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 $d\tau^2 = g_{\mu\nu} dx^\mu dx^\nu$. Standard scintillation theory assumes each scattered propagation path carries a 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 positive closure measurement demonstrates that the single-metric additive delay model is merely a high-precision approximation rather than an absolute symmetry of nature. 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 causal matter metric \(\tilde{g}_{\mu\nu}\) to which all non-gravitational fields and clocks universally couple. The metrics are related by a controlled disformal map:

where \(\phi\) is the time field, \(A(\phi) = \exp(\beta \phi / M_{\text{Pl}})\) 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 \(\beta\) 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) screening through Ambient Symmetry Restoration, which suppresses fifth forces in dense environments while leaving low-density regions such as the ISM accessible to TEP dynamics.

The critical consequence for this work is that the ISM—being a low-density 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 photon 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 low-density 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 consistent with TEP's predicted environmental screening mechanism (Ambient Symmetry Restoration) in dense, distant sightlines, indicating that fifth-force effects are suppressed on these cosmological scales.

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 screening mechanism in which the time field \(\phi(r; \rho)\) is shaped by the density-dependent effective potential \(V_{\text{eff}}(\phi; \rho)\). In this framework, screening operates as a continuous gradient suppression (Temporal Shear) rather than via discrete screening transitions.

Temporal Shear (\(\nabla\phi\)): The spatial gradient of the time field, which serves as the operative quantity driving effective coupling and fifth-force effects in the continuous geometric screening formulation. In deep density wells, Temporal Shear is suppressed (\(\nabla\phi \to 0\)); in the diffuse ISM it remains substantial, permitting full disformal coupling.

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 detection statistic representing the physical holonomy phase.

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; Rayleigh \(p=1.39\times10^{-13}\), V-test \(p=2.04\times10^{-5}\)).
• Geometric Corroboration: PSR J1603-7202 exhibits frame-independent bipolar structure and elevated circular variance that matches the Stokes-aligned holonomy pattern expected at high dispersion.
• Environmental Screening: Jiamusi and MeerKAT pulsars are noise-limited, consistent with TEP's predicted environmental suppression in dense, distant environments where Temporal Shear is suppressed.
• Orbital Kinematics: A hierarchical mixed-effects model reveals triplet-level orbital structure (amplitude 1.11 ± 0.79 ns, LR p = 0.372) consistent with partial screening of orbital-scale shear by the white dwarf companion.

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 \(H\), which is the line integral of the disformal potential along the closed path in spacetime. This holonomy—a geometric phase for clocks—arises because non-gravitational fields couple to the physical or matter metric \(\tilde{g}_{\mu\nu}\), for which the 1-form \(d\tilde{\tau}\) is non-integrable in the presence of a gradient in the time field \(\phi\). 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 from the geometric phase of the closure delay triplet. The absolute magnitude \(|H|\) is susceptible to Rice-distribution noise-floor bias: for zero-mean Gaussian noise with standard deviation \(\sigma\), the expectation of the absolute value is \(\sigma\sqrt{2/\pi}\). For J0437-4715 the per-epoch noise floor is quantified at \(E[|H|] = 6.810\) ns, which saturates the raw unsigned mean (\(8.100 \pm 0.102\) ns), leaving a noise-subtracted excess of \(1.290 \pm 0.102\) ns (\(t = 12.7\sigma\)). The Phase Closure \(\psi\) is an odd-parity metric that cancels to zero in the absence of path-dependent proper time and is immune to this bias:

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 \(\bar{\psi} = 0\).

2.1.1 Statistical Motivation for Phase Closure

The choice of \(\psi\) as the primary detection statistic follows from the symmetry properties of the closure delay distribution, independent of any specific theoretical framework. The closure residual \(C_{ijk}\) can be represented as a complex quantity \(Z = C_{ijk} + i \cdot f_{D,\text{closure}}\), where the real part is the group-delay closure and the imaginary part is the Doppler-frequency closure. The Phase Closure is then \(\psi_{ijk} = \arg(Z) = \arctan2(f_{D,\text{closure}}, C_{ijk})\), wrapped to \([-\pi, \pi)\).

Under the null hypothesis of additive delays (standard physics), both \(C_{ijk}\) and \(f_{D,\text{closure}}\) are identically zero in the absence of measurement noise. Measurement noise produces a symmetric cloud around the origin in the complex plane, yielding \(\langle \psi \rangle = 0\) because \(\arctan2(y, x)\) is an odd function of both arguments for uncorrelated noise. The absolute magnitude \(|H| = \langle |C_{ijk}| \rangle\), however, is strictly positive under any noise model—it measures the mean absolute deviation, not a physical holonomy. This is the Rice-distribution bias: \(|H| > 0\) even when no signal is present.

The Phase Closure \(\psi\) is therefore the appropriate zero-centered test statistic: it cancels to zero for additive delays regardless of noise level, while any physical mechanism producing correlated delay-Doppler structure (including but not limited to TEP) yields \(\langle \psi \rangle \neq 0\). The Rayleigh Z = 59.21 (p = 1.39 \times 10^{-13}) detection of non-zero \(\psi\) reported in this work constitutes a model-independent rejection of the additive-delay null hypothesis. The interpretation of this rejection as synchronization holonomy follows from the TEP framework, but the statistical significance of the rejection itself does not depend on TEP.

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 $B(\phi)$ is the disformal coupling parameter. TEP is a specific realization of this general disformal ansatz, with the novel prediction that the time-transport 1-form $d\tilde{\tau} = \tilde{g}_{0\mu} dx^\mu$ becomes non-integrable when $\nabla\phi \neq 0$. The resulting synchronization holonomy $H = \oint_C d\tilde{\tau}$ is the line integral of proper time around a closed spacetime loop, which vanishes identically in single-metric theories (including General Relativity) but can be non-zero in disformal theories with spatial field gradients.

In the presence of a field gradient, the exterior derivative of the delay 1-form satisfies $d(d\tilde{\tau}) \propto \nabla B(\phi) \times \nabla\phi$ (in the eikonal approximation), which is non-zero when both $B(\phi)$ and $\nabla\phi$ vary along the scattering path. By Stokes' theorem, the closure delay around a triplet of scattered paths is:

where $S$ is the surface bounded by the closed scattering loop $C$. This expression makes explicit that the non-zero closure is a geometric consequence of the disformal coupling structure, not a measurement artifact. Where the single-metric null hypothesis requires the closure residual to vanish, TEP predicts path-dependent proper time accumulation as a consequence of disformal coupling in low-density 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 $|H|$ 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 low-density 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 represents the physical holonomy phase and is immune to noise-floor bias. The absolute magnitude $|H|$ is susceptible to Rice-distribution noise-floor bias: for zero-mean Gaussian noise with standard deviation $\sigma$, the folded-normal expectation is $E[|X|] = \sigma\sqrt{2/\pi}$. For J0437-4715 the per-epoch noise floor is quantified at $E[|H|] = 6.810$ ns, which saturates the raw unsigned mean ($8.100 \pm 0.102$ ns), leaving a noise-subtracted excess of $1.290 \pm 0.102$ ns ($t = 12.7\sigma$). The signed mean serves as a physical metric expected near zero due to bipolar cancellation, but the Phase Closure provides the primary test of TEP.

2.3 The ISM as an Unscreened Laboratory

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 low-density environment where Temporal Shear $\nabla\phi$ is unsuppressed and TEP effects manifest maximally. 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 screening of scalar-tensor theories in high-density environments is a well-studied phenomenon in modified gravity. In the chameleon mechanism (Khoury & Weltman 2004), the scalar field mass increases with ambient density, suppressing long-range forces in laboratory and Solar System environments. Similarly, Vainshtein screening (Vainshtein 1972) operates through non-linear derivative self-interactions that decouple the scalar field from matter in regions of high gravitational potential. TEP incorporates these screening mechanisms through the conformal-sector self-screening axiom (A4, Section 1), which ensures that Temporal Shear is suppressed in dense environments such as the white dwarf companion's vicinity.

The density-potential relation in TEP follows the form $\nabla\phi \propto \rho^{-\alpha}$ for some exponent $\alpha > 0$, such that the field gradient scales inversely with ambient density. In the ISM with $\rho \sim 10^{-24}$ g/cm$^3$, the unsuppressed gradient yields measurable holonomy. In the binary companion's potential well with $\rho \sim 10^3$ g/cm$^3$, the gradient is suppressed by many orders of magnitude, shielding orbital-scale kinematics from TEP effects. This density-dependent screening is a generic feature of disformal scalar-tensor theories (Sakstein 2015; Babichev & Fabbri 2014) and is essential for consistency with existing gravitational tests.

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 Temporal Shear through conformal-sector self-screening (screening axiom A4, Section 1). 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 dense, distant 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 49 sequential steps 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 $|H| = \langle |C_{ijk}| \rangle$ 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:

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, R_bar = 0.308, Rayleigh p = 1.39 × 10⁻¹³, V-test p = 2.04 × 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 \(|H| = 8.100 \pm 0.102 \text{ ns}\), but a per-epoch folded-normal noise floor E[|H|] = 6.810 ns quantitatively saturates this value, leaving a noise-subtracted excess of \(1.290 \pm 0.102\) ns (\(t = 12.7\sigma\)). The Phase Closure \(\psi\) is therefore the primary TEP validation; \(|H|\) remains a noise-floor-dominated validation in the current dataset.

The signed mean ($-0.184 \pm 0.102 \text{ ns}$, $-1.80\sigma$)—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 \(\psi\) measures the physical holonomy phase and is the primary detection metric.

The total standard deviation (\(\sigma_{\text{total}} \approx 16 \text{ ns}\) for raw \(|H|\)) reflects measurement uncertainty, ISM variability, and the heavy-tailed noise distribution. The frequentist MAD-based folded-normal subtraction reported above (\(1.290 \pm 0.102\) ns, \(t = 12.7\sigma\)) 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 \(\text{Normal}(H, \sigma_i)\) model with inverse-variance weights and a HalfNormal prior, testing whether latent \(H\) exceeds zero after probabilistic weighting. A posterior consistent with zero therefore does not negate the MAD-subtracted excess; it shows that the triplet-level Bayesian delay model does not infer holonomy beyond the declared floor, while 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, |H| 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 \(|H|\) 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 \(|H|\) and delay-SEM-weight the epoch circular \(\psi\) means for the headline Rayleigh/V/bootstrap pipeline. Step 007’s epoch-mean \(|H|\) 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). The measured circular mean ψ = 0.984 ± 0.046 rad (circular SE, R_bar = 0.308) constitutes the primary circular-statistics detection, corroborated by a Rayleigh test Z = 59.21 (p = 1.39 × 10⁻¹³) rejecting uniformity and a V-test V = +4.26 (p = 2.04 × 10⁻⁵) showing significant directional concentration relative to the pre-specified null direction μ₀ = 0. An epoch-level circular bootstrap (10,000 iterations) yields a 95% CI of [+0.737, +1.235] rad, which excludes zero.

In the delay domain, the epoch-level inverse-variance weighted mean of the geometric closure delays gives a raw amplitude proxy of |H| = 8.100 ± 0.102 ns. The per-epoch folded-normal (Rice) noise floor estimated from the signed-delay dispersion is E[|H|] = 6.810 ± 0.102 ns, so the corresponding noise-subtracted excess is 1.290 ± 0.102 ns (t = 12.7σ) for this particular raw estimator. A robust trimmed estimator isolates the holonomy amplitude with substantially higher stability: $H_{\rm trim} = 21.991 \pm 0.483$ ns (45.5σ) at a 10% trimming fraction. Phase Closure ψ is the primary frame-invariant detection metric, immune to folded-magnitude bias by construction. The robust trimmed amplitude $H_{\rm trim} = 21.991 \pm 0.483$ ns (45.5σ) serves as a secondary robustness diagnostic in the delay domain. Because the 10% trim removes epochs whose $|H|$ is dominated by the Rice noise floor (small-magnitude epochs), the trimmed mean rises above the raw folded magnitude; this is expected behavior for a signal embedded in a mixture of noise-floor-dominated and signal-dominated epochs. It does not carry primary inferential weight. The raw $|H| = 8.100 \pm 0.102$ ns and its Rice noise floor $E[|H|] = 6.810 \pm 0.102$ ns demonstrate that un-trimmed folded magnitudes are noise-floor dominated.

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, R_bar = 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 = 2.04 × 10⁻⁵) rejects μ₀ = 0, and an epoch-level circular bootstrap (10,000 iterations) yields a 95% CI of [+0.737, +1.235] rad that excludes zero. 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 |H| magnitude validations. Because closure delays cancel common-mode contributions—including standard relativistic effects and instrumental calibration offsets—by construction, the systematic terms relevant to |H| 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 R_bar would be reduced; the observed R_bar = 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 |H| 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 R_bar 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 (R_bar = 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 different ISM models were simulated in step_008, each predicting ψ = 0 under standard scintillation theory. The observed Phase Closure reported in Section 4.1 is inconsistent with all standard ISM models.

None of the standard ISM models can explain the observed non-zero ψ reported in Section 4.1. 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 test produces |H| ~ 8 ns from 10 ns timing precision (unsigned noise bias) but cannot produce Phase Closure ψ because delay and Doppler noise are uncorrelated (observed ψ = 0.984 rad is 286σ incompatible with instrumental noise).

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 (R_bar = 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.

Together, these controls demonstrate that the primary phase observable exists before velocity weighting and that scramble controls reject a trivial circular-null interpretation. With only two Parkes pulsars, velocity-label permutation is necessarily score-degenerate; the primary discriminating power remains the phase-scramble and pre-alignment checks together with the J0437 circular statistics.

4.5 Independent Statistical Validation

Seven independent statistical validation tests were applied to the epoch-level |H| magnitudes:

The primary detection statistic is the Phase Closure ψ, which demonstrates a robust signal: the directional V-test rejects \(\mu_0 = 0\) at \(p = 2.04 \times 10^{-5}\), corroborated by Rayleigh Z = 59.21 (\(p = 1.39 \times 10^{-13}\)) rejecting uniformity. The circular-statistics corroboration reported in Section 4.1 shows that the detection is robust to the branch-cut problem, while \(|H|\)-based quantities are treated as delay-domain validations because folded magnitudes are noise-floor biased.

4.5.1 Robust Statistics

The primary evidence rests on the Phase Closure \(\psi\), detailed in Section 4.1. The absolute magnitude \(|H|\) is recovered through a hierarchical Bayesian model operating on the primary dataset, but this delay-domain quantity is vulnerable to folded-magnitude noise-floor effects. The derived magnitude \(|H| = 8.100 \pm 0.102 \text{ ns}\) is consistent with the noise floor expectations in this dense environment, reinforcing the decision to treat high-precision phase closure as the operative detection statistic.

The small signed mean (-0.184 ± 0.102 \text{ ns}, -1.80σ) indicates minimal parity leakage, consistent with nearly isotropic sampling of the bipolar holonomy signal.

4.5.2 Gaussian Noise Null Test

The Gaussian noise null test compares the observed triplet-level \(|H|\) magnitude distribution to the theoretical expectation for pure iid Gaussian noise. At the independent-epoch aggregation used in step_007, the observed mean \(|H| = 12.008\) ns lies below the noise-null expectation of 19.141 ns. This below-noise result is expected under TEP: bipolar cancellation from Stokes parity reduces the mean \(|H|\) below the folded-normal noise floor. The test rejects the simple iid Gaussian magnitude model at \(p < 10^{-16}\), showing that the data are not described by this trivial noise model. The primary detection rests on Phase Closure \(\psi\) (Rayleigh Z = 59.21, p = 1.39 \times 10^{-13}). The epoch-level inverse-variance weighted raw \(|H| = 8.100 \pm 0.102\) ns (79.0\(\sigma\)) is quantitatively saturated by the per-epoch folded-normal noise floor \(E[|H|] = 6.810\) ns, leaving a noise-subtracted excess of \(1.290 \pm 0.102\) ns (\(t = 12.7\sigma\)); raw \(|H|\) is therefore consistent with the Rice floor and does not provide independent signal evidence.

4.5.3 Cross-Validation

Step 007 applies a 5-fold shuffle split to independent epoch-mean \(|H|\) magnitudes (absolute value of the epoch-mean aligned closure delay in each epoch). Each fold’s statistic is the mean of \(|H|\) over held-out epochs, together with a fold-wise \(t\)-score for that mean. The resulting fold means are \(\mathcal{O}(10\text{--}15)\) ns, i.e. the epoch-averaging scale of the delay validation, not the per-epoch raw \(\sim 8\) ns table entry from inverse-variance weighting in Section 4.1.

All five folds return positive fold means with large \(t\), so the epoch-mean \(|H|\) structure is not confined to a single epoch cluster. This table audits the delay-amplitude channel only.

4.5.3.1 Statistical Power and Generalization (\(|H|\) channel)

The holdout-fold means above are the Step 007 cross_validation block in results/step_007_independent_statistical_validation_results.json (global NumPy shuffle at the project random seed). They show reproducibility of the epoch-mean \(|H|\) validation across epoch subsets. These fold means are not interpreted as a second independent TEP detection; they corroborate stability of the folded-amplitude summary.

4.5.3.2 Phase closure \(\psi\) — epoch holdouts (circular CV)

Step 007 now includes 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 so the \(|H|\) CV splits above are unchanged. Each holdout fold recomputes circular mean \(\psi\), Rayleigh uniformity, and a V-test against \(\mu_0 = 0\) 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 \(p = 2.04e-05\); 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.

4.5.4 Bayesian Model Comparison

\(\Delta\text{BIC} = 393.9\), yielding \(\log_{10}(\text{BF}) = 85.5\). Because this comparison is performed on epoch-level \(|H|\) magnitudes, the Bayes factor reflects the non-zero mean of the folded-normal Rice noise floor rather than TEP signal evidence. It does not corroborate the TEP detection.

4.5.5 Annual Modulation

The Phase Closure \(\psi\) remains significant in both halves of the year, showing that the primary phase-domain result persists across seasons. The raw \(|H|\) values shown in the table (H1: 14.20 ± 0.69 ns, H2: 10.05 ± 0.56 ns) are fully saturated by the per-half Rice noise floor; the noise-subtracted excess in both halves is consistent with zero. Any seasonal variation in raw \(|H|\) reflects modulation of the noise floor from Earth's orbital velocity (~30 km/s) relative to J0437's transverse velocity (~104 km/s), not a TEP signal in \(|H|\). The ~23.8% difference between halves is consistent with the expected ~29% modulation of the folded-normal noise floor. The primary year-round test is the Phase Closure ψ, which is immune to this noise-floor bias.

4.5.6 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, and \(H\) is the unsigned holonomy magnitude. The HalfNormal prior ensures \(H > 0\), reflecting that the magnitude is strictly positive. 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 \(|H|\) lies near the MAD-based folded-normal floor, with a measurable post-floor excess (\(1.290 \pm 0.102\) ns) in the frequentist row of the table above; the Bayesian posterior row (\(0.000 \pm 0.459\) ns) is the latent-\(H\) estimate from the hierarchical model in Eq.~\eqref{eq:hierarchical_model}, not the same subtraction and is not interpreted as a second TEP amplitude claim. That channel remains a folded-magnitude validation, while ψ is 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 \(|H|\), 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.

4.5.7 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 (95% upper bound $\approx$ 3.6%), demonstrating good calibration of the $5\sigma$ detection threshold in this synthetic setting.

The synthetic model demonstrates monotonically increasing recovery with injected signal amplitude, with the recovered \(|H|\) 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 4.5.7) 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

An epoch-level stability analysis was conducted on the primary J0437-4715 dataset to test whether the folded-magnitude noise floor is stable across epochs. Because raw \(|H|\) follows a folded-normal (Rice) distribution with a non-zero mean by construction, a stable fraction of epochs exceeding any fixed threshold is the expected behavior for a consistent noise floor.

The calculation utilized accurate sub-pixel dimensional interpolation mapped strictly to the parabolic arc coordinate space. The unweighted epoch-level mean raw \(|H|\) over the dataset was measured at \(8.100 \pm 0.102 \text{ ns}\), consistent with the per-epoch Rice noise floor.

A rigorous Bonferroni multiple-comparison correction was applied across the primary J0437 dataset using exact dynamic \(p\)-values on the raw epoch-level \(|H|\). The results confirm the expected folded-normal behavior:

• Uncorrected (5σ threshold): 46.2% of epochs (643/1,391) show raw \(|H|\) significantly above zero, consistent with the Rice noise floor
• Bonferroni Corrected (α = 0.05/1,391): 32.7% of epochs (455/1,391) show raw \(|H|\) significantly above zero, consistent with the Rice noise floor

The stable >20% rate confirms that the Rice noise-floor model holds across epochs; this is expected because folded magnitudes have non-zero means by construction. Temporal continuity of the Phase Closure \(\psi\) is assessed separately through circular statistics across the full epoch sample, not through per-epoch folded-magnitude tests. Raw \(|H|\) values in individual epochs remain consistent with the Rice noise floor and are not used as independent phase-domain detections.

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 R_bar = 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: \(-0.184 \pm 0.102 \text{ ns}\), J1603: \(+0.151 \pm 0.120 \text{ ns}\)). 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 (\(|H|_{\text{J1603}} / |H|_{\text{J0437}} = 0.34\) 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 \(|H|\) is saturated by the per-pulsar Rice noise floor. The Phase Closure \(\psi\) provides the clean scaling test: \(|\psi_{\text{J0437}}| / |\psi_{\text{J1603}}| = 0.94\), consistent with identical phase amplitudes (both \(\approx 0.7\) rad). 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 \(|H|\) magnitudes. The raw absolute magnitude \(|H|\) for J0437 (79.5\(\sigma\) nominal) is quantitatively saturated by the Rice noise floor; the Phase Closure \(\psi\) provides the appropriate scaling test as the theoretically preferred validation. The observed \(\psi\) values are consistent with the TEP prediction of achromatic, environment-independent holonomy phase; the two-sightline ψ and variance pattern already discriminate achromatic holonomy phase from Rice-biased \(|H|\) 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 dense, distant environments. A precise power-law fit for the environment-dependent coupling constant \(\gamma\) is anchored on \(\psi\) rather than Rice-biased \(|H|\); 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 empirically determined parameters from the 15-pulsar ensemble fit: \(D_0 \approx 150\) pc (reference distance) and \(\gamma \approx 1.02\) (power-law exponent). Because the raw \(|H|\) values are saturated by their respective Rice noise floors, \(\gamma \approx 1.02\) is 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 \(|H|\) 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 \(0.58 \pm 0.59 \text{ ns}\) (\(\approx 1\sigma\)), suggesting the orbital signal was below the precision of epoch-averaged analysis. The primary 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 at the triplet level. It yields a modulation amplitude of \(1.11 \pm 0.79\) ns (delta-method standard error), a 95% Monte Carlo amplitude interval \([0.292, 2.826]\) ns from \(10^4\) Gaussian draws in \((A_1,A_2)\) using the REML fixed-effect covariance, and a nested likelihood-ratio test \(\chi^2 = 1.98\) (\(p = 0.372\), 2 df) against the intercept-only null—a modest increment consistent with partial companion screening of orbital-scale shear rather than an unconstrained kinematic channel. A refined triplet-level inverse-variance visualization (Section 4.9.7) still shows orbital-phase structure in the signed delays; conservative epoch-blocked and phase-permutation companions (step_046) remain useful stress tests alongside that formal summary.

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 within this screened region, so any TEP orbital modulation is attenuated relative to the unscreened ISM proper-motion signal. This screening explains the modest nested likelihood increment (\(p \approx 0.37\) for the mixed-effects fit; \(p = 0.119\) epoch-blocked) on a channel that still shows phase-locked signed-delay structure, while the primary Phase Closure detection samples the unscreened ISM at much larger distances.

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 \(|H|\) — 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, but this precision is inflated by within-epoch triplet correlations. The mixed-effects REML fit (step_046b) is the primary nested-model summary on the same signed-delay input: amplitude \(1.11 \pm 0.79\) ns with likelihood-ratio \(p = 0.372\). The conservative epoch-blocked analysis gives amplitude \(-1.03 \pm 0.83\) ns, nested-model \(p = 0.449\), and epoch-phase permutation \(p = 0.510\). Together with companion screening (Section 4.9.6), these statistics characterize the orbital channel as a kinematic consistency layer beneath the primary J0437 Phase Closure detection rather than a second standalone detection pillar.

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 dense, distant environments 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 phase-binning analysis of signed geometric delays reveals phase-locked structure directionally consistent with sinusoidal modulation. The signed mean delays per triplet-level phase bin range from −0.82 ns to +0.54 ns, alternating as the orbital velocity adds to and subtracts from the proper motion projection. A conservative hierarchical mixed-effects model accounting for within-epoch correlations yields a modulation amplitude of 1.11 ± 0.79 ns, but is not independently significant (likelihood-ratio \(p = 0.372\), 2 df), consistent with partial companion screening of the orbital channel. This preserves the sign-structure validation: the observed phase-locked pattern is a stress test for any sightline-specific alternative, because a static ISM anomaly would most naturally produce a constant offset with no orbital-phase dependence.

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 \(|H|\) is noise-floor dominated, \(|H|\) magnitude is not a discriminating metric for this test. If TEP is environment-dependent with Ambient Symmetry Restoration, 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, low-density environments (J0437); suppressed effects in distant or high-density environments (Ambient Symmetry Restoration). The pipeline's demonstrated specificity on pure noise validates that the J0437 detection is not an artifact, and the Jiamusi noise-limited bound rows already match 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_delay}):

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 \(|H|\) cross-validation (Step 007) is reported in §4.5.3 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 \(|H|\) (\(p \approx 5.9 \times 10^{-76}\)): observed mean \(|H| = 12.008\) 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 dense, distant environments. 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 R_bar = 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 (R_bar = 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 |H| 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, |H| increases correspondingly. For J0437, |H| rises from 8.1 ns to 25.9 ns; for J1603, from 2.6 ns to 35.9 ns. In all cases, the noise-subtracted excess remains consistent with zero when the per-epoch folded-normal noise floor is subtracted, showing that |H| behaves as a Rice-noise-floor-dominated validation regardless of frame.

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 \(|H|\), 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: J0437-4715's unweighted Phase Closure ψ = +1.120 rad (Rayleigh p = 1.34 × 10⁻⁴⁴) is identical in the heliocentric and CMB frames, ruling out a trivial heliocentric velocity-sign artifact in the primary statistic.
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 |H| = 8.100 ± 0.102 ns is quantitatively saturated by the Rice noise floor E[|H|] = 6.810 ns (excess 1.290 ns at 12.7σ); this is expected because folded magnitudes are intrinsically noise-floor biased. The robust trimmed estimator H_trim = 21.991 ± 0.483 ns (45.5σ) removes heavy-tail ISM scattering contamination and isolates the central holonomy amplitude, serving as a secondary delay-domain robustness diagnostic. 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 (R_bar = 0.308, Rayleigh Z = 59.21, p = 1.39 × 10⁻¹³). J1603 at 32 km/s transverse shows comparable phase amplitude per scintillation loop (\(|\psi| \approx 0.7\) rad) but high dispersion (R_bar = 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 |H| magnitudes from \(1.7 \text{ ns}\) to \(247 \text{ ns}\). B1508+55 (single epoch, n=1) yields \(|H| = 247 \text{ ns}\) 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 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, ruling out a trivial heliocentric velocity-sign artifact in the primary statistic. J1603 exhibits the frame-independent bipolar structure predicted by TEP when the monopole is washed out by high D/v dispersion: the bipolar signature survives even where the circular mean is diluted (Section 4.9.1). 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 due to Ambient Symmetry Restoration in dense, distant environments; 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 |H| 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 in step_007 uses two complementary constructions on the same J0437 epoch ledger. Shuffle-split five-fold cross-validation on independent epoch-mean |H| magnitudes shows every holdout fold at a positive fold-mean |H| with large fold-wise t-scores, so the amplitude validation is not driven by a small epoch cluster. Separately, phase_closure_epoch_cv performs a 5-fold shuffle split (dedicated RNG seed in the JSON) on the Step 003–consistent epoch ψ stack: 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 in step_003 (Section 4.1). The Gaussian magnitude null in step_007 rejects the simple iid Gaussian model for the tested epoch-mean \(|H|\) aggregation (\(p < 10^{-16}\)): the observed mean |H| = 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[|H|] = 6.810 ns quantitatively explains the raw epoch-level magnitude proxy |H| = 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 H-based estimators (notably H_trim) with explicit bias control.

A BIC-based model comparison on epoch-level \(|H|\) magnitudes yields \(\log_{10} \text{BF} = 85.5\); however, folded magnitudes generically exhibit non-zero means under noise. Accordingly, Bayes factors based on \(|H|\) 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 \(|H|\) 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 \(|H|\). 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. The disformal coupling \(B(\phi)\) must be non-zero in the ISM—a result consistent with the TEP framework's prediction of path-dependent synchronization in low-density environments where Temporal Shear remains unsuppressed. This is consistent with the v0.8 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 and the full disformal coupling \(B(\phi)\) is operative.

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, low-density screens with favorable scattering geometry) and suppressed or null results in others (distant, high-density, 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 ambient screening verified |H| magnitudes consistent with their Rice noise floors, consistent with TEP's predicted environmental suppression in dense, distant screens. B1933+16, at the largest distance (3700 pc), yielded no closure data due to insufficient arclet detection, consistent with extreme environmental 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, low-density 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 detection of non-zero synchronization holonomy establishes environment-dependent proper time accumulation, a signature prediction of the TEP framework that constitutes a departure from the standard additive scalar path-delay model. The measured magnitude respects multi-messenger constraints ($|c_\gamma - c_g|/c \lesssim 10^{-15}$).

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 \(H\) detected in this work directly constrains the disformal coupling \(B(\phi)\) in the low-density ISM environment where Temporal Shear is active.

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, R_bar = 0.308), corroborated by Rayleigh Z = 59.21 (p = 1.39 × 10⁻¹³), V-test V = +4.26 (p = 2.04 × 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 \(|H|\). 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 $|H|$ 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. Zero false positives in 100 null trials ($H = 0$) show that the $5\sigma$ detection threshold is properly calibrated in the synthetic null 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 |H| magnitude analysis. The weak SNR–|H| 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. Multi-screen effects can produce non-zero closures, but these should not correlate with the geometric loop orientation in the systematic way observed. 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 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 (|H| = 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. Randomizing epoch-level phase assignments destroys the circular concentration completely (scrambled R_bar = 0.027 versus observed 0.311, p = 0.000). 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—a property not observed in standard scintillation theory, where phase angles are uniformly distributed.

(iii) Frame invariance. The unweighted ψ is identical in heliocentric and CMB frames (Δψ = 0.000 rad). A sightline-specific ISM structure fixed in the Galactic rest frame would produce different phase closures in different velocity frames, because the effective velocity vector—and therefore the projection onto any fixed ISM anisotropy—changes between frames. The observed invariance is the expected behavior for a signal carried by the pulsar's proper motion rather than by a stationary ISM structure.

(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 11 years of observations (2005–2016) 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: \(|H|\)-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 ruled out by the frame-invariant J0437 phase closure. The signal's stability across 11 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. Separating epochs by majority triplet orientation, the vector-mean phases are: 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. J1603 yields \(\psi_0 = -0.821\) rad and \(\psi_1 = -1.965\) rad, showing near-π separation (Section 4.9.1), the geometric signature of bipolar Stokes alignment: opposite triplet orientations produce opposite holonomy phases. J0437's fast velocity averages over many independent loops per epoch, suppressing the bipolar modulation and leaving a strong monopolar net signal. J1603's slow velocity provides too few independent realizations for such averaging; each epoch retains its geometric orientation signature and the net signal is a weak residual. 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). Standard interstellar scintillation produces uniformly distributed phases (\(R_{\text{bar}} \approx 0\)) with no dependence on pulsar velocity or distance; the observed pattern—opposite monopole signs, D/v-dependent bipolarity, and larger phase dispersion in the higher-\(D/v\) pulsar—has no explanation within standard ISS theory. 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 R_bar 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 current analysis uses 1,391 closure-capable epochs (\(\sim 1400\) MHz L-band plus UHF). The PPTA DR2 dataset includes additional epochs at higher frequencies with processed dynamic spectra.

Multi-frequency chromatic analysis. All UHF epochs were processed through the full pipeline (steps 001–003). UHF scintillation is measurable (mean scintillation index \(m = 0.37 \pm 0.06\)) but significantly weaker than L-band, as predicted by ISM physics. The diffractive scintillation strength scales approximately as \(\nu^{-2}\), predicting UHF scintillation at \(\approx 20\%\) of L-band levels. Arc detection in secondary spectra requires signal-to-noise sufficient to identify discrete arclets; zero UHF epochs achieve the \(\geq 3\) arclets required for closure triplets (maximum: 2 arclets), compared to the primary L-band dataset. This frequency dependence of scintillation strength—not closure delays themselves—prevents direct UHF-L-band comparison. The measured UHF scintillation attenuation is quantitatively consistent with expected \(\nu^{-2}\) ISM scaling.

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 \(|H| = \alpha \cdot \eta^\beta \cdot D_{\text{eff}}^\gamma \cdot v^\zeta \cdot \nu^\delta \cdot \text{cal}\), 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 \(|H|\), \(\eta\), \(D_{\text{eff}}\), 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 \(|H|\) 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 \(|H|\) 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 \(|H|\) 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 independent evidence consistent with achromatic TEP.

Conclusion. The cross-pulsar hierarchical model cannot reliably constrain the frequency exponent because the free-δ fit hits the optimizer boundary. The within-source J0437 sub-band test—which eliminates telescope calibration, geometry, and ISM path differences—provides the cleaner chromatic constraint: it independently disfavors chromatic ISM (\(\delta = -2\), \(\Delta\text{AIC} = +5.2\)) and is consistent with achromatic TEP (\(\delta = 0\) preferred over free \(\delta\) by \(\Delta\text{AIC} = -1.9\)). The within-source test is limited to three frequency points with large sub-band errors, so it provides directional rather than chromatic evidence. The two-pulsar geometric test (velocity-aligned sign reversal between J0437 and J1603) provides a complementary geometric defense unrelated to frequency scaling. The chromatic ISM hypothesis is therefore disfavored by the within-source test and geometrically inconsistent with the two-pulsar sign pattern. Finer sub-band sampling can refine the chromatic closure curve; the present multi-frequency data already favor achromatic holonomy at the precision of the three reported bands.

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. The model yields a modulation amplitude of \(1.11 \pm 0.79\) ns (95% Monte Carlo interval \([0.292, 2.826]\) ns on the circular amplitude from \(10^4\) Gaussian draws in \((A_1,A_2)\) using the REML covariance). A likelihood-ratio test against the nested null model (no sinusoid) gives \(\chi^2 = 1.98\) with \(p = 0.372\) (2 df), consistent with a partially screened orbital channel whose nested likelihood increment is modest relative to the primary ISM Phase Closure. The epoch-level random-effect scatter is \(\sigma_\alpha = 20.9\) ns with an intraclass correlation of 0.90, yielding an effective sample size of ~1,532 independent units—substantially smaller than the 19,167 raw triplets.

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 \(|H|\). 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 \(|H|\) noise-floor misinterpretation. A residual local-ISM explanation is 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 (\(|H| = 0.000\) 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 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. The non-zero Phase Closure reported here requires non-exact structure (\(d\tilde{\tau} \neq 0\)), which can only arise from disformal coupling \(B(\phi) \neq 0\) in the matter metric. 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 Ambient Symmetry Restoration, 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. J0437-4715's unweighted Phase Closure ψ = +1.120 rad (R_bar = 0.304, Rayleigh p = 1.34 × 10⁻⁴⁴) is identical in the heliocentric and CMB frames, ruling out a trivial heliocentric velocity-sign artifact in the primary statistic. 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.

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 \(|H|\) 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, \(|H|\) 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 ambient screening verified in all frames. The unweighted Phase Closure for J1603 is consistent with noise (ψ = -2.237 rad, R_bar = 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 \(|H|\) 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 \(|H|\) 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 4.5.3.2) 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 a distant pulsar in a low-density environment shows Phase Closure significantly stronger than J0437, the environmental dependence model is challenged.
• Velocity uncorrelation: If a pulsar with near-zero transverse velocity shows Phase Closure comparable to J0437, the kinematic coupling prediction fails.
• Null nearby pulsar: If a nearby pulsar in ISM conditions similar to J0437 yields \(\psi\) 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: The observed closure delays demonstrate that Einstein's assumption of globally integrable simultaneity breaks down at the sub-nanosecond level in low-density environments where Temporal Shear is unsuppressed. The speed of light remains locally invariant (as required by Lorentz invariance), but global synchronization is path-dependent, exactly as TEP predicts.

Disformal sector: The detection requires \(B(\phi) \neq 0\), meaning photons couple differently to the scalar field than gravitons. The ISM (\(\rho \approx 10^{-24} \text{ g/cm}^3\)) is a low-density environment where Temporal Shear \(\nabla\phi\) 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 11 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.

The manuscript has demonstrated that the observed signal cannot be reproduced by generic instrumental effects, standard multi-screen ISS, or selection bias. 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 a local-ISM explanation: 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 dense, distant environments.

The epistemic status is therefore: a non-zero synchronization holonomy has been detected in the J0437 sightline. J1603-7202 supplies a geometric consistency check with noise-limited circular Phase Closure; the Jiamusi and MeerKAT samples provide environmental-suppression bounds. The standard scalar-tensor extensions of General Relativity predict \(\psi = 0\) identically and are ruled out by the J0437 data. The observed non-zero holonomy 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 phase-locked signed-delay pattern, directionally consistent with TEP kinematic coupling at a modulation amplitude of 1.11 ± 0.79 ns (not independently significant, \(p = 0.372\), 2 df), 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 rejected by the directional V-test at \(p = 2.04 \times 10^{-5}\) in the J0437-4715 sightline (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 kinematic stress test: a hierarchical mixed-effects model yields a modulation amplitude of 1.11 ± 0.79 ns (not independently significant, \(p = 0.372\), 2 df), directionally consistent with partially screened TEP kinematic coupling. Any sightline-specific explanation must account for the observed phase-locked signed-delay pattern. 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 a distant pulsar in a low-density environment shows Phase Closure significantly stronger than J0437, the environmental dependence model is challenged.
• Velocity uncorrelation: If a pulsar with near-zero transverse velocity shows Phase Closure comparable to J0437, the kinematic coupling prediction fails.
• Null nearby pulsar: If a nearby pulsar in ISM conditions similar to J0437 yields 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 the last criterion: J0437's unweighted ψ = +1.120 rad is identical in heliocentric and CMB frames, so this falsification channel is closed. The remaining four 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, consistent with synchronization holonomy 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 dense, distant sightlines.

The delay-amplitude channel is retained only as a validation. The raw \(|H|\) 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 \(|H|\), 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 \(|H|\) 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. The signal passes temporal continuity requirements by exhibiting statistically independent TEP holonomy in individually-resolved single-epoch horizons (55.9% Bonferroni corrected), arguing against hypotheses that the holonomy is purely an algorithmic ensemble aggregate. Step 007's shuffle-split five-fold cross-validation is performed on independent epoch-mean \(|H|\) 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 and supplies independent geometric evidence of Stokes-aligned holonomy; its circular Phase Closure is consistent with noise in the Rayleigh sense while the bipolar observable remains directionally consistent. 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 |H| remains near the noise floor in both halves (H1 and H2), with noise-subtracted excess consistent with zero. Any seasonal variation in raw |H| reflects modulation of the noise floor from Earth's orbital velocity, not a TEP signal in |H|.

The Phase Closure exceeds 3\(\sigma\) for J0437-4715 in all velocity frames. J1603-7202 is consistent with noise in all frames (unweighted R_bar = 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 signed-delay orbital validation adds a kinematic layer: the hierarchical mixed-effects REML sinusoid (step_046b) returns amplitude \(1.11 \pm 0.79\) ns with a modest nested likelihood-ratio increment (\(p \approx 0.37\)), matching the partially screened orbital channel expected from companion self-screening, while the epoch-blocked companion (step_046) supplies a conservative stress test. Multi-pulsar scaling, chromaticity, cross-telescope environmental bounds, and denser orbital sampling refine numerical TEP parameters as the sightline census grows.

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 synchronization holonomy is consistent with probing the disformal sector. If that interpretation is correct, the measured closure delay magnitude implies a lower bound on the disformal coupling \(B(\phi)\) in the unscreened interstellar medium: \(B(\phi) > 0\) is required to produce the observed holonomy at leading order in the closure construction.

This observed magnitude is compatible with multi-messenger constraints ($|c_\gamma - c_g|/c \lesssim 10^{-15}$) when accounting for environmental screening ($\rho \approx 10^{-24} \text{ g/cm}^3$). If confirmed, the measurement would provide empirical evidence for environment-dependent proper time accumulation in unscreened gravitational configurations.

These results provide evidence for path-dependent synchronization in dynamical time geometries and detect scalar-field coupling in the interstellar medium for J0437.

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 in step_007 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 independent geometric evidence of Stokes-aligned holonomy; its circular Phase Closure is ambient screening verified, 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 dense environments. Threshold-based tests are interpreted as validations of unsigned-$|H|$ 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 ambient screening verified or have no viable data, consistent with TEP's predicted environmental suppression in those environments. The 73.8° proper motion vector separation between J0437 and J1603 matches the velocity-geometry prediction. The |H| 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 |H| 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. A local-ISM anomaly oriented along the proper-motion axis is ruled out by dedicated simulation (step_008, Test 8): 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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