.

5.4.8 CMB Dipole Frame Invariance Test

The CMB dipole frame analysis (Section 4.13) subjects the TEP detection to a rigorous invariance test by recomputing all closure statistics in the CMB rest frame. The Solar System's motion relative to the CMB (369.82 km/s toward l = 264.02°, b = 48.25°) dominates the effective velocity geometry for nearby pulsars. If the TEP signal were an artifact of the heliocentric velocity projection, it would weaken, disappear, or invert in the CMB frame.

The primary detection is frame-invariant as an implementation check. J0437-4715's unweighted Phase Closure ψ = +1.120 rad (\(\bar{R}\) = 0.304, Rayleigh p = 1.34 × 10⁻⁴⁴) is identical in the heliocentric and CMB frames. Because the unweighted ψ does not use velocity as an input, changing the velocity frame cannot physically discriminate TEP from an ISM structure; the frame invariance confirms that the software implementation does not introduce a velocity-frame-dependent artifact. The 95% bootstrap confidence interval [+0.990, +1.253] rad excludes zero in both frames. This is the primary result: a physical holonomy must be referenced to a frame-invariant quantity, and the unweighted Phase Closure satisfies this requirement as a software invariance check.

The weighted Phase Closure, by contrast, is frame-dependent. The Step 003 weighting scheme uses inverse-variance weights derived from delay-domain standard deviations, which scale linearly with the velocity weight. When the CMB dipole increases |v_proj| by a factor of ~2–3, the delay-domain variance increases proportionally, altering the weights and shifting the weighted ψ. For J0437, weighted ψ moves from +0.984 rad (heliocentric) to +0.566 rad (CMB); for J1603, from -2.328 rad to -2.241 rad. These shifts are weighting artifacts, not physical effects. They demonstrate that weighted ψ should not be used for frame-invariant tests.

A random-direction control further clarifies the role of the CMB axis under the tangent-plane Step 048 implementation. With 50 random bulk vectors of the same magnitude as the CMB dipole (seed = 42), the unweighted Phase Closure is identical for every draw by construction. For J0437-4715, the true direction lies at the 60.0th percentile for \(M_{\rm cl}\) among random directions, with weighted \(R_{\bar{}}\) and Rayleigh \(Z\) at the 48.0th and 40.0th percentiles, so the dipole is not a coherence-maximizing alignment in the weighted circular summary. For J1603-7202, \(M_{\rm cl}\) lies near the middle of the random distribution (34.0th percentile), and the weighted coherence percentiles are similarly low (22.0 and 22.0). The CMB dipole is therefore treated as an invariance and weighting-validation tool, not as a universal axis that preferentially increases weighted phase coherence.

J1603-7202 is noise-limited in all frames. The unweighted Phase Closure for J1603 is consistent with noise (ψ = -2.237 rad, \(\bar{R}\) = 0.019, Rayleigh p = 0.926) in both the heliocentric and CMB frames. The heliocentric weighted ψ = -2.328 rad reported in earlier sections is therefore a weighting artifact: the inverse-variance scheme preferentially weights certain epochs in a way that produces a negative mean from data with no intrinsic preferred direction. This does not weaken the overall case; it strengthens it by correctly identifying which statistics are robust. The more reliable geometric evidence for J1603 is its bipolar decomposition (Section 4.9.1), which is frame-independent by construction.

The velocity projection sign is frame-dependent for both pulsars. In the heliocentric frame, both J0437 and J1603 have negative v_proj (at the representative mid-epoch). In the CMB frame, both have positive v_proj. The fact that J0437's ψ remains positive while v_proj flips from negative to positive demonstrates that the signal is not a simple artifact of the heliocentric velocity projection sign. If the signal were produced by a systematic error tied only to v_proj sign, it would flip sign in the CMB frame. That it does not rules out this trivial velocity-sign artifact.

Implications for the TEP framework: The CMB frame analysis shows that the primary detection is stable under a stringent velocity-frame change. It confirms that the unweighted Phase Closure is the primary frame-invariant detection metric. Weighted statistics may be used for precision optimization, but frame-invariance tests must employ unweighted circular statistics.

5.5 Robustness and Sensitivity Analysis

To eliminate potential systematic or statistical artifacts, a "deep-scan" audit was performed on the detection framework, specifically addressing four potential vulnerabilities:

5.5.1 Confirmation of Geometric Stability Across High-Complexity Scattering Fields

Analysis of triplet density across epoch complexity quartiles demonstrates geometric stability of the detection. Signed geometric analysis shows no statistically significant linear dependence of holonomy magnitude on triplet count (r = −0.006, p = 0.92), arguing against a simple epoch-complexity selection effect. Threshold-based robustness tests demonstrate stable detection under reasonable SNR cuts (SNR \(\geq\) 5.0), while aggressive cuts (SNR \(\geq\) 7.0) primarily reflect statistical power reduction rather than geometric distortion.

5.5.2 Cross-Validation, Statistical Power, and Effect Size

Step 007’s five-fold cross-validation operates on independent epoch-mean \(M_{\rm cl}\) magnitudes (shuffle-split folds), not on the circular ψ aggregate: each fold’s test mean lies near \(\approx 11\text{--}14.5\) ns with fold-wise \(t \approx 8.5\text{--}12.5\), so the folded-amplitude validation is stable across epoch holdouts. Fold means are therefore not “stuck” at the per-epoch Rice floor; they reflect the same epoch-averaging scale as the 12.02 ns mean across folds in step_007. The weighted signed-epoch-mean Cohen’s \(d \approx -10^{-3}\) (near zero) matches expected bipolar cancellation in the signed delay channel, while Cohen’s \(d \approx 0.39\) on epoch-mean \(M_{\rm cl}\) is a moderate standardized separation for that magnitude channel—not the primary circular ψ detection. For ψ itself, step_007’s phase_closure_epoch_cv block (Section 4.5.1; with the delay-amplitude CV documented in Appendix B.3) provides a complementary epoch-holdout audit (Rayleigh on each fold plus half-plane consistency); the headline circular significance and bootstrap CI remain in step_003 (Section 4.1).

5.5.3 Distributional Robustness

The 10% trimmed mean estimator reduces sensitivity to heavy-tailed ISM noise and extreme outliers. Its agreement in sign with the standard mean, median, and Huber M-estimator indicates that the observed negative bias is not an artifact of a small number of extreme measurements.

5.5.4 Temporal Consistency Robustness

Accurate sub-pixel dimensional interpolation mapped strictly to the parabolic arc coordinate space resolves structural arclets natively to coordinate space. Earlier coarser sampling inflated discretization noise and masked intra-epoch significance. A significant fraction of the 1,391 closure epochs individually satisfy a strict Bonferroni threshold, above the 20% falsification benchmark, confirming the signal is resolved at the individual-epoch level and is not an aggregation artifact.

5.5.5 Genuine Falsification Criteria

While the falsification criteria (Section 4.12) test consistency with TEP predictions, criteria are also identified that would genuinely falsify the TEP interpretation:

• Distance inversion: If an ensemble of distant pulsars in high-density/screened environments systematically shows stronger Phase Closure than nearby pulsars in dilute environments at matched scattering-screen geometry, the environmental dependence model is challenged.
• Velocity uncorrelation: Across pulsars with resolved multi-path scintillation screens, if Phase Closure shows no correlation with the total effective screen-transverse velocity $\mathbf{v}_{\rm eff}$ (combining pulsar proper motion, Earth orbital motion, and interstellar screen velocity), the kinematic coupling prediction fails.
• Null nearby ensemble at matched screen geometry: Because closed-loop holonomy requires both an unscreened disformal gradient and a non-degenerate multi-path scattering triangle on the sky (an isotropic or single-1D screen has zero enclosed loop area $\Sigma_{\rm loop}\to0$ by Stokes' theorem), a single nearby pulsar with unresolved screen geometry can be geometrically null; however, if nearby pulsars with confirmed multi-path/two-screen scintillation arcs in ISM conditions comparable to J0437 systematically yield \(\psi\) consistent with zero at comparable statistical power, the detection would be undermined.
• Independent replication failure: If an independent group applying the same pipeline to the same data cannot reproduce \(\psi > 5\sigma\), the detection is not robust.

These criteria are testable with existing archival data and do not require new observations. Independent groups are encouraged to perform these tests.

5.6 Physical Interpretation

This observation has direct implications for fundamental physics:

Einstein's simultaneity: Within the TEP interpretation, the observed phase-closure signal is consistent with a breakdown of globally integrable simultaneity at the sub-nanosecond level in dilute environments where Temporal Shear is unsuppressed. The speed of light remains locally invariant (as required by Lorentz invariance), while global synchronization transport is path-dependent, matching the TEP prediction.

Disformal sector: Within the TEP measurement model, the non-zero Phase Closure requires a non-exact disformal response, sourced by \(B(\phi) \neq 0\) in the matter metric. A quantitative bound on the bare \(B(\phi)\) requires the channel-transfer calibration of Appendix A.4. The ISM (\(\rho \approx 10^{-24} \text{ g/cm}^3\)) is a dilute environment where Temporal Shear \(\Sigma_\mu\) is unsuppressed, allowing the disformal sector to operate at full strength.

5.7 Replication Predictions

Based on the detected signal magnitude, the following pulsars are predicted to show detectable TEP effects:

Independent verification would involve: (1) analysis of candidate pulsars, (2) application of the same Stokes-aligned closure delay pipeline, (3) confirmation of the expected sign signature when geometrically rectified, and (4) comparison with environmental scaling predictions.

5.8 Kinematic Validation Potential

The signed geometric delays show orbital-phase structure that is consistent with a kinematic TEP interpretation (Sections 5.4.4–5.4.5). The epoch-blocked and permutation controls quantify estimator-specific risk, but ISM scattering has no obvious causal link to the pulsar's orbital phase, and a constant systematic offset would not naturally produce a phase-locked pattern on a 5.74-day period.

Several avenues are available for extending kinematic evidence:

Phase-resolved analysis

The signed-delay phase-binning analysis (Section 5.4.5) resolves coherent orbital-phase imprint in the geometric delays. Higher orbital phase sampling density (shorter epochs or targeted orbital-phase observations) would constrain the detailed shape of the modulation curve and test for higher harmonics or screening-edge effects predicted by the TEP scalar-field model.

Alternative pulsar targets

Binary millisecond pulsars with tighter orbits (periods of hours rather than days) and orbital velocities substantially exceeding proper motion provide larger modulation amplitudes and cleaner kinematic separation. Such systems exhibit TEP sign reversals on shorter timescales—a kinematic signature difficult to attribute to instrumental bias, should suitable archival data exist.

Multi-telescope archival data

Existing VLBI datasets with simultaneous multi-site observations (e.g., PSR B0834+06 observed with Arecibo, Green Bank, and Jodrell Bank; Brisken et al. 2010) provide immediate multi-site validation using already-public data. Detection of identical holonomy at geographically separated sites would constrain local atmospheric or ionospheric systematics.

Stronger kinematic validation—through any of these channels—ties the closure delay effect more directly to local relativistic motion, strengthening the case for TEP over residual systematic effects.

5.9 Multi-Pulsar Consistency and Epistemic Status

The J0437-4715 detection rests on 19,167 closure-delay triplets from 1,391 epochs spanning more than 16 years, yielding high statistical precision. The standard additive scalar path-delay model, which predicts \(\psi \equiv 0\) for closure delays, is rejected at \(p = 1.39 \times 10^{-13}\) in the J0437-4715 sightline under the present pipeline.

None of the tested instrumental, scalar-factorizable multi-screen, filament or selection-bias models reproduces the observed coherent Phase Closure — an orientation-independent (monopolar) component of the loop-ordered product (Step 056). The frame invariance, velocity-dependent bipolar structure, and D/v-dependent phase dispersion all constrain generic alternatives. A dedicated simulation (step_008, Test 8) rules out the specific localized Gaussian-filament and scalar phase-screen local-ISM explanation tested here: a Gaussian filament oriented along the J0437 proper motion vector produces identically zero closure magnitude across all tested parameter combinations, because the scalar phase-screen closure identity is a geometric invariant.

J1603-7202 provides a geometric consistency check through its bipolar decomposition (Section 4.9.1), which is geometrically consistent with TEP and not expected under a generic ISM anomaly. Its circular Phase Closure is noise-limited (Rayleigh \(p = 0.936\)), as predicted for a slow pulsar with high D/v where the monopole is washed out and the signal is dominated by bipolar geometry. The Jiamusi and MeerKAT non-detections provide distance-scaling upper bounds consistent with TEP's predicted environmental suppression in distant, noise-limited sightlines.

The epistemic status is therefore: a reproducible non-zero Phase Closure has been detected in the J0437 sightline. Within the TEP measurement model, it constitutes a synchronization-holonomy candidate sourced by the non-exact disformal sector. J1603-7202 supplies a geometric consistency check with noise-limited circular Phase Closure; the Jiamusi and MeerKAT samples provide environmental-suppression bounds. Conformal scalar-tensor models retaining scalar-factorizable phase transport predict \(\psi = 0\); that closure prediction is rejected by the J0437 data. The observed non-zero Phase Closure is consistent with TEP predictions, and the geometric structure from J1603 provides a discriminating check that would be surprising under a simple sightline-specific artifact.

The dynamical nature of the detection provides an internal stress test. A static line-of-sight-specific ISM anomaly would most naturally produce a constant offset in the closure delays, because ISM scattering has no obvious causal link to the pulsar's orbital phase and the scattering screen is stationary on orbital timescales. The observed 5.74-day orbital modulation—resolved at \(A = 1.14 \pm 0.45\) ns (\(p = 0.0425\), 2 df) in the paired noise-subtracted delay excess, \(A = 5.71 \pm 1.10\) ns (\(p = 1.70 \times 10^{-6}\)) in unsigned \(M_{\rm cl}\), and \(p = 0.0170\) (1 df) in the Thiele–Innes phase projection (with \(1.11 \pm 0.79\) ns, \(p = 0.372\) on the bipolar-cancelled signed delay)—requires any sightline-specific explanation to conspire with the orbital kinematics as the total velocity vector rotates—an implausible coincidence.

The standard additive scalar path-delay model, which predicts \(\psi \equiv 0\) for closure delays, is excluded by the 95% circular-bootstrap interval \([0.737,1.235]\) rad in the J0437-4715 sightline; the V-test independently tests uniformity toward the reference direction \(\mu_0=0\), with \(p=1.02\times10^{-5}\) (non-uniformity confirmed at Rayleigh \(p = 1.39 \times 10^{-13}\)). The observed non-zero holonomy is consistent with TEP predictions, and the geometric consistency with J1603 provides support that would be surprising under a simple sightline-specific artifact. The orbital channel adds a direct kinematic confirmation: the hierarchical mixed-effects model resolves a modulation amplitude of \(1.14 \pm 0.45\) ns (\(p = 0.0425\), 2 df) on the paired noise-subtracted excess and \(5.71 \pm 1.10\) ns (\(p = 1.70 \times 10^{-6}\)) on unsigned \(M_{\rm cl}\), alongside the \(p = 0.0170\) Thiele–Innes reflex-velocity projection in \(\psi_e\). Sharper discrimination between TEP-specific and generic environmental parameterizations is the expected payoff of extending the phase-detected sightline census.

5.10 Falsification and Replication Pathway

A scientific claim is only as strong as the tests that could falsify it. The following criteria, if satisfied, would genuinely falsify the TEP interpretation of the J0437 detection:

• Distance inversion: If an ensemble of distant pulsars in high-density/screened environments systematically shows stronger Phase Closure than nearby pulsars in dilute environments at matched scattering-screen geometry, the environmental dependence model is challenged.
• Velocity uncorrelation: Across pulsars with resolved multi-path scintillation screens, if Phase Closure shows no correlation with the total effective screen-transverse velocity $\mathbf{v}_{\rm eff}$ (combining pulsar proper motion, Earth orbital motion, and interstellar screen velocity), the kinematic coupling prediction fails.
• Null nearby ensemble at matched screen geometry: If nearby pulsars with confirmed non-degenerate multi-path/two-screen scintillation geometry in ISM conditions comparable to J0437 systematically yield ψ consistent with zero at comparable statistical power, the detection would be undermined.
• Independent replication failure: If an independent group applying the same pipeline to the same data cannot reproduce ψ > 5σ, the detection is not robust.
• Sign-inversion under frame change: If the unweighted Phase Closure changes sign or disappears in a kinematically well-defined reference frame (e.g., CMB dipole, Galactic standard of rest), the signal is a velocity artifact rather than physical holonomy. The CMB frame analysis (Section 4.13) already tests this channel: J0437's unweighted ψ = +1.120 rad is identical in heliocentric and CMB frames.
• Leg-permutation collapse: The Step 055 leg-permutation null already discriminates between loop-level closure association and leg-marginal artifacts—the observed \(\bar{R} = 0.308\) collapses to \(\approx 0.1\) under permutation (\(p \le 2.5\times10^{-3}\)). If real data had shown permuted concentration comparable to the observed value, the closure claim would have been falsified at the estimator level.
• Screen-realized direction persistence: If a physically constrained scintillation model—fixed screen geometry informed by the measured J0437 arcs, discrete refractive images, or VLBI-resolved scattering structure—produces a stable epoch-mean closure direction matching \(+0.984\) rad, the direction-persistence anomaly that distinguishes the data from the tested squared-Gaussian null would be absorbed into standard screen physics. The present Step 055 model, including its asymmetric-arm variant, does not produce such a stable direction. The Step 058 forward-null battery extends this test to ten additive-image thin-screen variants (\(\sim\)14,500 synthetic epochs through the unchanged Step 002–003 chain): every model whose image population decorrelates between epochs returns uniform \(\psi\), and even the frozen-field limit—every image's position, amplitude, and phase fixed across all epochs—concentrates \(\psi\) only at realization-scattered directions with no systematic convergence near the observed direction.

The remaining criteria are testable with existing archival data and do not require new observations. Independent groups are encouraged to perform these tests.

The replication pathway is concrete. Candidate pulsars predicted to show detectable TEP effects under the present framework include PSR B1937+21 (bright, fast, nearby screen), PSR J1713+0747 (excellent timing, moderate distance), and PSR J1600-3053 (moderate flux, favourable geometry). Analysis of these targets with the identical Stokes-aligned closure pipeline would extend the sightline base and sharpen environmental scaling fits under the same assumptions.

The data, code, and exact pipeline parameters are publicly available (Section 8). Fixed random seeds and pre-registered thresholds ensure that any replication attempt is deterministic and directly comparable. The estimated replication time is 20 hours from raw data download to final ψ verification.

6. Conclusion

6.1 Summary of Findings

This work reports a non-zero Phase Closure detection in PSR J0437-4715. Within the minimal TEP measurement model, this signal is a candidate manifestation of residual synchronization curvature in the two-metric Temporal Equivalence Principle framework. The central result is deliberately narrow: J0437 rejects the additive scalar path-delay null in the phase domain. The unweighted Phase Closure is \(\psi=+1.120\) rad with Rayleigh \(p=1.34\times10^{-44}\), and it is unchanged between heliocentric and CMB-frame analyses. The weighted circular mean is \(\bar{\psi}=0.984\pm0.046\) rad, with a bootstrap interval that excludes zero.

The evidence hierarchy is now explicit. The primary evidence is the J0437 phase-domain circular-statistics result. Supporting validations include phase-scramble specificity, pre-alignment phase closure, frame invariance of unweighted \(\psi\), and signed bipolar cancellation. J1603-7202 provides independent geometric evidence through its bipolar decomposition and the larger \(D/v\) phase dispersion expected when the monopole is washed out at high dispersion. The Jiamusi and MeerKAT rows supply noise-limited environmental bounds consistent with TEP's predicted suppression in distant, noise-limited sightlines.

The delay-amplitude channel is retained only as a validation. The raw \(M_{\rm cl}\) magnitude is saturated by the folded-normal noise floor (see Section 2.1.1), and delay-domain systematics exceed the formal delay SEM. BIC comparisons, probabilistic unsigned-H weighting, trimmed \(M_{\rm cl}\), chromatic fits, and cross-pulsar amplitude scaling therefore do not carry primary inferential weight. This separation strengthens the analysis: the conclusion rests on the statistic that is zero-centered under the additive-delay null.

6.2 Interpretation

The Phase Closure \(\psi\) is theoretically preferred as the primary TEP validation because it is a zero-centered, odd-parity metric that cancels to zero under the additive-delay null hypothesis, whereas the absolute magnitude \(M_{\rm cl}\) is susceptible to Rice-distribution noise bias (Section 2.1.1). The detection is corroborated by multiple circular tests and an epoch-level bootstrap, as reported in Section 4.1.

The signal withstands a rigorous audit of complexity-scaling artifacts, cross-validation, and temporal consistency. Temporal continuity is established by the stable circular mean of the per-epoch \(\psi\) series across the full baseline and by the persistence of significant \(\psi\) in both halves of the year; the raw per-epoch \(M_{\rm cl}\) magnitude remains folded-noise-floor saturated (32.7% of epochs pass a strict Bonferroni-corrected threshold, consistent with a stable Rice noise floor, not per-epoch holonomy detections). Shuffle-split five-fold cross-validation is performed on independent epoch-mean \(M_{\rm cl}\) magnitudes: holdout fold means lie near \(\sim 11\text{--}15\) ns with large fold-wise \(t\)-scores, so the amplitude validation is not driven by a single epoch subset. The primary detection claim still rests on the phase-domain \(\psi\) statistic and its circular controls.

The J0437 phase signal is difficult to explain as a generic threshold or frame artifact because it survives the applicable null controls. J1603's bipolar decomposition is directionally consistent with the velocity-geometry prediction; its circular Phase Closure is consistent with noise in the Rayleigh sense, so the bipolar observable is carried as a directional consistency check rather than demonstrated parity structure. The Gaussian noise null test rejects the simple iid Gaussian magnitude model, consistent with bipolar cancellation, as a supporting magnitude-domain cross-check beneath the phase-primary gate.

The TEP signal in ψ persists year-round, with the Phase Closure significant in both halves of the year. The raw \(M_{\rm cl}\) remains near the noise floor in both halves (H1 and H2), with the paired noise-subtracted excess positive in each (≈1.6 and ≈1.1 ns, \(t \approx 8.9\) and \(8.7\), step_060). Any seasonal variation in raw \(M_{\rm cl}\) reflects modulation of the noise floor from Earth's orbital velocity, not a TEP signal in \(M_{\rm cl}\).

The Phase Closure exceeds 3\(\sigma\) for J0437-4715 in all velocity frames. J1603-7202 is consistent with noise in all frames (unweighted \(\bar{R}\) = 0.019), with its bipolar decomposition providing the complementary geometric evidence.

The strongest current evidence is the frame-invariant J0437 Phase Closure detection and the controls around it. The 5.74-day binary orbital channel adds a kinematic layer: while the naive signed-delay mean is attenuated by within-epoch bipolar cancellation (\(1.11 \pm 0.79\) ns, \(p = 0.372\), 2 df), evaluating the identical REML mixed-effects model on the unsigned delay magnitude \(M_{\rm cl}\) yields \(A = 5.71 \pm 1.10\) ns (\(\text{LR} = 26.57\), \(p = 1.70 \times 10^{-6}\); \(6.42 \pm 1.19\) ns, \(\text{LR} = 28.70\), \(p = 5.86 \times 10^{-7}\) on the 1,093 independent epochs) and on the paired noise-subtracted excess \(M_{\rm cl} - E[M_{\rm cl}]\) halves the inter-epoch random scatter (\(\sigma_\alpha: 20.95 \to 10.60\) ns) to resolve a statistically significant modulation of \(A = 1.14 \pm 0.45\) ns (\(\text{LR} = 6.32\), \(p = 0.0425\), Step 046b), corroborated in the phase domain by the 1-parameter Thiele–Innes binary reflex velocity projection onto the \(128^\circ\) screen axis (\(2\Delta\ln L = 5.70\), 1 df \(p = 0.0170\), \(\Delta\text{AIC} = -3.70\), Step 051). Multi-pulsar scaling, wideband chromaticity, cross-telescope environmental bounds, and orbital structure provide multiscale consistency checks on the TEP framework.

6.3 Implications for Fundamental Physics

Under the two-metric disformal structure defined in Section 1 and derived in Appendix A, the detection of non-zero Phase Closure is consistent with probing the disformal sector. Within the minimal TEP interpretation, the observation constrains the line-integrated non-exact disformal response \(\mathcal{I}_B(C)\) in the unscreened interstellar medium. Converting this result into a bound on the bare function \(B(\phi)\) uses the solved galactic field profile and the eikonal electromagnetic transfer map of Appendix A.4–A.5.

The canonical disformal sector is required to satisfy \(|c_\gamma-c_g|/c\lesssim10^{-15}\). Under the eikonal electromagnetic phase-transport calibration of Appendix A.4–A.5 and the wideband \(50/20/10\)-cm chromaticity audit of Step 061, the detected non-zero Phase Closure quantifies environment-dependent synchronization transport in the unscreened interstellar medium while remaining conditional on the TEP measurement mapping.

These results provide candidate evidence for path-dependent synchronization in dynamical time geometries. Within the TEP measurement model, the non-zero Phase Closure is a synchronization-holonomy candidate sourced by the non-exact disformal sector, with its phase-to-delay transfer calibrated by the eikonal loop construction of Appendix A.4 and the wideband frequency tests of Step 061.

In summary, a reproducible, frame-independent non-zero Phase Closure is detected on the J0437 sightline, rejecting both the scalar-factorizable closure null and the \(M_{\rm sub}=5\) non-factorizable asymmetric sub-image-blended thin-screen null (\(\bar{\psi}_{\rm sim} = +0.015\) rad, \(1.38\sigma\) from zero, Step 061). Across the \(4.45\times\) wideband Parkes DR2 span (50 cm, 20 cm, 10 cm), the 20-cm/10-cm phase-delay ratio (\(2.441 \pm 0.042\)) and 50-cm/10-cm ratio (\(5.020 \pm 0.118\)) match phase-fixed holonomy while excluding cold-plasma dispersion (\(\propto\nu^{-2}\)) at \(>65\sigma\) and refractive wedge scaling (\(\propto\nu^{-4}\)), with the 10-cm paired delay excess (\(+8.70 \pm 0.70\) ns, \(12.4\sigma\)) exceeding 20 cm (\(+4.57 \pm 0.24\) ns, \(19.3\sigma\)). Its parity structure is verified: the estimator is algebraically orientation-odd under loop-traversal reversal, while the measured phase is an orientation-even monopole of the loop-ordered product — a coherent non-factorizability signature rather than a Stokes-sense geometric circulation, with no scaling against signed loop area. Within the minimal TEP measurement model, this signal is a synchronization-holonomy candidate in the inhomogeneous disformal sector: on the measured screen-triangle geometry (Step 057), the canonical disformal connection supplies \(\approx 1.47\times 10^{-12}\) s at the \(B_0=+1\) unit benchmark (\(1.3\%\) of \(H_{\rm implied}\approx 1.13\times 10^{-10}\) s on the median leg, \(5.0\%\) on the \(p_{95}\) leg) and reaches \(1.14\times 10^{-10}\) s (\(100.5\%\) on the median leg; \(B_0^{\rm req}=20.1\) on the \(p_{95}\) leg) at the corpus's GW170817 galactic-profile envelope \(B_0\lesssim 77.7\) (Paper 0, Step 50), while remaining within the direct time-domain delay bound (\(\lesssim 0.46\) ns).

6.4 Analytical Boundaries and Methodological Constraints

The detection is robust in the phase domain, not across all possible amplitude estimators. The weighted signed epoch-mean effect size is Cohen's \(d \approx -10^{-3}\) (near zero), compatible with bipolar cancellation where opposite velocity projections partially cancel in the signed mean. The frame-invariant unweighted Phase Closure rules out a simple heliocentric velocity-weighting artifact. J1603's bipolar decomposition provides directionally consistent geometric structure; its circular Phase Closure is noise-limited, so the 73.8° proper-motion separation is interpreted through the joint velocity–geometry pattern rather than through a second Rayleigh statistic alone.

No significant Parkes/PPTA-to-Jiamusi offset is detected; the Jiamusi sample is noise-limited, consistent with TEP's predicted environmental suppression in distant, noise-limited sightlines. Threshold-based tests are interpreted as validations of unsigned-\(M_{\rm cl}\) sensitivity rather than independent TEP detections, because high-SNR cuts substantially change folded-amplitude summaries. The phase-domain result remains the primary defensible observable.

The 15-pulsar sample provides broad coverage across distances and telescopes, with J0437-4715 yielding a robust circular Phase Closure detection and J1603-7202 providing complementary geometric evidence through bipolar structure and variance decomposition. The remaining 13 are noise-limited or have no viable data; they provide bounds but do not independently diagnose environmental suppression. The 73.8° proper motion vector separation between J0437 and J1603 matches the velocity-geometry prediction. The \(M_{\rm cl}\) magnitude ratio between J1603 and J0437 (0.32 for weighted means, 0.55 for trimmed means) is not captured by simple distance-velocity scaling models, consistent with \(M_{\rm cl}\) being a secondary validation subject to Rice bias while \(\psi\) carries the scaling inference.

The frame-invariant J0437 detection is not reproduced by a simple additive-delay model, and the J1603 geometry is directionally consistent with the TEP interpretation. The tested Gaussian-filament local-ISM anomaly aligned with the proper-motion axis is ruled out by dedicated simulation: a Gaussian filament produces identically zero closure magnitude across all tested parameter combinations because the scalar phase-screen closure identity is a geometric invariant. The current result is a robust detection of non-zero Phase Closure consistent with TEP, with independent geometric evidence from J1603 and environmental-suppression consistency from the distant sample.

6.5 Methodological Contributions

Several methodological advances are introduced. Stokes-theorem alignment using continuous velocity-projection weighting (\(v_{\perp} / v_0\) where \(v_0 = 50\) km/s) preserves kinematic modulation information, improving sensitivity over binary sign-only rectification. Robust estimators, such as the 10% trimmed mean, provide heavy-tail-resistant summary statistics alongside standard means. Multi-method validation using six independent statistical tests—including effect size, Gaussian noise null test, cross-validation, and Bayesian comparisons—supports the stability of the signal across distinct validations. The use of fixed random seeds ensures exact reproducibility, enabling rigorous TEP testing with additional pulsar scintillation datasets.

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Appendix A. Derivation of Synchronization Holonomy from Disformal Metric Geodesics

This appendix provides a self-contained derivation of the observable synchronization holonomy from the disformal metric structure. The derivation proceeds from the geodesic equation in the matter metric, showing explicitly that non-integrability of the synchronization 1-form leads to a non-zero closure when \(B(\phi) \neq 0\).

A.1 Disformal metric structure

TEP posits that all non-gravitational fields couple to the matter metric \(\tilde{g}_{\mu\nu}\), related to the gravitational metric \(g_{\mu\nu}\) by the disformal map (Bekenstein 1993; Koivisto & Zumalacárregui 2012):

In the weak-field, static limit the gravitational metric is \(g_{00}=-(1+2\Phi)\), \(g_{ij}=(1-2\Phi)\delta_{ij}\) with \(|\Phi|\ll 1\). Assuming the scalar field is time-independent (\(\nabla_0\phi=0\)) at fixed cosmic epoch (\(A_{\rm sp}(t_0)=1\)), the canonical matter-metric components are \(\tilde g_{00}=-A_{\rm loc}^2(u_{\rm grav})(1+2\Phi)\) and \(\tilde g_{ij}=(1-2\Phi)\delta_{ij}+B(\phi)\partial_i\phi\partial_j\phi\) (or in the isotropic-conformal shorthand,

both sharing the identical leading projected disformal synchronization connection \(\delta\tilde\sigma_i\) below).

A.2 Residual synchronization connection and projected disformal correction

Following the Jakarta construction (Paper 0, Section 3.3 and Appendix A3), the physical observable is not the raw loop integral of a proper-time 1-form but the GR-subtracted residual. Let \(\tilde{\sigma}^{(\tau)}\) denote the matter-frame synchronization transport one-form and \(\sigma_{\rm GR}^{(\tau)}\) the corresponding GR reference one-form. The residual connection is:

and the time-valued residual holonomy is:

This is invariant under synchronization re-gaugings because both \(\tilde{\sigma}^{(\tau)}\) and \(\sigma_{\rm GR}^{(\tau)}\) shift by the same exact one-form. In the purely conformal subclass (\(B = 0\)), the scalar contribution is the exact one-form \(\omega^{(A)} = d\ln A\), so \(\oint_C d\ln A = 0\) for any closed loop. A leading-order nonzero \(H_{\rm resid}^{(\tau)}\) requires the disformal sector.

To leading order in the disformal correction, let \(u^\mu\) denote the four-velocity of the clock congruence and \(P_\mu{}^\nu = \delta_\mu{}^\nu + u_\mu u^\nu\) the spatial projector. The projected disformal contribution to the synchronization connection is (Jakarta, Appendix A3):

In a hypersurface-orthogonal \(3+1\) slicing (\(u^\mu = n^\mu\)):

Unlike \(d\ln A\), the disformal contribution is not generically exact. Its curvature satisfies \(d(\delta\tilde{\sigma}^{(\tau)}) \neq 0\) whenever the prefactor varies independently around the loop — through time dependence, motion through \(\nabla\phi\), anisotropic boundary conditions, inhomogeneous field structure, or spatial variation of the disformal response.

A.3 Closure holonomy and the physical condition for non-vanishing signal

At leading order beyond the GR-subtracted reference model:

A static scalar gradient in a hypersurface-orthogonal frame does not generate the leading holonomy. The pulsar observable requires \(u \cdot \nabla\phi \neq 0\): relative motion through the field, time dependence, mixed temporal-spatial structure, or independently varying lapse/shift and disformal response around the effective loop. For J0437−4715, the pulsar's proper motion (\(v_\perp \simeq 104\) km/s), Earth's orbital motion (\(v_{\rm orb} \simeq 30\) km/s), and screen motion provide the kinematic ingredients that generate \(u \cdot \nabla\phi \neq 0\). These enter the theoretical prediction through the \(u \cdot \nabla\phi\) factor in equation $\eqref{eq:app_disformal_sigma}$, not merely through the later Stokes-alignment algorithm.

The leading-order residual holonomy for a loop \(C\) bounding a smooth surface \(\Sigma\) is:

Equation $\ref{eq:app_holonomy_final}$ is the central result: the residual holonomy vanishes in the conformal-only subclass (\(B = 0\)), where the scalar contribution is exact. A non-zero \(H_{\rm resid}^{(\tau)}\) at leading order is possible when \(B(\phi) \neq 0\) and \(u \cdot \nabla\phi \neq 0\) combine to produce nonzero residual curvature. The precise necessary condition is:

over the loop region. This is stronger than merely requiring \(B \neq 0\) and \(u \cdot \nabla\phi \neq 0\) individually, because it specifies exactly when the curvature of the residual connection — and therefore the holonomy — is nonzero. The conformal sector alone cannot generate a closed-loop residual; it can affect local rates and open-path redshift comparisons but not the synchronization holonomy.

A.4 Why ordinary ray phases close but TEP synchronization transport need not

A central question is whether the measured secondary-spectrum cross-term phases are the edges of a genuine spacetime synchronization loop, or merely differences between three complete source-observer ray phases that would trivially cancel. The following measurement model identifies the observed cross-terms with synchronization-transport links. A complete electromagnetic transfer derivation is required to determine the corresponding channel coefficient and effective phase frequency \(\omega_{\rm eff}\); the identification below is a measurement-model postulate, not a derivation from the matter-coupled electromagnetic field.

For three single-ray complex amplitudes \(X_i, X_j, X_k\) at the observer, rank-one pairwise cross-terms \(Q_{ab} = X_a X_b^*\) obey the algebraic identity \(Q_{ij}Q_{jk}Q_{ki} = |X_i X_j X_k|^2\), whose phase vanishes identically regardless of the scalar phase \(\Theta_a\) accumulated along each individual ray: \((\Theta_i - \Theta_j) + (\Theta_j - \Theta_k) + (\Theta_k - \Theta_i) = 0\). In the secondary-spectrum measurement pipeline, however, each extracted arclet peak \(Q_{ab}(\tau_{ab}, f_{D,ab})\) is not a point-wise product of isolated single-ray amplitudes; it is the two-dimensional Fourier transform of the dynamic intensity spectrum \(I(\nu,t) = |E(\nu,t)|^2\) evaluated over a finite delay–Doppler pixel bin integrating a coherent bundle of sub-rays across the scattering screen (as demonstrated explicitly by the non-factorizable \(\chi^2\) screen simulation of Step 055, where secondary-spectrum cross-terms are intensity convolutions that break rank-one factorization). While conventional multi-ray screen convolutions produce non-zero per-epoch closure phases whose epoch-mean direction averages to zero over time (Step 055: synthetic \(\bar{R}_{\rm dir}=0.09\), Rayleigh \(p=0.55\)), a stationary non-exact spacetime synchronization connection \(\Delta\sigma^{(\tau)}\) couples to the finite-baseline interference transport across each screen-pair link \(\Gamma_{ab}\), breaking rank-one factorization with a persistent directional bias.

TEP modifies the operational comparison itself. The pairwise observable is not merely the difference of endpoint phases, but the transport of synchronization between the path-defined events associated with rays \(a\) and \(b\):

where \(\Gamma_{ab}\) is the comparison path in spacetime induced by the scattering geometry: the spatial separation between the points where rays \(a\) and \(b\) intersect the scattering screen, transported along the observer's worldline. The endpoint phases \(\varphi_a, \varphi_b\) are state functions and cancel in the closure; the residual synchronization transport \(\int_{\Gamma_{ab}} \omega_{\rm eff}\,\Delta\sigma^{(\tau)}\) is path-dependent and generically non-exact when \(B(\phi) \neq 0\).

The three measured cross-terms \((ij), (jk), (ki)\) correspond to three comparison paths \(\Gamma_{ij}, \Gamma_{jk}, \Gamma_{ki}\) that form a genuine closed spacetime loop:

The endpoint phases cancel telescopically, while the residual synchronization curvature remains. Thus the null \(\psi = 0\) is recovered whenever \(\Delta\sigma^{(\tau)}\) is exact — including the conformal-only sector — whereas the disformal non-exact sector permits \(\psi \neq 0\). The measured cross-term is non-factorizable under TEP precisely because the synchronization transport along each edge is an independent path integral, not a difference of state functions.

In the scintillation geometry, each arclet \(i\) in the secondary spectrum corresponds to a discrete scattered ray arriving at the observer with delay \(\tau_i\) and Doppler shift \(f_{D,i}\). The physical events and paths are:

• \(S_i, S_j, S_k\): the three scattering-screen events where rays \(i, j, k\) are redirected by plasma inhomogeneities.
• \(O_i, O_j, O_k\): the corresponding observer detection events (co-located at the receiver, but distinguished by their arrival directions and delays in the secondary spectrum).
• \(u^\mu\): the physical clock congruence (the observer's four-velocity, including Earth orbital motion and station velocity).
• \(\Gamma_{ij}\): the oriented synchronization-comparison link between the scattering events \(S_i\) and \(S_j\), transported along the observer's worldline.
• \(\Gamma_{jk}, \Gamma_{ki}\): the corresponding links for the other two pairs.
• \(C = \Gamma_{ij} + \Gamma_{jk} + \Gamma_{ki}\): the closed synchronization-comparison loop.

The cross-term \(\hat{\tau}_{ab}\) is measured from the interference of rays \(a\) and \(b\) in the secondary spectrum, and its phase \(\phi_{ab}\) is extracted from the complex cross-spectrum. The three cross-terms \((i,j), (j,k), (k,i)\) are measured independently from three distinct pairs of arclets, and their closure \(\phi_{ij} + \phi_{jk} - \phi_{ik}\) corresponds to the loop \(C = \Gamma_{ij} + \Gamma_{jk} + \Gamma_{ki}\) in the scattering geometry. The minus sign on \(\phi_{ik}\) reflects the reversal of the \((i,k)\) leg to close the loop. Under the scalar-delay null, each \(\phi_{ab}\) is a state-function difference and the closure vanishes; under TEP, each \(\phi_{ab}\) carries an additional path-dependent synchronization transport that does not cancel. The measured complex cross-term acquires \(\int_{\Gamma_{ab}} \omega_{\rm eff}\,\Delta\sigma^{(\tau)}\) through the electromagnetic phase transport between the scattering events, calibrated by the emitter and receiver clock congruence; photon propagation is null, so the phase is formulated through clock calibration and electromagnetic phase transport, not photon proper time.

Amplitude closure of the measurement model is evaluated in Step 057 (Section 5.4.3): the canonical connection integrated over the measured screen-triangle geometry on the admissible \(B \geq 0\) branch yields \(\approx 1.47\times10^{-12}\) s at the \(B_0=+1\) unit benchmark (\(1.3\%\) of \(H_{\rm implied}\approx 1.135\times10^{-10}\) s on the median leg, \(5.0\%\) on the \(p_{95}\) leg) and reaches \(1.141\times10^{-10}\) s (\(100.5\%\) on the median leg; \(B_0^{\rm req}=20.1\) on the \(p_{95}\) leg) at the corpus's GW170817 galactic-profile bound \(B_0\lesssim 77.7\) (Paper 0, Step 50), while remaining below the direct time-domain delay bound \(|H|\le 0.459\) ns (Section 4.4).

A.5 Mapping to the measured closure observables

The measured Phase Closure is the complex cross-term closure product \(\mathcal{B}_{ijk} = Q_{ij} Q_{jk} Q_{ik}^*\) (Equation $\ref{eq:closure_product}$), where each \(Q_{ab}\) is extracted from the complex secondary spectrum at the cross-term peak position. The TEP measurement operator modifies each cross-term:

Within the TEP measurement model, the measured cross-term is represented as above. Appendix A.4 defines the associated events and paths; electromagnetic transfer determines the channel coefficient. The closure product is then:

and therefore:

The endpoint amplitudes and phases cancel, leaving only the residual synchronization holonomy converted to phase through \(\omega_{\rm eff}\). Under the additive-delay null hypothesis (\(\Delta\sigma^{(\tau)} = 0\)), \(\psi_{ijk} = 0\) up to measurement noise. When \(B(\phi)\neq 0\) and the curvature condition (Equation $\ref{eq:app_curvature_condition}$) is satisfied, the nonfactorizable synchronization transport imprints a non-zero mean on \(\psi\), providing the circular statistic used as the primary detection test in this work. The delay-domain folded magnitude \(M_{\rm cl}\) is a separate quantity, computed from the real-valued closure delay \(\tau_{\text{closure}}\); it is not the real part of a complex delay-Doppler vector and is not used to define \(\psi\).

Appendix B. Limitations of Delay-Domain Amplitude Estimators

While the Phase Closure \(\psi\) is a zero-centered odd-parity circular statistic, unsigned delay-domain amplitude estimators of the form \(M_{\rm cl} = |\langle \tau_{\text{closure}} \rangle_{\text{aligned}}|\) are strictly non-negative and therefore subject to folded-normal (Rice-distribution) noise-floor bias. For zero-mean Gaussian noise with standard deviation \(\sigma\), the expectation of the folded magnitude is \(E[|X|] = \sigma\sqrt{2/\pi} > 0\). Consequently, raw unsigned delay magnitudes and information-criterion comparisons evaluated on \(M_{\rm cl}\) against a zero-mean null primarily measure the non-zero expectation of the folded noise distribution rather than a physical synchronization holonomy. This appendix documents the complete delay-domain amplitude audit from Step 007, Step 052, and Step 060 for pipeline completeness.

B.1 Raw, Noise-Subtracted, and Symmetrically Trimmed Delay Estimators

In the primary PSR J0437-4715 dataset (1,093 independent epochs), the inverse-variance-weighted epoch-level unsigned closure magnitude is \(M_{\rm cl} = 8.100 \pm 0.26\) ns (epoch-level bootstrap SEM; the internal measurement-noise SEM is \(0.102\) ns). Estimating the per-epoch folded-normal noise expectation from the within-epoch signed-delay dispersion yields \(E[M_{\rm cl}] = 6.810 \pm 0.24\) ns, demonstrating that the raw unsigned estimator is dominated by the Rice noise floor. Subtracting the per-epoch floor in paired epoch-level comparisons leaves a statistically positive residual delay excess of \(M_{\rm cl,excess} = 1.290 \pm 0.11\) ns (\(t \approx 12.2\sigma\) under independent-epoch resampling, and \(3.8\sigma\)–\(10.8\sigma\) across 15–240-epoch MJD-ordered contiguous block bootstraps; Steps 052 and 060).

A secondary delay-domain diagnostic, the symmetric 10% trimmed amplitude \(M_{\rm cl,sym10} = 21.991 \pm 0.75\) ns (epoch-level iid bootstrap SEM, rising to \(\pm 2.0\text{--}4.9\) ns under 15–240-epoch contiguous blocks; Step 052), removes the lowest and highest 10% of unweighted per-epoch folded magnitudes. Because \(M_{\rm cl,sym10}\) is an unweighted trimmed summary of per-epoch folded magnitudes whereas \(M_{\rm cl} = 8.100\) ns is an inverse-variance-weighted global mean, their numerical scales reflect their distinct weighting conventions rather than two estimates of the same parameter, and neither carries primary inferential weight.

B.2 Step 007 Delay-Amplitude Validation Ledger

Step 007 applied seven validation checks to the unweighted independent-epoch \(M_{\rm cl}\) summary prior to the adoption of circular Phase Closure \(\psi\) as the primary test statistic:

B.3 Five-Fold Holdout Cross-Validation of Epoch-Mean \(M_{\rm cl}\)

In the Step 007 cross_validation block (results/step_007_independent_statistical_validation_results.json), a 5-fold shuffle split on unweighted epoch-mean \(M_{\rm cl}\) values confirms that the delay-amplitude scale (\(\mathcal{O}(10\text{--}15)\) ns before inverse-variance weighting) is uniform across epoch subsets: