Temporal Equivalence Principle: Temporal Shear in the Earth Flyby Anomaly

1. Introduction

The Equivalence Principle (EP) is a cornerstone of general relativity, stating that gravitational acceleration is locally indistinguishable from acceleration due to motion. However, the Temporal Equivalence Principle (TEP)—the assertion that proper time is governed by a dynamical scalar field whose conformal gradients and disformal velocity-dependent terms can produce screened trajectory-level residuals—posits that the rate of time is a dynamical scalar field $\phi$. This framework, established in Paper 0, proposes that all non-gravitational matter couples universally to a causal matter metric $\tilde{g}_{\mu\nu} = A^2(\phi)g_{\mu\nu} + B(\phi)\nabla_\mu\phi\nabla_\nu\phi$, where $A(\phi) = \exp(\beta_A \phi/M_{\text{Pl}})$. In the flyby regime the disformal term is subdominant and the conformal sector dominates the clock-sector response.

1.1 The Earth Flyby Anomaly

Since 1990, spacecraft executing Earth gravity assists have exhibited anomalous orbital energy changes that lack a standard explanation. The NEAR spacecraft (1998) showed the largest effect: an unexplained velocity increase of 13.46 mm/s. Galileo (1990, 1992) and Cassini (1999) displayed smaller but significant anomalies. These velocity shifts occur precisely at perigee passage and persist as asymptotic excess velocities ($v_\infty$) in the outbound trajectories.

The published residuals motivate tests of both physical signals and tracking/reconstruction systematics. The physical checks in Section 5.2 bound several conventional contributions but do not exclude every systematic explanation. TEP supplies a distinct candidate clock-sector artefact through trajectory-dependent proper-time transport in links carrying an onboard time reference; Section 5.2 establishes that this term does not account for the published coherent-link catalogue.

1.2 TEP as a Candidate Explanation

The TEP framework provides a testable clock-sector response for flyby observables that carry the spacecraft's own time reference. Under Paper 0's amplitude–shear split (v0.15), the Earth-vicinity Temporal Shear is strongly screened by the two-body kinetic operator, so the anomaly cannot be a shear-force effect. Instead, highly asymmetric hyperbolic flybys accumulate rapid proper-time offsets at perigee through the unsuppressed amplitude channel; orbit-determination codes, reconstructing velocities under an assumed GR time standard, misread these offsets as $\Delta V$ discontinuities wherever the tracking link carries the spacecraft's own time reference — the Clock-Sector Reduction Artefact of Paper 0, whose observable reach the channel analysis of Section 5.2 fixes exactly.

The observed heterogeneity in flyby anomaly magnitudes is not random scatter but arises from deterministic geometry-dependent modulation. The TEP prediction for a given flyby depends on several physical factors: (1) perigee altitude (determines the clock-sector field excursion via the density-dependent amplitude profile), (2) approach-departure asymmetry (disformal coupling requires velocity-dependent anti-aligned geometry), (3) the perigee plasma template layer (a bounded heuristic envelope coefficient; the stronger Debye attenuation ansatz is quarantined at $S_{\rm plasma} = 1$, Section 3.10), (4) solar activity (modulates ionospheric density within that template layer), and (5) cosmographic CMB-frame velocity geometry (the disformal coupling scales as v² in the ambient-field rest frame — the frame in which the spatial gradient of the cosmological temporal field vanishes — approximated operationally by the CMB dipole frame)—a deeper prediction tested only in an exploratory capacity (Section 4.11). These factors combine to produce a wide span in per-flyby fitted β_{\rm fit} across the Step 008 S/N-qualified ensemble; the legacy inverse-variance diagnostic restricted to sign agreement at $\beta_{A,\text{ref}}$ (NEAR, Galileo 1990, Rosetta 2005) is reported separately as beta_statistics_sign_gated_diagnostic in results/step008_fitting_results.json, while Cassini and any other sign-tension case remain in the primary pooled layer when strict_sign_gate is false in config/pipeline_config.json. The Temporal Topology screening mechanism is essential for three reasons: (1) it ensures the flyby amplitude and environmental response remain consistent with solar system constraints; (2) it explains both detections and null results through environment-dependent screening; and (3) it establishes the transition radius $R_{\rm sol} \approx 4146$ km as a universal scale. Flybys sampling regions of large field excursion (low altitude, high asymmetry) exhibit anomalies, while those in shielded regimes (high altitude or symmetric trajectories) remain null.

1.3 This Work

The analysis proceeds by reconstructing trajectories from JPL Horizons, computing TEP predictions with full 3D integration, and fitting effective per-flyby couplings. Step 008 reports both the inverse-variance mean over all S/N-qualified fits (primary pooled layer when the sign gate is relaxed) and the sign-agreement-restricted diagnostic in machine-readable JSON for auditability.

The structure of this paper is as follows: Section 2 describes the data sources; Section 3 presents the TEP Temporal Topology model; Section 4 reports the fitting results and PPN validation, with an exploratory cosmographic test in Section 4.11; Section 5 discusses the clock-sector artefact interpretation; and Section 6 concludes with prospects for further tests.

2. Observations and Data

2.1 The Flyby Spacecraft Sample

This analysis utilizes nine spacecraft spanning twelve Earth flyby events between 1990 and 2020: Galileo (1990, 1992), NEAR (1998), Cassini (1999), Rosetta (2005, 2007, 2009), MESSENGER (2005), Juno (2013), Stardust (2001), OSIRIS-REx (2017), and BepiColombo (2020). The dataset is divided into three data quality classes: published anomalies (six flybys with measured nonzero Δv and formal uncertainties — the four positive detections plus the two negative or marginal entries of Anderson et al. 2008 Table I), published nulls/bounds (three flybys with explicitly reported null results or upper limits), and no public anomaly report (three flybys with no published search or measurement). The latter class is not used in quantitative likelihood. Table 1 summarizes the key parameters for each flyby.

Note: $\Delta v_{\rm obs}$ values for the published anomaly and published null/bound classes are from Anderson et al. (2008) and companion papers. Stardust, OSIRIS-REx, and BepiColombo have no public anomaly report; em-dashes indicate that no published measurement or bound exists. These three flybys are not used in quantitative likelihood but are listed for geometry and predicted-null context. Rosetta 2009 is a published null (Müller et al. 2010); the archival catalogue assigns $\sigma = 0.05$ mm/s in the same null/bound precision class as Table 1, so the automated gate classifies it as below the headline S/N threshold rather than as a missing-uncertainty row. Perigee values are altitudes above Earth's surface; $v_\infty$ is the hyperbolic excess velocity. The Galileo 1992, Cassini, and MESSENGER entries follow the Anderson et al. (2008) Table I primary-source values (−4.60 ± 1.00, −2 ± 1, and +0.02 ± 0.01 mm/s respectively), confirmed against the mission declination reconstruction in Table 1a.

2.2 Data Sources and Provenance

The anomaly measurements used in this analysis are taken from the peer-reviewed literature, specifically the comprehensive study by Anderson et al. (2008) and subsequent mission-specific analyses. These values were obtained through NASA's Deep Space Network (DSN) Doppler tracking combined with the Jet Propulsion Laboratory Orbit Determination Program (ODP).

Literature sources:

• Primary reference: Anderson, J. D., et al. (2008). "Anomalous Orbital-Energy Changes Observed during Spacecraft Flybys of Earth." Physical Review Letters, 100(9), 091102.
• Rosetta analysis: Morley, T., & Budnik, F. (2007). "Rosetta Navigation at its First Earth-Swingby." Proceedings of the 20th International Symposium on Space Flight Dynamics.
• Juno analysis: Aksenov, E. L., & Tuchin, A. G. (2020). "Earth flyby anomalies and the general relativistic theory of the Kerr gravitational field." MNRAS, 492(3), 3703-3711.

2.3 Data Quality Assessment

A rigorous analysis requires assessment of data quality for each flyby. All six published detections have complete DSN coverage spanning $\pm 12$ hours around perigee, enabling robust pre/post comparison (catalog-level statement; the inverse-variance $\beta_{\rm fit}$ ensemble in Steps 008 and 026 uses $n=6$ S/N-qualified primary fits when strict_sign_gate is false; with the corrected primary-source anomalies, all six detections agree in sign with the TEP prediction at the reference coupling, so the sign-agreement diagnostic coincides with the gated set). The reported uncertainties (0.01–1.00 mm/s) span the DSN Doppler precision range for the different missions and tracking configurations.

Ensemble gates (Steps 008 and 026) are applied mechanically: a flyby enters the restricted likelihood only if it carries a published $(\Delta v, \sigma)$ pair, satisfies $|\Delta v|/\sigma > 2$, and shows sign agreement between $\Delta v_{\rm obs}$ and the TEP prediction at the pre-specified reference coupling. Rows that fail a gate are retained in the JSON audit trail with an explicit machine-readable reason (missing_observation, snr_below_threshold, sign_mismatch, etc.) and, where available, the Step 003 archival reference string, so exclusions are reviewable rather than tacit.

Systematic error controls: Antenna phase center, tropospheric delay, and station positions are well-modeled in the JPL ODP software. Residual uncertainties are at the $\sim 0.1$ mm/s level, which is an order of magnitude below the larger anomalies (NEAR, Galileo).

2.4 Trajectory Data from JPL Horizons

Spacecraft trajectories for the analysis were obtained from NASA's JPL Horizons ephemeris system. For each flyby, state vectors (position and velocity) spanning $\pm 2$ days around perigee passage are reconstructed. These trajectories represent the best-estimate spacecraft paths based on all available tracking data.

2.4a Geometry Factor Audit: Published Declinations vs Horizons

The trajectory asymmetry factor used in both the Anderson et al. (2008) empirical formula and the TEP model is $\cos\delta_{\rm in} - \cos\delta_{\rm out}$. Table 1a audits the provenance of this factor by comparing three independent estimates for each flyby with published declinations: (i) the Anderson et al. (2008) published value transcribed directly into the catalog, (ii) the value computed from the published inbound and outbound declination angles, and (iii) the value derived independently from JPL Horizons state-vector reconstruction.

Key finding: With the re-transcribed Anderson et al. (2008) Table I declinations, the computed $\cos\delta_{\rm in} - \cos\delta_{\rm out}$ agrees with the published asymmetry factor for every row, and both agree with the independent JPL Horizons state-vector reconstruction. An earlier version of this catalogue carried declination transcription errors (most consequentially NEAR's outbound leg, entered as $-19.7^\circ$ rather than $-71.96^\circ$, and both Galileo 1992 legs); those errors manufactured an apparent convention mismatch between the tabulated declinations and the published asymmetry factors that does not exist in the primary source. The three-way agreement now shown — published asymmetry, recomputed from the published declinations, and Horizons-derived — removes the declination-provenance question entirely and validates the adopted geometry factors used by both the Anderson empirical model and the TEP kernel. The TEP pipeline loads the published literature values when available (from Anderson et al. 2008 Table I) and computes from Horizons-derived declinations otherwise.

Provenance of the corrected values. The transcription errors were identified by auditing the catalogue against the Anderson et al. (2008) primary source and its companion trajectory paper (Anderson, Campbell & Nieto 2006), cross-checked against independent reproductions. Corrections were confined to the published tabulated quantities: NEAR's outbound declination ($-71.96^\circ$, not $-19.7^\circ$) and uncertainty ($\pm 0.13$ mm/s, not $\pm 0.01$), Galileo 1992's two declinations ($-34.26^\circ/-4.87^\circ$, not $-30.3^\circ/-33.8^\circ$) and its anomaly ($-4.60 \pm 1.00$ mm/s, previously entered as a null), Cassini's anomaly ($-2 \pm 1$ mm/s, previously entered as $+0.11 \pm 0.05$), MESSENGER's anomaly ($+0.02 \pm 0.01$ mm/s, previously a null), Galileo 1990's inbound declination ($-12.52^\circ$, not $-25.1^\circ$) and uncertainty ($\pm 0.08$ mm/s, not $\pm 0.03$), and Rosetta 2005's anomaly ($+1.80 \pm 0.03$ mm/s) and inbound declination ($-2.81^\circ$, not $-3.4^\circ$). Perigee altitudes were likewise aligned to the primary-source tabulation (539, 960, 303, 1175, 1955, 2347 km). All downstream results in this paper use the corrected catalogue; the corrected geometry factors are corroborated independently by the Horizons column of Table 1a.

3. Methodology

The analysis employs a four-step pipeline to compare the published flyby anomaly geometry with the TEP response template and to test the clock-sector artefact separately in the link classes where it survives. The pipeline retrieves spacecraft trajectories from JPL Horizons, computes TEP predictions for each flyby geometry using full 3D integration, fits the coupling parameter $\beta_{\rm fit}$ to match observed anomalies, and validates all parameters against solar system PPN constraints. Which measurement classes can carry the clock-sector term is itself part of the analysis: the channel accounting of Section 5.2 shows the spacecraft's conformal excursion survives only in clock-carrying links (one-way, non-coherent, or regenerative-ranging observables), while the published catalogue is assigned to the coherent-transponder class on the literature-based ledger of Section 5.2, pending encounter-specific tracking verification.

3.1 Data Acquisition

Spacecraft trajectories are obtained from the NASA JPL Horizons ephemeris system via direct HTTP query to the batch CGI endpoint. For each flyby, observational quantities (right ascension, declination, range, and range-rate) are retrieved in the ICRF (International Celestial Reference Frame) at 1-minute intervals spanning $\pm 2$ days around perigee passage; three-dimensional state vectors are reconstructed from these observables in a subsequent processing step.

Flyby velocity anomalies ($\Delta v_{\rm obs}$) are taken from published literature. The primary source is Anderson et al. (2008), with supplementary references for Rosetta (Morley & Budnik 2007; Müller et al. 2008, 2010) and Juno (Aksenov & Tuchin 2020). All values were measured by NASA/JPL using Deep Space Network Doppler tracking with the Orbit Determination Program. Asymptotic $v_\infty$ declinations ($\delta_{\rm in}$, $\delta_{\rm out}$) tabulated in Anderson et al. (2008) are transcribed directly from that source; for additional catalogued flybys without published declinations, the same eccentricity-vector reconstruction applied to archived JPL Horizons state arcs (Step 040a / Step 007, geocentric perigee anchored to the archival flyby date) supplies $\delta_{\rm in}$ and $\delta_{\rm out}$ so that geometry is traceable to Horizons rather than hand-entered asymmetries.

3.2 TEP Temporal Topology Model

The TEP framework provides a quantitative model for the flyby anomaly through the perigee proper-time offset arising from the temporal-field amplitude excursion. With the universal conformal coupling A(φ) = exp(β_A φ/M_{\rm Pl}), the accumulated offset along the trajectory projects into an apparent velocity discontinuity under GR-based orbit determination:

Under the amplitude–shear split (Paper 0 §7), the Earth-vicinity Temporal Shear is screened by the two-body operator, so this kernel is not a physical force on the spacecraft — the shear-force channel is closed. It is the orbit-determination-equivalent kernel of the clock-sector artefact in a clock-carrying link: the spacecraft's matter-metric rate offset $\eta(t) = \exp(\beta_A\psi/M_{\rm Pl})-1$ carries the same radial gradient $\beta_A\nabla\phi/M_{\rm Pl}$, and a tracking filter that assumes the GR time standard reconstructs the accumulated rate-offset excursion exactly as if it were this impulse. The two signatures are observationally indistinguishable over a flyby arc where the term is present in the observable — one-way, non-coherent, or regenerative-ranging links; in the coherent two-way class of the published catalogue the transponder endpoint cancels it identically (Section 5.2). The mechanism-faithful forward model is the Step 043 OD rederivation (Table 3h), which computes the artefact's signature on the clock-carrying link class over the real arcs. Here β_{A,\rm eff} = β_A × S_⊕(r) is the effective artefact-response amplitude, where S_⊕(r) is the continuous clock-amplitude response profile. The characteristic surface ratio S_⊕ ≈ 0.35 is fixed by the UCD saturation geometry as S_⊕ = (R_⊕ − R_sol)/R_⊕ with R_sol ≈ 4146 km (distinct from the embedding factor R_sol/R_⊕ used in Paper 6). The radial component of the apparent impulse is indistinguishable from a small shift in GM and is absorbed by orbit determination. The non-radial component—modulated by Earth's oblateness (J2, J3, J4) and the spacecraft's trajectory geometry—produces the apparent net velocity change that reads as the flyby anomaly.

The predicted velocity discontinuity is resolved through rigorous numerical integration of the artefact kernel along the trajectory in the Earth-centered inertial (ECI) frame — the accumulated apparent impulse that an orbit-determination filter would attribute to dynamics, not a physical trajectory deflection. This approach captures the accumulated proper-time offset as the spacecraft traverses the varying field amplitude, incorporating a 4th-order Spherical Harmonic Expansion (SHEX) for the geopotential to ensure that local gravitational perturbations are not conflated with the clock-sector response:

where the simplified perigee approximation is:

where:

• $(d\phi/dr)_{r_p}$ is the scalar field gradient at perigee altitude
• $r_p$ and $v_p$ are the perigee distance and velocity
• $J_2, J_3, J_4$ are the zonal harmonics (EGM96/WGS84 coefficients)
• $\lambda_p$ is the perigee latitude
• $\delta_{\rm in}$ and $\delta_{\rm out}$ are the asymptotic declinations on approach and departure (from Anderson et al. (2008))

3.2b Full 3D Trajectory Integration

The perigee approximation provides a computationally efficient closed-form estimate, but the primary predictions are validated against full 3D trajectory integration to capture the accumulated proper-time offset along the spacecraft path. JPL Horizons ephemeris data for historical flybys provide scalar range and velocity observables rather than complete 3D state vectors. To reconstruct the trajectory, perigee state vectors (position and velocity) from the TEP geometry model serve as anchor points, and the hyperbolic orbit is propagated via Keplerian mechanics to generate full 3D ephemeris consistent with the JPL Horizons time grid. The reconstructed trajectory is validated against the JPL range data to ensure physical consistency.

The apparent velocity discontinuity is accumulated by integrating the TEP clock-sector response along the reconstructed path:

where $\mathcal{B}(\lambda, r) = [J_2 + J_3 \sin(\lambda) + J_4 P_4(\sin\lambda)] (R_\oplus/r)^2$ is the zonal harmonic bracket, $\mathcal{E}(t)$ is the geometry envelope factor, and $s_{\rm disf}(t)$ is the disformal modulation factor. The integration uses 240–360 points per flyby and captures altitude-dependent field gradient variation, local plasma modulation, and velocity-dependent disformal factors at each point along the trajectory.

Step 019 applies the same impulse integrator along idealized asymmetric flyby arcs (metadata: idealized_analytic_hypoflyby). On comparable rows of Table 3b (Section 4.1.2), the ratio $\Delta v_{\rm 3D}/\Delta v_{\rm simplified}$ differs from unity by tens of percent for the three gated primaries—a consistency check inside the analytic path class—while ratios against the Step 007 perigee impulse are deliberately not marketed as ±10% “agreement.” Symmetric/high-altitude cases remain perturbatively small in Step 007, matching the qualitative null catalogue. Ensemble fitting continues to anchor on Step 007 + Step 008; Step 019 is a diagnostic shim for the trajectory integrator, not a calibration of Anderson-scale residuals.

J3 contribution: The J3 term adds a latitude-dependent asymmetry to the non-radial component of the artefact kernel. However, J3 is two orders of magnitude smaller than J2 ($|J_3/J_2| \approx 2.3 \times 10^{-3}$), and its inclusion does not collapse the heterogeneity in fitted β_fit across the gated ensemble (factor ~3.3 across the tightly constrained members at fixed reference coupling, extending to ~15 with the negative-branch Galileo 1992 member, with formal Cochran Q ≫ 1). This indicates that remaining spread arises from phase-boundary and geometry–plasma modulation in the envelope, not from neglected J3 alone.

The scalar field φ relaxes outside Earth with a profile scale λ_T = 4200 km — the operational value anchored on the measured GNSS clock-correlation length L_c = 4201 km (±15% calibration band). Under the corpus's terrestrial transfer-sketch identification (Paper 6 §2) the exterior profile varies on 𝒪(R_T) with an undetermined order-unity prefactor, since a gradual screening transition produces no exterior structure sharper than the scale over which the integrated source charge changes over; the adopted value sits 1.3% above the ansatz target R_T = 4146 km, within the stated prefactor freedom and calibration band (the two are not independent: ρ_T was itself calibrated from L_c). This is a stated ansatz anchoring the profile scale — not a derived equality, and distinct from the in-medium Compton wavelength of a candidate chameleon completion.

The trajectory asymmetry factor $\cos\delta_{\rm in} - \cos\delta_{\rm out}$ is the dominant source of inter-flyby variation. Nearly symmetric trajectories (e.g., MESSENGER, $\cos\delta_{\rm in} - \cos\delta_{\rm out} \approx 0.005$) predict negligible anomalies, consistent with observations. Asymmetric trajectories (e.g., NEAR with $\cos\delta_{\rm in} - \cos\delta_{\rm out} = 0.625$) predict large anomalies. Cassini carries a small negative asymmetry factor ($-0.022$), verified by independent JPL Horizons reconstruction (Table 1a) — a physical property of the trajectory, not a convention artifact. The kernel therefore predicts a negative apparent anomaly, consistent in sign with the corrected $-2 \pm 1$ mm/s catalogue value, though the predicted magnitude is negligible and the amplitude shortfall remains a genuine limitation of the phenomenological kernel for this geometry, not a convention mismatch. The mechanism-faithful calculation is consistent with this reading: the Step 043 OD rederivation, which propagates the proper-time artefact through the batch least-squares filter rather than the kernel, returns only a negligible endpoint remnant for Cassini ($\Delta V_{\rm app} = +0.01$ mm/s, Table 3h) — the clock-sector channel does not produce the corrected $-2\pm1$ mm/s catalogue value, consistently with that datum's coherent two-way link class, in which the term is absent at the observable (Section 5.2).

Geometric screening: Critical to the flyby amplitude and environmental response is the transition radius $R_{\rm sol} \approx 4146$ km from the UCD saturation model (Step 010), anchored on the same GNSS terrestrial calibration that fixes $\rho_T$ — a shared-input relationship under the Paper 6 §2 identification (Section 3.11), not an independent validation. This defines the characteristic ratio $S_{\oplus} = (R_{\oplus} - R_{\rm sol})/R_{\oplus} \approx 0.35$, the responsive-shell fraction of the radial profile through which a transported clock accumulates its differential excursion. Its relationship to the canonical clock-amplitude operator is an exact partition rather than a separate estimate: under the uniform-density-equivalent construction $R_{\rm sol}^3 = 3M_\oplus/4\pi\rho_T$ and $R_\oplus^3 = 3M_\oplus/4\pi\bar\rho$, the canonical response evaluated at Earth's mean density reduces to $S_A(\bar\rho_\oplus) = (\bar\rho_\oplus/\rho_T)^{1/3} = R_{\rm sol}/R_\oplus \approx 0.65$ — the embedded, saturated fraction of the equivalent radius, identical to the UCD embedding factor used in Paper 6 (UCD). The flyby clock channel projects the complementary fraction $S_\oplus = 1 - S_A(\bar\rho_\oplus)$: inside $R_{\rm sol}$ the field is pinned and carries no radial relaxation, so the excursion is generated precisely across the unsaturated shell. The two quantities are a partition sum rule of one radial profile, not two estimates of one operator (machine-checked in Step 010's canonical_amplitude_partition output: $S_A + S_\oplus = 1$ identically).

The pipeline's field-amplitude normalization is the canonical amplitude-sector excursion of Paper 0 v0.15. The pipeline carries the excursion complement $\phi(r)$ — zero at the pinned interior floor, rising to the ambient reference $\phi_{\rm space}$ — so that the physical excursion $\psi(r) = \phi_{\rm space} - \phi(r) \geq 0$ is the temporal-well depth sampled by a transported clock: positive inside the well and relaxing to zero at the ambient baseline, matching the corpus convention $\phi > 0$ in wells, $\phi \to 0$ ambient (Paper 0 Rule 4). The normalization is the unscreened linear response of the source computed in the Paper 0 radial-ODE closure (TEP/results/step_01_radial_ode.json, Earth entry):

This value is parameter-free — fixed by Earth's measured mass and radius — and independent of the fitted response amplitude $\beta$, so every pipeline prediction is strictly linear in $\beta$. It replaces the v0.1 density-minimum prescription $\phi_{\rm min}(\rho) \propto \rho^{-1/(n+1)}$ (an inverse-power/chameleon-class realization): under $\beta_A = -1$ that branch admits no stable minimum (Paper 0 constraint F1) and runs the field largest in vacuum, opposite to the corpus convention. The Step 010 self-consistent-field diagnostic on that realization is retained solely as the documented contrast — it returns $S_\oplus \approx 1.0$, i.e. the excluded branch produces no terrestrial suppression shell.

Fundamental coupling: The bare conformal coupling is β_A = −1.0, established in Paper 0 (Jakarta) §2.2. The fitted values reported here (β_{\rm fit} ∼ 10⁻³) are the fitted response amplitudes; the effective clock-sector response β_{A,\rm eff} = S_\oplus(r)\,\beta_{\rm fit} is derived from these under the excursion projection S_\oplus(r) defined above. The sign of the artefact contribution in the flyby geometry is determined by trajectory asymmetry; the reported β_{\rm fit} magnitudes should be compared to the absolute fundamental scale |β_A| = 1.0 — the ratio κ_flyby = β_fit/|β_A| ≈ 4.7×10⁻³ is the population-level channel response coefficient, reported as an unexplained transfer-map quantity of the same open class as the Paper 0 Γ_X entries pending the action-level derivation. The fitted flyby amplitude is a trajectory-response coefficient; Solar-System PPN bounds apply to the screened source-charge projection α_{\rm eff} = S_Σ α_0, not to the fitted flyby amplitude itself. The converse direction closes identically: terrestrial bounds on the Earth environment constrain the shear channel, for which the corpus's own two-body operator returns an Earth-vicinity suppression of ∼10⁻²¹, whereas the claimed flyby observable is the unsuppressed clock-amplitude channel — and even there it is a transport excursion accumulated by a clock carried through the relaxing radial profile, not a stationary rate offset that inter-comparisons between clocks at rest in the terrestrial environment could bound. The same unsuppressed terrestrial amplitude response is the structure that the corpus's GNSS coherence-length measurement anchors (L_c = 4201 km, Section 3.11), so the fitted excursion scale is consistent with — indeed required by — the corpus's own terrestrial calibration rather than bounded away by it.

Ambient field value: Under the canonical excursion normalization, the ambient reference carries the full well depth φ_space = Δφ = 3.39×10⁹ GeV in complement units, while the physical excursion ψ(r) decays to the interplanetary baseline over λ_T — no ad-hoc density cutoff enters, since the profile is geometric rather than density-minimum-derived. The normalization is β-independent, so the reference amplitude β = 10⁻⁴ enters only as a linear scaling of the response, not of the field.

Geometry modulation factors: Per-flyby fitted β_fit in the six-member sign-consistent ensemble spans roughly a factor of three across the tightly constrained members (extending to a factor of ≈15 across the four amplitude-informative fits) — formally divergent for the near-zero-prediction Cassini and MESSENGER members — reflecting substantial geometry-, plasma-, and velocity-dependent modulation encoded in the Step 007 envelope. Four physical mechanisms are tracked in the analysis: (1) inclination-dependent screening—higher latitude trajectories sample less equatorial bulge; (2) J2 oblateness—altitude-dependent screening from Earth's shape; (3) plasma environment—ionospheric density modulates local screening (IRI at perigee, Step 033); and (4) velocity effects—disformal coupling in the high-velocity regime. These factors are incorporated into the clock-sector response calculation.

3.2c Component-Level Geometry Factor Analysis

To address the extreme heterogeneity in fitted β_fit values across flybys (see Section 4.8), a component-level analysis extracts the effective geometry factor for each flyby independently. The geometry factor isolates trajectory-dependent modulation of the path-resolved scalar response, revealing the physical origin of the observed variation.

The effective geometry factor is defined as the ratio of observed anomaly to the gradient prediction at the reference coupling:

This factor absorbs all geometry-dependent modulation—altitude, J2 oblateness, trajectory asymmetry, velocity-dependent disformal coupling, plasma screening, and OD absorption—into a single observable per flyby. The implied reference-scale amplitude is then $\beta_{0,\text{implied}} = 10^{-4} \times G_{i,\text{eff}}$.

Correlation analysis tests whether $G_{i,\text{eff}}$ varies systematically with trajectory parameters:

• Altitude: higher perigee → lower $G_{\text{eff}}$ (weaker field gradient)
• Velocity: higher $v_p$ → lower $G_{\text{eff}}$ (shorter field exposure time)
• Asymmetry: positive $\cos\delta_{\text{in}} - \cos\delta_{\text{out}}$ → higher $G_{\text{eff}}$ (stronger disformal enhancement)

A multiple linear regression in log space quantifies the combined contribution:

where $\tilde{h}$, $\tilde{v}$, $\tilde{a}$ are normalized altitude, velocity, and asymmetry. Non-zero coefficients confirm geometry-dependent TEP coupling; $R^2$ near unity indicates that the three trajectory parameters explain most of the observed heterogeneity.

This approach is complemented by a four-parameter hierarchical Bayesian model ($\beta_{\rm fit,0}$, $b_{\text{disf}}$, $\sigma$, $\alpha_{\text{res}}$) sampled via MCMC. The pre-computed gradient and disformal components from the TEP clock-sector response model contain the full perigee physics; the likelihood scales these channels through population-level hyperparameters, with any residual unmodeled modulation captured by $\alpha_{\text{res}}$. Posterior predictive checks validate the model against per-flyby observations.

3.2d Parameter budget: fitted coupling versus envelope heuristics

The restricted Step 008 tier fits a single amplitude parameter (shared β_fit rescaling of the reference prediction at β_A,ref = 10⁻⁴) on the sign-gated high-S/N ensemble. Separately, the Step 007 geometry envelope carries seven nominal deterministic coefficients (inclination scale, J2/altitude modulation, plasma-density power-law parameters, velocity-screening exponent, and near-symmetry coherence threshold) documented in scripts/utils/tep_geometry_envelope.py. These coefficients are not varied by the nonlinear least-squares map in Step 008; they are nevertheless physical knobs whose ±50% stress band must be machine-checked because skeptical readers will count them toward an “epicycle” budget.

The pipeline therefore emits an explicit parameter audit in results/step008_fitting_results.json (parameter_budget_audit), including a defensibility score \(\mathrm{defensibility} = k_{\rm fit}/(k_{\rm fit}+k_{\rm heur})\) for the restricted tier, and a conservative BIC/AIC companion row that adds \(k_{\rm heur}\) to the effective parameter count as a stress test (TEP_heuristic_pessimistic inside enhanced_validation.model_comparison). Independent ±50% Monte Carlo sweeps over all seven coefficients are archived in results/step041_envelope_heuristic_sensitivity.json (Step 041), including one-at-a-time leverage rankings for the inverse-variance pooled β_fit.

Ionospheric screening uses a bounded phenomenological ansatz family compared under identical IRI perigee densities in results/step017_plasma_modulation.json (plasma_ansatz_robustness). A first-principles scalar–plasma coupling derived from the TEP action remains the correct long-term replacement for these ansätze.

3.3 Ensemble Selection Protocol

The gated ensemble for inverse-variance β_fit fitting and Bayesian model comparison is constructed via two a priori criteria applied before any fitting. These criteria are encoded in scripts/utils/flyby_ensemble.py and contain no mission-specific exclusions.

Flybys failing either gate are excluded from the fitted ensemble but remain in the full catalog for hierarchical diagnostics, fixed-β_ref cross-catalog stress tests, and literature comparison. The criteria are applied automatically by the pipeline; no human decision enters after the criteria are specified.

Application to the current dataset:

• NEAR (1998): S/N = 103.5 > 2; sign agreement (+13.46 mm/s vs +0.30 mm/s predicted at β_{A,ref}). Passes both gates.
• Galileo 1990: S/N = 49 > 2; sign agreement (+3.92 mm/s vs +0.03 mm/s predicted at β_{A,ref}). Passes both gates.
• Rosetta 2005: S/N = 60 > 2; sign agreement (+1.80 mm/s vs +0.03 mm/s predicted at β_{A,ref}). Passes both gates.
• Cassini (1999): S/N = 2.0 ≥ 2; passes sign agreement at β_{A,ref} on the re-transcribed catalogue (−2.00 mm/s observed vs a negative, near-zero predicted total, sign product > 0). The negative predicted total arises because Cassini's high perigee velocity (19.02 km/s > v_trans ≈ 16.8 km/s) places it on the template's mixed-regime branch, where the conformal-gradient term (−1.1×10⁻⁵ mm/s) and disformal term (+5×10⁻⁷ mm/s) nearly cancel. Because the reference prediction is numerically negligible, the closed-form β_fit rescaling is formally divergent (β_fitted ≈ 19 with an uncertainty of comparable size); Cassini enters the Step 008 pool and the Step 026 gated likelihood but contributes negligible weight to the pooled amplitude.
• Galileo 1992: S/N = 4.6 > 2 on the re-transcribed catalogue (−4.60 ± 1.00 mm/s vs −0.007 mm/s predicted at β_{A,ref}, sign agreement). Passes both gates; the prediction is negative but underpredicts the observed magnitude by an order, so the fit contributes a positive residual rather than an amplitude anchor.
• MESSENGER (2005): S/N = 2.0 ≥ 2 (+0.02 ± 0.01 mm/s vs +6×10⁻⁸ mm/s predicted at β_{A,ref}, sign agreement). Passes both gates; the reference prediction is numerically zero, so its closed-form rescaling is formally divergent and contributes negligible weight.
• Rosetta 2007: S/N = 0.4 < 2 (+0.02 ± 0.05 mm/s). Fails the S/N gate. Excluded.
• Rosetta 2009: S/N = 0 (Δv = 0.00 mm/s bound). Fails the S/N gate. Excluded.
• Juno (2013): Δv = 0.00 mm/s. Fails S/N gate (S/N = 0). Excluded.

Distinction between caveat and exclusion. The manuscript acknowledges that Galileo 1990's high-gain antenna failure and spin-rate changes introduce additional spacecraft-specific uncertainty (Section 5.13). This is a caveat—a flag for cautious interpretation—not an exclusion criterion. Galileo 1990 remains in the fitted ensemble because it satisfies the pre-specified objective gates. Conflating a stated caveat with an operational exclusion would be a methodological error.

3.4 Deterministic Factor Computation

3.4a Deterministic Geometry Modulation Analysis (Step 009)

With n = 6 sign-consistent detections, formal variance decomposition into structural, observational, environmental, and residual components remains statistically underpowered: the standard error on a sample variance estimate with ddof = 1 is a large fraction of the estimate at this sample size, rendering any reported percentage unreliable. Step 009 therefore does not produce an ANOVA-style partition. Instead, it reports three complementary analyses that are statistically defensible at small n:

1. Beta scatter statistics. The raw span and log-standard deviation of fitted β_fit are reported as descriptive statistics, with an explicit note that the Step 007 prediction already includes the geometry envelope, so residual scatter reflects genuine coupling heterogeneity or model incompleteness rather than unmodeled trajectory geometry.
2. Full-catalog detection pattern. Across all n = 12 catalogued flybys, the Step 007 deterministic prediction is classified against the published observation as true positive (predicted and observed detection), true negative (predicted and observed null), false negative, or false positive. This yields a classification accuracy that tests whether the model correctly identifies which flybys should show anomalies independent of the small fitted-β_fit sample.
3. Rank correlation. A Spearman rank correlation between predicted and observed anomaly magnitudes across the full catalog tests whether the model captures the relative ordering of anomaly sizes, again independent of the n = 6 β-fitting sample.

The legacy four-stage chained-heuristic output (variance_decomposition.stages) is retained in the JSON for backward compatibility but is explicitly marked DEPRECATED and must not be used for manuscript inference.

3.5 Disformal Transition Criterion

A disformal transition criterion Ξ is defined to classify flybys into conformal-dominated, mixed, or disformal-dominated regimes of the fitted response template. This provides a formal regime label for Cassini's velocity-activated branch (the status of the underlying scale is assessed below).

where:

• v_trans ≈ 16.8 km/s is the transition velocity (an empirical scale of the response template, see below)
• v_i is the flyby perigee velocity
• p = 2 is the velocity exponent
• q = 1 is the gradient exponent
• ∇φ_⊕ is the field-gradient excursion at Earth's surface
• ∇φ_i is the field-gradient excursion at flyby altitude
• sgn indicates aligned (positive) vs anti-aligned (negative) disformal response

Status of the Transition Scale $v_{\rm trans}$

The transition velocity enters the model as a fixed scale of the velocity-dependence response template — a pipeline constant carrying an assigned $\pm 20\%$ uncertainty — not as an output of the TEP field equations. The distinction matters and is stated explicitly here, because an earlier draft presented $v_{\rm trans}$ as a derived quantity. The disformal term of the TEP matter metric,

would produce a genuine velocity-dependent response at the scale where the disformal contribution becomes comparable to the conformal perturbation, $B(\phi)\,(\mathbf{v}\cdot\nabla\phi)^2/c^2 \sim A^2(\phi)-1$. Evaluated on the solved Earth profile under the canonical envelope $B(\phi)\propto B_0\varphi^2$, however, that balance is not realized at flyby velocities: the microscopic coefficient required is $B_0 \sim 1.6\times10^{18}$, exceeding both the calibrated Paper-0/28 envelope normalization by $\sim 21$ orders of magnitude and the projected $10^{-19}$-fractional holonomy bound ($|B_0|\lesssim5\times10^{-8}$) by $\sim 26$ orders (Step 011 audit). Under the canonical action, therefore, no kilometre-per-second disformal transition exists, and flyby phenomenology at these velocities is conformal-sector — consistent with the fitted disformal amplitude of the hierarchical layer, $b_{\rm disf} = 0.026 \pm 0.043$, which is consistent with zero and is the expected outcome on the admissible branch (Section 4.2.2).

The coefficient $B_{\rm eff} = 4|\beta_A|R_\oplus/\lambda_T \approx 6.1$ quoted in earlier drafts is accordingly disclosed for what it is: a bookkeeping identity of the template rather than a derived coefficient. Writing $B_{\rm eff}\equiv (A^2-1)/(v_{\rm trans}/c)^2$ defines it through the balance condition itself, so inserting it back into the balance returns the assumed scale; it is not an action-level or shell-averaged property of the solved field. For the same reason the closed form

where $\epsilon_\phi \equiv |\nabla\phi_\oplus|\,\lambda_T/M_{\rm Pl}$ is the dimensionless field excursion across one relaxation length of the screened profile (the Helmholtz solution of $\nabla^2\phi - \lambda_T^{-2}\phi = -(\beta_A/M_{\rm Pl})\rho$ for the uniform-sphere source), is retained here only as a dimensional anchor. Evaluated at the UCD-pinned surface combination $\epsilon_\phi = |\beta_A|\rho_T\lambda_T^2/M_{\rm Pl}^2 \approx 6.6\times10^{-9}$ (the field pinned at the saturation-density source inside $R_{\rm sol}$; the documented $\pm40\%$ GNSS uncertainty on $\rho_T$ spans $\epsilon_\phi \approx (4.0\text{--}9.2)\times10^{-9}$), the combination evaluates to $14\text{--}16.5$ km/s, numerically bracketing the $16.8$ km/s template value used below (Step 011 audit). This proximity — the geometric combination of the clock-offset velocity $c\sqrt{2\epsilon_\phi}$ diluted by the layer aspect ratio $\sqrt{\lambda_T/4R_\oplus}$ — is recorded as a consistency observation that motivates the template scale; it is not a derivation, and no claim of a field-theoretic transition is made. The UCD-derived characteristic suppression $S_\oplus = (R_\oplus - R_{\rm sol})/R_\oplus \approx 0.35$ is connected to the same field solution through the surface-to-transition geometry and continues to supply the amplitude prior (Step 010).

Within the template, $v_{\rm trans}$ therefore marks the empirical scale separating the velocity-suppressed and velocity-unsaturated response regimes of the fitted envelope. Cassini's high perigee velocity (19.02 km/s $> v_{\rm trans}$) places it on the velocity-activated branch of the template; the physical anomaly mechanism claimed in this paper remains the clock-sector channel (Section 4.6.3a), and the regime labels below are phenomenological descriptors of the fitted response, not statements about a propagating disformal sector.

Classification (by |Ξ|):

• |Ξ| < 0.05: Conformal-dominated
• 0.05 ≤ |Ξ| ≤ 0.10: Mixed
• |Ξ| > 0.10: Disformal-dominated

The sign of Ξ encodes the orientation of the template's disformal-term response: positive for aligned trajectories and negative for anti-aligned trajectories. Cassini, with its high perigee velocity (19.02 km/s) and negative asymmetry, falls into the mixed regime with a negative sign — on the template reading, it occupies the anti-aligned branch where the conformal-gradient and disformal terms partially cancel. As established above, this classification is a phenomenological descriptor of the fitted envelope rather than a derived field-theoretic regime assignment.

Velocity shift formula: The predicted velocity anomaly combines four physical effects:

Each factor has a distinct physical origin:

1. Field amplitude excursion $d\phi/dr = (\Delta\phi / \lambda_T)\, e^{-h/\lambda_T}$: the clock-sector response strength at perigee altitude $h$, decaying exponentially with the GNSS-established relaxation length. Lower flybys experience stronger field excursions.
2. Perigee dwell time $r_p / v_p$: the effective duration of the close encounter. Slower, lower flybys accumulate larger proper-time offsets.
3. $J_2$ oblateness $J_2 (R_\oplus/r_p)^2$: the non-radial component of the offset arising from Earth's oblateness. The radial component is absorbed into the orbit determination program's estimate of $GM$; only the non-radial residual produces an apparent net velocity change.
4. Trajectory asymmetry $\cos\delta_{\rm in} - \cos\delta_{\rm out}$: the difference in approach and departure $v_\infty$ declinations (from Anderson et al. (2008)). This factor determines how asymmetrically the spacecraft samples the oblate field. For symmetric trajectories ($\delta_{\rm in} \approx \delta_{\rm out}$), the non-radial impulse cancels and the predicted anomaly vanishes—correctly predicting null results for flybys such as Galileo 1992 and MESSENGER.

3.6 Robust Bayesian Fitting

The primary Step 008 coupling estimate is an inverse-variance scaling fit on the sign-gated detections, and the Step 026 model-comparison layer uses Gaussian weighted least-squares likelihoods. A Student's t-distribution likelihood with degrees of freedom $\nu = 3$ is used only in the auxiliary robust Bayesian/hierarchical checks, where it tests whether the conclusions are sensitive to outlier treatment in the small sample.

Primary fit (Step 008). After the pre-specified S/N and sign gates, each retained flyby maps the Step~007 prediction at $\beta_{\rm ref} = 10^{-4}$ to a single effective amplitude using the linear rescaling $\beta_{\rm fit} = \beta_{\rm ref}\,(\Delta v_{\rm obs}/\Delta v_{\rm TEP})$ implied by the canonical $\beta$-independent excursion normalization (Section~3.3). This is a deterministic per-flyby consistency check, not an iterative maximisation of a heavy-tailed likelihood; it replaces the v0.1 $(\Delta v)^{4/3}$ map, which traced to the $\beta$-dependence of the excluded inverse-power density minimum rather than to the canonical action.

Hierarchical population model (Step 015). The four-parameter MCMC layer uses independent Gaussian residuals for each flyby with variance $\sigma_{\rm pop}^2 + \sigma_{{\rm obs},i}^2$ in the log-likelihood (see scripts/steps/step_015_hierarchical_bayesian.py). Sampling targets the log-posterior $\ln p(\theta \mid {\rm data})$ under the stated priors; reported medians and credible intervals are standard MCMC summaries.

Auxiliary tail diagnostics. Cochran's $Q$, $I^2$, and Student-$t$ critical values are used only when summarising scatter among the per-flyby $\beta_{A,i}$ and when constructing heterogeneity-aware error bars in Step~008 (analyze_fit_quality); they do not replace the primary scaling law above.

PPN constraint validation is reported in Section 3.9 and Section 4.6.

3.7 Statistical Analysis

The weighted mean $\beta_{\rm fit}$ across all detections is:

with inverse-variance weights derived from propagated measurement uncertainties. The weighted standard error is:

The NEAR detection dominates the pooled estimate through its fitted-$\beta$ uncertainty ($\sigma_\beta \approx 2.3\times10^{-5}$, versus $4.9\times10^{-5}$ for Rosetta 2005, $2.4\times10^{-4}$ for Galileo 1990, and formally divergent values for the near-zero-prediction members).

Heterogeneity assessment: Following meta-analysis conventions (Higgins & Thompson, 2002), heterogeneity is quantified using:

An $I^2 > 75\%$ indicates extreme heterogeneity, justifying uncertainty inflation by $\sqrt{Q/(n-1)}$ to account for model scatter beyond measurement error.

Robustness verification: Step 008 parametric bootstrap ($10^4$ draws) yields median $\beta \approx 4.77 \times 10^{-3}$ with 95% interval $[4.42 \times 10^{-3},\,1.47 \times 10^{-2}]$, and leave-one-out recomputations $5.88 \times 10^{-3}$ (without NEAR), $4.58 \times 10^{-3}$ (without Galileo 1990), and $4.64 \times 10^{-3}$ (without Rosetta 2005). The stability coefficient $\approx 0.092$ is below the 0.5 robustness guideline, indicating moderate leave-one-out stability on the gated trio.

1. Smooth bootstrap ($n = 10\,000$): Resampling with replacement combined with Gaussian noise injection validates the weighted mean distribution and provides confidence intervals.
2. Leave-one-out cross-validation: Systematically excluding each detection verifies that no single flyby dominates the conclusion. Stability coefficient < 0.5 indicates robustness.

3.8 Orbit Determination Filtering Mechanism (Hypothesis)

Modern orbit determination (OD) employs a multi-stage processing pipeline that may inadvertently filter TEP-like signals. Understanding this potential mechanism is relevant for interpreting why some flybys show null results despite TEP predictions. This remains a hypothesis requiring independent verification through raw DSN data analysis.

Standard OD processing chain:

1. Raw Doppler measurements: Two-way/3-way Doppler tracking from DSN stations, typically at X-band (8.4 GHz) or Ka-band (32 GHz), with sampling rates of 1-60 Hz.
2. Cycle-slip detection and correction: Automated algorithms detect discontinuities in phase measurements and correct them to maintain phase continuity.
3. Outlier rejection: Measurements deviating by more than 3σ from the expected trajectory are flagged and removed as erroneous data points.
4. Smoothing and averaging: Raw measurements are typically averaged over 10-60 second intervals to reduce noise and computational load.
5. Bias estimation and removal: Systematic biases (e.g., station clock offsets, media delays) are estimated and subtracted from the measurements.
6. Empirical acceleration estimation: To absorb unmodeled forces, OD fits empirical accelerations (constant, once-per-revolution, stochastic) that absorb any residual systematic errors.
7. Residual analysis: Final residuals are examined; large residuals trigger additional data editing or model refinement.

Hypothesized filtering of TEP signals: TEP produces a sudden velocity shift precisely at perigee passage (±2 hours), characterized by:

• Sharp temporal structure (not gradual acceleration)
• Correlation with gravitational potential gradient
• Consistent amplitude across multiple spacecraft geometries
• Occurrence at a predictable location (perigee)

These characteristics could cause TEP signals to be treated as systematic errors in the OD pipeline:

1. Outlier rejection: The sharp perigee anomaly could appear as an outlier in the Doppler residuals and be removed by the 3σ threshold.
2. Empirical acceleration absorption: The sudden velocity shift could be absorbed by empirical acceleration terms, effectively modeling it as a force rather than a clock rate effect.
3. Smoothing: Averaging over 10-60 second intervals could dilute the sharp perigee signal, reducing its amplitude.
4. Bias estimation: The perigee anomaly could be partially absorbed into station bias estimates.

Proposed minimal OD approach for validation: To test whether TEP signals can be recovered from raw data, a minimal OD pipeline is recommended:

• Use reduced gravity field (10×10 instead of 50×50 or higher)
• Disable empirical acceleration estimation
• Disable outlier rejection (or use relaxed threshold)
• Use raw Doppler without smoothing
• Fit only initial state and solar radiation pressure coefficient

This minimal approach would preserve TEP signals while still providing adequate orbit determination for anomaly extraction. Where the pipeline reports perigee-window statistics from sequential pairwise Doppler differences (Steps 006 and 030), those quantities are explicitly not commensurate with published post-OD $\Delta v$ anomalies unless a batch OD residual chain is bound; model comparison for the flyby ensemble (Step 026) uses a fractional misspecification floor as the headline likelihood. Each signal-bearing model carries a per-flyby systematic uncertainty $\sigma_{{\rm sys},i} = CV_\beta\,|\hat y_i|$, where $CV_\beta = \tau/\bar\beta \approx 0.47$ is the measured between-flyby amplitude heterogeneity (DerSimonian–Laird $\tau \approx 2.23\times10^{-3}$ on the Step 008 scale factors; $\bar\beta \approx 4.74\times10^{-3}$) and $\hat y_i$ is the model's own fitted prediction, iterated to a fixed point. The floor is dimensionless and therefore invariant under the arbitrary reference-amplitude anchor $\beta_{\rm ref}$ — a property the earlier constant geometry-spread floor lacked, since the spread of predictions evaluated at $\beta_{\rm ref}$ rescales with that anchor and can alter the model ordering without any change to the data. The Null model predicts zero signal and accordingly receives no floor: its published uncertainties are taken at face value, so the large anomaly detections penalize it directly. Per-flyby floors under the fitted models span $\approx 0$–$14$ mm/s, concentrated where a model claims a large signal. The former constant geometry-spread floor and a published-uncertainties-only block ($\sigma_{\rm sys} = 0$) are retained as labelled sensitivity analyses together with a constant-floor sweep, not the primary evidence claim.

3.9 PPN Constraints and Cassini Solar Conjunction

For scalar-tensor theories with conformal coupling, the PPN parameter deviation is bounded by

where $\gamma$ is the PPN parameter, $\alpha_0 = \sqrt{2}\,\beta_A = -\sqrt{2}$ is the DEF-normalized coupling ($\beta_A = -1$ frozen; the deviation is linear in the screened source charge because the photon probe is unscreened), and $S_\Sigma^{(\odot)}$ is the Solar-System source-charge screening factor. The Cassini solar conjunction experiment provides the tightest bound on the post-Newtonian light-propagation sector. It measured the gravitationally induced frequency shift of radio photons exchanged with the spacecraft and obtained $\gamma = 1 + (2.1 \pm 2.3) \times 10^{-5}$. The Earth-vicinity flyby response geometry $S_\oplus(r)$ is a distinct projection of the same environment-dependent scalar configuration; it governs the flyby impulse but is not the Cassini source-charge parameter.

Cassini constrains the reciprocity-even radio light-time observable in the screened solar-system environment. In the TEP decomposition, this constrains three specific sectors:

A. Gravitational/light-propagation sector (directly constrained): Cassini requires that any unscreened solar scalar charge, any long-range conformal/disformal coupling affecting the radio link, or any deviation in the solar-system Shapiro sector be smaller than roughly the measured $\gamma$ uncertainty: $|\gamma - 1| \lesssim 2.3 \times 10^{-5}$.

B. Conformal clock-sector structure (not directly tested): A purely conformal transformation $\tilde g_{\mu\nu} = A^2(\phi)g_{\mu\nu}$ preserves null cones. Therefore, a conformal clock-sector field can evade a direct Cassini light-cone constraint only if it does not create an observable solar-system $\gamma$ shift or anomalous clock/redshift signature.

C. Screening sector (boundary condition): For the solved high-acceleration solar configuration, where the kinetic response suppresses Temporal Shear, then Cassini becomes a boundary condition: $\Sigma_\mu = \nabla_\mu \ln A \approx 0$ in the solar-system Shapiro regime. This is not a weakness but exactly how the theory must be formulated.

Therefore Cassini should be treated not as irrelevant to TEP, but as a stringent boundary condition: a viable TEP model must reduce to the GR PPN light-propagation limit near the Sun while reserving its discriminating predictions for observables outside the Cassini measurement class (spatial clock covariance, one-way residual shear, low-density temporal-shear recovery).

The deep potential well of the Sun suppresses Temporal Shear toward zero, providing screening in the solar environment. The UCD-derived characteristic ratio $S_{\oplus} \approx 0.35$ at Earth's surface calibrates the clock-sector response profile along flyby trajectories, while the solar-screening calculation (Section 4.6.1a) shows that the effective coupling along the Cassini radio path also remains below the Cassini bound.

3.10 Plasma Modulation

The Cassini flyby exhibits a near-cancellation regime where the conformal-gradient term is negative (−5.6×10⁻⁵ mm/s) and the disformal term is positive (+2.5×10⁻⁶ mm/s), yielding a numerically negligible negative total (−5.3×10⁻⁵ mm/s) at the reference coupling. The historical sign mismatch was a catalogue transcription artifact: the primary-source anomaly is itself negative (−2 ± 1 mm/s, Anderson et al. 2008), so the predicted sign agrees and the residual gap is one of magnitude — corrected in the present re-transcribed catalogue. Plasma-dependent attenuation is treated as a secondary modulation effect.

The plasma density along the flyby trajectory is computed using:

where the ionospheric component is obtained from the International Reference Ionosphere (IRI) empirical model (Step 033), which provides continuous electron density profiles along spacecraft trajectories using historical F10.7 solar flux data. The IRI model replaces the Chapman layer approximation with real ionospheric data, improving accuracy for plasma environment reconstruction (Step 020). For theoretical reference, the Chapman layer model is:

with $h_{\rm max} = 300$ km, $H_{\rm scale} = 50$ km, and $n_{\rm max} = 10^6$ cm$^{-3}$ (solar maximum). The magnetospheric component scales with L-shell as $n_{\rm mag} \propto L^{-4}$.

A Debye-like plasma attenuation ansatz is used as a phenomenological proxy for ionospheric screening:

where $n_e$ is the electron density in cm$^{-3}$ and $n_{\rm ref} = 10^4$ cm$^{-3}$ is a reference density. Plasma attenuation is retained only as a heuristic sensitivity layer; the primary TEP prediction uses $S_{\rm plasma} = 1$, so the headline result does not depend on the phenomenological plasma ansatz.

Plasma attenuation does not cause sign reversal—it only modulates the magnitude of the scalar field. The primary mechanism for sign reversal is disformal coupling (Section 3.5), which produces velocity-dependent effects for high-velocity anti-aligned trajectories.

Solar activity data for plasma density estimation are obtained from documented historical records: F10.7 solar flux from NOAA/SWPC and the Kp geomagnetic index from the GFZ German Research Centre for Geosciences. The current implementation uses continuous International Reference Ionosphere (IRI) model electron density data fetched for the exact historical trajectories of each flyby (Step 033) and ingested by the plasma environment reconstruction step (Step 020). The IRI model is a well-validated empirical model based on decades of ionospheric measurements. Step 033 is configured so its mission keys mirror the cached JPL Horizons trees under data/raw/jpl_horizons/<mission>, and the Step 017 lookup table (iri_mission_map) maps catalogue names to those keys; together this prevents a silent plasma-path gap for a flyby that already has a reconstructed Horizons arc.

The IRI electron density and computed phenomenological screening factor at perigee show that the ansatz predicts stronger attenuation at lower altitudes (higher plasma density), which is physically intuitive. Rosetta 2007 at 5430 km has the weakest attenuation ($S_{\rm plasma} \approx 0.96$) because it samples the most tenuous plasma environment, while NEAR at 539 km has the strongest ($S_{\rm plasma} \approx 1.3 \times 10^{-6}$) due to the dense F-region ionosphere.

Because the Debye ansatz is phenomenological and not derived from the TEP action, the primary predictions in Table 3 use $S_{\rm plasma} = 1$ (no plasma suppression): the ansatz is computed and recorded for every flyby but is never applied to a prediction. Applying it literally would suppress NEAR’s predicted anomaly to $\sim 5 \times 10^{-6}$ mm/s, far below the observed 13.46 mm/s and inconsistent with the detection. A weaker, bounded power-law plasma factor $f_{\rm plasma} = (1 + \rho/\rho_{\rm crit})^{-0.3}$ on IRI-derived perigee densities is distinct from the quarantined ansatz: it remains an active nominal coefficient inside the Step 007 deterministic geometry envelope — one of the seven documented heuristic knobs carrying the machine-checked $\pm 50\%$ stress band (Steps 007, 041) — contributing modulation factors of 0.19–1.00 across the catalogue and 0.18–0.91 on the gated ensemble. It is carried explicitly as a template-layer coefficient, not as an action-derived scalar–plasma coupling, whose derivation remains an open refinement.

3.11 UCD-Motivated Temporal Topology Derivation

To eliminate systematic bias from phenomenological suppression factors, $R_{\rm sol}$ and the characteristic suppression $S_{\oplus}$ are derived from the Temporal Topology Saturation Scale (UCD) saturation model. The saturation radius is calculated from the UCD ansatz using Earth's total mass and the saturation scale $\rho_T \approx 20$ g/cm³, a scale established across astrophysical systems from dwarf-galaxy cores to galaxy-cluster halos (Paper 6; Schive et al. 2014; Mocz et al. 2018). GNSS atomic-clock correlation analysis (Step 016) yields a covariance decay length $L_c \approx 4201$ km. Under the corpus's terrestrial calibration ansatz — the Paper 6 §2 transfer sketch, which identifies $L_c$ with the network-projected scale of the exterior clock-transport field at $\mathcal{O}(1)\times R_T(M_\oplus)$ — inverting the saturation formula returns $\rho_T \approx 19.2$ g/cm³, within 3.9% of the adopted 20 g/cm³. Because the adopted $\rho_T$ is itself anchored on $L_c$, this agreement is a shared-input consistency check (the global anchor recovered at its own terrestrial calibration point), not an independent validation; what it does establish is that the density scale carries no tuning to Earth flyby data.

This yields the UCD-motivated saturation estimate, consistent with the GNSS terrestrial anchor under the same identification ($L_c = 4201$ km → $\Delta R/R = 0.34$, 2% agreement — a calibration-internal check rather than an independent validation):

The systematic uncertainty on $\rho_T = 20 \pm 7$ g/cm³ (35%) from Paper 6 (UCD) — a formal scaling constant of the $R_T(M)$ mass relation, not a literal interior density — propagates to $\Delta R_{\rm sol} \approx \pm 540$ km ($\sim$13%) and $\Delta S_{\oplus} \approx \pm 0.09$ ($\sim$25%). The GNSS comparison ($L_c = 4201 \pm 1967$ km, Step 016) verifies that the adopted global value remains consistent with its own terrestrial calibration anchor under the Paper 6 §2 identification; it is not an independent determination. The independent empirical content entering the prior is instead supplied by the cross-scale astrophysical consistency of $\rho_T$ (Paper 6) and the flyby altitude-threshold check (Step 010, approach 5). Together, these constraints establish $S_{\oplus} = 0.35^{+0.09}_{-0.09}$ as a rigorously derived prior.

3.12 Cosmographic Temporal Shear Modulation Analysis

A deeper TEP prediction is that the disformal coupling term depends on the total velocity in the ambient-field rest frame — the frame in which the spatial gradient of the cosmological temporal field vanishes. If the CMB dipole frame approximates this frame, the Solar System's ~370 km/s bulk motion toward (RA, Dec) = (167.94°, −6.93°) provides a cosmographic modulation of the effective coupling strength. Step 040 tests this using full 3D spacecraft state vectors from JPL Horizons, computing heliocentric distance, CMB dipole projection, and disformal enhancement proxies. With only n = 9 flybys carrying both reconstructed 3D state vectors and a catalogued anomaly value or published bound (the three modern passes lack a published anomaly), all cosmographic tests remain exploratory; the full extraction protocol is documented in the Step 040 pipeline output.

4. Results

4.1 Individual Flyby Fits

The TEP clock-sector response model with J2/J3/J4 multipole contributions, disformal coupling, perigee plasma factors (Step 017 / IRI, Step 033), and Temporal Topology screening quantitatively reproduces the gated primary detections as apparent velocity discontinuities at a reference coupling β_{A,ref} = 10⁻⁴, then rescales to a universal β_fit by per-flyby fitting subject to pre-fit gates (S/N ≥ 2, sign agreement). On the primary-source catalogue of Table 1 the sign gate is satisfied by every published detection — the corrected Cassini entry ($-2 \pm 1$ mm/s) and the previously mis-entered Galileo 1992 detection ($-4.60 \pm 1.00$ mm/s) both carry the same sign as their TEP reference predictions — so the sign-gated ensemble is the full six-member detection set rather than a three-member subset. Cassini and MESSENGER have reference predictions of order $10^{-4}$–$10^{-6}$ mm/s at β_{A,ref}; their per-flyby rescalings are therefore formally unconstrained (fractional uncertainty $\gtrsim 0.7$) and contribute negligible weight to the pooled amplitude, while remaining in the gated ensemble, the Step 039 classification, and the hierarchical diagnostics (Step 015). Table 3 lists per-flyby predictions at β_{A,ref} and fitted β_{\rm fit} for the ensemble members.

The four amplitude-informative gated fits span $4.43 \times 10^{-3}$ to $6.65 \times 10^{-2}$, with the three tightly constrained members (NEAR, Galileo 1990, Rosetta 2005) clustering within a factor of about 3.3 ($4.43 \times 10^{-3}$ to $1.44 \times 10^{-2}$), consistent with geometry-dependent modulation of the effective coupling (the plasma term is a bounded heuristic envelope coefficient; the stronger Debye attenuation ansatz is quarantined at $S_{\rm plasma}=1$, Section 3.10). The inverse-variance weighted mean is $\beta_{\rm fit} = 4.74 \times 10^{-3} \pm 5.83 \times 10^{-4}$ (formal uncertainty from Step 008); the random-effects uncertainty is $1.28 \times 10^{-3}$. Cross-validation indicates moderate robustness (stability coefficient 0.092 < 0.5). Residuals on the six-member gated set are consistent with normality (Shapiro–Wilk $p \approx 0.95$).

Table 3a shows the component-level breakdown for the six gated detections.

4.1.2 Full 3D Trajectory Integration Validation

The perigee approximation used in the primary analysis (Table 3) is cross-checked against full 3D trajectory integration reconstructed from JPL Horizons scalar data via Keplerian orbit propagation (Section 3.2b). Table 3b compares the perigee estimate $\Delta v_{\rm peri}$ with the integrated velocity shift $\Delta v_{\rm int}$ for the analyzed flybys.

† Ratios are not reported where the perigee estimate is numerically zero; they carry no information there.

For the three tightly constrained detections (NEAR, Galileo 1990, Rosetta 2005), the 3D integrated velocity shift exceeds the perigee estimate by factors of 6.2–18.6 at the pooled β_{\rm fit} = 4.74×10⁻³ (a factor of ~47.4 in coupling relative to β_{A,ref}), with trajectory-curvature and altitude-dependent modulation partially suppressing the signal along the path. NEAR shows the largest ratio (18.6), Galileo 1990 is intermediate (14.6), and Rosetta 2005 shows the smallest ratio (6.2), consistent with its higher perigee altitude (1955 km) where the field gradient varies more significantly along the trajectory. The perigee approximation therefore captures the dominant physics at β_{A,ref} but should not be naively rescaled to the fitted β_{\rm fit} without the geometry envelope. Ratios are not reported where the perigee estimate is numerically zero, since they carry no information there.

Rosetta 2007, Rosetta 2009, and MESSENGER 2005 predict negligible anomalies in both methods, consistent with their published sub-threshold results; Rosetta 2007 is a high-altitude suppression case (5301 km perigee). Galileo 1992 is the informative negative-asymmetry case: the perigee estimate reproduces the observed sign ($-0.007$ mm/s versus the published $-4.60 \pm 1.00$ mm/s), while the integrated 3D shift changes sign along the extended trajectory (+0.30 mm/s at the pooled coupling), so the two layers bound the response differently and both underpredict the observed magnitude. BepiColombo's 12,697 km pass integrates stably to a numerically negligible shift ($\Delta v_{\rm int} = 7.8\times10^{-9}$ mm/s), consistent with the suppression expected at its altitude.

The overall conclusion is that the perigee approximation captures the dominant physics at the reference coupling β_{A,ref} = 10⁻⁴, while full 3D integration at the pooled β_{\rm fit} = 4.74×10⁻³ provides a cross-check that incorporates trajectory curvature, altitude-dependent field-gradient variation, and the complete geometry envelope. The ensemble fitting retains the perigee approximation as the primary computational tool, with 3D integration serving as a validation layer where numerically stable.

4.2 Hierarchical Bayesian Model Results

4.2.1 Per-Flyby Geometry Factor Extraction

The component-level analysis extracts the effective geometry factor $G_{i,\text{eff}} = \Delta v_{\text{obs}} / \Delta v_{\text{grad}}(\beta_{\rm fit,0} = 10^{-4})$ for each flyby. This factor represents the multiplicative scaling between the observed anomaly and the gradient prediction at the reference coupling, isolating geometry-dependent modulation from the universal coupling strength.

The geometry factor spans a wide dynamic range across the gated ensemble. Rows marked † divide by a reference prediction of order $10^{-4}$ mm/s or smaller, so their $G_{\rm eff}$ is a numerical artefact of the near-zero denominator rather than a constrained amplitude. Among the four amplitude-informative members, the implied couplings span $\beta_{0,\rm implied} \sim 5.2 \times 10^{-3}$–$3.1 \times 10^{-2}$, bracketing both the Step 008 weighted mean ($4.74 \times 10^{-3}$) and the hierarchical Bayesian median ($5.27 \times 10^{-3}$) within the ensemble spread (the $G_{\rm eff}$ normalization divides by the gradient component only, so it sits above the pooled amplitude). Correlations against the trajectory parameters are reported in the Step 015 output: at $n=6$ the only approach to significance is against velocity ($r = 0.90$, $p = 0.011$); altitude ($r = -0.13$) and asymmetry ($r = -0.28$) show none.

4.2.2 MCMC Hierarchical Inference

The four-parameter hierarchical Bayesian model ($\beta_{\rm fit,0}$, $b_{\text{disf}}$, $\sigma$, $\alpha_{\text{res}}$) is sampled via MCMC. The log-$\beta_{\rm fit,0}$ prior is centered on the Step 008 weighted mean with a heterogeneity-aware width ($\beta_{\rm fit,0} \sim \text{LogNormal}(\ln(4.74 \times 10^{-3}), 1.08)$, sourced from the random-effects scatter so the prior stays diffuse on the scale of the observed flyby-to-flyby scatter); the layer therefore functions as an independent population-level estimate rather than an echo of the pooled fit. The pre-computed gradient and disformal components from Step 007 contain the perigee physics; the likelihood scales these components by inferred universal couplings, with residual modulation captured by $\alpha_{\text{res}}$.

Posterior parameter estimates (Step 015, current pipeline):

• $\beta_{\rm fit,0} = 5.27 \times 10^{-3} \pm 2.55 \times 10^{-3}$ (16th–84th: $3.57$–$7.57 \times 10^{-3}$)
• $b_{\text{disf}} = 0.026 \pm 0.043$
• $\sigma = 1.65 \pm 0.62$ mm/s (flyby-to-flyby scatter)
• $\alpha_{\text{res}} = +0.022 \pm 0.286$ (consistent with zero)

The hierarchical posterior median $\beta_{\rm fit,0} = 5.27 \times 10^{-3}$ is consistent with the Step 008 inverse-variance weighted mean ($4.74 \times 10^{-3}$) within the heterogeneity width, and the agreement is not prior-imposed: the stored likelihood profile over $\beta_{\rm fit,0}$ — evaluated with the remaining parameters at their posterior medians and no prior weighting — peaks at $5.57 \times 10^{-3}$, with the Step 008 centre within $\Delta\ln\mathcal L \approx 0.75$ of the maximum. The two estimation layers thereby reconcile on a common $\mathcal O(10^{-3})$ amplitude. Posterior predictive checks track NEAR and Rosetta 2005 well; Galileo 1990, Galileo 1992, and Cassini retain residuals of 2.5, 4.0, and 2.0 mm/s respectively in this hierarchical layer, motivating continued envelope and OD work.

The posterior median $\beta_{\rm fit,0} = 5.27 \times 10^{-3}$ agrees with the Step 008 inverse-variance weighted mean ($4.74 \times 10^{-3}$) within the heterogeneity width; the width is driven by the population scatter $\sigma \approx 1.65$ mm/s absorbed by the hierarchical layer rather than by any residual offset between the two estimators. The factor-of-order-unity rescaling from the reference coupling ($\beta_{A,\rm ref} = 10^{-4}$) is a computational anchor only: because the canonical excursion normalization is $\beta$-independent, the response model is strictly linear ($\Delta v_{\rm TEP} \propto \beta$), so the closed-form inversion $\beta_{\rm fit} = \beta_{A,\rm ref}\,(\Delta v_{\rm obs}/\Delta v_{\rm TEP})$ is algebraically independent of $\beta_{A,\rm ref}$ — the fitted amplitude is set by the data, and re-evaluating the same flyby at $\beta_{A,\rm ref} = 10^{-5}$ or $10^{-3}$ returns the identical fitted value. The residual bookkeeping is therefore a single channel quantity: the population response coefficient $\kappa_{\rm flyby} = \beta_{\rm fit}/|\beta_A| \approx 4.7 \times 10^{-3}$ — an unexplained transfer-map coefficient of the same open class as $\kappa_{\rm Cep}$ and the Paper 0 $\Gamma_X$ entries, reported pending the action-level derivation rather than absorbed into the theory. The Step 043 clock-channel forward model carries the same bookkeeping in its own normalization: the OD-level rescaling $C_A \approx 4.2$ between reconstructed apparent shifts and published anomalies is the clock-channel analogue of $\kappa_{\rm flyby}$ and is reported as such.

4.3 Deterministic Geometry Modulation Analysis

With only n = 6 sign-gated detections, formal variance decomposition into structural, observational, environmental, and residual components remains underpowered: the standard error on a sample variance estimate with ddof = 1 exceeds 100% of the estimate for n < 5 and stays near 50% at n = 6. Step 009 therefore does not report ANOVA-style percentages. Instead, three complementary analyses that are defensible at small n are presented.

4.3.1 Beta Scatter Statistics

Across the six gated fits, fitted β spans $4.43 \times 10^{-3}$ to $3.40 \times 10^{1}$ (log STD = 1.77 dex, n = 6), dominated by the two unconstrained near-zero-prediction rows; the four amplitude-informative members span $4.43 \times 10^{-3}$ to $6.65 \times 10^{-2}$. Because the Step 007 prediction already includes the geometry envelope, the residual scatter among the constrained members reflects genuine coupling heterogeneity or model incompleteness, not unmodeled trajectory geometry. With n = 6, no formal partitioning of this scatter into structural, observational, or environmental buckets is attempted.

4.3.2 Full-Catalog Detection Pattern

Across all n = 12 catalogued flybys, the Step 007 deterministic prediction at β_{A,ref} = 10⁻⁴ correctly classifies 8/12 flybys (accuracy 66.7%): 0 true positives, 8 true negatives, 4 false negatives, and 0 false positives. Under the canonical field normalization no flyby exceeds the detection threshold at the reference coupling — NEAR itself is predicted at only $+0.30$ mm/s versus the observed $13.46$ mm/s. The four false negatives at β_{A,ref} are the entire published detection set: NEAR ($+0.30$ mm/s predicted), Galileo 1990 ($+0.03$ mm/s), Rosetta 2005 ($+0.03$ mm/s), and Galileo 1992 ($-0.01$ mm/s) — all four carry the correct sign, with magnitudes below threshold because the reference coupling sits a factor $\sim 50$ below the fitted pooled amplitude. Cassini's corrected $-2$ mm/s entry sits at the adopted detection boundary and is classified as a true negative; the sign of its reference prediction nevertheless agrees with observation. No false positives are present: no published null or sub-threshold row is predicted as a detection.

4.3.3 Rank Correlation

A Spearman rank correlation between predicted and observed anomaly magnitudes across the full catalog (n = 12) yields ρ = 0.56 (p = 0.059). This indicates moderate rank agreement: the model captures the relative ordering of anomaly sizes, with the largest predicted anomalies corresponding to the largest observed anomalies. The companion ordering diagnostic between |trajectory asymmetry| and |observed anomaly magnitude| gives ρ = 0.56 (p = 0.11, n = 9 flybys with published asymmetries) and ρ = 0.68 (p = 0.090, n = 7 nonzero anomalies); within the gated n = 6 subset the ordering is positive but not decisive (ρ = +0.66, p = 0.156): NEAR carries both the largest asymmetry and the largest anomaly in every subset.

The legacy four-stage variance-decomposition output in results/step009_variance_analysis.json is retained for backward compatibility but is explicitly marked DEPRECATED and must not be used for manuscript inference. The dominant formal heterogeneity statistic remains Cochran Q / I² on the six gated β fits (Step 008).

4.4 Disformal Transition Criterion Results

The disformal transition criterion Ξ classifies flybys into conformal-dominated, mixed, or disformal-dominated regimes of the fitted response template, using velocity, asymmetry, and altitude. The definition Ξ = (v/v_trans)² × |asym| × (|∇φ|/|∇φ_⊕|) × sgn(asym) uses the empirical template scale v_trans ≈ 16.8 km/s (status: phenomenological ansatz, Section 3.5 — under the canonical B(φ) envelope no kilometre-per-second disformal transition exists, the required microscopic coefficient being excluded by ~26 orders of magnitude); the analyzed flybys span multiple template regimes.

Cassini, with its high perigee velocity (19.02 km/s) and weakly negative asymmetry (cos_asymmetry = -0.022), lies in the template's mixed regime where gradient and disformal contributions partially cancel at β_{A,ref}, yielding a small negative total prediction ($-5.3 \times 10^{-5}$ mm/s). With the corrected catalogue entry the published anomaly is also negative ($-2 \pm 1$ mm/s), so Cassini passes the sign gate; its residual tension is in amplitude only, since the reference prediction is far below the observation and the unconstrained per-flyby rescaling is absorbed into the negligible-weight fit reported in Table 3. Consistent with the canonical branch, the hierarchical layer's fitted disformal amplitude is null ($b_{\rm disf} = 0.026 \pm 0.043$), so the velocity-template terms act as phenomenological descriptors rather than a detected propagating sector.

4.4a Full-Catalog Raw Stress Test

Before restricting attention to the sign-gated detections, the universal-$\beta_{\rm fit}$ prediction is tested against all published rows with explicit raw TEP predictions in Step 039. This raw-layer stress test includes NEAR, Galileo 1990, Rosetta 2005, Cassini, Galileo 1992, MESSENGER, Rosetta 2009, Juno, and Rosetta 2007 ($n=9$). Flybys with no public anomaly report are also excluded. The uncertainty is $\sigma_{\rm tot}^2=\sigma_{\rm obs}^2+\sigma_{\rm raw}^2$, combining the published measurement uncertainty with the propagated universal-$\beta_{\rm fit}$ prediction uncertainty.

This table is the headline full-catalog stress test: once the random-effects prediction uncertainty is included, the raw universal-$\beta_{\rm fit}$ model improves over the null by $\Delta\log L \approx 43.03$. The improvement reflects the large NEAR, Galileo 1990, and Rosetta 2005 signals, while still exposing the remaining stress cases: Galileo 1992 is classified as a raw surplus (correct sign, underpredicted magnitude; the uncertainty-aware scheme flags it as an observed detection with an uncertain prediction), and Juno remains the explicit raw-tension null (predicted $+0.12$ mm/s at the pooled $\beta_{\rm fit}$, a $\sim 3\sigma$ tension under the propagated uncertainty). These results are therefore stronger than a pure three-row fit, but they are not a post-OD mission likelihood because the $F_{\rm OD}$ columns remain withheld until real OD configuration data are available.

4.5 Bayesian Model Comparison

The four-tier model comparison below evaluates Null, Anderson, TEP restricted, and TEP flexible models on the full catalog of nine flybys with published observations (n = 9), using Gaussian likelihoods and systematic uncertainty from the Step 026 heterogeneity budget. The TEP restricted amplitude β_{\rm fit} is fitted on the six-member sign-gated ensemble (all published detections now pass the sign gate on the corrected catalogue), then applied to the full n = 9 catalog for the model-comparison likelihood. This tests whether a coupling inferred from the detections also compresses the full catalog.

4.5.1 Model Definitions and Parameter Status

4.5.2 Log-likelihoods and Information Criteria

The archived Step 026 models are trajectory-geometry templates, not distinct clock-sector predictions for the coherent catalogue. Their iteratively weighted least-squares scores are retained below for reproducibility. Because the fractional variance depends on the fitted prediction, this fixed-point procedure is not a certificate of maximum likelihood for the full heteroscedastic model; the fitted scatter and template choices also require explicit nuisance accounting. The following log-likelihood, AIC and BIC values are therefore archived convention-dependent scores, not calibrated evidence for the clock mechanism:

• Null: log L = -8360.96, AIC = 16721.9, BIC = 16721.9
• Anderson: log L = -11.59, AIC = 27.2, BIC = 27.6
• TEP restricted: log L = -15.09, AIC = 32.2, BIC = 32.4
• TEP flexible: log L = -30.88, AIC = 67.8, BIC = 68.4
• TEP restricted (pessimistic): k_eff = 8 (1 fitted β_fit + 7 heuristic envelope coefficients), BIC = 47.8

Sign-gated ensemble (n = 6) log-likelihoods under the same fractional floor: Null log L = -8368.02, Anderson log L = -6.55, TEP restricted log L = -16.97, TEP flexible log L = -35.42 (BICs 16736.0, 16.7, 35.7, 76.2 respectively). The gated sensitivity at $\sigma_{\rm sys} = 0$ — published uncertainties taken at face value — favours TEP restricted over Anderson by $\Delta{\rm BIC} \approx +848$ (log L $-582.0$ versus $-1005.2$), the same direction as the full-catalog $\sigma_{\rm sys}=0$ result ($+2767$); the nominal Anderson preference exists only under a nonzero systematic floor whose scale is a modelling convention rather than a measurement.

4.5.3 Information Criteria and Model Comparison

Information criteria for the full n = 9 catalog:

• Anderson vs Null: $\Delta$BIC $\approx 16694.3$
• TEP restricted vs Null: $\Delta$BIC $\approx 16689.5$
• TEP flexible vs Null: $\Delta$BIC $\approx 16653.6$
• TEP restricted vs Anderson: $\Delta$BIC $\approx -4.8$ (parity; Anderson nominally first)

Interpretation. The three geometric templates organize the catalogue with archived score differences $\Delta$BIC $\gtrsim 1.66\times10^{4}$ against the zero-signal Null under the adopted uncertainty convention. This is not support for the clock-sector mechanism: that term is absent from the coherent observable by Section 5.2. The comparison also assigns the Null no additional fractional scatter, so it does not exclude tracking-systematic alternatives. On the direct head-to-head, the Anderson empirical model attains the smallest BIC ($27.6$ versus $32.4$ for TEP restricted) — a $\Delta$BIC $\approx 4.8$ gap that lies in the weak-evidence regime and constitutes statistical parity rather than a decisive preference. Two context points matter. First, the Anderson tier is fitted on the very catalogue it is scored against (amplitude $A$ and offset $B$), whereas TEP restricted carries a single amplitude on top of geometry pre-specified from independent measurements. Second, the ordering between the two tiers is systematic-convention dependent: under the published-uncertainties-only sensitivity block ($\sigma_{\rm sys} = 0$), TEP restricted leads Anderson by $\Delta$BIC $\approx +2767$, because its geometry tracks the dominant NEAR amplitude while the linear asymmetry law does not; the former constant geometry-spread floor placed Anderson ahead by $\approx 98$ only through its dependence on the arbitrary $\beta_{\rm ref}$ anchor, and the constant-floor sweep in Step 026 shows the ordering crossing zero near $\sigma_{\rm sys}\approx5$ mm/s. The headline fractional floor sits inside this range and yields parity. The TEP flexible tier (BIC $= 68.4$) does not improve on the restricted tier under the canonical linear response. The BIC-approximated Bayes factor ($BF \approx e^{\Delta{\rm BIC}/2}$) is a large-sample result. For n < 10 and extreme signal-to-noise it is unreliable; when $\log_{10}(BF) > 100$ the value is driven by formal uncertainties and should not be read as a literal probability ratio (Step 026 flags all comparisons in this regime). The displayed scores are arithmetic functions of the archived objective values and parameter counts. Their probabilistic interpretation additionally requires a verified likelihood optimum and treatment of all data-estimated nuisance quantities; those checks are distinct from the channel theorem. Even under the pessimistic parameter count ($k_{\rm eff} = 8$), the TEP restricted BIC = 47.8 still beats the Null by $\Delta$BIC $\approx 16674.1$, confirming that the catalog compression is not an artifact of aggressive parameter counting; under that penalty the Anderson tier's margin over TEP restricted widens to $\approx 20$ BIC units.

Akaike weights (Step 026, full $n = 9$ catalog, AICc): Anderson $\approx 0.86$, TEP restricted $\approx 0.14$, TEP flexible $\approx 0$, Null $\approx 0$. The weight distribution quantifies the same parity-level ordering as the BIC comparison — neither tier dominates under the fractional floor.

The restricted model remains the scientifically important tier because every quantity except $\beta_{\rm fit}$ is pre-specified from independent measurements or stated response templates. Its large $\Delta$BIC separation on the full n = 9 catalog therefore reflects predictive compression relative to the Null with a catalog-independent geometry, while the Anderson tier measures how much additional compression is purchasable with a second catalog-fitted parameter.

The fitted $\beta_{\rm fit}$ values are flyby trajectory-response amplitudes, not PPN $\gamma$ predictions. Cassini PPN compliance is evaluated through the solar source-charge projection $\alpha_{\rm eff}^{(\odot)} = S_\Sigma^{(\odot)}\alpha_0$ (Section 4.6.1a). The ensemble weighted mean yields $\beta_{\rm fit} = 4.74 \times 10^{-3} \pm 5.83 \times 10^{-4}$. The solar source-charge projection in Section 4.6.1a adopts the Paper 0 v0.15 convention $S_\Sigma^{(\odot)} \sim 2\times10^{-6}$, inside the Cassini requirement $S_\Sigma^{(\odot)} \lesssim 5.8\times 10^{-6}$ (canonical linear mapping); the kinetic-operator benchmark evaluates the solar-field screening far deeper, $\sim 10^{-15}$ at the 1 AU path. This supports Temporal Topology screening in the solar environment without overstating the margin.

4.6 PPN Constraints and Validation

4.6.1 PPN Constraint Derivation

The PPN (Parametrized Post-Newtonian) formalism characterizes deviations from General Relativity. For scalar-tensor theories with conformal coupling $A(\phi) = \exp(\beta_A \phi/M_{\rm Pl})$, the PPN parameter $\gamma$ relates to the coupling strength:

Derivation (Paper 0, Sec. 7): in the screened-source limit the photon probe is unscreened while the source charge is suppressed, so $\gamma_{\rm PPN} - 1 = -2\alpha_0\,\alpha_{\rm eff}/(1 + \alpha_0\,\alpha_{\rm eff}) = -4\beta_A^2 S_\Sigma/(1 + 2\beta_A^2 S_\Sigma)$, linear in $S_\Sigma$, with $\alpha_{\rm eff} = S_\Sigma\,\alpha_0$ and $\alpha_0 = \sqrt{2}\,\beta_A$ the DEF-normalized microscopic coupling. The Cassini source-charge parameter is $\alpha_{\rm eff}^{(\odot)} = S_\Sigma^{(\odot)}\alpha_0$, giving $|\gamma - 1| \approx 4\beta_A^2 S_\Sigma^{(\odot)}$ for magnitude comparisons to Cassini (the measured $\gamma - 1$ is negative in the DEF convention). The Earth-vicinity flyby response geometry $S_\oplus$ is a distinct projection and is not the Cassini source-charge parameter.

The fitted $\beta_{\rm fit}$ values are flyby trajectory-response amplitudes for Earth-vicinity geometry, distinct from the Cassini source-charge parameter $\alpha_{\rm eff}^{(\odot)} = S_\Sigma^{(\odot)}\alpha_0$. The Earth-screened flyby response amplitudes $\beta_{\rm fit} \times S_\oplus$ are:

• NEAR: $\beta_{\rm fit} \times S_\oplus = 1.55 \times 10^{-3}$
• Galileo 1990: $\beta_{\rm fit} \times S_\oplus = 5.03 \times 10^{-3}$
• Galileo 1992: $\beta_{\rm fit} \times S_\oplus = 2.33 \times 10^{-2}$
• Rosetta 2005: $\beta_{\rm fit} \times S_\oplus = 1.80 \times 10^{-3}$
• Cassini and MESSENGER: formally gated but unconstrained (reference prediction $\approx 0$); Cassini PPN compliance is evaluated through the solar source-charge projection $\alpha_{\rm eff}^{(\odot)} = S_\Sigma^{(\odot)}\alpha_0$ (Section 4.6.1a), not through the flyby amplitudes.

These are trajectory-response amplitudes, not PPN $\gamma$ predictions. Cassini PPN compliance is evaluated through the solar source-charge projection $\alpha_{\rm eff}^{(\odot)} = S_\Sigma^{(\odot)}\alpha_0$ (Section 4.6.1a), not through the fitted flyby amplitudes.

4.6.1a Solar-Screening PPN Check for Cassini

The Cassini Shapiro-delay measurement constrains the scalar field along the radio path during solar conjunction, not at Earth's surface. Cassini PPN compliance is evaluated through the solar source-charge projection, not through the fitted flyby amplitudes. Following the Jakarta mapping (Paper 0, §7):

where $\alpha_0 = \sqrt{2}\,\beta_A$ is the DEF-normalized bare microscopic coupling (linear in the source charge because the photon probe is unscreened) and $S_\Sigma^{(\odot)}$ is the solar Temporal-Topology screening factor. With $\beta_A = -1$ (Jakarta §2.2), the Cassini bound $|\gamma - 1| < 2.3\times 10^{-5}$ requires (Paper 0, Jakarta v0.15 §7, canonical linear source-charge mapping):

Cassini constrains the solar exterior source-charge projection $S_\Sigma^{(\odot)}$, not the Earth-vicinity flyby response $S_\oplus$. Paper 0 v0.15 adopts the Solar-System source-charge convention $S_\Sigma^{(\odot)} \sim 2\times10^{-6}$, which satisfies the Cassini requirement $S_\Sigma^{(\odot)} \lesssim 5.8\times 10^{-6}$ comfortably; the kinetic-operator evaluation on the solar field returns far deeper screening, $S_\Sigma \sim 1.6\times10^{-15}$ at the 1 AU Cassini path (Paper 0 Appendix E, static-branch benchmarks), with the microscopic uniqueness of that charge within the wider $K(X)$ class remaining open. The flyby response and the solar PPN source charge are therefore treated as distinct environmental projections of the same scalar configuration.

4.6.2 Sensitivity Analysis

To assess robustness, the TEP model is tested against variations in key parameters. Table 3f shows how results change when parameters are varied within physically plausible ranges:

Robustness conclusion: The TEP model maintains fit stability across a broad range of parameter values. The phase-boundary factor can vary by ±32% (0.25 to 0.45) and all fitted β_fit values remain within the fitted range. This suggests that the fit stability is not fine-tuned but is a feature of the screening mechanism. The relaxation length has moderate impact on predicted Δv but does not affect fit stability because the fitted β values adjust to compensate.

4.6.3 OD Filter Simulation: Suppression Hypothesis Validation

Step 021 now withholds all mission-specific OD survival factors until real mission OD configuration data are available. The earlier synthetic Step 012 batch least-squares experiment is retained only as a diagnostic development artifact: its empirical-acceleration implementation is numerically unstable in the current 3D form, and the generated result is not valid for computing $F_{\rm OD}$ or for supporting quantitative claims about modern OD suppression.

Interpretation: The OD-suppression mechanism remains a falsifiable hypothesis rather than a calibrated correction. It is physically plausible that empirical acceleration states and residual editing can absorb unmodeled perigee-local forces, but this paper does not assign numerical survival fractions without mission-specific OD settings.

Connection to observations: Step 039 classifies fixed-amplitude raw predictions (Step 008 pooled $\beta_{\rm fit}$) with the Step 007 geometry envelope (3 deterministic true positives, 4 deterministic true nulls, 1 sign-consistent raw surplus, and 1 deterministic fixed-amplitude warning case across the 9 published rows). After propagating Step 008 random-effects amplitude scatter, the uncertainty-aware raw layer retains 1 raw-tension case (Juno) and moves Galileo 1992 to "observed detection, prediction uncertain". Post-OD survival factors are withheld until mission OD configuration yields defensible $F_{\rm OD}$ estimates.

Juno tension: The Juno non-detection (universal-$\beta_{\rm fit}$ prediction $+0.12 \pm 0.03$ mm/s, observed $0.00 \pm 0.02$ mm/s) is the most serious raw-tension case and motivates independent raw DSN re-analysis. In the current pipeline run, the DSN ingestion layer emits mission discovery and request artifacts but does not yet provide indexed raw Doppler products for most missions (ingest status no_indexed_products), so this re-analysis remains a defined falsification pathway rather than a completed reprocessing result.

4.6.3a Clock-Sector Artefact: End-to-End Orbit-Determination Rederivation

The clock-sector mechanism has now been propagated through the full reduction chain rather than asserted (Step 043). For each of the twelve flybys with reconstructed JPL Horizons state-vector arcs, the canonical field excursion $\psi(r) = \phi_{\rm space}-\phi(r)$ of the pipeline's screened field solution is evaluated along the real trajectory; the matter-metric rate offset $\eta(t) = \exp(\beta_A\,\psi/M_{\rm Pl})-1$ ($\beta_A=-1$; the unsuppressed amplitude channel $S_A$, not the screened shear channel) accumulates the proper-time offset $\delta\tau(t)=\int\eta\,dt$ and injects the apparent range-rate offset $c\,\eta(t)$ that the excursion produces in a clock-carrying link — a spacecraft-local oscillator term entering once wherever the downlink is referenced to an onboard time standard (one-way, non-coherent, or regenerative-ranging observables). In the fully coherent two-way class of the published catalogue the term is absent at the observable: the transponder multiplies the received carrier by a fixed ratio, and the endpoint conformal factors cancel identically (Section 5.2). The step therefore computes what the clock excursion reconstructs as on a clock-carrying link over the real arcs — the mechanism's falsifiable prediction — together with the postfit-residual signature that quantifies how much of it survives reduction into the orbit solve. The identical batch least-squares orbit-determination filter used in the OD-suppression study (six-state estimator, point-mass dynamics, assumed GR time standard) is then run on clean and corrupted synthetic Doppler, and the differential of the fitted end-minus-start speed change isolates the clock-induced apparent $\Delta V$. Truth dynamics are pure point-mass gravity — consistent with the screened shear channel contributing nothing to the trajectory — while $\eta(t)$ is sampled on the real arcs, preserving the physical in/out asymmetry.

Three findings follow, all conditional on the link class. First, the amplitude regime is correct: the canonical field excursion ($\Delta\phi = 3.4\times10^{9}$ GeV, $\eta_{\rm surface}=-1.39\times10^{-9}$) produces accumulated proper-time offsets of $0.1$–$2\,\mu$s; the raw signature reaches $c\eta \sim$ m/s at perigee, so the excursion would be an unmissable feature of any one-way, non-coherent, or onboard-referenced flyby measurement. Second, the discriminating observable is the postfit residual, not the reconstructed endpoint $\Delta V$: the six-state fit absorbs the bulk of the perigee-concentrated offset into the estimated state, but a structured remnant always survives — $0.2$–$7.7$ mm/s RMS across the catalogue (peaks up to $\sim 83$ mm/s), i.e. $4$–$25\%$ of the injected signature leaks through orbit determination at $\sim 180$ s cadence and remains detectable at DSN noise floors. The endpoint $\Delta V_{\rm app}$ column is reported for completeness but is an absorption remnant rather than a signal amplitude — it shrinks by a factor of $\sim 15$ as the tracking window widens from $\pm 6$ to $\pm 24$ h (NEAR: $+0.68 \to +0.05$ mm/s) — while the residual signature is window-stable (the linearized filter response reproduces the nonlinear result within 10–40%). Third, the recovered artefact is controlled by the projection of the $\eta(t)$ profile through the tracking geometry into the fit basis, not by the leg-integrated proper-time asymmetry: across the nine published rows the reconstructed apparent shift correlates only moderately with the catalogue (Pearson $r = 0.17$, Spearman $\rho = 0.58$), so the artefact is intrinsically geometry-projection-dependent, consistent with the observed anomaly's documented resistance to simple energy-change parameterizations.

Amplitude calibration: evaluated against the detection subset, the raw canonical normalization under-produces the detected anomalies, and the global clock-response coefficient — the clock-channel analogue of the Step 008 response amplitude $\beta_{\rm fit}$ — returns $C_A = 4.2$ on the sign-inclusive $>3\sigma$ detection subset (NEAR, Galileo 1990, Galileo 1992, Rosetta 2005; $n=4$, Pearson $r=0.08$ at this normalization) with $\chi^2 = 1.17 \times 10^{4}$ on 3 dof: no single global clock-channel scaling describes the detections, so the coefficient is reported rather than adopted. The per-flyby ordering of $\Delta V_{\rm app}$ does not, by itself, reproduce the catalog pattern: the same geometry envelope and response-amplitude structure fitted in §4.3–4.5 remains load-bearing, and the clock-sector rederivation supplies the physical channel through which that response acts in a clock-carrying link rather than a replacement fit. Because the catalogued anomalies were recorded in the coherent two-way class (Section 5.2), the coefficients calibrated here characterize the forward model's response scale for clock-carrying observables, not an attribution of the published values. Limitations are explicit: a generic three-complex network rather than each mission's historical tracking schedule, point-mass truth dynamics, and single-channel (clock-only) injection. The per-mission tracking-schedule reconstruction and joint clock-plus-environment injection are identified refinements.

4.6.4 Leave-One-Out Cross-Validation

To verify that the weighted mean β is not dominated by any single detection, the analysis is repeated excluding each flyby successively:

The stability coefficient (relative standard deviation of LOO estimates divided by their mean) is 0.092, indicating moderate robustness (values < 0.5 are considered robust). Even when the high-S/N NEAR detection is excluded, the remaining five flybys yield β = 5.88×10⁻³, within the 95% bootstrap interval. This indicates that the TEP conclusion does not depend on any single detection.

4.6.5 Enhanced Statistical Validation

Artefact-kernel impulse consistency: The clock-sector response model's apparent velocity predictions integrate the field excursion along 3D trajectories while preserving the TEP metric structure. For each flyby, the predicted $\Delta v_{\rm TEP}$ is computed via path integration of the artefact kernel $\mathbf{F}_\phi = \beta_{A,\rm eff} c^2 \nabla\phi/M_{\rm Pl}$ — the OD-equivalent impulse of the proper-time rate-offset excursion, not a physical force — along the actual spacecraft trajectory from JPL Horizons ephemeris. The open-path impulse $\mathcal{I} = \int \mathbf{F}_\phi \cdot d\mathbf{r}$ is consistently mapped to observable velocity shifts. This geometric consistency check distinguishes TEP from phenomenological force laws that lack field-theoretic structure.

Effect size analysis: Cohen's d compares each detection to the null-result population mean, using the pooled standard deviation of the two groups. The null population comprises the three published sub-threshold flybys (Rosetta 2007, Rosetta 2009, Juno) with mean $\Delta v = 0.007 \pm 0.012$ mm/s. The detection population ($n=6$) has mean $\Delta v = 2.10 \pm 6.30$ mm/s. The pooled standard deviation is $\sigma_{\rm pooled} = 5.33$ mm/s. Cohen's d for each detection vs. the null population:

• NEAR: $d = +2.53$ — large effect ($d \gg 0.8$)
• Galileo 1990: $d = +0.73$ — medium effect
• Galileo 1992: $d = -0.87$ — large effect on the negative branch
• Cassini: $d = -0.38$ — small effect on the negative branch
• Rosetta 2005: $d = +0.34$ — small effect
• MESSENGER 2005: $d = +0.003$ — negligible effect

NEAR shows the largest effect, and the corrected catalogue places Galileo 1992 on the negative branch with a large $|d|$ — consistent with the negative sign the TEP kernel predicts for both it and Cassini. The two strongest detections (NEAR and Galileo 1990) provide the bulk of the positive-branch statistical separation.

Bayesian model comparison: Stable four-tier model comparison (Step 026) on the full n = 9 catalog favors TEP restricted over Null ($\Delta{\rm BIC}\approx16689.5$) under the fractional misspecification floor, while the Anderson empirical model — whose two parameters are fitted on this same catalogue — attains the nominally smallest BIC at parity-level separation (TEP restricted minus Anderson $\Delta{\rm BIC}\approx-4.8$; the ordering reverses in TEP restricted's favour under the published-uncertainties-only sensitivity). See Section 4.5 for tier definitions. The physics-based single-amplitude model therefore compresses the full catalog far beyond the null, with the residual gap to the empirical tier quantifying the amplitude-level heterogeneity documented in Section 4.2.

Prediction accuracy: On the Step 008 primary comparison, $R^2 \approx 0.850$, Pearson $r \approx 0.930$, MAE $\approx 1.67$ mm/s, RMSE $\approx 2.24$ mm/s, and MAPE $\approx 62.5\%$. NEAR dominates the variance fraction; small-anomaly rows inflate percentage errors.

Residual analysis: Shapiro–Wilk normality on the gated prediction residuals gives $p \approx 0.95$ (consistent with Gaussian tails at $n=6$).

4.6.6 Characteristic Suppression from UCD Saturation Model

The characteristic suppression $S_{\oplus} \approx 0.35$—critical to the flyby amplitude and the magnitude of the flyby anomaly—is derived from the UCD saturation model in Step 010. The derivation uses Earth's total mass and the saturation scale $\rho_T = 20$ g/cm³, yielding a transition radius $R_{\rm sol} \approx 4146$ km and suppression factor $S_{\oplus} = (R_{\oplus} - R_{\rm sol})/R_{\oplus} \approx 0.35$. Under the corpus's terrestrial identification $L_c \sim \mathcal{O}(1)\,R_T(M_\oplus)$ (Paper 6 §2 transfer sketch), the GNSS covariance scale returns $S_\oplus = 0.34$ — a shared-input consistency check at the 2% level, since the same measurement anchors $\rho_T$, rather than an independent validation. The independent empirical checks — the flyby altitude threshold and the astrophysical cross-scale consistency of $\rho_T$ across dwarf-galaxy cores and cluster halos (Paper 6) — converge on $S_{\oplus} \in [0.34, 0.39]$. See Step 010 for the complete derivation and cross-scale consistency arguments.

Distinction from the canonical amplitude operator: The EFA uses $S_{\oplus} = (R_{\oplus} - R_{\rm sol})/R_{\oplus}$ as the field-excursion response ratio at the surface. Its relation to the corpus's canonical clock-amplitude operator $S_A(\mathcal E) = \min[1, (\bar\rho/\rho_T)^{1/3}]$ (Paper 0/Paper 26) is an exact identity, not a second estimate: under the uniform-density-equivalent construction, $S_A(\bar\rho_\oplus) = (\bar\rho_\oplus/\rho_T)^{1/3} = R_{\rm sol}/R_\oplus \approx 0.65$ — the saturated (embedded) fraction of the equivalent radius, coinciding with the UCD embedding factor used in Paper 6 (UCD). The flyby clock channel projects the complementary responsive-shell fraction $S_\oplus = 1 - S_A \approx 0.35$, because a transported clock's differential excursion relative to the ground accumulates only across the unsaturated shell where the field relaxes; the pinned interior contributes no radial excursion. The Step 010 output records the machine-checked partition sum $S_A + S_\oplus = 1$ (canonical_amplitude_partition). For strongly centrally condensed objects the enclosed mean density exceeds $\rho_T$, in which case $S_A$ saturates at unity and the excursion projection clamps at $S_\oplus \to 0$ (a fully pinned profile) rather than going negative — the uniform-density-equivalent partition applies to bodies whose enclosed mean density remains below $\rho_T$.

4.6.7 Systematic Uncertainty Budget

A comprehensive uncertainty budget quantifies the contribution of each uncertainty source to the fitted $\beta_{\rm fit}$ parameters. The corrected uncertainty analysis (Step 025) distinguishes between variance contributions and total relative uncertainty:

Variance Contributions:

• Statistical: 0.8%
• Systematic: 3.4%
• Heterogeneity: 95.8%

Total Relative Uncertainty:

• Statistical: 14.5%
• Systematic: 29.2%
• Heterogeneity: 155.4%
• Total: 158.8%

Systematic Breakdown:

• Trajectory reconstruction: 1.0%
• Characteristic suppression (UCD): 25.0% (from ρ_T = 20 ± 7 g/cm³, Paper 6) ← DOMINANT
• Multipole coefficients: 0.1%
• Relaxation length (UCD): 15.0% (SCF theoretical prior)

Interpretation: The total relative uncertainty of 152.7% is dominated by heterogeneity (149.4%), which reflects genuine geometry-dependent physical variation in the effective coupling across flybys. This is expected in the TEP framework where $\beta_{A,\rm eff}$ varies with altitude, latitude, velocity, plasma environment, and trajectory asymmetry (the plasma term is a bounded heuristic envelope coefficient, Section 3.10). The systematic uncertainty (29.2%) is dominated by characteristic suppression uncertainty (25.0%) from the UCD saturation model (ρ_T = 20 ± 7 g/cm³, Paper 6), with relaxation length uncertainty (15.0%) from the SCF theoretical prior as the second-largest source.

4.7 Model Predictions for All Flybys

Table 4 presents the full prediction set evaluated at the universal weighted-mean coupling constant ($\beta_{\rm fit} = 4.74 \times 10^{-3}$), scaled from the reference predictions ($\beta_{\rm fit,0} = 10^{-4}$) via the linear response established in Step 008 (the canonical excursion normalization is $\beta$-independent, so $\Delta v \propto \beta_{\rm fit}$). Each row reports the raw TEP prediction at that universal coupling, the universal-$\beta_{\rm fit}$ residual $\Delta v_{\rm obs} - \Delta v_{\rm TEP}^{\rm raw}$, and the raw classification from Step 039. Mission-specific OD survival factors $F_{\rm OD}$ and post-OD predictions are emitted only when Step 021 supplies defensible mission OD configuration data; otherwise those columns are withheld. The raw classification uses a $3\sigma$ detection threshold relative to the published uncertainty.

Summary: At the universal weighted-mean $\beta_{\rm fit}$, Step 039 classifies three published anomalies as raw true positives (NEAR, Galileo 1990, Rosetta 2005). Cassini is a raw true null (corrected $-2 \pm 1$ mm/s entry versus a near-zero prediction), Galileo 1992 is a raw surplus (correct sign, underpredicted magnitude at fixed universal amplitude), MESSENGER and the three published null/bound cases are consistent with universal-$\beta_{\rm fit}$ predictions under the Step 007 geometry envelope, and one raw-tension case remains (Juno). Stardust, OSIRIS-REx, and BepiColombo lack public anomaly reports and are not used in quantitative likelihood.

Falsifiability criterion: Raw-tension cases define model stress tests independent of era-based OD survival factors. Juno remains the sole deterministic fixed-amplitude warning case at the refit weighted-mean $\beta_{\rm fit}$ ($+0.12$ mm/s raw prediction vs. $0.00 \pm 0.02$ mm/s observed) with random-effects prediction uncertainty $\pm 0.03$ mm/s. Even after propagating Step 008 random-effects scatter, the uncertainty-aware Step 039 layer retains Juno as a raw-tension case ($\sim 3\sigma$), while Galileo 1992 moves to the uncertainty-aware class "observed detection, prediction uncertain" — the model assigns the correct sign but insufficient magnitude at the pooled amplitude. Post-OD false-negative counts are reported only when mission-specific $F_{\rm OD}$ values are available from Step 021; with current OD configuration data, those columns are withheld.

Honest assessment: The clock-sector response model reproduces the three largest published anomalies at universal $\beta_{\rm fit}$ with sub-millimetre to millimetre residuals, while high-altitude or symmetric trajectories remain consistent with null predictions. Galileo 1992 — now correctly catalogued as a $-4.60 \pm 1.00$ mm/s published detection — and Juno define the priority raw DSN reanalysis targets. Cassini is no longer a sign-tension case: the corrected $-2 \pm 1$ mm/s entry carries the same sign as the TEP prediction, leaving only an amplitude shortfall that requires independent DSN OD or envelope refinements, not only a universal rescaling of $\beta_{\rm fit}$.

4.8 Heterogeneity and Robustness Analysis

Heterogeneity assessment: The six gated fitted β_fit values span $4.43 \times 10^{-3}$ to $3.40 \times 10^{1}$ (Step 009), the upper end driven by the two unconstrained near-zero-prediction rows; the four amplitude-informative members span $4.43 \times 10^{-3}$ to $6.65 \times 10^{-2}$. Step 009 does not perform a formal variance decomposition because n = 6 remains underpowered for ANOVA-style partitioning. Instead, Step 009 reports three complementary deterministic geometry modulation analyses: (1) beta scatter statistics (log STD = 1.77 dex, n = 6); (2) full-catalog detection-pattern classification (67% accuracy, n = 12: 0 true positives, 8 true negatives, 4 false negatives, 0 false positives); and (3) Spearman rank correlation between predicted and observed anomaly magnitudes (ρ = 0.56, p = 0.059, n = 12) together with the asymmetry-ordering diagnostic (ρ = 0.56, p = 0.11, n = 9). The formal heterogeneity statistics on the gated ensemble are dominated by sub-percent measurement precision:

The elevated $I^2$ on the six gated $\beta_{\rm fit}$ fits reflects formal tension between sub-percent per-flyby uncertainties and a multiplicative spread in fitted coupling, amplified by the two formally unconstrained rows. The metric is designed for meta-analyses where true effects are identical; here, geometry, velocity, and heuristic envelope structure intentionally modulate $\beta_{A,\rm eff}$, so large $I^2$ is expected until additional physics is folded into a single generative curve or $n$ grows.

Bootstrap resampling: To assess uncertainty given the small sample ($n = 6$ gated detections), parametric bootstrap resampling with $10\,000$ iterations is performed in Step 008:

• Bootstrap median: $\beta_{\rm fit} \approx 4.77 \times 10^{-3}$ (tracks the inverse-variance weighted mean)
• 68% interval: $[4.54 \times 10^{-3},\, 5.93 \times 10^{-3}]$
• 95% confidence interval: $[4.42 \times 10^{-3},\, 1.47 \times 10^{-2}]$

The bootstrap median reproduces the weighted mean while the interval is widened by Galileo 1990’s higher per-flyby $\beta_{\rm fit}$; the occasional resampled divergent row inflates the bootstrap mean, so the median and percentile intervals are the robust summaries.

Leave-one-out (Step 008): Inverse-variance $\beta_{\rm fit}$ is recomputed excluding each gated detection:

• Exclude NEAR: $\beta_{\rm fit} \approx 5.88 \times 10^{-3}$ (+24%)
• Exclude Galileo 1990: $\beta_{\rm fit} \approx 4.58 \times 10^{-3}$ (-3%)
• Exclude Rosetta 2005: $\beta_{\rm fit} \approx 4.64 \times 10^{-3}$ (-2%)
• Exclude Cassini, Galileo 1992, or MESSENGER: $\beta_{\rm fit} \approx 4.74 \times 10^{-3}$ (<0.1%)

The stability coefficient is $0.092$, indicating moderate robustness (values $< 0.5$ are treated as acceptable in Step 008). NEAR is the dominant lever on the pooled scale; the three near-zero-prediction rows are negligible levers, as expected from their formal uncertainties.

Effect size: Cohen's $d$ compares each detection to the null-result population using the pooled standard deviation of the two groups:

The null population comprises the published flybys with sub-threshold anomalies ($n_{\rm null}=3$: Rosetta 2007, Rosetta 2009, Juno; $\mu_{\rm null} = 0.007$ mm/s, $s_{\rm null} = 0.012$ mm/s). The detection population ($n_{\rm det}=6$, $\mu_{\rm det} = 2.10$ mm/s, $s_{\rm det} = 6.30$ mm/s) yields $\sigma_{\rm pooled} \approx 5.33$ mm/s. The resulting Cohen's $d$ values are:

• NEAR: $d = +2.53$ (large effect)
• Galileo 1990: $d = +0.73$ (medium effect)
• Galileo 1992: $d = -0.87$ (large effect, negative branch)
• Cassini: $d = -0.38$ (small effect, negative branch)
• Rosetta 2005: $d = +0.34$ (small effect)
• MESSENGER 2005: $d = +0.003$ (negligible effect)

NEAR remains strongly distinguishable from the null population, and Galileo 1992 now appears as a large negative-branch effect — both consistent with the sign structure the TEP kernel assigns. Rosetta 2005 shows a small effect and MESSENGER a negligible one, reflecting their proximity to the null-population mean. The spread in $d$ values is consistent with the spread in gated fitted $\beta_{\rm fit}$, confirming geometry-dependent modulation rather than a perfectly universal effective coupling at fixed envelope.

4.9 Resolution of Beta Heterogeneity

The multiplicative spread in gated fitted $\beta_{\rm fit}$ values is partially summarized through a four-stage decomposition (Step 009). This unified analysis consolidates structural physics modulation, observational pipeline effects, environmental modulation, and statistical limitations into a coherent framework. The apparent scatter is not treated as pure noise: it is the object of the envelope construction. See Section 4.3 for the detailed variance decomposition analysis.

4.10 PPN Compliance and Global State

Model comparison: Stable four-tier model comparison (Step 026) compares the Null, Anderson empirical, TEP restricted, and TEP flexible models on the full n = 9 catalog. All non-null tiers compress the catalog by $\Delta$BIC $\gtrsim 1.66\times10^{4}$ against the Null (TEP restricted $\approx 16689.5$, TEP flexible $\approx 16653.6$, Anderson $\approx 16694.3$), confirming that trajectory asymmetry carries genuine signal. The Anderson empirical model — fitted with two free parameters on this same catalogue — attains the nominally smallest BIC, at parity-level separation from TEP restricted ($\Delta$BIC $\approx 4.8$); that ordering and its systematic-convention dependence are discussed in Section 4.5.

Formal correlation analysis: Pearson correlation tests quantify relationships between the per-flyby effective geometry factor $G_{\rm eff}$ (Section 4.2.1) and the physical parameters, as computed in Step 015 on the six-member ensemble:

With n = 6 the correlation analysis is exploratory: the velocity trend ($r = 0.90$, $p = 0.011$) is the only coefficient approaching significance and is consistent with velocity-modulated response in the Temporal Topology screening framework, but a six-point sample cannot establish it. A multiple regression of $\ln G_{\rm eff}$ on altitude, velocity, and asymmetry reaches $R^2 = 0.97$ (Step 015), dominated by the two divergent low-prediction rows and reported for completeness only.

Prediction intervals: Uncertainty propagation yields prediction intervals for additional flybys:

• Representative β = $4.74 \times 10^{-3} \pm 5.83 \times 10^{-4}$ (Step 008 weighted mean and formal uncertainty; random-effects uncertainty $1.28 \times 10^{-3}$)
• 68% bootstrap interval (Step 008): $[1.86 \times 10^{-3},\, 2.52 \times 10^{-3}]$
• 95% bootstrap interval (Step 008): $[1.81 \times 10^{-3},\, 8.97 \times 10^{-3}]$

The prediction intervals bracket the amplitude-informative gated fitted $\beta_{\rm fit}$ values and illustrate residual width driven largely by Galileo 1990's high per-flyby coupling.

Sensitivity analysis: All model parameters show stable results across plausible variation ranges:

Model adequacy tests: Shapiro–Wilk on the gated prediction residuals yields $p \approx 0.95$ in Step 008 (n = 6; consistent with normality, small-$n$ caution). Heterogeneity diagnostics (Cochran Q, $I^2$) dominate the interpretation relative to classical normality tests.

The preceding sections have established that the TEP model reproduces the observed anomalies and satisfies PPN constraints. The following section tests a deeper prediction: that the residual discrepancy between observation and prediction should correlate with the geometry of velocity in the ambient-field rest frame, approximated by the CMB dipole frame.

4.11 Cosmographic Temporal Shear Modulation Analysis

A key prediction of the TEP framework is that temporal shear should depend on the total velocity of the Earth-Moon system relative to the ambient-field rest frame, not merely the spacecraft velocity relative to Earth. If the cosmic microwave background (CMB) dipole frame approximates this frame, the ~370 km/s bulk motion of the Solar System toward (RA, Dec) = (167.94°, −6.93°) would provide a cosmographic modulation of the velocity-activated response branch. Additionally, Earth's elliptical orbit produces a heliocentric distance-dependent modulation via solar scalar topology. Under the canonical $B(\phi)$ envelope the propagating-sector amplitude at flyby velocities is negligible (§3.5), so this section tests these predictions as template-level directional hypotheses rather than as signals of a detected disformal channel, using full three-dimensional spacecraft state vectors extracted from JPL Horizons archival ephemeris.

4.11.1 3D State Vector Extraction

Raw JPL Horizons ephemeris files were parsed for each flyby mission, extracting geocentric apparent right ascension, declination, range, and range-rate at 1-minute intervals. Cartesian position and velocity vectors were reconstructed in the J2000 equatorial frame and rotated to the ecliptic frame using the obliquity of the ecliptic ε = 23.439281°. Perigee state vectors were identified by minimum geocentric range. All twelve catalogued flybys carry reconstructed 3D state vectors (Step 038 refuses declination-only fallbacks); the correlation sample of this section is the $n = 9$ subset that also carries a catalogued anomaly value or published bound — Stardust, OSIRIS-REx, and BepiColombo have no published anomaly and so do not enter the ratio tests. Earth heliocentric position and velocity were computed via a low-precision analytical ephemeris with proper elliptical orbit mechanics, yielding non-zero radial velocity components up to ±0.5 km/s consistent with Earth's orbital eccentricity e = 0.0167.

4.11.2 Cosmographic Modulation Factors

For each flyby, three classes of modulation proxies were computed:

1. Heliocentric distance modulation: The solar scalar field density scales as r^{-2}, yielding a modulation proxy M_⊙ = 1/r²_AU.
2. Solar scalar wind factor: Earth's orbital speed relative to the Sun modulates the scalar wind experienced by the spacecraft, approximated as v_orb/29.78 km/s.
3. CMB dipole projection: The total velocity of the spacecraft in the CMB rest frame is v_total = v_CMB + v_Earth + v_sc. The component along the CMB dipole direction n_CMB defines the modulation factor M_CMB = (v_total · n_CMB) / 369.82 km/s.

The TEP disformal coupling scales as v² in the ambient-field rest frame. The CMB-rest-frame disformal enhancement factor is f_enh = |v_total|² / |v_sc|², ranging from ~350 to ~1300 across the sample. (Under the canonical $B(\phi)$ envelope this channel carries negligible absolute amplitude at flyby velocities — §3.5 — so the modulation explored here is a template-level directional check, not a predicted propagating-sector signal.) Because the 370 km/s CMB bulk velocity is nearly constant, the dominant variation in the effective coupling comes from the direction of the spacecraft velocity relative to the CMB dipole, quantified by cos θ_SC-CMB = (v_sc · n_CMB) / |v_sc|.

4.11.3 Results

4.11.4 Correlation Analysis

Pearson correlation tests were performed between the observed-to-predicted ratio and each cosmographic modulation factor (n = 9). With the corrected catalogue, several ratio correlations are formally significant — but the residual ratio is dominated by two divergent rows (Cassini and MESSENGER, whose reference predictions are $\approx 0$ and whose ratios reach $\sim 4\times10^{4}$ and $\sim 7\times10^{4}$), so these correlations are outlier-driven and reported for transparency only. The strongest individual correlations are:

4.11.5 Directional Consistency: The Both-Aligned Test

Within the fitted response template, the velocity-activated branch depends on the total CMB-frame velocity, which is the vector sum of the Earth's orbital velocity and the spacecraft velocity, both projected onto the CMB dipole direction. When both the spacecraft velocity and Earth's orbital velocity are aligned with the CMB dipole apex (cos θ_SC-CMB > 0 and v_Earth · n_CMB > 0), the two velocity components add constructively in the ambient-field rest frame, boosting the template's velocity factor. When one or both are anti-aligned, the components partially cancel, suppressing the template factor. This is a directional hypothesis about the phenomenological envelope only: the CMB-rest-frame enhancement multiplies a canonical-branch amplitude that is negligible at flyby velocities (§3.5), so even the largest f_enh factors remain far below any propagating-sector signal.

This template hypothesis was tested by defining a binary "both-aligned" flag for each flyby, equal to 1 when both projections are positive and 0 otherwise. The correlation between this flag and the residual ratio is:

Both-aligned flag: Pearson r = −0.57, p = 0.11 (n = 9)
Mann-Whitney U (aligned > unaligned): U = 7.5, p = 0.79 (exact test)

On the corrected catalogue the both-aligned flag shows no positive association with the residual ratio — the formal sign is negative and the Mann–Whitney test is null (p = 0.79). NEAR and Galileo 1990 retain the largest non-divergent observed-to-predicted ratios in the sample, but the directional flag does not separate them. The test remains exploratory pending additional flybys with published anomalies and full 3D trajectory reconstructions.

4.11.6 Multivariate Geometric Regression

A multivariate ordinary least squares regression was fitted to test whether a linear combination of geometric alignment factors can explain the residual ratio:

ratio = b₀ + b₁ cos θ_SC-CMB + b₂ (v_Earth · n_CMB / 30) + b₃ (SC-orbital alignment) + ε

The fitted coefficients are b₀ = +3.28 × 10⁴, b₁ = +1.42 × 10⁴, b₂ = −3.76 × 10⁴, b₃ = −1.58 × 10⁴. The model achieves R² = 0.96 (adjusted R² = 0.94) and reduces the residual standard deviation by 80.7%. These figures are reported for completeness only: the response variable contains divergent ratios from the two near-zero-prediction rows (Cassini $\approx 3.8\times10^{4}$, MESSENGER $\approx 6.8\times10^{4}$), so a four-parameter fit can score a high R² by tracking the two outlier points while leaving the physical content empty. The regression does not constitute evidence for a CMB-frame coupling at this sample size.

Table 9b lists observed-to-predicted ratios for the nine flybys with usable 3D vectors. The multivariate fit is reported for transparency; it should not be over-interpreted at this sample size.

4.11.7 Optimal Weighted Combination

The relative weighting of the spacecraft and Earth CMB projections was determined by scanning the coefficient w in the linear combination E = cos θ_SC-CMB + w (v_Earth · n_CMB / 30) and selecting the value that maximizes |r(E, ratio)|. On the corrected catalogue the optimum lies at the grid boundary (w = −2.0 of the scanned range [−2, +2]), yielding:

Optimal combination: E = cos θ_SC-CMB − 2.0 (v_Earth · n_CMB / 30) (grid boundary)
Pearson r = +0.96, p = 5.5\times10^{-5} (n = 9); permutation p = 0.003

The apparently strong optimum is not evidence of a calibrated coupling: the maximizing weight sits at the boundary of the scanned grid, and the correlation is driven by the divergent residual ratios flagged in Table 9a. The artifact's own sensitivity exclusion of the near-zero-prediction Rosetta 2007 row leaves the heliocentric-distance correlation essentially unchanged ($r = 0.89$, $p = 0.003$, $n = 8$), confirming that the signal is carried by the Cassini/MESSENGER divergent-ratio points rather than by a generic property of the set. The scan is reported as an exploratory diagnostic only.

4.11.8 Interpretation

The corrected catalogue changes the exploratory picture in two ways. First, several formal correlations are now nominally significant, but they are carried by the divergent residual ratios of Cassini and MESSENGER (predictions $\approx 0$) rather than by a directional property of the detection population. Second, the directional tests proper are null or negative:

1. Both-aligned flag (r = −0.57, p = 0.11; Mann–Whitney U = 7.5, p = 0.79): the flag shows no positive association with the residual ratio; NEAR and Galileo 1990 retain the largest non-divergent ratios but are not separated by alignment.
2. Multivariate regression (R² = 0.96, adjusted R² = 0.94): the nominal fit is outlier-driven — four parameters track the two divergent-ratio rows — and is reported for transparency only.
3. Optimal weighted combination (r = +0.96, p = 5.5\times10^{-5}, permutation p = 0.003): the optimum lies at the scanned grid boundary and is likewise carried by the divergent rows; it does not define a calibrated CMB-frame coefficient.

Caveats: With n = 9 flybys, cosmographic modulation remains exploratory. Neither the both-aligned test nor the directional correlations support a CMB-rest-frame claim on the corrected catalogue, and the formally significant ratio correlations are artefacts of near-zero denominators rather than directional evidence. Additional flybys with published anomalies and full 3D trajectory reconstructions are required before elevating this sector above the core altitude–asymmetry law.

5. Discussion

5.1 Physical Interpretation: The Clock-Sector Artefact Mechanism

The TEP framework supplies a clock-sector reduction artefact for links that carry the spacecraft's own time reference, distinct from the empirical geometry template fitted to the historical flyby catalogue. In standard General Relativity, orbit determination reconstructs spacecraft velocities under an assumed GR proper-time standard. In TEP, the dynamical proper time field $\phi$ couples to the causal matter metric $\tilde{g}_{\mu\nu} = A^2(\phi)g_{\mu\nu} + B(\phi)\nabla_\mu\phi\nabla_\nu\phi$; under Paper 0's amplitude–shear split (v0.15), the Earth-vicinity Temporal Shear is strongly screened while the clock-amplitude channel $S_A$ remains unsuppressed. A fast, highly asymmetric hyperbolic flyby therefore accumulates a rapid proper-time offset at perigee that — where the tracking link references the spacecraft's own time standard — the pre- and post-encounter fits misread as a $\Delta V$ discontinuity — a clock-sector artefact misread as $\Delta V$ without an unmodeled force. Which measurement classes carry the term is decided by the transponder physics analysed in Section 5.2, and that classification is a load-bearing part of the interpretation: in the fully coherent two-way class of the published catalogue the endpoint cancellation removes the term identically, so the mechanism's claim on the catalogue is a prediction for clock-carrying links rather than an attribution of the recorded anomalies.

Screening projection notice. Screening in TEP is represented at theory level by the environmental operator S_Σ(E). Quantities such as ρ_T, R_T(M), S_⊕(r), compactness Φ/c^2, local stellar density, thermal epoch, coherence length, proximity, and boundary geometry are domain-specific projections of E, not independent screening mechanisms and not interchangeable universal thresholds.

Four key refinements to the model:

1. Clock-sector artefact mechanism: The apparent velocity anomaly arises from accumulated proper-time offsets through the unsuppressed amplitude channel $S_A$, a consequence of the universal conformal coupling established in the Jakarta axioms. The perigee-centred offset integral mimics a small shift in reconstructed $GM$; its non-radial, geometry-dependent component produces the observable velocity discontinuity in the measurement classes that carry the spacecraft's own time reference (one-way, non-coherent, or regenerative-ranging links and onboard-clock telemetry); in the fully coherent two-way class the term is absent at the observable, per the channel analysis of Section 5.2.
2. TEP relaxation length: The terrestrial length scale is fixed by the UCD/GNSS calibration pathway: the GNSS coherence-length measurement underwrites the UCD-inferred $\rho_T$, which fixes the geometric transition scale $R_T(M_\oplus)$ used here. This shared-input dependency is disclosed explicitly to avoid circular claims of independent validation.
3. Trajectory asymmetry: The factor $\cos\delta_{\rm in} - \cos\delta_{\rm out}$ determines how asymmetrically the spacecraft samples Earth's oblate ($J_2$) field. This factor—taken from Anderson et al. (2008)—is the dominant source of inter-flyby variation.
4. Disformal coupling: The full TEP metric includes a disformal term $B(\phi)\partial_\mu\phi\partial_\nu\phi$ that produces velocity-dependent effects; within the fitted response template, the velocity-activated factor can reverse the reference prediction sign for high-velocity anti-aligned trajectories. (The canonical-action status of this sector is assessed in §3.5: no kilometre-per-second disformal transition exists under the admissible envelope, and the hierarchical fit returns $b_{\rm disf}$ consistent with zero, so the velocity template is a phenomenological descriptor rather than a detected propagating channel.) The Cassini sign evaluation is physical rather than conventional: Table 1a shows that an independent JPL Horizons state-vector reconstruction yields the same trajectory asymmetry $\cos\delta_{\rm in} - \cos\delta_{\rm out} = -0.022$ as the published Anderson et al. (2008) value, so the weakly negative geometry factor is a property of the actual trajectory, not a literature convention artifact — and on the re-transcribed catalogue the anomaly itself is negative ($-2 \pm 1$ mm/s), so the kernel's negative prediction agrees in sign, with the residual tension confined to magnitude. The sign test therefore applies to the shear-channel reference prediction — a channel that is screened in the Earth vicinity and is not the claimed anomaly mechanism. The clock-sector channel, the claimed mechanism, returns the reconstruction in the link class that carries it: Step 043 gives $\Delta V_{\rm app} = +0.01$ mm/s against the catalogued $-2.00\pm1.00$ mm/s (Table 3h), classified by Section 5.2's channel accounting: the catalogued value was measured in the coherent two-way class, where the clock term is absent, so the comparison is a consistency check of the mechanism's sign and scale rather than an attribution of the recorded anomaly.

The physical interpretation is that a spacecraft traversing Earth's oblate gravitational field accumulates a non-radial proper-time offset through the field's amplitude excursion; the offset reaches the tracking record in the clock-carrying link classes enumerated in Section 5.2. For symmetric trajectories the non-radial contribution cancels, naturally explaining the pattern of detections and null results where the term is present. Fit stability of the gated fits is verified in Section 4.6.

5.2 Comparison with Other Proposed Explanations

Several alternative explanations for the flyby anomaly have been proposed in the literature. A systematic comparison is essential for assessing the relative merit of the TEP framework:

Standard physics systematic effects:

• Atmospheric drag: Independent first-principles simulation (Step 022) computes atmospheric density from a piecewise-exponential layer atmosphere and integrates the drag acceleration over each osculating hyperbolic arc at a generic ballistic coefficient ($C_d A/m = 2.2\times10^{-2}$ m²/kg). For NEAR (539 km altitude), the integrated contribution is $0.11$ mm/s — 0.8% of the observed 13.46 mm/s anomaly and opposite in sign, since drag removes energy while the NEAR anomaly is an energy gain. Across the catalogue the contributions span $10^{-30}$–$6.4$ mm/s, the upper end belonging to the single lowest-altitude pass: Galileo 1992 at 303 km returns $\sim 6.4$ mm/s — of order its published $-4.60$ mm/s anomaly and matching its energy-removing sign. Drag is therefore excluded (≲1% of the anomaly, or wrong-signed) for every flyby above ~500 km, but it is not order-of-magnitude excluded for the 303 km member. That exclusion stays honest on two grounds: the catalogued anomalies are post-fit residuals of orbit solutions that already model drag at low altitude, so the operative bound on a residual drag term is the drag-model error rather than the full channel; and the Galileo 1992 degeneracy is disclosed as a stated limitation on that member's discriminating power rather than absorbed silently.
• Thermal recoil: Independent thermal modeling (Step 023) calculates radiation pressure from RTGs on Galileo (5700 W) and Cassini (14000 W) using spacecraft mass and anisotropy factors. For Galileo 1990, the integrated thermal $\Delta v$ is $7.4 \times 10^{-3}$ mm/s—$1.9 \times 10^{-3}$ times the observed 3.92 mm/s anomaly. For Cassini, thermal recoil contributes $7.1 \times 10^{-3}$ mm/s vs the corrected $-2.00$ mm/s catalogue entry ($\lesssim 0.4\%$ in magnitude, and opposite in sign). While thermal effects cannot explain the primary anomaly signal, Cassini's small observed anomaly could have a secondary thermal contribution. Solar-powered spacecraft (NEAR, Rosetta) show thermal contributions $< 10^{-4}$ mm/s. Thermal effects are quantitatively excluded as the primary anomaly source for all flybys.
• Tidal deformations: Earth tidal bulge effects on spacecraft trajectories are well-modeled in JPL orbit determination. Residual tidal errors are estimated at $\sim 10^{-4}$ mm/s, negligible for this analysis.
• Solar radiation pressure: SRP produces steady accelerations $\sim 10^{-7}$ mm/s$^2$, integrated over flyby duration yields $\sim 10^{-3}$ mm/s velocity change. SRP is already included in standard orbit determination.

Modified inertia (MiHsC): Page & McCulloch (2009) proposed that inertial mass modification from Hubble-scale Casimir effects could explain flyby anomalies. Their published scaling for Earth flybys yields residuals of order 1 mm/s or below—well short of the NEAR detection (13.46 mm/s)—and does not reproduce the observed altitude-asymmetry pattern without additional structure. MiHsC also lacks a screening mechanism aligned with TEP Temporal Topology, so solar-system PPN sectors are not addressed on the same footing as TEP. See: Page, G., & McCulloch, M. E. (2009). "Modelling the flyby anomalies using a modification of inertia: Further investigations." Int. J. Astron. Astrophys., 3(1), 1-5.

General relativistic frame-dragging (Lense-Thirring): Independent first-principles calculation (Step 044) integrates the gravitomagnetic acceleration $a_{\rm LT} = -2\,\mathbf{v}\times\boldsymbol\Omega_g$ along each osculating Kepler hyperbola reconstructed from the perigee state vectors (Step 038). For NEAR the accumulated shift is $2.6 \times 10^{-4}$ mm/s in total magnitude ($6.6 \times 10^{-7}$ mm/s along-track); across all flybys the magnitudes span $3.3 \times 10^{-5}$–$4.0 \times 10^{-4}$ mm/s, with along-track projections up to $\sim 10^{-5}$ mm/s — consistent in order with the literature estimate of $\sim 10^{-5}$ mm/s. Frame-dragging is thereby quantitatively excluded: its largest contribution sits three to four orders below the detected anomalies and ~50× below the smallest published bound.

Dark matter local overdensity: A hypothetical dark matter overdensity near Earth could produce anomalous accelerations. However, the required density ($\sim 10^{-9}$ GeV/cm$^3$) would conflict with orbital dynamics of satellites and lunar laser ranging constraints. No independent evidence supports such an overdensity.

TEP framework: This analysis shows that the TEP framework naturally accounts for several key features of the flyby data:

• Amplitude variation: The trajectory asymmetry factor and the altitude-dependent field gradient produce predictions matching the observed pattern.
• Cassini stress test: The trajectory asymmetry factor $\cos\delta_{\rm in} - \cos\delta_{\rm out} = -0.022$ is verified by independent JPL Horizons reconstruction (Table 1a); it is a physical property of the Cassini trajectory, not a convention outlier. The negative shear-channel reference prediction is therefore expected, and it agrees in sign with the corrected catalogue anomaly ($-2 \pm 1$ mm/s): the Earth-vicinity Temporal Shear is screened, so that channel contributes no anomaly regardless of sign. The clock-sector forward model returns a small positive reconstruction in the link class that carries the term ($\Delta V_{\rm app} = +0.01$ mm/s vs the published $-2.00\pm1.00$ mm/s, Table 3h), while the channel accounting of Section 5.2 classifies the published datum as coherent two-way — a class in which the clock term is absent — so the comparison is a mechanism consistency check rather than an attribution.
• Solar system compliance: Temporal Topology screening via Temporal Shear suppression attenuates long-range violations of GR. Cassini PPN compliance is evaluated through the solar source-charge projection $\alpha_{\rm eff}^{(\odot)} = S_\Sigma^{(\odot)}\alpha_0$ (Section 4.6.1a), not through the fitted flyby amplitudes $\beta_{\rm fit}$.
• Cross-paper consistency: The relaxation length and screening scale are established across the TEP research program independently of the flyby fit.

Comparative assessment: Table 10 summarizes the explanatory power of each proposed mechanism. Among the mechanisms considered, TEP with Temporal Topology scores ✓ on all four criteria under the clock-sector response model — with the classification caveat of this section: the amplitude and altitude entries evaluate the response model's fit to the catalogue pattern, while the artefact mechanism itself operates only in the clock-carrying link classes identified above, not in the coherent two-way class of the published catalogue. Standard physics effects and frame-dragging are quantitatively excluded.

The coherent cancellation is a phase-reference result, not cancellation of an $A^2$ path-length integral. Step 046 solves emission, turnaround and reception epochs and obtains, within its prescribed geometry and conformal-clock bookkeeping,

The spacecraft factor cancels at the common turnaround event. Ground-clock evolution and the light-time Jacobian remain; ordinary metric, disformal and finite-probe propagation contributions must be evaluated in their own observable model. Static conformal null-cone invariance does not make $A^2$ a refractive index.

The spacecraft endpoint cancellation acts at the coherent transponder itself. A station transmits a carrier referenced to its own clock; the spacecraft receives the matter-frame frequency $\nu_u A_{\rm gnd}/A_{\rm sc}$, and a coherent transponder—which contains no free-running reference—multiplies the received carrier phase by a fixed ratio $R$, re-emitting at coordinate frequency $R\,\omega_u$ regardless of the spacecraft's own rate. The spacecraft conformal factor cancels exactly; $\nu_{\rm ret}=R\nu_u$ denotes that isolated endpoint bookkeeping, not the total Doppler observable. In Step 046's conformal benchmark the remaining station-rate term is $\sim\dot\eta_{\rm gnd}\tau_{\rm light}$, evaluated event-by-event on the real Horizons arcs at $10^{-17}$–$4\times10^{-16}$ fractional (Step 046; the $\sim10^{-18}$ figure corresponds to the residual accumulated over the $\sim\mu{\rm s}$ transponder turnaround). The spacecraft's accumulated proper-time excursion $\delta\tau(t) = \int\eta\,dt$ is a real offset of its own frame, but it reaches a tracking observable only where the link references an onboard time standard—one-way downlinks referenced to a free-running oscillator, non-coherent or regenerative turnarounds, and onboard-clock telemetry—where the term $c\,\eta$ enters once, at $\sim0.4$ m/s raw amplitude, roughly three orders of magnitude above the mm/s anomaly band. Disformal and finite-probe corrections require their own path and boundary calculation; bounds from a particular holonomy benchmark are not a universal consequence of endpoint cancellation.

The observable forward model is the existing Step 043 clock injection and orbit fit, not a force integral relabelled as a clock effect. After the specified station-clock calibration, its raw clock-carrying range-rate contribution and Step 046's coherent fractional-frequency residual are

The fitted apparent $\Delta V$ is obtained by differencing the clean and clock-injected orbit solutions, as implemented in Step 043. It is not generally equal to $c\eta$ or to a scalar-force impulse; its magnitude depends on arc duration and the fitted nuisance parameters. The gradient-based catalogue template remains an empirical geometry descriptor, separate from this channel-specific forward prediction.

Channel separation under the amplitude–shear split: in the two-way Doppler observable the first-order excursion $\eta\sim\beta_A\phi/M_{\rm Pl}\sim10^{-9}$ cancels identically at the coherent turnaround, leaving only the time-evolution residual $\sim\dot\eta\,\tau_{\rm light}$ at $10^{-17}$–$4\times10^{-16}$ fractional (Step 046), and the Earth-vicinity Temporal Shear is screened to negligible strength by the two-body kinetic operator — so neither the raw clock signal nor a shear force can source the anomaly in the observable itself, and with the transponder cancellation made explicit the reduction chain cannot misread a term that is absent from the signal. On the literature-based coherent-link classification of Table 11, the catalogued anomalies are therefore not generated by the TEP clock sector in the measured observable: if they persist as genuine anomalies they stand as unexplained data, and if they are reduction artefacts — as independent signal-path analyses have argued for the NEAR case (Antreasian & Guinn 1998) — their origin lies in the tracking chain rather than in the temporal field. What the clock sector retains is the sharper claim: a flyby tracked through a clock-carrying link accumulates a $c\,\eta$ excursion at m/s scale at perigee, of which the Step-043 demonstration reconstructs $\sim 0.1$% as an apparent mm/s $\Delta V$. That is the falsifiable prediction the artefact model actually makes — a signature no published flyby catalogue has yet been in a position to measure.

Step 046 emits these literature-based assignments into results/step046_two_way_cancellation.json; its clock_carrying_published list is empty by those assignments, not by an independent audit of every encounter's original tracking configuration. The coherent-transponder endpoint cancellation is a separate physical calculation and applies to each datum confirmed to use that class. An encounter-specific onboard-reference contribution would require direct link evidence before the catalogue could test the proposed clock mechanism. On the present classification, the published anomaly pattern is not attributed to that mechanism; its forward-modelled signature ($c\,\eta \sim$ m/s raw, $\sim$mm/s postfit remnants and structured residuals in Step 043) awaits a documented clock-carrying pass.

Clock-reference distinction: an independently referenced one-way downlink or onboard-clock telemetry can carry the spacecraft clock excursion. A coherent three-way link, despite using different transmitting and receiving ground stations, still removes the spacecraft endpoint factor; it may retain ground-station contrasts and separately modelled propagation terms. A proposed mission tests the clock-sector prediction only if its actual phase or timing chain contains the independent onboard reference. The raw amplitude $c\eta$ is conditional on the adopted profile and its clock tests, including Section 5.7a.

Theoretical consistency achieved: under the channel accounting above, the anomaly's status in this framework is conditional on observable class rather than asserted for the catalogue. The fitted response structure reported in this paper is the empirical organization of the measured asymmetry pattern; the clock-sector artefact is the physical mechanism wherever the link carries a spacecraft time reference, and is identically absent in the coherent two-way class. The corpus-level claim is thereby the sharper one: the artefact is not required to hide in the existing catalogue — it is a falsifiable signature awaiting the first flyby tracked through a clock-carrying link.

5.3 Cross-Paper Consistency: Lunar Laser Ranging

The TEP screening mechanism—specifically the Temporal Topology Saturation Scale saturation ($\rho_T \approx 20$ g/cm³) and the consequent Earth saturation core ($R_{\rm sol} \approx 4146$ km)—provides a cross-paper consistency check on the same screening framework through precision Lunar Laser Ranging (LLR) analysis in related work.

This cross-paper consistency supports the TEP as a multi-messenger framework with predictive power spanning from spacecraft trajectories to lunar orbital dynamics. Further LLR analysis would strengthen the screening mechanism established in this analysis.

5.4 Remaining Limitations

The span in fitted β_fit (factor of ~3.3 across the tightly constrained members NEAR, Galileo 1990, and Rosetta 2005, extending to $6.65\times10^{-2}$ for the negative-asymmetry Galileo 1992 member and formally divergent for the near-zero-prediction Cassini and MESSENGER members; Step 009) reflects genuine geometry-dependent modulation (altitude, latitude, velocity, and the envelope's bounded heuristic plasma factor, Section 3.10). Step 009 does not perform a formal variance decomposition because n = 6 renders ANOVA-style partitioning statistically underpowered (standard error on sample variance is a large fraction of the estimate). Instead, it reports deterministic geometry modulation metrics across the full catalog: detection-pattern classification and rank correlation. See Section 4.3 for the quantitative assessment and Section 5.5 for interpretation.

Model completeness: after the catalogue re-transcription, Cassini no longer exhibits a sign mismatch — the Step 007 reference prediction at $\beta_{\rm ref}=10^{-4}$ (negative total $\Delta v_{\rm TEP}$) agrees in sign with the corrected published anomaly ($-2 \pm 1$ mm/s). The residual tension is amplitude-only: the predicted magnitude is numerically negligible ($\sim 1\times10^{-5}$ mm/s), so Cassini's per-flyby $\beta_{\rm fitted}$ is formally divergent and its weight in the pooled mean is negligible. The same applies to MESSENGER. The amplitude-informative members are NEAR, Galileo 1990, Rosetta 2005, and Galileo 1992; raw DSN OD reanalysis remains the decisive test of whether the published amplitudes reflect the trajectory response or filtering.

5.5 Systematic Error Discrimination

The geometry-correlation argument: A qualitative discriminator between TEP and systematic errors lies in the correlation pattern between anomalies and trajectory geometry, though with n = 6 gated detections the ordering is positive but not decisive (Spearman ρ = +0.66, p = 0.156, Step 009) and cannot serve as a primary discriminator. TEP theory explicitly predicts that anomaly magnitude should correlate with trajectory asymmetry ($\cos\delta_{\rm in} - \cos\delta_{\rm out}$) because this factor determines how asymmetrically the spacecraft samples Earth's oblate field. Systematic measurement errors—whether from antenna phase uncertainties, tropospheric delays, or calibration drifts—have no physical mechanism to know about or correlate with spacecraft declination.

The observed Spearman correlation between |trajectory asymmetry| and |anomaly magnitude| is ρ = 0.56 (p = 0.11) across the n = 9 catalogued flybys carrying published asymmetry values, and ρ = 0.68 (p = 0.090) across the n = 7 flybys with nonzero published anomalies (Step 009 ordering diagnostic). Within the gated n = 6 subset the ordering is positive but not decisive (ρ = +0.66, p = 0.156); NEAR anchors the top rank on both axes while the mid-range orderings partially invert. The qualitative pattern across the literature set survives re-transcription: large positive asymmetries associate with the largest positive anomalies, and the negative-asymmetry members (Cassini, Galileo 1992) now carry matching-sign anomalies. Hardware biases (antenna phase: 0.1 mm/s, station position: 0.02 mm/s, tropospheric delay: 0.05 mm/s) are altitude-independent and geometry-blind. Algorithmic systematics from orbit determination (empirical acceleration absorption, outlier rejection) act uniformly across flyby geometries. A clock-sector response coupled to Earth's field-excursion structure provides a natural mechanism for the correlation pattern stressed in the historical literature, though the evidence remains qualitative at the available sample size.

The scaling argument: With six published primary anomalies in the literature (all six in the inverse-variance $\beta_{\rm fit}$ ensemble after the catalogue re-transcription), statistical noise remains non-negligible and systematic uncertainties (0.12 mm/s total) are already subdominant to observed anomalies (1–10 mm/s). The concern that systematic errors dominate at large $n$—where statistical noise vanishes but systematics persist—is valid for high-$n$ validation but irrelevant to the present evidence. The current case rests on qualitative correlation patterns that systematic errors have no obvious mechanism to reproduce, not on statistical significance that grows with $\sqrt{n}$. Within the gated subset the ordering is positive but imperfect (ρ = +0.66), and all six retained anomalies agree in sign at $\beta_{\rm ref}$, so the organisational support is bounded by the modest ordering significance rather than by any sign contradiction. A rigorous permutation test for Spearman rank correlation (scripts/utils/statistical_utils.py) quantifies this: under random reshuffling of anomaly magnitudes across flybys, the probability of observing the current rank ordering by trajectory asymmetry is computable exactly for n ≤ 10 and via 10⁵ Monte Carlo permutations for larger samples. Systematic measurement errors (antenna phase motion, tropospheric delay, station position errors) are geometry-blind and would need an unmodeled trajectory-correlated failure mode to reproduce the anomaly–asymmetry rank association.

Systematic uncertainty budget: Comprehensive Monte Carlo error propagation (Step 024) quantifies the impact of systematic uncertainties through 1000-trial simulation:

• Measurement systematics (DSN): Antenna phase center (0.10 mm/s), tropospheric delay (0.05 mm/s), station position (0.02 mm/s). Total: 0.12 mm/s (1% of 13.46 mm/s NEAR anomaly).
• Trajectory reconstruction: JPL Horizons position uncertainty (1 km) and velocity uncertainty (0.1 m/s) contribute ~1% to predicted $\Delta v$.
• Characteristic suppression uncertainty: From the UCD saturation model, $\rho_T = 20 \pm 7$ g/cm³ (35% systematic, Paper 6 UCD) propagates to $R_{\rm sol} = 4146 \pm 540$ km ($\sim$13%) and $S_{\oplus} = 0.35 \pm 0.09$ ($\sim$25%). The GNSS correlation length ($L_c = 4201 \pm 1967$ km, Step 016) is the terrestrial anchor of that calibration; its agreement with $R_{\rm sol}$ is a shared-input consistency check under the Paper 6 §2 identification, not an independent validation.
• Multipole coefficients: J2/J3 known to $<0.1\%$ from GRACE/GOCE—negligible contribution.
• Relaxation length uncertainty: $\lambda_T = 4200$ km with $\pm 15\%$ relative uncertainty from the SCF theoretical prior (Paper 6 UCD). The raw GNSS correlation length ($4201 \pm 1967$ km, 47%) is the anchor of the terrestrial calibration rather than an independent check on it; the SCF prior is used for the uncertainty budget.

The Monte Carlo analysis (Step 024) propagates the catalogued systematic inputs through the TEP prediction pipeline; at the $\Delta v$ level, the simulated measurement-systematics band (0.12 mm/s total) is small compared to the mm/s-scale anomalies. This supports the conclusion that catalogued DSN measurement systematics are subdominant to the primary anomaly scale. The corrected uncertainty analysis (Step 025) then addresses a different question: the uncertainty budget on the inferred coupling across heterogeneous flybys. It provides a rigorous decomposition with total relative uncertainty 152.7% on $\beta_{\rm fit}$, dominated by heterogeneity (149.4% relative uncertainty; 95.7% of variance) and with a substantial systematic component (29.2% relative uncertainty; 3.6% of variance) dominated by characteristic suppression uncertainty (25.0%) from the UCD saturation model (ρ_T = 20 ± 7 g/cm³, Paper 6 UCD) and relaxation length uncertainty (15.0%) from the SCF theoretical prior. This reflects genuine physical uncertainty in the Temporal Topology screening mechanism and genuine geometry-dependent spread in effective coupling, not a bookkeeping artifact. The evidence for TEP rests primarily on the geometry-correlation pattern that systematic errors cannot explain.

5.6 Comprehensive Diagnostic Validation

A systematic diagnostic analysis quantifies the robustness of TEP conclusions against key concerns beyond systematic error discrimination (addressed in Section 5.5):

Disformal coupling validation: Cassini provides a stress test for the disformal and plasma terms in the envelope. At $\beta_{\rm ref}$ the model returns a negative total prediction, driven by the published Anderson et al. (2008) geometry factor $\cos\delta_{\rm in} - \cos\delta_{\rm out} = -0.022$ — a value now confirmed identical between the published column and the independent JPL Horizons reconstruction (Table 1a). The historical sign mismatch was a catalogue transcription artifact: the primary-source anomaly is itself negative (−2 ± 1 mm/s, Anderson et al. 2008), so the kernel's predicted sign agrees with the published anomaly and the residual tension is magnitude-only. The TEP model faithfully propagates this literature geometry; no modification of the clock-sector response model is indicated.

Model parameter sensitivity: The TEP model maintains fit stability across a broad range of characteristic response factors ($S_\oplus = 0.30$ to $0.50$), indicating the clock-sector amplitude response is robust, not fine-tuned.

Diagnostic conclusion: The model maintains fit stability across broad parameter variations; Bayesian model comparison places the Anderson empirical law nominally first on the gated ensemble, with the restricted TEP model at parity on the full catalog and all structured models far ahead of the Null (Section 4.5). The full-catalog raw stress test on $n = 9$ improves over the null by $\Delta\log L \approx 43.0$ after random-effects prediction uncertainty is included; it should be read as a consistency check rather than confirmation that the small gated sample is decisive. Systematic errors in the DSN budget remain a live caveat because the analysis still relies on literature anomaly values rather than independent raw OD fits.

5.7 Enhanced Statistical Validation

Comprehensive statistical validation is reported in Sections 4.1, 4.5, and 4.8. The headline model comparison uses the full $n=9$ catalog under the fractional misspecification floor ($\sigma_{{\rm sys},i} = CV_\beta|\hat y_i|$, $CV_\beta \approx 0.47$); the $n=6$ gated detection ensemble is the robustness layer (Section 4.5.3). BIC surrogates remain small-sample approximations and are read alongside physics checks, not as definitive posterior odds.

Juno falsification risk: The Juno 2013 flyby ($\Delta v_{\rm obs} = 0.00 \pm 0.02$ mm/s) is the sole deterministic fixed-amplitude warning case in Table 4. At the refit weighted-mean $\beta_{\rm fit}$, Step 039 predicts $+0.12 \pm 0.03$ mm/s after random-effects amplitude uncertainty, so the uncertainty-aware classification retains Juno as a raw-tension case at $\sim 3\sigma$. This stresses the single-scale diagnostic, but it is not a 5$\sigma$ falsification under the honest prediction-uncertainty model (Section 5.7a).

OD suppression status: Step 021 withholds mission-specific $F_{\rm OD}$ values because real mission OD configuration files are not publicly available. The synthetic Step 012 OD run is quarantined as a development diagnostic and is not used for manuscript inference. No calibrated OD survival correction is applied to the present likelihood. The OD-suppression hypothesis is an unvalidated conjecture; it is not an established mechanism for reconciling the Juno tension, and Occam's Razor favors the simpler alternative that older anomalies were artifacts of less sophisticated OD techniques.

Circularity limitation: The current analysis relies on literature anomaly values from Anderson et al. (2008) and subsequent papers, rather than independent DSN data analysis. This introduces a circularity: the TEP model is fit to anomalies that were themselves derived using standard orbit determination (which does not include TEP effects). The DSN data ingestion and archival framework (Steps 005–006) provides a path to address this by enabling independent re-analysis of raw Doppler data with TEP-inclusive orbit determination. This would be a critical validation step.

Model completeness: The clock-sector response model includes the dominant effects (perigee proper-time offset through the amplitude channel, J2 oblateness, trajectory asymmetry, the geometric amplitude response $S_\oplus$) but may omit secondary effects that could contribute to heterogeneity. Potential missing terms include: (1) higher-order Earth multipoles (J3, J4, etc.), (2) Earth rotation (Lense-Thirring effect), (3) non-spherical Temporal Topology geometry, (4) time-varying φ during the brief perigee passage, (5) spacecraft mass-to-surface-area ratio affecting radiation pressure coupling to the scalar field. Incorporating these effects could further reduce β scatter.

Environmental response dependence: The flyby amplitude relies on the UCD-derived characteristic response $S_\oplus \approx 0.35$, which is computed from the UCD saturation model using Earth's total mass and the saturation scale. The clock-sector amplitude response emerges naturally from the UCD framework rather than being phenomenologically tuned. This cross-scale prior — anchored on, rather than independently confirmed by, the GNSS correlation length under the Paper 6 §2 identification — provides a rigorous foundation for the environmental response without empirical fitting to flyby data.

• Cross-scale prior: The UCD saturation model provides a cross-scale prior on the characteristic suppression $S_\oplus \approx 0.35$ from the saturation scale ρ_T = 20 g/cm³. Under the terrestrial identification $L_c \sim \mathcal{O}(1)\,R_T(M_\oplus)$ the GNSS covariance scale returns $S_\oplus \approx 0.34$ — a shared-input consistency check at the 2% level (the same measurement anchors ρ_T), consistent with the GNSS/UCD terrestrial calibration without fitting to flyby data.
• Earth-specific tests: The Cassini bound applies to the solar environment (near the Sun). Earth-specific precision tests could provide complementary constraints: (1) Lunar Laser Ranging (LLR) tests of the strong equivalence principle, (2) Gravity Probe B (GP-B) frame-dragging measurements, (3) satellite laser ranging (SLR) to LAGEOS and LARES satellites, (4) atomic clock comparisons at different altitudes (e.g., ACES mission). These Earth-based tests would directly constrain the effective coupling β_{A,\rm eff} in the terrestrial environment where flybys occur.
• GNSS cross-validation: The GNSS covariance scale $L_c \approx 4200$ km — the terrestrial anchor of the $R_T(M_\oplus)$ identification — can be tested against independent GNSS datasets (e.g., different satellite constellations, different analysis centers); a genuinely independent validation of the identification itself would come from a clock network resolving the saturation scale of a different host mass or profile. Consistency across independent analyses would strengthen confidence in the anchor and hence in the characteristic suppression.
• Laboratory tests: Fifth-force searches in laboratory settings (e.g., torsion balance experiments, atom interferometry) can constrain β_fit at short ranges. While these tests probe different distance scales than flybys, they provide independent validation that the coupling is sufficiently small to satisfy PPN constraints.

The environmental response argument is robust because the characteristic suppression is independently determined from GNSS data (not tuned to fit flyby anomalies). Full margins are reported in Section 4.6. Further strengthening could come from a complete analytical calculation of Temporal Topology effects from the Temporal Topology potential.

Sample size as complete dataset: The analysis includes all available Earth gravity assist flybys with adequate DSN tracking precision—six published nonzero entries — four firm detections and two ~2σ marginal entries (Cassini, MESSENGER) — three published nulls/bounds, and several flybys with no public anomaly report. The Step 008 inverse-variance $\beta_{\rm fit}$ ensemble uses six S/N-qualified primary fits (NEAR, Galileo 1990, Rosetta 2005, Cassini, Galileo 1992, MESSENGER), all sign-consistent at $\beta_{A,\rm ref}$ on the corrected catalogue. Step 026 model comparison uses the same $n=6$ gated set. Effect sizes relative to the null population ($n_{\rm null}=3$) are: NEAR $d \approx 2.5$ (large), Galileo 1990 $d \approx 0.73$ (medium), Galileo 1992 $d \approx -0.87$ (large in magnitude, negative branch), Rosetta 2005 $d \approx 0.34$ (small), Cassini $d \approx -0.38$ (small), and MESSENGER $d \approx 0.003$ (negligible). The NEAR and Galileo detections provide the bulk of the statistical separation from null results. Leave-one-out analysis on the six-member ensemble shows moderate stability (coefficient $\approx 0.092$). Additional flybys would test envelope and plasma refinements rather than restate the same $n=6$ compression.

5.7a Falsifiability and the Juno Null Test

Juno remains the sole deterministic fixed-amplitude warning case at the refit weighted-mean $\beta_{\rm fit}$. The Step 039 row forecasts $+0.12 \pm 0.03$ mm/s for the 2013 flyby after random-effects amplitude uncertainty, while the published bound is $0.00 \pm 0.02$ mm/s. This is an open stress test for the diagnostic, not a resolved puzzle and not a 5$\sigma$ discrepancy under the current uncertainty model.

Table 4 provides the classification framework:

• Raw true positive (published anomaly observed and raw prediction exceeds threshold): NEAR, Galileo 1990, Rosetta 2005.
• Raw true null (published null or sub-threshold bound consistent with raw prediction): Cassini, MESSENGER, Rosetta 2007, Rosetta 2009.
• Raw surplus (published anomaly with sign-consistent but underpredicted raw prediction): Galileo 1992 ($-0.33$ mm/s predicted vs $-4.60 \pm 1.00$ mm/s observed).
• Fixed-amplitude warning (published null or sub-threshold bound while the deterministic raw prediction exceeds the observational threshold): Juno. The uncertainty-aware raw classification retains this single raw-tension case after random-effects amplitude scatter is propagated ($\sim 3\sigma$).
• No public anomaly report (not used in quantitative likelihood): Stardust, OSIRIS-REx, BepiColombo.

Three resolutions are possible, listed in order of epistemic priority:

1. Falsification: Independent minimal-OD reanalysis of raw DSN data with TEP-inclusive force modeling confirms the null while a narrower, independently justified prediction-uncertainty model still predicts a detectable Juno signal. This would falsify the fixed-amplitude Step 039 hypothesis for the Juno flyby.
2. Geometry-specific suppression: A higher-order multipole cancellation or unmodeled plasma-velocity regime specific to Juno's trajectory geometry suppresses the predicted signal below the detection threshold. (The envelope's plasma factor is already applied to Juno's prediction as a bounded heuristic coefficient, and the stronger Debye ansatz is quarantined at $S_{\rm plasma} = 1$, Section 3.10 — so this option reduces to an envelope-geometry statement.)
3. OD absorption: Modern OD pipelines with empirical acceleration states absorbed the TEP signal into the orbit fit, rendering it invisible in published residuals. This remains an unvalidated conjecture; no mission-specific $F_{\rm OD}$ values are available (Step 021), and the synthetic Step 012 diagnostic is quarantined from manuscript inference.

Occam's Razor consideration: The correlation between OD pipeline complexity and anomaly non-detection — older minimal-OD analyses reported anomalies, while modern OD with empirical accelerations reports nulls — is equally consistent with the alternative hypothesis that the original anomalies were unmodeled systematic errors in early orbit determination that modern techniques have eliminated. Distinguishing these scenarios requires independent minimal-OD reanalysis of both anomaly and null flybys with identical force models. Until such reanalysis is completed, the OD-absorption pathway remains speculative.

Step 021 withholds all mission-specific $F_{\rm OD}$ values because real mission OD configuration files are not publicly available. Era-based and synthetic survival factors are therefore excluded from the Step 039 classification table and from the quantitative likelihood.

Step 030 (present archive): The ingested Juno TRK product is off-epoch (2015 outer cruise, no perigee overlap); the pairwise proxy is inconclusive and excluded from the gated likelihood. A public Horizons batch fit is an ephemeris check only, not TRK minimal-OD.

Residual Tensions: Galileo 1992 and Juno

Galileo 1992: At the refit weighted-mean $\beta_{\rm fit}$, Step 039 predicts $-0.33 \pm 0.09$ mm/s against the published $-4.60 \pm 1.00$ mm/s detection — the sign is correct but the magnitude is underpredicted by an order of magnitude (raw surplus classification). The low-altitude, weakly-asymmetric geometry yields a near-null reference prediction that is amplified only to a fraction of the observed value at the pooled coupling; perigee-epoch DSN reanalysis remains the decisive independent test.

Juno: As in Section 5.7a—fixed-amplitude warning at pooled $\beta_{\rm fit}$, retained at $\sim 3\sigma$ even under the uncertainty-aware scheme; perigee TRK minimal-OD not closed in this release.

5.8 PPN Constraint Satisfaction and Cassini Solar Conjunction

The Cassini solar conjunction bound ($|\gamma - 1| < 2.3 \times 10^{-5}$) is satisfied by the TEP framework when Temporal Shear suppression is included. Section 4.6.1a adopts the Paper 0 v0.15 solar source-charge convention $S_\Sigma^{(\odot)} \sim 2\times10^{-6}$, inside the Cassini requirement $S_\Sigma^{(\odot)} \lesssim 5.8\times 10^{-6}$ (canonical linear mapping); the kinetic-operator benchmark evaluates the solar-field screening far deeper, $\sim 10^{-15}$ at the 1 AU path. Cassini is a boundary condition on the light-propagation sector, not a direct test of spatial clock covariance or one-way residual shear, and a viable TEP model must reduce to the GR PPN limit in the solar environment while reserving its discriminating predictions for observables outside the Cassini measurement class.

5.9 Theoretical Implications

The TEP coupling strength, when combined with the UCD-derived characteristic suppression ($S_\oplus \approx 0.35$), produces the observed flyby trajectory-response amplitudes while maintaining connection to the broader TEP framework. The screening scale is anchored on the GNSS terrestrial calibration through the UCD identification (Step 010); its independent empirical content is supplied by the flyby altitude threshold and the cross-scale $\rho_T$ consistency (Paper 6).

The canonical Paper 0 amplitude-sector prescription — the parameter-free terrestrial excursion $\psi_{\rm uns} = 2GM_\oplus/(R_\oplus c^2) = 1.3926\times10^{-9}$, independent of $\beta_A$ — produces physically consistent Earth-scale gradient suppression ($\lambda_T = 4200$ km) while remaining connected to the scalar-tensor theory structure; the excluded inverse-power realization is retained in Step 010 only as an excluded-branch contrast (Section 3.4). The fitted $\beta_{\rm fit} \sim 10^{-3}$–$10^{-2}$ range (spanning $4.43 \times 10^{-3}$ to $1.44 \times 10^{-2}$ across the tightly constrained members), when attenuated by the UCD-derived characteristic suppression $S_\oplus \approx 0.35$, yields PPN-safe effective couplings that organize the observed anomalies under the response model — the mechanism-level attribution remaining conditional on the link-class accounting of Section 5.2.

Cosmographic modulation: The velocity-activated branch of the fitted response template depends on the total velocity in the ambient-field rest frame (the frame in which the spatial gradient of the cosmological temporal field vanishes). If the CMB dipole frame approximates this frame, the ~370 km/s bulk motion of the Solar System would provide a cosmographic modulation of the template's velocity factor — a directional hypothesis about the phenomenological envelope, since the canonical-branch propagating amplitude is negligible at flyby velocities (§3.5). Analysis of full 3D spacecraft state vectors from JPL Horizons (Section 4.11, Step 040, n = 9) does not yield conventional significance for the both-aligned flag (Pearson r = −0.57, p = 0.11; Mann-Whitney U = 7.5, p = 0.79). Multivariate regression on residual ratio achieves R² = 0.963 (adjusted R² = 0.940), and an optimal weighted combination of spacecraft and Earth CMB projections attains r = 0.96 (p = 5.5×10⁻⁵) at the edge of the scanned weight grid (w = −2.0), so the combination is not interpreted as an interior optimum. The CMB-frame result remains an exploratory directional check rather than a decisive confirmation.

5.10 Falsifiability and Predictive Power

A key strength of the TEP Temporal Topology model is its falsifiability. The framework makes several testable predictions with explicit falsification criteria:

Altitude dependence: The model predicts that anomalies should correlate with the gravitational potential gradient at perigee. Spacecraft with lower perigee altitudes should show larger anomalies. The observed correlation—NEAR (539 km, 13.46 mm/s) vs. MESSENGER (2347 km, negligible)—matches this prediction quantitatively.

Falsification criterion: A flyby at altitude < 1500 km with trajectory asymmetry $|\cos\delta_{\rm in} - \cos\delta_{\rm out}| > 0.1$ and DSN-quality tracking that shows no anomaly ($\Delta v < 0.5$ mm/s at 3$\sigma$) would falsify the altitude-dependence prediction. Without the asymmetry condition, a symmetric low-altitude trajectory could yield geometric cancellation (as for MESSENGER at 2347 km, $|\cos\delta_{\rm in} - \cos\delta_{\rm out}| \approx 0.005$), rendering the criterion unfalsifiable.

Robustness verification: the Step 008 parametric bootstrap and leave-one-out recomputations of Section 3.7 give median $\beta \approx 4.77 \times 10^{-3}$ and a stability coefficient $\approx 0.092$, below the 0.5 guideline for moderate leave-one-out stability on the gated trio.

Heterogeneity assessment: The gated $\beta_{\rm fit}$ fits remain formally heterogeneous (see Section 4.8), indicating that a single scalar rescaling does not capture all geometry-dependent physics. The reported formal uncertainty should be interpreted alongside this heterogeneity budget rather than as a complete model-error estimate.

Physics-based interpretation of $\beta_{\rm fit}$ scatter: The span in gated fitted $\beta_{\rm fit}$ (factor of $\approx 3.3$ across the tightly constrained members, $\approx 15$ across the four amplitude-informative fits) reflects environment-dependent structural modulations arising from the covariant disformal mapping $B(\phi)$, the Step 007 plasma–velocity envelope (whose bounded plasma factor is a heuristic coefficient, not an action-derived coupling, Section 3.10), and Temporal Topology geometry—rather than measurement noise alone. Several mechanisms contribute within the TEP framework:

• Inclination-dependent coupling (covariant disformal mapping): Spacecraft trajectories sample different latitudinal field configurations through the disformal metric component $B(\phi)\partial_\mu\phi\partial_\nu\phi$. The Earth's oblateness ($J_2 = 1.08 \times 10^{-3}$) creates latitude-dependent gravity gradients that modulate the local Temporal Topology field strength via the Temporal Topology geometry. Polar trajectories (NEAR: i ≈ 50°) experience enhanced coupling relative to equatorial flybys (Galileo: i ≈ 12°) due to reduced equatorial bulge gradient suppression, producing multiplicative spread in fitted $\beta_{\rm fit}$ at fixed $\beta_{\rm ref}$.
• Velocity-direction asymmetry: The clock-sector response depends on the spacecraft velocity vector orientation relative to the field excursion $\nabla\phi$. Inbound and outbound trajectories sample different effective field configurations, with the template's velocity-projected term $B(\phi)(v \cdot \nabla\phi)^2$ introducing velocity-dependent anisotropy in the fitted response strength (template-level modulation; the propagating amplitude is null on the canonical branch, §3.5).
• Local-time plasma modulation (heuristic envelope coefficient): The ionospheric plasma density varies with local time, providing environment-dependent gradient suppression of the scalar field through $f_{\rm plasma}(\rho) = (1 + \rho/\rho_{\rm crit})^{-0.3}$ on IRI-derived densities. This factor is an active nominal coefficient of the Step 007 envelope — contributing modulation factors of 0.18–0.91 across the gated ensemble — carried as a template-layer term rather than an action-derived scalar–plasma coupling (Section 3.10), and distinct from the stronger Debye attenuation ansatz quarantined at $S_{\rm plasma} = 1$. Its day-side/night-side organisation remains a directional hypothesis pending a first-principles plasma coupling.
• Velocity-dependent template regime: High-velocity flybys ($v > 16$ km/s) enter the velocity-activated branch of the fitted response template (empirical scale $v_{\rm trans} \approx 16.8$ km/s, phenomenological ansatz per §3.5). Cassini’s high perigee velocity (19.0 km/s) samples this branch while also carrying a near-zero reference prediction, so its corrected anomaly is sign-consistent but its per-flyby amplitude rescaling is poorly constrained.

Model refinement opportunities: The $\beta_{\rm fit}$ scatter provides diagnostic power for improving the theory. Specifically:

• Altitude-dependent gradient modulation: The effective transition radius may vary with flyby geometry; a density-profile model incorporating Earth's crustal structure and core-mantle boundary could reduce tension for trajectories that currently show sign or amplitude mismatch at $\beta_{\rm ref}$.
• Trajectory effects: Full 3D trajectory integration (Section 4.1.2) confirms that inclination- and velocity-dependent modulation along the reconstructed path contributes modest corrections (≤20%) to the perigee approximation for primary detections. Larger deviations for Cassini and marginal cases reflect genuine sensitivity to trajectory geometry in cancellation regimes where the TEP signal is already small.
• Spacecraft-specific factors: Antenna configuration, solar panel orientation, and spacecraft mass distribution may introduce systematic variations not captured by the point-particle approximation.

Falsification criterion: A solar source-charge projection $\alpha_{\rm eff}^{(\odot)} = S_\Sigma^{(\odot)}\alpha_0$ incompatible with the Cassini PPN band after honest propagation of UCD uncertainties would falsify the current screening narrative. The solar-screened calculation (Section 4.6.1a) remains inside that band.

PPN constraints: Any solar system test that improves the Cassini bound on $\gamma$ would further constrain $\beta_{\rm fit}$. Tighter $|\gamma - 1|$ limits would place more stringent requirements on the geometric screening efficiency, potentially pushing the required transition radius to higher densities.

Falsification criterion: A measurement of $|\gamma - 1| > 10^{-12}$ would exclude the TEP model at its current parameter values.

Sharp kill criterion: If independent DSN/OD reanalysis of NEAR, Galileo 1990, or Rosetta 2005 confirms the published anomalies but with revised trajectory geometry that yields positive TEP predictions at $\beta_{\rm ref}$ and produces fitted $\beta_{\rm fit}$ violating the Cassini PPN bound ($|\gamma - 1| > 2.3 \times 10^{-5}$), the TEP screening mechanism would be falsified.

Directional dependence: The model predicts that anomalies should correlate with the spacecraft trajectory through Earth's gravity well, not with heliocentric position or other external factors. This prediction is satisfied: anomalies appear only during Earth gravity assists, not during interplanetary cruise.

Falsification criterion: Detection of anomalous velocity shifts during interplanetary cruise (far from any planetary gravity well) would falsify the TEP explanation, which requires proximity to massive bodies.

Null results: The TEP framework explains two distinct categories of null results observed in the data: (1) High-altitude gradient suppression — flybys above ~2500 km (Stardust, OSIRIS-REx, BepiColombo, Rosetta 2007, and the published Rosetta 2009 null/bound) where the field gradient is too small to produce detectable effects; and (2) Geometric cancellation — flybys with nearly symmetric trajectories where the non-radial component of the artefact kernel cancels (MESSENGER at 2347 km, $|\cos\delta_{\rm in} - \cos\delta_{\rm out}| \approx 0.005$; Rosetta 2007 and 2009 are comparably weak-geometry cases). Galileo 1992, formerly grouped here under the transcribed null entry, is on the corrected catalogue a genuine negative detection and is treated with the detections. Both categories are supported by existing data.

Consistency test: A flyby at altitude < 1500 km with symmetric trajectory geometry showing a large anomaly (> 5 mm/s) would be inconsistent with the geometric cancellation mechanism and would require revisiting the model assumptions.

Eccentric-orbit clock bound: The canonical excursion $\psi_{\rm uns} = 2\Phi_N$, extended on the adopted exterior profile scale $\lambda_T = 4200$ km, would imply an orbit-modulated clock-rate offset for an eccentric satellite traversing the terrestrial temporal well — for example the Galileo disposal-orbit pair GSAT-0201/0202 ($a \approx 2.8\times10^{4}$ km, $e \approx 0.16$). Step 045 computes the predicted lapse-level modulation on this orbit directly from the production model: $\Delta\eta \approx 2.1\times10^{-11}$, exceeding the measured bound on anomalous eccentric-orbit redshift modulation ($\alpha = (0.19 \pm 2.48)\times10^{-5}$ relative to the GR eccentric signal $\Delta\Phi_N \approx 5.3\times10^{-11}$; Delva et al. 2018) by a factor of $\approx 7.7\times10^{3}$. If the clock-amplitude excursion contributed to a transported clock's rate as an ordinary lapse-level term at this profile scale, it would be excluded. The channel accounting of Section 5.2 must therefore be read in one of two consistent ways: either the exterior vertical relaxation scale is $\lambda_T \le 1.30\times10^{3}$ km — the computed bound, suppressing the excursion at GNSS altitudes while retaining a substantial fraction at flyby altitudes, and decoupled from the lateral correlation scale $L_c$ — or the excursion enters transported-clock observables only through non-integrable transport (the holonomy sector), in which case it is not a lapse term and the Step 043 forward model would be the wrong observable. Deriving which realization the action produces is a corpus-level open item (register 15-11); the discriminating computation is bounded and falsifiable either way.

Testable predictions: The TEP framework makes falsifiable predictions that can be tested with additional Earth flyby data. Based on the gated fitted $\beta_{\rm fit}$ values (order $10^{-3}$), the model predicts:

• Flybys at perigee altitude < 2000 km should show detectable anomalies (1–10 mm/s)
• Flybys at perigee altitude 2000–3000 km should show marginal anomalies (0.1–5 mm/s)
• Flybys at perigee altitude > 5000 km should show no detectable anomaly (< 0.1 mm/s)

These predictions assume spacecraft velocity profiles similar to historical flybys. Precise predictions require detailed trajectory data from mission navigation teams. Any flyby with adequate DSN-quality tracking provides an opportunity for independent validation or falsification of the TEP framework.

5.11 Addressing the $\beta_{\rm fit}$ Parameter Scatter

A critical concern for physical interpretation is the formal heterogeneity in gated β (Cochran Q ≈ 1.18×10³, reduced χ² ≈ 236, I² ≈ 99.6%). In a standard meta-analysis, such an extreme χ² would be interpreted as model failure or outlier contamination. In the TEP framework, however, this heterogeneity is expected because the clock-sector response amplitude is trajectory-dependent: altitude modulates the field gradient, velocity modulates the disformal coupling, and the trajectory-asymmetry factor $\cos\delta_{\rm in} - \cos\delta_{\rm out}$ directly controls how much of the non-radial impulse survives. A single universal rescaling at fixed envelope is therefore an oversimplified null hypothesis that TEP itself predicts will be rejected. The extreme χ² is not evidence against TEP; it is evidence that geometry-dependent modulation must be included in any honest forecast. Step 009 does not produce a formal ANOVA-style variance decomposition because n = 6 renders percentage-based partitioning statistically underpowered; instead it reports defensible descriptive statistics and rank correlations (Section 4.3).

Testable predictions: With detailed trajectory reconstruction (velocity vectors at perigee), the following can be tested:

• $\beta_{\rm fit} \propto 1/|v_\perp|$ (anticorrelation with perpendicular velocity)
• $\beta_{\rm fit} \propto |\cos(i)|$ (correlation with equatorial inclination)
• $\beta_{\rm fit} \propto \cos({\rm latitude})$ (correlation with equatorial perigee)

Within the six-member detection ensemble, Galileo 1990 carries the largest amplitude-informative fitted $\beta_{\rm fit}$ ($1.44\times10^{-2}$) while NEAR and Rosetta 2005 sit at $4.4$–$5.1\times10^{-3}$ and Galileo 1992 occupies the negative-asymmetry branch at $6.7\times10^{-2}$; Cassini and MESSENGER carry near-zero reference predictions, so their per-flyby rescalings are unconstrained and contribute negligible pooled weight. Full 3D trajectory integration (Section 4.1.2) validates the perigee approximation for the primary detections and confirms that trajectory curvature contributes only modest corrections (≤20%). Raw DSN OD reanalysis remains the decisive path to test the residual amplitude shortfalls at Cassini and Galileo 1992.

5.12 Model Assumptions and Domain of Validity

The TEP Temporal Topology model relies on several explicit assumptions that define its domain of validity:

Assumption 1: Scalar-tensor gravity framework. The field-amplitude prescription uses the canonical Paper 0 amplitude-sector normalization — a conformally coupled scalar field $\phi$ whose terrestrial excursion $\psi_{\rm uns} = 2GM_\oplus/(R_\oplus c^2) = 1.3926\times10^{-9}$ is parameter-free and $\beta_A$-independent (Section 3.4). The earlier inverse-power realization ($V(\phi) \propto \Lambda^{4+n}/\phi^n$) is excluded by the Paper 0 F1 constraint under $\beta_A = -1$ and is retained in Step 010 only as an excluded-branch contrast. Alternative amplitude-sector completions would yield different predictions.

Assumption 2: Geometric screening via Temporal Shear suppression. This mechanism requires that Earth develops a continuous spatial profile (Temporal Topology) where the scalar field gradient is suppressed by the environmental-state operator $\mathcal{S}_\Sigma(\mathcal{E})$ — the density projection being the relevant coordinate in this near-Earth application. This transition radius is computed from the field equation and depends on the assumed density profile (5515 kg/m$^3$ for Earth interior, 2700 kg/m$^3$ for crust, 1.225 kg/m$^3$ for atmosphere). Different density profiles would modify the relaxation length by $\sim 10\%$.

Assumption 3: Instantaneous coupling. The model assumes the TEP effect manifests instantaneously during perigee passage, with no memory or hysteresis effects. This is consistent with the field equation structure but could be violated if the scalar field has dynamical relaxation times longer than the flyby duration ($\sim$hours).

Assumption 4: Negligible spacecraft mass. The model treats spacecraft as test particles, ignoring their self-gravity. This is justified as spacecraft masses ($\sim 500$–5000 kg) are 21 orders of magnitude smaller than Earth mass.

Assumption 5: Spherical Earth symmetry. The Disformal Temporal Topology field is computed assuming spherical symmetry. Earth's oblateness ($J_2 = 1.08 \times 10^{-3}$) introduces $\sim 0.1\%$ corrections to the gravitational potential, negligible compared to the three-order-of-magnitude anomaly amplitude variation.

Domain of validity: The model is valid for flybys with perigee altitudes below the transition region ($\sim 2500$ km) and velocities in the range 10–20 km/s. Extrapolation outside this parameter space requires caution. High-altitude or near-threshold null cases such as Rosetta 2007 (5430 km), Rosetta 2009 (2572 km), Stardust, OSIRIS-REx, and BepiColombo provide limited constraint on the fitted coupling but test that the anomaly does not persist where the geometry envelope predicts strong suppression or no public detection exists.

5.13 Limitations and Caveats

A rigorous assessment of this analysis requires explicit acknowledgment of several limitations, their impact on conclusions, and mitigation strategies:

1. Data provenance and independence:

• Issue: The analysis relies on published anomaly values from Anderson et al. (2008) and companion publications rather than independent reanalysis of raw DSN tracking data.
• Impact: Systematic errors in the original orbit determination (e.g., unmodeled spacecraft maneuvers, antenna offset corrections) would propagate directly to this analysis. The reported uncertainties (0.01–0.05 mm/s) may not fully capture all systematic contributions.
• Mitigation: The literature values are derived from NASA/JPL orbit determination using the same software (ODP) employed for interplanetary navigation, with established systematic error budgets. Cross-validation between independent analyses (JPL vs. ESA/ESOC for Rosetta) shows consistency at the 0.1 mm/s level.
• Validation: Direct access to DSN tracking archives would enable independent orbit fits with explicit systematic error modeling. The pipeline implements Steps 005–006 and 028–031 for that purpose; ingest is deferred in the present release (Step 028: no perigee-matched products; Step 030: inconclusive). Re-analysis when data are available is the decisive validation step.

2. Sample size and selection effects:

• Issue: Six flybys enter the Step 008 inverse-variance $\beta_{\rm fit}$ ensemble (NEAR, Galileo 1990, Rosetta 2005, Cassini, Galileo 1992, MESSENGER), all sign-consistent at $\beta_{\rm ref}$ on the corrected catalogue. Cassini and MESSENGER carry near-zero reference predictions, so their fitted $\beta_{\rm fit}$ are poorly constrained and contribute negligible weight; the amplitude-informative core remains four flybys. This yields modest statistical power for distinguishing fine-grained geometry hypotheses.
• Impact: Small sample size increases susceptibility to over-interpretation of single-proxy decompositions and reduces ability to test alternative screening functional forms.
• Justification: The accessible set of low-altitude, well-tracked Earth flybys is small by nature of the mission cadence.
• Statistical robustness: Despite small $n$, the effect sizes are substantial for NEAR and Galileo, while Rosetta 2005 and Cassini are weak under the null-population contrast. Step 026 on the corrected catalogue places the Anderson empirical law nominally first by BIC, with the restricted TEP model at parity-level separation and all structured models far ahead of the Null (Section 4.5); the BIC map remains small-sample and surrogate. A full-catalog raw stress test on all nine flybys with published observations and explicit TEP predictions improves over the null after random-effects prediction uncertainty is included ($\Delta\log L\approx43.0$). Leave-one-out shows the pooled $\beta_{\rm fit}$ is moderately robust, with a stability coefficient of 0.092 across the six members.
• Sample expansion: Additional Earth flybys with adequate tracking precision would strengthen the statistical analysis and enable tests of model variations. Approximately $n \approx 74$ primary detections would be required to achieve 80% power to distinguish between geometry-dependent modulation of effective coupling and a single pooled amplitude held fixed across rows (conservative estimate: $n \approx 153$).

3. Trajectory reconstruction uncertainties:

• Issue: Trajectories from JPL Horizons are post-fit ephemerides that already include the anomalous velocity shifts in their reconstruction. This introduces circularity: the trajectory used to compute TEP predictions incorporates the anomaly being modeled.
• Impact: The perigee altitude and velocity values may have systematic offsets of $\sim 1$ km and $\sim 1$ m/s respectively, propagating to $\sim 1\%$ uncertainty in TEP predictions.
• Mitigation: The TEP model depends primarily on the ratio of gravitational potential gradients, which is insensitive to small trajectory perturbations. A 1% trajectory error produces $\sim 1\%$ error in predicted $\Delta v$, negligible compared to the three-order-of-magnitude amplitude variation between flybys.
• Previously unavailable flybys: Rosetta 2007 (Δv = 0.02 mm/s reported) was initially unavailable in JPL Horizons due to spacecraft identifier conflicts (JPL ID -85 returns no ephemeris for these dates). This flyby is now included in the analysis using ESA SPICE kernels, which provide independent trajectory data. Rosetta 2009 is a published null/bound case (Δv = 0.00 mm/s reported), but it lacks explicit geometry in the Step 039 prediction table and is therefore not used in the fitted likelihood.

Assumption 1: Post-fit trajectory independence: The analysis uses JPL Horizons ephemerides, which are post-fit trajectories incorporating all available tracking data including the anomalous velocity shifts. This introduces a potential circularity concern: if the orbit determination process absorbed the anomaly into the trajectory fit, the TEP predictions would be based on trajectories that already contain the effect under investigation. However, several factors mitigate this concern:

• Scale separation: The flyby anomalies are velocity shifts of order 1-10 mm/s, whereas the perigee velocities are order 10 km/s. The anomaly represents a fractional change of $10^{-7}$ to $10^{-6}$ in the velocity vector. Orbit determination processes typically converge to solutions with residuals at the mm/s level, meaning the anomaly is comparable to the solution precision rather than being absorbed into the trajectory.
• Global fit constraint: JPL Horizons trajectories are constrained by tracking data spanning years, not just the flyby epoch. The global fit includes pre-flyby and post-flyby arcs that are not affected by the anomaly. The perigee geometry (altitude, velocity) is determined by the global orbit solution, which is dominated by the long-arc data rather than the short perigee passage where the anomaly manifests.
• Independent verification: The Rosetta 2005 and 2007 trajectories were obtained from ESA SPICE kernels, which use independent orbit determination software and tracking networks. The consistency between JPL and ESA trajectory solutions for these flybys supports the validity of using post-fit trajectories.
• Null-result flybys: The eight null-result flybys use the same orbit determination methodology yet show no anomalies. If the circularity concern were severe, all flybys would show apparent anomalies due to trajectory fitting artifacts. The selective detection pattern (detections at low altitude, nulls at high altitude) is not an artifact of the orbit determination process.

While the circularity concern cannot be entirely eliminated without independent raw DSN data analysis, the scale separation, global fit constraints, and independent ESA verification provide sufficient justification for using JPL Horizons trajectories in this analysis.

4. Phenomenological gradient suppression model:

• Issue: The screened field model uses parameterized density-dependent field values rather than a full first-principles calculation from a specific scalar-tensor action.
• Issue: The screened field model anchors the terrestrial excursion on the canonical amplitude-sector normalization ($\psi_{\rm uns}$, Section 3.4) with a phenomenological relaxation profile, rather than a full first-principles profile calculation from the microscopic scalar action.
• Mitigation: The canonical normalization is parameter-free — fixed by Earth's measured mass and radius and independent of $\beta_A$ — and the excluded inverse-power branch is retained only as a contrast diagnostic (Step 010). One free phenomenological parameter remains ($\beta_{\rm fit}$, a trajectory-response amplitude distinct from the bare coupling $\beta_A = -1$), preserving predictive power.
• Validation: Comparison with numerical Temporal Topology field solvers would validate the phenomenological approximation. Additionally, a plasma-model sensitivity analysis (scripts/utils/plasma_screening.py) tests three functional forms for plasma attenuation—exponential, power-law, and linear—quantifying the coefficient of variation across flybys under each form. If the CV and spread ratio are similar across forms, the conclusions are robust to the specific plasma ansatz choice.

5. Systematic error budget:

• DSN measurement systematics: Antenna phase center motion ($\sim 0.1$ mm/s), tropospheric delay modeling ($\sim 0.05$ mm/s), and station position errors ($\sim 0.02$ mm/s) contribute to the anomaly uncertainty budget. These are partially correlated across flybys, potentially affecting the weighted mean calculation.
• Spacecraft-specific systematics: Galileo's high-gain antenna failure and spin-rate changes introduce additional uncertainty not captured in the 0.08 mm/s formal error. The Galileo 1990 anomaly should be interpreted with caution. This is a caveat, not an exclusion criterion. Galileo 1990 satisfies both a priori ensemble gates (S/N = 49 > 2; sign agreement with TEP prediction) and is retained in the fitted ensemble. The stated caution means its published uncertainty may underestimate true systematic error, not that the datum should be discarded. Arbitrary exclusion would reduce the ensemble from n = 6 to n = 5 without methodological justification and would violate the pre-specified selection protocol (Section 3.3). The leave-one-out diagnostic confirms that removing Galileo 1990 shifts the pooled β by only 3.5% (4.58×10⁻³ vs 4.74×10⁻³), so the headline conclusion is insensitive to its inclusion.
• Orbit determination methodology: The pre-perigee to post-perigee residual comparison assumes constant systematic errors. Time-varying systematics (e.g., thermal expansion) could produce spurious velocity signatures.

If the Juno null is confirmed by independent minimal-OD reanalysis: The fixed-amplitude Step 039 cross-catalog hypothesis is stressed for this flyby. It fails only if the prediction-uncertainty model can be independently tightened enough that Juno remains a detectable forecast. The original detections in older analyses would then require re-evaluation as possible systematic errors in less sophisticated OD pipelines.

If the Juno null is overturned by minimal-OD reanalysis: A TEP signal recovered from raw DSN data with TEP-inclusive force modeling would demonstrate that modern OD absorbed the anomaly, validating the OD-absorption conjecture for this mission.

Current status: The OD-absorption hypothesis is not established. A statistical power analysis (scripts/utils/statistical_utils.py, juno_falsification_power_analysis) quantifies the detectability of the TEP-predicted effect: at the Step 008 pooled β with random-effects prediction uncertainty (~0.03 mm/s) and DSN measurement precision (~0.02 mm/s), the combined σ_total ≈ 0.039 mm/s gives Cohen's d ≈ 3.2 for the predicted +0.12 mm/s effect, yielding >90% power for a one-sided test at α = 0.05. The minimum detectable effect at 80% power is ~0.05 mm/s. This means a genuine TEP signal of the predicted magnitude should be readily detectable with existing DSN precision; its absence in published residuals is therefore informative and motivates the OD-absorption hypothesis, though it does not prove it. A definitive test requires MONTE- or ODP-class minimal orbit determination with TEP-inclusive force modeling, which is outside the scope of the current pipeline. Until such reanalysis is completed, the Juno tension remains an open falsification test.

5.14 Evidence Ledger for Review

The following evidence ledger documents the core model-comparison and cross-check entries required for independent assessment:

• Restricted TEP vs Null: large information-criterion separation ($\Delta$BIC $\approx 16690$ on full n = 9 catalog under the fractional misspecification floor); TEP restricted is strongly favoured over the Null, while the Anderson empirical law fitted on the same catalogue attains the nominally smallest BIC overall.
• Restricted TEP vs Anderson: the Anderson empirical model attains the nominally smaller BIC ($\Delta$BIC $\approx -4.8$ on full n = 9 catalog — parity-level separation, not a decisive preference; under the published-uncertainties-only sensitivity the ordering reverses, $\Delta$BIC $\approx +2767$ in TEP restricted's favour). The restricted model retains the pre-specified-parameter advantage.
• External anchors: GNSS atomic-clock correlation anchors $\lambda_T = 4200$ km under the Paper 6 §2 identification (Step 016 records the bookkeeping consistency of the adopted constant with its own anchor); UCD saturation model gives $S_\oplus \approx 0.35$ (Step 010); Cassini solar conjunction constrains $|\gamma - 1|$ (Section 4.6).
• Null-catalog stress test: full n = 9 raw-catalog likelihood improves by $\Delta\log L \approx 43.0$ over the null after random-effects prediction uncertainty is included (Step 039).
• Failure-mode accounting: velocity-template regime classification via the empirical scale $v_{\rm trans} \approx 16.8$ km/s (phenomenological ansatz, §3.5) and sign-rule consistency checks (Step 007/017).
• Machine-checked manuscript: this document is generated from site/components/*.html and audited against pipeline JSON outputs by Step 027.

5.15 Summary

These limitations are explicitly acknowledged to ensure intellectual honesty. They do not invalidate the central conclusion—that TEP with Temporal Shear suppression within continuous Temporal Topology provides a quantitative explanation for the flyby anomaly—but indicate areas requiring additional scrutiny. The framework makes falsifiable predictions that can be tested with additional flyby data.

6. Conclusions

This study investigated whether the Temporal Equivalence Principle (TEP), through the clock-sector reduction-artefact channel identified in Paper 0 (v0.15) and scoped to its observable classes in Section 5.2, can organize the Earth flyby anomaly—unexplained velocity shifts observed during spacecraft gravity assists. The analysis of twelve Earth flyby events spanning nine spacecraft (Galileo 1990/1992, NEAR, Cassini, Rosetta 2005/2007/2009, MESSENGER, Juno, Stardust, OSIRIS-REx, BepiColombo) yields the following key findings:

1. Six-member TEP ensemble: Inverse-variance fitting (Step 008) enters NEAR ($13.46 \pm 0.13$ mm/s), Galileo 1990 ($3.92 \pm 0.08$ mm/s), Rosetta 2005 ($1.80 \pm 0.03$ mm/s), Cassini ($-2.00 \pm 1.00$ mm/s), Galileo 1992 ($-4.60 \pm 1.00$ mm/s), and MESSENGER ($+0.02 \pm 0.01$ mm/s) after pre-fit gates (S/N ≥ 2, sign agreement at $\beta_{\rm ref}=10^{-4}$) on the re-transcribed catalogue. All six reference predictions agree in sign with the published anomalies. At $\beta_{\rm ref}$, Rosetta 2005 predicts $\Delta v_{\rm TEP} \approx 0.03$ mm/s vs $1.80$ mm/s observed; the full-model fit yields $\beta_{\rm fitted} \approx 5.15\times10^{-3}$. Cassini and MESSENGER carry near-zero reference predictions, so their per-flyby rescalings are poorly constrained and contribute negligible pooled weight. The amplitude-informative gated $\beta_{\rm fit}$ values span roughly a factor of 3.3 ($4.43\times10^{-3}$ to $1.44\times10^{-2}$, with Galileo 1992 at $6.7\times10^{-2}$), consistent with geometry-dependent modulation (the plasma term is a bounded heuristic envelope coefficient; the stronger Debye attenuation ansatz is quarantined at $S_{\rm plasma}=1$, Section 3.10). The flyby trajectory-response amplitudes $\beta_{\rm fit}$ are distinct from the bare conformal coupling $\beta_A = -1.0$ and from the Cassini source charge $\alpha_{\rm eff}^{(\odot)} = S_\Sigma^{(\odot)}\alpha_0$. Cassini PPN compliance is evaluated through the solar source-charge projection (Section 4.6.1a), not through the fitted flyby amplitudes.
2. TEP parameter estimate: The inverse-variance weighted mean $\beta_{\rm fit} \approx 4.74 \times 10^{-3} \pm 1.28 \times 10^{-3}$ (random-effects standard error; formal inverse-variance error $5.83 \times 10^{-4}$) summarizes the gated ensemble. This is the flyby trajectory-response amplitude, distinct from the bare conformal coupling $\beta_A = -1.0$ and from the Cassini source charge $\alpha_{\rm eff}$. Formal heterogeneity (reduced $\chi^2 \approx 236$, $I^2 \approx 99.6\%$) signals that a single rescaling does not exhaust the geometry physics carried by the envelope (the plasma term is a bounded heuristic envelope coefficient). Bootstrap resampling ($10^4$ draws) and leave-one-out recomputations in Step 008 show moderate stability (coefficient $\approx 0.092$), with NEAR as the dominant lever.
3. PPN compliance via Temporal Topology screening: The Cassini solar conjunction experiment provides the tightest bound on the post-Newtonian light-propagation sector, measuring $\gamma = 1 + (2.1 \pm 2.3) \times 10^{-5}$. This constrains the solar-system Shapiro/light-propagation sector but does not directly test spatial clock-sector covariance, one-way residual shear, or low-density temporal-shear recovery. The fitted $\beta_{\rm fit}$ values are flyby trajectory-response amplitudes, not the Cassini source charge $\alpha_{\rm eff}^{(\odot)} = S_\Sigma^{(\odot)}\alpha_0$. Cassini PPN compliance is evaluated through the solar source-charge projection (Section 4.6.1a): the adopted Paper 0 v0.15 convention gives $S_\Sigma^{(\odot)} \sim 2\times10^{-6}$, inside the Cassini requirement $S_\Sigma^{(\odot)} \lesssim 5.8\times 10^{-6}$ (canonical linear mapping, Paper 0 §7), with the kinetic-operator benchmark evaluating the solar-field screening at $\sim 10^{-15}$ along the 1 AU path. This supports the claim that TEP can reduce to the GR PPN light-propagation limit in the screened solar environment while reserving its discriminating predictions for observables outside the Cassini measurement class.
4. TEP suppression by modern orbit determination: Analysis of the expanded dataset reveals that published null results (MESSENGER, Rosetta 2007) are consistent with universal-$\beta_{\rm fit}$ null predictions, while Juno is the sole raw-tension case at universal $\beta_{\rm fit}$. Post-OD survival factors are withheld until mission OD configuration yields defensible $F_{\rm OD}$ estimates. Rosetta 2009 is treated as a published null/bound case with insufficient explicit geometry for the universal-$\beta_{\rm fit}$ prediction table; Stardust, OSIRIS-REx, and BepiColombo have no public anomaly report and are not used in quantitative likelihood.
5. Multiple independent lines of evidence: Altitude-dependent anomaly pattern (see point 6), historical timeline and the OD filtering mechanism motivate the hypothesis that modern orbit determination can treat TEP-like signals as systematic errors. However, mission-specific survival factors are not computed in this paper: Step 021 withholds $F_{\rm OD}$ until real OD configuration data are available, and the synthetic Step 012 OD run is quarantined from manuscript inference.
6. Temporal Topology screening support: The response template organizes the catalogue pattern — null results on high-altitude flybys, where the conformal excursion is shallow, and large anomalies on low-altitude encounters — conditional on observable class: in the coherent two-way class of the published catalogue the organization measures how strongly the recorded anomalies track the trajectory geometry the artefact kernel encodes, while the mechanism-level signature is confined to clock-carrying links (Section 5.2). Altitude alone is not the relevant TEP discriminator; the pattern is governed by the full trajectory-geometry response, which combines altitude, asymmetry, velocity and field structure.
7. Systematic uncertainty compression: Transitioning from empirical characteristic suppression factors to a UCD-derived estimate via the Self-Consistent Field (SCF) solver and the corrected uncertainty analysis (Step 025) provides a rigorous uncertainty budget. The total relative uncertainty is 152.7%, with heterogeneity contributing 149.4% and systematic uncertainty contributing 29.2%. The systematic component is dominated by characteristic suppression uncertainty (25.0%, Paper 6 UCD) and relaxation length uncertainty (15.0%, SCF theoretical prior). This shift from "parameter fitting" to "systematic prediction" with proper variance decomposition strengthens the theoretical foundation of the TEP analysis.
8. Robust statistical checks: The primary gated fit and model-comparison layer use inverse-variance/Gaussian weighted likelihoods, while Student's t likelihoods are retained in auxiliary Bayesian checks to test sensitivity to outliers. Residual diagnostics on the gated ensemble is consistent with normality (Shapiro–Wilk $p \approx 0.95$, $n = 6$), so heterogeneity diagnostics, not normality alone, dominate the statistical interpretation. (NEAR, Galileo 1990, Rosetta 2005, Cassini, Galileo 1992, MESSENGER).
9. Cosmographic CMB-frame directional consistency: Full 3D spacecraft state vectors from JPL Horizons were tested for CMB-frame velocity geometry (Step 040, n = 9). The exploratory both-aligned flag yields Pearson r = −0.57, p = 0.11, and Mann-Whitney U = 7.5, p = 0.79. Multivariate regression on residual ratio achieves R² = 0.96 (adjusted R² = 0.94), and an optimal weighted combination of spacecraft and Earth CMB projections gives r = 0.96, though the optimum sits at the boundary of the scanned weight grid (w = −2.0) and is read as a boundary-limited exploratory diagnostic, not a calibrated coefficient. These results are sample-limited and remain exploratory.

Significance

The TEP interpretation of the Earth flyby anomaly provides a coherent theoretical framework connecting spacecraft dynamics to fundamental physics. The flyby trajectory-response amplitude $\beta_{\rm fit} \approx 4.74\times10^{-3}$ (weighted mean) is a trajectory-response coefficient, not the bare coupling $\beta_A$ or the Cassini source charge $\alpha_{\rm eff}$. Interpreted as the clock-sector amplitude response under the amplitude–shear split, the flyby catalogue's geometric organization is consistent with solar system constraints and organizes the anomalous velocity-shift pattern. What that organization attests is conditional on the tracking-link class: the published catalogue is assigned to the coherent-transponder class on the literature-based ledger of Section 5.2, pending encounter-specific tracking verification; in that class the clock-sector term is absent at the observable (Section 5.2), so the catalogue pattern constrains the response template's structure while the mechanism's claim is carried by the falsifiable prediction of an m/s-scale signature on the first clock-carrying flyby pass.

Unlike ad hoc modifications to gravity, the TEP framework preserves all successes of general relativity in solar system tests while supplying a structured account of the anomaly pattern in the specific regime of planetary gravity assists — an account whose mechanism-level claim is scoped to the clock-carrying link classes of Section 5.2 rather than asserted for the recorded catalogue. The geometric amplitude response, calibrated by independent UCD saturation analysis, is essential for the flyby amplitude and environmental response: without it, the required $\beta_{\rm fit}$ would not produce the observed trajectory-dependent residuals.

Statistical evidence strength: The validation analysis provides substantial statistical support for TEP:

• Effect sizes: Cohen's $d$ relative to the published null population ($n_{\rm null}=3$) yields a large effect for NEAR ($d \approx 2.5$), medium for Galileo 1990 ($d \approx 0.73$), large-in-magnitude negative for Galileo 1992 ($d \approx -0.87$), and small effects for Rosetta 2005 ($d \approx 0.34$) and Cassini ($d \approx -0.38$). The coefficient of variation CV $\approx 149\%$ across the six gated $\beta_{\rm fit}$ fits is inflated by the near-zero-prediction members and reflects genuine geometry-dependent modulation at fixed envelope.
• Model comparison: On the full n = 9 catalog under the fractional misspecification floor, Step 026 gives a large information-criterion separation between TEP restricted and Null ($\Delta{\rm BIC}\approx16690$); the Anderson empirical law, whose two parameters are fitted on the same catalogue, attains the nominally smallest BIC overall but only at parity-level separation ($\Delta{\rm BIC}\approx4.8$), an ordering that reverses in TEP restricted's favour under the published-uncertainties-only sensitivity
• Akaike model weight favours the Anderson empirical tier ($\approx 0.86$ versus $\approx 0.14$ for TEP restricted, full $n=9$ catalogue under the fractional misspecification floor) — a parity-level ordering that reverses under the published-uncertainties sensitivity; the TEP restricted and flexible tiers remain far above the Null and are documented in Step 026 outputs.
• Robustness: Bootstrap resampling, leave-one-out recomputations, and auxiliary robust checks are reported in Step 008/013.
• Prediction accuracy: Step 008 reports $R^2 \approx 0.85$ (correlation $r \approx 0.93$) between pooled-$\beta$ predicted and observed anomalies across the six-member ensemble; prediction intervals are bootstrap-derived.
• Residual analysis: Shapiro–Wilk $p \approx 0.95$ on the gated six-member ensemble; heterogeneity diagnostics dominate formal tests.
• Sensitivity analysis: All parameters stable across plausible ranges; fit stability maintained

The catalog of Earth flyby events (six published nonzero entries — four firm detections and two ~2σ marginal entries, Cassini and MESSENGER — plus three published nulls/bounds and several without public anomaly reports) provides the empirical substrate. Step 026's headline likelihood uses the full n = 9 catalog with a fractional misspecification floor ($\sigma_{{\rm sys},i} = CV_\beta|\hat y_i|$, $CV_\beta \approx 0.47$, measured from the catalogue's own amplitude heterogeneity and invariant under the field-normalization anchor). The archived comparison gives $\Delta{\rm BIC}\approx1.67\times10^{4}$ for the restricted geometric template relative to Null and near-parity with the catalogue-fitted Anderson tier. These scores describe catalogue organization under the documented fitting convention, not support for the clock-sector mechanism. Section 4.5 states the outstanding likelihood-optimum and nuisance-accounting checks; the physical clock-sector test is a clock-carrying pass, with the profile independently constrained by clock observations.

Robustness Assessment

Several potential concerns have been investigated and addressed through rigorous statistical analysis (Step 024, Step 025, Step 026):

Systematic error discrimination: trajectory asymmetry ($\cos\delta_{\rm in} - \cos\delta_{\rm out}$) organizes the adopted catalogue template, but this correlation does not exclude geometry-dependent tracking or reconstruction errors. Within the gated n = 6 ensemble the ordering is positive but imperfect (Spearman ρ = +0.66, p = 0.156), a qualitative consistency check rather than a primary discriminator. All six retained anomalies agree in sign with the reference prediction on the re-transcribed catalogue, and the geometry envelope carries seven nominal deterministic coefficients plus one fitted amplitude against six anomalies, so the archived scores characterize template organization under the adopted convention, not the clock-sector mechanism. The broader catalogue rank correlations are complementary diagnostics on overlapping data: asymmetry-vs-magnitude gives ρ = 0.56 (p = 0.11, n = 9) and ρ = 0.68 (p = 0.090, n = 7 nonzero anomalies) in the Step 009 ordering diagnostic. See Section 5.5 for comprehensive systematic uncertainty budget.

Data provenance: The analysis relies on published anomaly values from Anderson et al. (2008) rather than independent DSN re-analysis. This is addressed by: (a) cross-referencing multiple literature sources for consistency, (b) demonstrating that TEP predictions match the observed anomaly pattern (altitude dependence, trajectory geometry), (c) providing a framework for raw DSN data re-analysis to independently test the suppression hypothesis. The current repository state provides mission discovery, structured requests, and an ingestion scaffold; in typical runs most missions do not yet have indexed local raw Doppler products available for reprocessing (pipeline ingest status no_indexed_products). The DSN pathway is therefore an explicit falsification route, not a completed independent reanalysis across the catalog.

β_fit scatter as physical modulation: Fitted β_A,eff spans roughly a factor of 3.3 across the tightly constrained members ($4.43\times10^{-3}$ to $1.44\times10^{-2}$), extending to $6.7\times10^{-2}$ for Galileo 1992 and formally divergent for the near-zero-prediction Cassini and MESSENGER members (Step 009), reflecting geometry-dependent modulation: altitude ($J_2$ gradient suppression), perigee latitude (inclination-dependent coupling), and velocity (the response template's disformal branch); the plasma term is a bounded heuristic envelope coefficient (Section 3.10). Step 009 does not report a formal variance decomposition because n = 6 renders ANOVA-style partitioning statistically underpowered. Instead, it reports deterministic geometry modulation metrics: beta scatter statistics, full-catalog detection-pattern classification, and Spearman rank correlation (ρ = 0.56 over the n = 9 catalogue; ρ = +0.66 in the gated ensemble). Leave-one-out recomputation on the six-member ensemble reports stability coefficient ≈ 0.092. The UCD-derived characteristic suppression $S_\oplus \approx 0.35$ provides a cross-scale prior. See Section 4.3 for the quantitative geometry modulation assessment and Section 5.5 for detailed interpretation.

Cassini status: At $\beta_{\rm ref}=10^{-4}$ the Step 007 envelope yields a negative $\Delta v_{\rm TEP}$ of negligible magnitude, and the corrected published anomaly is $-2.00 \pm 1.00$ mm/s, so Cassini is sign-consistent at $\beta_{\rm ref}$ and enters the gated ensemble (n = 6) — its residual tension is amplitude-only. Galileo 1992, transcribed as a null in the earlier catalogue, is a genuine negative detection ($-4.60 \pm 1.00$ mm/s) whose magnitude is underpredicted ($\approx -0.33$ mm/s at the pooled amplitude). Addressing these amplitude shortfalls requires independent DSN OD or envelope refinements, not only a universal rescaling of $\beta_{\rm fit}$.

Juno falsification pressure: At the refit weighted-mean $\beta_{\rm fit}$, Step 039 predicts $\Delta v_{\rm TEP}^{\rm raw}\approx +0.12$ mm/s with uncertainty $\sim 0.03$ mm/s while the published bound is $0.00\pm 0.02$ mm/s. This raw-tension case is the headline stress test on universal $\beta_{\rm fit}$ before mission-specific OD survival factors are supplied (Step 021). Independent raw DSN re-analysis with TEP-inclusive orbit determination is required to adjudicate the tension.

Sample size as complete dataset: The analysis includes all accessible Earth gravity assist flybys with adequate DSN tracking between 1990–2020. The rarity of suitable flyby events (low altitude, Doppler tracking, no major maneuvers) means only a handful of literature anomalies exist; six enter the gated $\beta_{\rm fit}$ ensemble. Step 008 bootstrap 95% CI $[4.42\times10^{-3},\,1.47\times10^{-2}]$ brackets the gated fits (bootstrap median $4.77\times10^{-3}$). Additional flybys would test envelope refinements rather than restate the same $n=6$ compression.

PPN compliance: The UCD-derived characteristic suppression $S_\oplus \approx 0.35$ is determined from the UCD saturation model. The flyby trajectory-response amplitudes $\beta_{\rm fit}$ are distinct from the Cassini source charge $\alpha_{\rm eff}^{(\odot)} = S_\Sigma^{(\odot)}\alpha_0$; Cassini PPN compliance is evaluated through the solar source-charge projection (Section 4.6.1a), where the adopted Paper 0 v0.15 convention gives $S_\Sigma^{(\odot)} \sim 2\times10^{-6}$, inside the Cassini requirement $S_\Sigma^{(\odot)} \lesssim 5.8\times 10^{-6}$.

Model comparison: Step 026 on the full n = 9 catalog gives a large information-criterion separation between TEP restricted and Null ($\Delta{\rm BIC}\approx16690$). The Anderson empirical model, fitted on the same catalogue, attains the nominally smallest BIC overall ($\Delta{\rm BIC}\approx4.8$ ahead of TEP restricted — parity-level; $\Delta{\rm BIC}\approx16694$ ahead of Null), confirming that trajectory asymmetry carries genuine signal. The TEP flexible model (3 parameters) is penalized by its parameter count and does not outperform the restricted model on BIC. Residual diagnostics on the gated ensemble are consistent with normality (Shapiro–Wilk $p\approx0.95$, $n=6$).

Independent validation pathways: Several approaches can independently test the TEP hypothesis without relying on the published anomaly values:

• Raw DSN data re-analysis: Analysis of raw DSN tracking archives from NASA's Planetary Data System using minimal orbit determination (reduced gravity field expansion, unfiltered Doppler, no continuity penalties) would test whether TEP signals are filtered by modern orbit determination methods. This would provide an important test of the suppression hypothesis.
• Additional flyby analysis: Earth gravity assist missions provide opportunities for independent detection. Analysis with both standard and minimal orbit determination methods would test the suppression prediction.
• GNSS clock correlation: GNSS atomic clock correlation analysis provides a cross-consistency check with the GNSS/UCD terrestrial calibration on the transition radius ($R_{\rm sol} \approx 4200$ km). This shared-input dependency is disclosed explicitly to avoid circular claims of independent validation.
• Lunar Laser Ranging: Precision LLR analysis in related work reports a synodic-phase signal consistent with the screening mechanism and Temporal Topology Saturation Scale (UCD) framework. Independent LLR validation would strengthen the screening mechanism established in this analysis.

Data Availability

Spacecraft trajectories are available through the JPL Horizons ephemeris service. Literature anomaly values are from Anderson et al. (2008) and companion publications. Analysis code and processed data products are available at https://github.com/matthewsmawfield/TEP-EFA with archived DOI at 10.5281/zenodo.19454862.

Acknowledgments

The NASA Deep Space Network and Jet Propulsion Laboratory provided the precision Doppler tracking that enabled flyby anomaly detection. The JPL Horizons system provided trajectory reconstruction. This work utilizes published literature values from the Orbit Determination Program analyses by Anderson et al. and collaborators. This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors. The author declares no conflicts of interest.

Additional Considerations

Several avenues for extending this analysis are identified:

Raw DSN data re-analysis: Analysis of raw DSN tracking archives from NASA's Planetary Data System using minimal orbit determination (reduced gravity field expansion, unfiltered Doppler, no continuity penalties) would test whether TEP signals are filtered by modern orbit determination methods. This provides an important test of the suppression hypothesis.

Extended spacecraft sample: Additional flyby events would increase the sample size beyond the current six published nonzero entries (all six sign-consistent in the inverse-variance $\beta_{\rm fit}$ ensemble, with Cassini and MESSENGER marginal at ~2σ). A sample of $n \approx 74$ primary detections would provide sufficient statistical power to distinguish between geometry-dependent modulation of $\beta_{\rm fit}$ and a single universal coupling constant at 80% power (conservative estimate: $n \approx 153$).

• Full numerical Temporal Shear Suppression solver: Implementation of a numerical Temporal Topology field solver (e.g., using the shooting method or relaxation techniques) validates the phenomenological gradient suppression model used in this analysis. This enables prediction of the Temporal Topology profile without the phase-boundary approximation and could explain the observed $\beta_{\rm fit}$ scatter through detailed density-dependent effects.
• Inclination-dependent modeling: Incorporation of Earth's oblateness ($J_2$) and latitude-dependent density variations into the TEP model could explain part of the observed $\beta_{\rm fit}$ scatter. Spacecraft with different orbital inclinations sample different gravitational field geometries, which modulate the Temporal Topology field strength.
• Disformal coupling exploration: Extension to scalar-tensor theories with disformal coupling terms introduces velocity-dependent gradient suppression that could explain the correlation between fitted $\beta_{\rm fit}$ and flyby velocity. This provides a more general framework for understanding the geometry-dependence of the TEP effect.
• Local-time plasma effects: A first-principles scalar–plasma coupling derived from the TEP action would test whether ionospheric density variations with local time modulate the clock-sector response. At present plasma enters only as a bounded heuristic envelope coefficient, with the stronger Debye attenuation ansatz quarantined at $S_{\rm plasma}=1$ (Section 3.10); this bullet records the question as an open refinement, not a current mechanism.

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TEP Research Series

Smawfield, M. L. Temporal Equivalence Principle: Dynamic Time & Emergent Light Speed. Preprint v0.15 (Jakarta). Zenodo. DOI: 10.5281/zenodo.16921911

Smawfield, M. L. Global Time Echoes: Distance-Structured Correlations in GNSS Clocks. Preprint v0.27 (Jaipur). Zenodo. DOI: 10.5281/zenodo.17127229

Smawfield, M. L. Global Time Echoes: 25-Year Analysis of CODE Precise Clock Products. Preprint v0.20 (Cairo). Zenodo. DOI: 10.5281/zenodo.17517141

Smawfield, M. L. Global Time Echoes: Raw RINEX Consistency Test. Preprint v0.8 (Kathmandu). Zenodo. DOI: 10.5281/zenodo.17860166

Smawfield, M. L. Temporal-Spatial Coupling in Gravitational Lensing: A Reinterpretation of Dark Matter Observations. Preprint v0.8 (Tortola). Zenodo. DOI: 10.5281/zenodo.17982540

Smawfield, M. L. Global Time Echoes: Empirical Synthesis. Preprint v0.7 (Singapore). Zenodo. DOI: 10.5281/zenodo.18004832

Smawfield, M. L. Temporal Topology Saturation Scale: Cross-Scale Consistency of ρ_T. Preprint v0.8 (New Delhi). Zenodo. DOI: 10.5281/zenodo.18064365

Smawfield, M. L. The Soliton Wake: Exploring RBH-1 as a Temporal Topology Candidate. Preprint v0.4 (Blantyre). Zenodo. DOI: 10.5281/zenodo.18059250

Smawfield, M. L. Global Time Echoes: Optical-Domain Consistency Test via Satellite Laser Ranging. Preprint v0.6 (Mombasa). Zenodo. DOI: 10.5281/zenodo.18064581

Smawfield, M. L. What Do Precision Tests of General Relativity Actually Measure?. Preprint v0.8 (Istanbul). Zenodo. DOI: 10.5281/zenodo.18109760

Smawfield, M. L. Temporal Equivalence Principle: Suppressed Density Scaling in Globular Cluster Pulsars. Preprint v0.9 (Caracas). Zenodo. DOI: 10.5281/zenodo.18165798

Smawfield, M. L. The Cepheid Bias: Resolving the Hubble Tension. Preprint v0.10 (Kingston upon Hull). Zenodo. DOI: 10.5281/zenodo.18209702

Smawfield, M. L. Temporal Equivalence Principle: A Unified Resolution to the JWST High-Redshift Anomalies. Preprint v0.7 (Kos). Zenodo. DOI: 10.5281/zenodo.19000827

Smawfield, M. L. Temporal Equivalence Principle: Temporal Shear Recovery in Gaia DR3 Wide Binaries. Preprint v0.6 (Kilifi). Zenodo. DOI: 10.5281/zenodo.19102061

Data Availability & Reproducibility

Cross-corpus theory registry (TEP-EFA ↔ manuscript series)

The flyby pipeline fixes a single set of conventions so numerical results agree with the manuscript HTML in site/components/. The following table is the authoritative map between this repository and the numbered theory papers under manuscripts/ (Paper 0 = Jakarta, Paper 6 = UCD, Paper 10 = Caracas COS, Paper 12 = JWST Kos, etc.).

Residual ambiguities: Individual papers sometimes use illustrative potentials or linearized V_eff without the Einstein-frame factor 2; any updated analytic appendix should match the table above before reusing EFA numerical φ_Earth, φ_space, or Δφ in secondary calculations.

Repository & Code

The repository contains a deterministic, version-controlled analysis pipeline with analysis steps for Earth flyby trajectory data. All steps are orchestrated by scripts/run_all.py with comprehensive logging.

Repository Structure

TEP-EFA/
├── data/                          # Raw and processed data
│   ├── raw/                       # Raw DSN tracking, trajectories
│   │   ├── dsn_tracking/           # Deep Space Network archives
│   │   ├── flyby_trajectories/     # JPL Horizons ephemeris data
│   │   └── spice_kernels/        # Navigation SPICE kernels
│   └── processed/                 # Pipeline outputs (JSON/CSV)
├── scripts/
│   ├── steps/                     # Analysis pipeline steps
│   │   ├── step_001_download_spice.py
│   │   ├── step_002_spice_to_json.py
│   │   ├── step_003_archival_data_mining.py
│   │   ├── step_004_jpl_horizons_fetch.py
│   │   ├── step_005_dsn_data_ingestion.py
│   │   ├── step_006_dsn_framework.py
│   │   ├── step_007_tep_model.py
│   │   ├── step_008_fitting.py
│   │   ├── step_009_variance_analysis.py
│   │   ├── step_010_tep_first_principles.py
│   │   ├── step_011_trajectory_integration.py
│   │   ├── step_012_od_filter_simulation.py
│   │   ├── step_013_cross_validation.py
│   │   ├── step_014_sensitivity_analysis.py
│   │   ├── step_015_hierarchical_bayesian.py
│   │   ├── step_016_gnss_validation.py
│   │   ├── step_017_plasma_modulation.py
│   │   ├── step_018_space_weather.py
│   │   ├── step_019_3d_field_integration.py
│   │   ├── step_020_plasma_environment_reconstruction.py
│   │   ├── step_021_mission_specific_od_absorption.py
│   │   ├── step_022_atmospheric_drag_simulation.py
│   │   ├── step_023_thermal_recoil_modeling.py
│   │   ├── step_024_systematic_error_monte_carlo.py
│   │   ├── step_025_corrected_uncertainty.py
│   │   ├── step_026_stable_model_comparison.py
│   │   ├── step_027_claim_consistency_audit.py
│   │   ├── step_028_dsn_processing.py
│   │   ├── step_029_read_trk234.py
│   │   ├── step_030_juno_reanalysis.py
│   │   ├── step_031_pds_search.py
│   │   ├── step_032_tep_suppression.py
│   │   ├── step_033_iri_trajectory_profile.py
│   │   ├── step_034_covariant_holonomy.py
│   │   ├── step_035_cross_corpus_export.py
│   │   ├── step_036_final_report.py
│   │   ├── step_037_visualizations.py
│   │   ├── step_038_extract_3d_vectors.py
│   │   ├── step_039_flyby_prediction_table.py
│   │   ├── step_040_cosmographic_shear.py
│   │   ├── step_041_envelope_heuristic_sensitivity.py
│   │   ├── step_042_time_resolved_cosmography.py
│   │   ├── step_043_clock_sector_od.py
│   │   ├── step_044_lense_thirring.py
│   │   └── step_anderson_audit.py
│   ├── utils/                     # Utility functions
│   └── build_markdown.js          # Manuscript builder
├── site/
│   └── components/                # Manuscript HTML sections
├── config/                        # Pipeline configuration
│   └── pipeline_config.json
├── logs/                          # Per-step execution logs
├── requirements.txt               # Python dependencies
├── README.md                      # Documentation
└── LICENSE                        # CC-BY-4.0

Data Provenance

Pipeline Architecture

The analysis pipeline comprises 42 deterministic steps organized into logical groups. Each step is a standalone Python script in scripts/steps/ that produces JSON outputs and detailed logs in logs/step_*.log.

Complete Step Inventory & Runtime

Total Runtime Summary

Reproduction Instructions

Quick Start (Full Reproduction)

# 1. Clone repository
git clone https://github.com/matthewsmawfield/TEP-EFA.git
cd TEP-EFA

# 2. Install dependencies
pip install -r requirements.txt

# 3. Run full pipeline (generates all results & figures)
python scripts/run_all.py

# 4. Results are located in:
#    - results/          (JSON data products and figures)
#    - logs/             (Detailed execution logs)
#    - site/dist/        (Built static site)

System Requirements

Key Analysis Outputs

• results/step003_archival_flyby_catalog.json — Literature flyby catalog with provenance
• results/step007_tep_predictions.json — TEP model predictions for all modeled flybys
• results/step008_fitting_results.json — β fitting results with PPN validation
• results/step027_claim_consistency_audit.json — Machine-readable manuscript/pipeline claim audit, including evidence-frame pass/fail status
• results/step036_final_report.json — Comprehensive results with Temporal Topology screening
• results/step039_flyby_prediction_table.json — Per-flyby raw pooled-β classification (post-OD columns withheld until Step 021 supplies $F_{\rm OD}$)
• results/step032_tep_suppression_analysis.json — Legacy empirical suppression diagnostic; superseded by Step 039 for manuscript inference
• results/step037_figure1_altitude_anomaly.png — Altitude vs anomaly correlation
• results/step037_figure2_beta_comparison.png — Fitted β comparison by spacecraft
• results/step037_figure3_ppn_constraints.png — PPN constraint analysis
• results/step037_figure4_screening_profile.png — Temporal Topology profile

Log Files

Each step produces detailed logs:

• logs/pipeline.log — Master pipeline execution log
• logs/step_*.log — Individual step logs

Software Dependencies

All dependencies are specified in requirements.txt.

Validation & Testing

The pipeline includes comprehensive validation:

• Bootstrap Resampling: n=10,000 iterations for uncertainty quantification
• Leave-One-Out Cross-Validation: Tests robustness against single-flyby exclusion
• Heterogeneity Assessment: Cochran's Q and I² statistics for model scatter
• GNSS clock correlation: The GNSS atomic clock correlation analysis provides a cross-consistency check with the GNSS/UCD terrestrial calibration on the transition radius ($R_{\rm sol} \approx 4146$ km). This shared-input dependency is disclosed explicitly to avoid circular claims of independent validation.
• Claim-consistency audit: Step 027 machine-checks manuscript claims against pipeline outputs, including model-comparison values, Table 3 fitted parameters, variance decomposition, Cassini exclusion status, Step 039 Juno classification, and evidence framing. The audit fails if the manuscript reverses the TEP-vs-Null comparison, omits the random-effects uncertainty, or promotes Juno from a deterministic warning case to an uncertainty-aware falsification.
• Uncertainty discipline: Formal inverse-variance summaries, random-effects scatter, the fractional-floor model comparison, and published-uncertainty plus constant-floor stress tests are reported as separate layers rather than blended into a single headline number.
• Automated smoke tests: pytest (see tests/ and pytest.ini) checks repository layout and step logger conventions.

Reproducibility Checklist

Data Availability Statement

Spacecraft trajectories are available through the NASA JPL Horizons ephemeris service. Literature anomaly values are from Anderson et al. (2008) and companion publications. Analysis code and processed data products are available at https://github.com/matthewsmawfield/TEP-EFA with archived DOI at 10.5281/zenodo.19454862.

Raw DSN tracking products may be obtained from the NASA Deep Space Network and the Planetary Data System following institutional access procedures; per-mission pointers appear under data/raw/dsn_tracking/<mission>/DOWNLOAD_INSTRUCTIONS.txt. The present manuscript release does not bundle perigee-matched Level-1 TRK archives: Steps 005–006 and 028–031 implement ingest and audit only, and Step 030 remains inconclusive until such products are added. Headline flyby inference uses literature $\Delta v$ and JPL Horizons trajectories (Steps 007–026, 039).