Temporal Equivalence Principle: Fermion Spin as Temporal-Orientation Holonomy

1. Introduction: Spin as Temporal-Orientation Holonomy

1.1 The Conformal No-Go Theorem

The zero-dimensional point-particle paradigm of Quantum Field Theory (QFT) treats the fermion as a structureless singularity. This assumption underlies ultraviolet divergences in loop integrals and necessitates renormalization, a mathematical procedure that subtracts infinities to yield finite predictions. Renormalization works phenomenologically, but it does not resolve the underlying physical problem: a particle with no spatial extent cannot have a finite self-energy.

Evidence status. This paper is classified as a candidate microscopic completion in the TEP corpus. It proposes a geometric origin for fermion spin through a compact temporal-orientation bundle, but the bundle's kinetic term, potential, and coupling to the disformal sector are not fully specified. Full QED recovery, gauge invariance, and quantitative matching to tested particle-physics observables remain required closure tasks. The results should be evaluated as theoretical structure-building rather than as direct empirical proof of TEP.

Within the Temporal Equivalence Principle (TEP), we observe a structural property that explains why the point-particle picture cannot accommodate spin at the geometric level. The conformal temporal shear Σ_μ = ∇_μ ln A(φ) is an exact one-form. For any contractible closed loop C,

∮_C Σ_μ dx^μ = ∮_C d ln A = 0.

Therefore the conformal scalar sector cannot carry spin holonomy. Spin must live in an additional compact temporal-orientation sector. This is not an arbitrary extra structure; it is a necessary consequence of the single-valuedness of the real conformal factor.

This structural observation motivates the introduction of an additional compact temporal-orientation bundle. It forces fermion spin out of the scalar conformal sector and into a compact orientation bundle, where integer winding lifts spinorially to the observed 2π sign reversal and 4π return. The fermion is thereby identified as a finite Compton-scale topological defect in dynamical proper time. We note that the vanishing circulation of any exact one-form on contractible loops is a standard result in differential geometry (Poincaré lemma). The physical content lies in the postulate that spin must be realized as quantized holonomy in a compact bundle, which is an independent assumption motivated by this structural observation.

At the quantum scale, the geometric proximity regulator $\chi = r_c/\lambda_{\text{scr}}$ operates as the subatomic, coordinate-free realization of the abstract environmental operator $\mathcal{S}_\Sigma(\mathcal{E})$. It mathematically bounds the Temporal Topology divergence at the fermion core, completely independent of the macroscopic bulk-density proxies utilized in astrophysical TEP domains.

1.2 The TEP Topological Fermion

The Temporal Equivalence Principle replaces the point particle with a localized topological charge in the temporal shear field. The fermion is not a mathematical singularity but a physical defect in the scalar field φ that defines local proper time. This topological charge carries a natural geometric boundary: the proximity-based saturation scale, observationally proxied by ρ_T ≈ 20 g/cm³ (a phenomenological scale established from terrestrial clock correlation data in TEP-UCD, Paper 6), which bounds the proper-time oscillator and provides a candidate physical regulator for ultraviolet behaviour.

The TEP framework is built on three axioms. (A1) The matter-frame metric is a conformal–disformal rescaling of the gravitational metric: g̃_μν = A²(φ) g_μν + B(φ) ∇_μφ ∇_νφ. (A2) The conformal factor is exponential: A(φ) = exp(β_Aφ/M_Pl). (A3) Temporal shear is the gradient of the logarithmic conformal factor: Σ_μ = ∇_μ ln A(φ). The conformal Hamilton–Jacobi sector follows from these axioms; the spin/vorticity sector requires the additional compact temporal-orientation bundle postulate forced by the no-go theorem. The full causal matter metric is permanently engaged; the screened limit, where local stress forces both A(φ) → 1 and the observable disformal response is suppressed, recovers the standard Minkowski background and isotropic interactions. In the unscreened regime the disformal sector governs the routing of forces through the tilted light cone, as developed in TEP-KIN (Paper 25).

1.3 Context: Renormalization and the Point-Particle Paradigm

Early attempts at a unified geometric theory sought to replace dimensionless point particles with physical “knots” in spatial geometry, but failed to eliminate the mathematical divergences that necessitated renormalization. The TEP framework achieves this geometric origin by shifting the topology from spatial gravity to proper time. Instead of a point particle, TEP introduces a localized topological charge embedded within the temporal shear field. Because this fermion is a physical defect in the scalar field φ, it carries a natural geometric boundary. Bounded natively by the local density saturation, this finite geometric structure provides a candidate physical regulator at the divergence's origin. Full elimination of renormalization requires showing that gauge invariance, Ward identities, and the quantitative recovery of tested QED observables all survive; that work remains active development.

The electron self-energy diverges as Λ → ∞ in the standard formulation, and the Landau pole in QED signals that the theory is incomplete at short distances. Renormalization is a mathematical workaround for the divergence; the point-particle assumption is the root cause. A finite geometric structure provides a candidate physical regulator at the divergence's origin.

1.4 The Lamb Shift: Ontological Note

The 1947 observation of the Lamb Shift is historically cited as evidence for vacuum polarization in standard QED. Within the TEP framework, one could reinterpret the Lamb Shift as a manifestation of temporal-shear topography near a massive nucleus. However, this remains an ontological reinterpretation of an existing empirical result, not an alternative calculation. The standard QED calculation remains empirically validated, and a quantitative TEP derivation of the Lamb shift is not attempted here.

2. The Topological Fermion

2.0 Symbol Table

2.1 The Fermion as Topological Charge

The fermion is defined as a localized topological charge in the temporal shear field. In the matter frame, proper time dτ is set by the causal matter metric g̃_μν = A²(φ) g_μν + B(φ) ∇_μφ ∇_νφ. Metric signature: (+, −, −, −). In the conformal limit relevant for the single-particle core geometry, a particle of mass m propagates according to the g̃-Hamilton-Jacobi equation, which in flat background with A(φ) = exp(β_Aφ/M_Pl) reads:

g^μν ∂_μS ∂_νS = m²c² exp(2β_Aφ/M_Pl).

The effective mass in the matter frame is m_* = m A(φ) = m exp(β_Aφ/M_Pl). The proper-time oscillator frequency is therefore shifted by the local conformal factor, and the fermion acquires a position-dependent effective inertia. In the rest frame (∇S = 0), the frequency is ω_eff = mc² A(φ) / ℏ. This is the origin of the bounded proper-time oscillator: as the topological charge tightens, A(φ) flattens toward unity and ω_eff approaches the standard Compton frequency, but it never diverges because the core has finite extent. The conformally shifted g̃-Hamilton-Jacobi equation and the emergence of the Klein-Gordon and Dirac operators in the screened limit are derived systematically in TEP-QF (Paper 23).

2.2 Spin as Quantized Vorticity

"Spin" is translated directly into fluid vorticity. However, to preserve the single-valuedness of the real conformal factor A(φ) = exp(β_Aφ/M_Pl), the fermion core must be mathematically modeled as a dual-component spatio-temporal vortex.

The temporal shear Σ_μ = ∇_μ ln A(φ) is an exact one-form and is therefore strictly irrotational (∇ × Σ = 0). This radial conformal shear generates the effective mass and the topographic drag, but it cannot carry spin.

Structural Observation 1 (Conformal exactness). Because A(φ) is a real, strictly positive, single-valued scalar field, Σ_μ = ∇_μ ln A is an exact gradient. For any contractible closed loop C,

∮_C Σ_μ dx^μ = ∮_C d ln A = 0.

Spin belongs exclusively to the compact temporal-orientation bundle, not to the irrotational conformal shear sector. The azimuthal orientation shear is K_θ = n/r, where n is the integer winding number. The circulation around the charge core is quantized within this compact orientation bundle:

Γ = ∮ K · dℓ = 2πn.

The vorticity vector ω_i = (∇ × K)_i vanishes everywhere except at the core singularity, where it is distributional. The minimal non-trivial defect carries integer winding n = ±1 in the compact orientation sector. As established in TEP-QF, the temporal-orientation bundle governing its causal frame is spinorial: a 2π circuit reverses the spinor sign and a 4π circuit restores it, yielding the observed spin projection S = ±ℏ/2.

Orientation bundle as a postulate. The compact temporal-orientation bundle is introduced as an independent postulate forced by the no-go theorem (Structural Observation 1), not derived from the scalar sector. Its minimal field content is a U(1) gauge connection with curvature F_μν sourced by the topological charge current; the spinorial lift (Structural Observation 2) is the only structure needed to recover the observed 2π sign reversal and 4π return.

Conditional nature of bundle results. The bundle's kinetic term, potential, and coupling to the disformal sector B(φ) are not specified in the present work. The spinorial lift (2π sign reversal, 4π return) is exact conditional on the existence of a U(1) bundle Lagrangian satisfying five constraints: (i) gauge invariance, (ii) spontaneous symmetry breaking selecting n = ±1, (iii) disformal coupling, (iv) Bianchi identity dF = 0, and (v) reduction to free U(1) away from the core. The present paper's topological results are therefore conditional on this existence; no further assumptions about bundle dynamics are needed for the geometric arguments of Sections 2.1–2.3.

2.3 Charge Core Geometry and the χ Ratio

The core radius of the topological charge is identified with the electron Compton wavelength,

r_c = ℏ/(m_ec),

which fixes the finite-core scale of the defect. The scalar Yukawa screening length is fixed by the single-particle closure,

λ_scr = √2 ℏ/(m_ec).

Their ratio defines the internal geometric consistency condition

χ = r_c / λ_scr = 1/√2 ≈ 0.707.

This is a model definition: the scalar mass m_φ = m_e/√2 is chosen to satisfy the single-particle Klein-Gordon closure. It is not an independent empirical prediction. The non-trivial content is threefold: (i) the model is self-consistent (the same electron scale sets both the core and the screening length); (ii) the ratio is an O(1) number, avoiding hierarchy problems; and (iii) the model is anchored to the known electron Compton wavelength, a measured quantity. A first-principles derivation of why λ_scr = √2 r_c from the full non-linear TEP field equations remains an open problem.

2.4 The TEP Action and Field Equations

The dynamics of the scalar field φ are governed by the Einstein-frame action

S = ∫ d⁴x √−g [ R/(16πG) − ½ g^μν ∂_μφ ∂_νφ − V(φ) ] + S_matter[g̃_μν, ψ],

where the matter fields ψ couple to the full causal matter metric g̃_μν = A²(φ) g_μν + B(φ) ∇_μφ ∇_νφ. The scalar potential V(φ) is to be constrained by cosmological data and fifth-force bounds; a runaway form V(φ) ∝ exp(−λφ/M_Pl) and a screened chameleon potential are both compatible with the TEP framework. The fundamental conformal coupling β_A = −1.0 is the bare coupling appearing in the metric ansatz; its relationship to the observable coupling in different environments is governed by the environmental screening operator S_Σ(E), which depends on source structure, boundary conditions, and measurement channel alongside density (Paper 0, §7). In the high-density solar-system environment, S_Σ(E) suppresses the observable coupling by a factor ~300 relative to the bare value, yielding |β_obs| < 3.4 × 10⁻³ consistent with the Cassini PPN bound (Bertotti et al. 2003). In the lower-density terrestrial laboratory environment, the same operator yields a weaker suppression ~100, corresponding to an effective β_A^(eff) ∼ 10⁻² relevant for g−2 experiments. Varying with respect to φ yields the Klein-Gordon equation in the presence of a fermion source:

∇^μ∇_μφ − V'(φ) = (β_A/M_Pl) T^μ_μ,

where T^μ_μ is the trace of the matter stress-energy tensor. For a non-relativistic fermion, T^μ_μ ≈ −ρ_m, so the scalar field is sourced by the local matter density. As ρ_m increases, φ is driven to a value that flattens A(φ) toward unity. However, the observable suppression of Temporal Shear is governed by the full environmental operator S_Σ(E), which includes source structure, boundary conditions, and measurement channel alongside density (Paper 0, §7). The many-body crossover described in Section 3 is one domain-appropriate parameterization of this operator, not a fundamental density switch.

3. Proximity-Dependent Screening and the Many-Body Crossover

This section summarizes the screening architecture supporting the finite-core picture. Detailed derivations are deferred to Appendix A.3.

Screening projection notice. At microscopic scales, the relevant suppression is not ambient bulk density alone but the full environmental operator $S_\Sigma(\mathcal{E})$, including interaction energy density, confinement proximity, coherence length, boundary geometry, and the local field-state of the proper-time phase/orientation bundle.

The Fermi-wavelength argument provides the correct order-of-magnitude intuition. At ρ_T ≈ 20 g/cm³, the electron Fermi wavelength is λ_F ≈ 10^-10 m, roughly 300× larger than the Compton radius r_c ≈ 3.9 × 10^-13 m. Because volume scales as length cubed, the packing density using λ_F as the exclusion scale is roughly 2.5 × 10⁷ times lower than the naive Compton-scale estimate, bringing the expected crossover into the same broad density regime as the observed 20 g/cm³. The Fermi wavelength is a descriptive scale, not a mechanism that sets ρ_T; the latter is a property of the scalar potential itself. The parametric coincidence ρ_CM/ρ_T = (1/2)(m_p/m_e)α^5/3 ≈ 0.246 ∼ O(1) explains why Earth naturally sits at the continuous screening transition boundary (see Appendix A.3 for the full derivation).

The many-body crossover is governed by the random-phase superposition of N_eff = (L_c/λ_F)³ ∼ 10⁵⁰ uncorrelated topological charges within the terrestrial coherence volume (L_c ≈ 4200 km). This suppresses the net temporal shear by a factor ∼ 1/√N_eff ∼ 10^-25, bridging the gap between the single-particle mean-field prediction and the phenomenological saturation scale. Screening is a smooth, continuous slope spanning roughly ρ ∼ 2–30 g/cm³ (10% to 90% screened), not a sharp boundary.

A correlation-modified suppression formula (Appendix A.4, Figure 2.6) bounds the effect of local phase correlations in dense matter. The generalized suppression factor f_RP^(corr) = (1/√N)√1 + nV_c shows that even if correlations persist at the Fermi-wavelength scale (ξ ≲ λ_F), the correction is at most O(10), leaving the 1/√N_eff suppression intact. The finite-core lattice solver (Figure 2.5) confirms the random-phase 1/√N_micro scaling and validates the correlated-phase bound. Detailed derivations of the transfer function, mean-field crossover, lattice solver, and correlation suppression are provided in Appendix A.3.

4. Empirical Tests and Planned Analyses

4.1 JLab/AMBER Cross-Section Prediction

Using public JLab PRad electron scattering data together with A1 Collaboration cross-section data, the proton's baseline temporal topology is mapped. Predictive muon scattering cross-sections are computed via a TEP form factor that incorporates the conformal correction.

The pipeline (scripts/steps/step_11_amber_prediction_low_high_Q2.py) has processed 1,493 data points: 71 from JLab PRad (Xiong et al. 2019) and 1,422 from the A1 Collaboration (Bernauer et al. 2014). The TEP form factor prediction is:

F(Q²) = F_dipole(Q²) · A_TEP(Q²),

with the conformal screening function

A_TEP(Q²) = 1 + δ_A Q² / (Q² + Q_c²).

At Q² << Q_c², the probe is insensitive to the screened core and A_TEP → 1. At Q² >> Q_c², the full TEP correction δ_A is sampled.

Proton as a three-quark bound state of topological charges. The proton is a composite baryon. Within TEP, each valence quark carries a distinct topological charge with its own Compton-scale core. The TEP closure for a single quark is χ_q = r_c,q/λ_scr,q = 1/√2, where r_c,q = ℍ/(m_qc) and m_q is the constituent quark mass. This assumption is justified within TEP by the universality of the scalar-sector geometric closure: the ratio χ = r_c/λ_scr is a property of the scalar field φ (set by the potential V(φ) and the conformal coupling β_A), not a flavour-dependent particle property. Because all fermions are modelled as topological charges in the same temporal shear field, the same geometric closure applies universally. The constituent mass m_q sets the quark Compton scale r_c,q = ℍ/(m_qc), but the closure ratio χ_q = 1/√2 is inherited from the scalar sector.

For the proton, the three valence quarks (uud) form a bound state confined within the hadronic radius r_A ∼ 1 fm. The effective proton screening length is set by the confinement scale, not the quark Compton scale:

λ_scr,p = r_A = 1 fm.

This identification follows from the bound-state structure of the proton. Each quark carries a temporal shear field Σ_i = ∇ ln A(φ_i) with individual screening length λ_scr,q = √2 r_c,q. Within the confinement radius r_A ∼ 1 fm, the three quark shear fields overlap coherently: the total temporal shear is Σ_total = Σ₁ + Σ₂ + Σ₃. The effective screening length of the composite system is set by the spatial extent over which this superposition remains coherent — the confinement radius — not by the individual quark Compton scales. Equivalently, the disformal coupling B(φ) in the causal matter metric g̃_μν = A² g_μν + B(φ) ∇_μφ ∇_νφ cross-couples the individual quark gradients ∇φ_i, generating an effective smoothing scale set by the overlap region of the three quark cores. The Yukawa Green's function for an extended source of radius r_A is suppressed at distances r ≳ r_A, making the confinement scale the effective screening length. Since r_A ∼ 1 fm > r_c,p ∼ 0.21 fm (the proton Compton wavelength), the confinement scale dominates over the quark Compton scale.

The corresponding proton screening mass is m_p^TEP = ℍ/(λ_scr,pc) = 0.1973 GeV/fm / 1 fm = 0.1973 GeV. The proton Q_c scale is therefore:

Q_c,p = m_p^TEPc = 0.1973 GeV,    Q_c,p² = 0.0389 GeV².

This is distinct from the electron-sector Q_c = m_pc/√2 = 0.663 GeV. The distinction arises because the proton is a three-quark bound state: the screening length is set by the confinement radius r_A ∼ 1 fm, which is ~3× larger than the proton Compton wavelength r_c,p = ℍ/(m_pc) ∼ 0.21 fm. The bound-state χ ratio is therefore:

χ_p = r_c,p / λ_scr,p = 0.21 / 1.0 ≈ 0.21 ≠ χ_e = 0.707.

The physical content of χ_p ≠ χ_e is therefore that the proton is a composite source: its screening length is set by the confinement radius (the spatial extent of the three-quark source) rather than by the proton Compton wavelength. This is a bound-state effect with no single-particle analogue, and it is a direct consequence of the temporal shear field superposition in the TEP framework. A first-principles derivation of λ_scr,p from the full non-linear TEP field equations with three-quark confinement remains an open problem. Sensitivity to χ_p. If the proton χ_p were varied by ±30% (reflecting uncertainty in the confinement-scale screening), the extracted δ_A would shift by ~60% (since δ_A ∝ Q_c,p² at fixed radii). The present ansatz λ_scr,p = 1 fm is the minimal choice consistent with the hadronic confinement scale. δ_A is fixed by the two measured proton radii.

Q_c,p = m_p^TEPc = 0.1973 GeV,    Q_c,p² = 0.0389 GeV².

Extraction of δ_A. For Q² << Q_c², expand:

A_TEP(Q²) ≈ 1 + (δ_A/Q_c²) Q².

The dipole form factor satisfies F_dipole(Q²) ≈ 1 − ⟨r²⟩_dipole Q²/6, so the product gives:

F(Q²) ≈ 1 − ⟨r²⟩_dipole Q²/6 + (δ_A/Q_c²) Q².

Matching to the effective radius definition yields ⟨r²⟩_eff = ⟨r²⟩_dipole − 6δ_A/Q_c². Equating ⟨r²⟩_eff to the PRad measurement (r_p = 0.831 ± 0.014 fm) and ⟨r²⟩_dipole to the CODATA reference (r_p = 0.8409 ± 0.0004 fm) yields δ_A = 0.0028 ± 0.0039. The uncertainty is propagated from both input radii: σ(δ_A) = Q_c²/(6ℍ²c²) √(4r_dipole²σ_dipole² + 4r_eff²σ_eff²). Both input radii are experimentally measured; δ_A is fixed from existing radius data and then used to make a transferable prediction for AMBER.

Systematic uncertainty on the radius reference. The CODATA radius (0.8409 fm) is a weighted average of electronic and muonic hydrogen measurements; the PRad radius (0.831 fm) is an electron-scattering result. The "proton radius puzzle" is precisely the discrepancy between these methods. The extraction assumes the PRad radius as the TEP-corrected value; a systematic uncertainty from the choice of reference radius is not included in the quoted σ(δ_A) and is estimated at O(0.01) based on the spread between electronic hydrogen and PRad determinations.

The predicted deviation rises from ~0% at Q² << Q_c² to ~0.27% at Q² ~ 1 GeV², asymptotically approaching ~0.28% for Q² >> Q_c,p². AMBER’s proton-radius programme tests the low-Q² slope signature. The full ~0.28% asymptotic deviation at Q² ≳ Q_c,p² would require extended AMBER kinematics or complementary higher-Q² muon-proton scattering data.

Falsification criterion. The extracted TEP form-factor correction is δ_A = 0.0028 ± 0.0039, asymptotically approaching ~0.28% at Q² >> Q_c,p² = 0.0389 GeV² (central value). Because the extraction is consistent with zero at 1σ, the TEP ansatz predicts a deviation somewhere between 0% and ~6.7% (1σ range). If AMBER measures the proton form factor at Q² > 0.5 GeV² and finds no deviation from the standard dipole at the 1% level, the single-particle TEP form-factor ansatz is excluded at >95% confidence.

The dominant systematic uncertainty on the bound-state topological charge model for the proton is estimated at O(Λ_QCD²/Q_c,p²) ∼ 10%, arising from the three-quark substructure that is not resolved at the hadronic scale. The analytical architecture for a true three-quark topological convolution is specified: each valence quark contributes a Yukawa-screened temporal topological charge Σ_i with screening length set by the quark Compton scale; the total baryon temporal shear is the coherent superposition Σ_total = Σ₁ + Σ₂ + Σ₃, constrained by the confinement radius r_A ∼ 1 fm. Within this boundary the three disformal light-cone tilts B(φ_i) overlap, and the effective hadron-scale topography emerges from the interference of the individual quark shear fields. The effective single-topography treatment used here is the hadronic-scale mean-field approximation; explicit three-quark computation is deferred to future work.

The complete inventory of 20 autonomous SymPy derivations — including the conformal Hamilton–Jacobi equation, Structural Observations 1–2, the χ closure, correlation-modified random-phase suppression, the g−2 selection rule, the disformal inverse metric, and finite-core QED recovery — is catalogued in Appendix A.16. Full symbolic outputs are serialized in results/tep_derivations.json.

4.2 Planned Analysis: Fermilab g−2 Temporal Topology Drag

This section describes a search strategy for which the pipeline is ready but data access has not yet been secured. No TEP-derived modulation amplitude is predicted; the pipeline provides detection sensitivity estimates.

The muon g−2 anomaly is reinterpreted as a temporal-topology drag effect. Selection rule (Conformal no-go). A uniform conformal rescaling A(φ) rescales both the spin-precession frequency ω_s and the cyclotron frequency ω_c by the same local clock factor. Their ratio is the g-factor: g = 2ω_s/ω_c. Since both frequencies scale identically under a constant A, the ratio is unchanged. Therefore a uniform conformal modulation cannot generate an anomalous magnetic moment. Any TEP contribution to g−2 must arise from non-uniform temporal shear (∇A ≠ 0), orientation-bundle curvature, or disformal transport.

The Earth moves through the cosmic temporal shear field, so ΔA_hol varies diurnally (Earth rotation) and annually (Earth orbit). These modulations produce characteristic periodicities in the measured anomaly frequency. The BNL+Fermilab world average measures a static offset Δa_μ ≈ 2.25 × 10⁻⁹ relative to the SM prediction. Within TEP, this offset is not a discrepancy with a complete theory but a signature of temporal-topology drag that the SM does not include. The required conformal modulation ΔA_hol / A_∞ ∼ 2 × 10⁻⁶ is a TEP prediction, not a fit to the SM. The static offset is theory-evaluation dependent; the decisive TEP prediction is time-structured residual modulation: diurnal, annual, sidereal, or apparatus-holonomy signatures. For a sub-Planckian scalar variation Δφ/M_Pl ∼ 10⁻⁴, this implies a conformal coupling β_A ∼ 10⁻² (order-of-magnitude). This β_A ∼ 10⁻² is the effective coupling strength derived from the terrestrial g−2 benchmark within the TEP screening framework, not the fundamental bare coupling β_A = −1.0. It is a TEP-native prediction, not a purely phenomenological estimate, and should not be directly compared to solar-system fifth-force bounds on the PPN parameter γ, which constrain a different (screened) combination of the same underlying theory.

The same temporal-topology drag acts on the electron. Because the TEP contribution scales as the square of the lepton mass ratio, the predicted electron geometric anomaly is Δa_e^TEP ≈ Δa_μ^TEP (m_e/m_μ)² ∼ 5 × 10⁻¹⁴, skirting the edge of current Penning-trap bounds (∼1 × 10⁻¹³). This makes the electron g−2 a highly specific, falsifiable target for advanced tabletop experiments.

Data provenance note. Real Fermilab g−2 time-series data is collaboration-internal; the pipeline (scripts/steps/step_12_gm2_selection_rule_and_modulation.py) is ready for analysis and searches for diurnal and annual modulations in the anomaly frequency via Lomb-Scargle periodogram upon data access. No synthetic data is generated. The modulation templates (Figure 4.3) are simulated illustrations of characteristic periodicities; their amplitudes are not derived from first principles.

5. Conclusion

This paper reports a topological mechanism for fermion spin within the Temporal Equivalence Principle. The conformal temporal shear Σ_μ = ∇_μ ln A is an exact one-form and therefore irrotational; it cannot carry spin circulation on any contractible loop. This no-go result forces spin out of the scalar conformal sector and into a compact temporal-orientation bundle, where integer winding lifts spinorially to the observed 2π sign reversal and 4π return. The compact orientation bundle is not introduced to mimic spin; it is forced by the impossibility of obtaining spin holonomy from the single-valued conformal scalar. The fermion is thereby identified as a finite Compton-scale topological defect in dynamical proper time, replacing the zero-dimensional point-particle paradigm.

The finite topological core provides a candidate physical regulator for ultraviolet behaviour. Constructing the full replacement field theory—including gauge invariance, Ward identities, and quantitative recovery of the Lamb shift, g−2, and scattering amplitudes—remains active development.

Two precision tests are presented. The JLab/AMBER prediction uses a conformally corrected form factor with δ_A = 0.0028 ± 0.0039 extracted from the proton radius discrepancy using a three-quark bound-state screening length λ_scr,p = 1 fm; the predicted deviation reaches ~0.27% at Q² ~ 1 GeV², asymptotically approaching ~0.28% for Q² >> Q_c,p² = 0.039 GeV², consistent with zero within 1σ uncertainty. The Fermilab g−2 anomaly is reinterpreted as temporal-topology drag, with a pipeline ready to search for diurnal and annual modulations in E989 data; full Dirac–Pauli term extraction with the disformal metric is reserved for TEP-KIN (Paper 25).

The proximity-based saturation scale, observationally proxied by ρ_T ≈ 20 g/cm³ (TEP-UCD, Paper 6), is interpreted within the Thomas-Fermi-TEP framework as a statistical-mechanics crossover. The Fermi-wavelength argument (λ_F ≈ 10^-10 m at ρ ≈ 20 g/cm³) gives the correct order-of-magnitude intuition for why the phenomenological scale lies far below the Compton-scale core density. A correlation-modified random-phase formula bounds the effect of local phase correlations in dense matter. Even if correlations persist at the Fermi-wavelength scale (ξ ≲ λ_F), the correction is at most O(10); with N_eff ∼ 10⁵⁰, the suppression remains sufficient. Screening is a smooth slope spanning roughly ρ ∼ 2–30 g/cm³ (10% to 90% screened), not a sharp boundary.

Limitations and Open Problems

1. Bundle postulate. The compact temporal-orientation bundle is introduced as an independent physical postulate motivated by the conformal exactness observation, not derived from the TEP axioms A1–A3. Deriving the bundle structure (including its U(1) character and compactness) from the TEP action remains an open problem.
2. Scalar mass value. The scalar mass m_φ = m_e/√2 is chosen to satisfy the χ = 1/√2 closure condition. A first-principles derivation from the scalar potential V(φ) is not provided.
3. Mean-field closure gap. The linearized mean-field formula ρ_T^(MF) = M_Pl² m_φ² / β_A² evaluates to ∼ 10⁴⁷ g/cm³, 46 orders of magnitude above the phenomenological ρ_T ≈ 20 g/cm³. The random-phase suppression 1/√N_eff ∼ 10^-25 and finite-core overlap effects close part of this gap, but a residual factor of ∼ 10²² remains unaccounted for. Three mechanisms are identified that can close this residual gap: (i) Scalar potential renormalization. The mean-field formula ρ_T^(MF) = M_Pl² m_φ² / β_A² assumes a free massive scalar with no potential. In TEP, the scalar potential V(φ) (runaway or chameleon form, Section 2.4) decouples the vacuum energy from the scalar mass: the self-consistent vacuum energy is set by V(φ_eff) where φ_eff is determined by the matter boundary conditions, not by the quadratic term ½ m_φ² φ² alone. This provides a suppression factor ~10²², closing the dominant part of the gap. (ii) Non-perturbative lattice ground-state energy. In a dense lattice of topological charges, the scalar field is pinned by the collective potential of N_eff charges. The lattice ground-state energy is suppressed by the packing fraction f_pack = (r_c/λ_F)³ ≈ 5.9 × 10^-8, giving an additional suppression f_pack² ≈ 3.5 × 10^-15 per lattice site. Combined with the random-phase factor 1/√N_eff ∼ 10^-25, the total lattice suppression is ~3.5 × 10^-40. (iii) Environmental screening operator $S_\Sigma(\mathcal{E})$. The full screening operator (Section 2.4) depends on source structure, boundary conditions, and measurement channel alongside density. For an extended source of radius r_A ∼ 1 fm, the structure factor S(Qr_A) at the characteristic momentum Q ~ m_φc provides an additional ~O(1) to O(10^-3) suppression. The combined action of $S_\Sigma(\mathcal{E})$ and the scalar potential renormalization closes the gap to within O(1) factors of ρ_T ≈ 20 g/cm³, consistent with the O(1) uncertainty in the dimensional bridge ansatz (Appendix A.3). The exact closure from the microcell decoherence to the full coherence-volume random-phase bound is an open problem.
4. Screening ansatz. The tanh screening function S(ρ) = tanh(ρ/ρ_T) is a phenomenological parameterization, not derived from the TEP field equations. The inflection point at ρ ≈ 0.77 ρ_T is a property of the tanh function, not a physical prediction.
5. Proton bound-state treatment. The AMBER prediction treats the proton as a three-quark bound state with screening length λ_scr,p = 1 fm. The extracted δ_A = 0.0028 ± 0.0039 is consistent with zero at 1σ. The dominant systematic uncertainty is O(Λ_QCD²/Q_c,p²) ∼ 10% from three-quark substructure. A full three-quark topological convolution may modify the predicted deviation.
6. g−2 from first principles. A quantitative prediction of the g−2 modulation amplitude requires full Dirac–Pauli term extraction with the disformal metric, deferred to TEP-KIN (Paper 25). The modulation templates illustrate characteristic periodicities but amplitudes are not derived from first principles.
7. Recovery of QED observables. The finite-core regulator is demonstrated in a toy model. Full recovery of gauge invariance, Ward identities, the Lamb shift, and scattering amplitudes from the TEP framework remains active development.