Temporal Equivalence Principle: A Topological Fermion Model for Spin and the g−2 Anomaly

1. Introduction: The Failure of Renormalization

1.1 The Infinite Point-Mass Catastrophe

Quantum Field Theory relies on a zero-dimensional point-particle paradigm that leads to ultraviolet divergences in loop integrals. Renormalization is a mathematical procedure that subtracts infinities to yield finite predictions. It works phenomenologically, but it does not resolve the underlying physical problem: a particle with no spatial extent cannot have a finite self-energy. 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 eliminates the divergence at its origin.

1.2 The TEP Alternative

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 eliminates the divergence at its origin, removing the need for artificial ultraviolet cutoffs.

The Temporal Equivalence Principle (TEP) 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_c ≈ 20 g/cm³ (a phenomenological scale established from terrestrial clock correlation data in TEP-UCD, Paper 6), which bounds the proper-time oscillator and eliminates the need for artificial ultraviolet cutoffs.

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). All results in this paper are derived from these axioms. 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 The Lamb Shift as Topography

The 1947 observation of the Lamb Shift is historically cited as the primary evidence for vacuum polarization via virtual particles. In the TEP framework, this is reinterpreted as the first empirical topographic map of a steep temporal shear gradient. A massive nucleus generates a dense temporal topology. As the electron vortex orbits, its geometric structure is continuously deformed by its proximity to this temporal shear wall ($\Sigma_\mu$). The Lamb Shift does not measure a boiling vacuum of ghost particles; it measures the macroscopic geometric drag of a fluid vortex navigating a non-isochronous landscape.

2. The Topological Fermion

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 governed by local temporal shear Σ_μ = ∇_μ ln A(φ) = (β_A/M_Pl) ∇_μφ. For a topological charge with azimuthal symmetry in cylindrical coordinates, the shear field is:

Σ_θ = (β_A/M_Pl) (n/r),

where n is the integer winding number of the compact phase/orientation variable associated with the charge core. The real conformal factor A(φ) = exp(β_Aφ/M_Pl) is single-valued but not itself periodic; quantization belongs to the phase/orientation bundle carried by the defect. The circulation around the charge core is quantized:

Γ = ∮ Σ · dℓ = 2πn (β_A/M_Pl).

The vorticity vector ω_i = (∇ × Σ)_i vanishes everywhere except at the core singularity (r = 0), where it is a delta function. This is the topological origin of spin: the minimal non-trivial defect carries integer phase winding n = ±1, while the local orientation bundle is spinorial. A 2π circuit reverses the spinor sign and a 4π circuit restores it, so the observed spin projection is S = ±ℏ/2 even though the scalar/phase winding is integer. The spin-statistics connection arises because exchange of two identical topological charges implements the same non-trivial spinor holonomy, introducing a phase factor of π and enforcing antisymmetry under exchange.

2.3 Charge Core Geometry and the κ Ratio

The core radius of the topological charge is fixed by the electron Compton wavelength, r_c = ℏ/(m_ec), a known quantity from quantum mechanics. The Yukawa screening length of the scalar field, derived from the single-particle energy density, is λ_scr = √2 ℏ/(m_ec). Their ratio defines a geometric consistency condition:

κ = r_c / λ_scr = 1/√2 ≈ 0.707.

The consistency condition κ = 1/√2 is a geometric constraint: the framework predicts both the core radius and the screening length from the scalar field dynamics, and their ratio is fixed by the topology. It anchors the TEP charge geometry to an established quantum scale.

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. 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. Environmental Screening and the Many-Body Crossover

3.1 The Single-Particle Core Density

As the topological charge tightens, the conformal factor A(φ) mechanically flattens, forcing temporal shear to zero at the core. From the Klein-Gordon energy density ρ_φ ∼ (∇φ)²/2 and the relation ∇φ = (M_Pl/β_A) Σ, the single-particle core density evaluates to:

ρ_core ∼ m_e⁴c³ / ℏ³ ∼ 10⁴ g/cm³.

This is white-dwarf-scale density, a direct dimensional estimate from the Compton-wavelength cutoff. It is not the phenomenological saturation scale.

3.2 The Many-Body Saturation Scale and the Fermi-Wavelength Crossover

The saturation proximity scale ρ_c ≈ 20 g/cm³ is the phenomenological scale at which collective many-body screening in bulk matter suppresses observable temporal shear (TEP-UCD, Paper 6). It is determined from terrestrial clock correlation data, not derived from first principles. The goal of this section is to provide a physical interpretation for why this scale is many orders of magnitude below the single-particle core density ρ_core ∼ 10⁴ g/cm³. The naive packing argument failed because it used the wrong length scale: fermions do not pack at their Compton wavelength r_c = ℏ/(m_ec); they pack at their Fermi wavelength λ_F = 2π/k_F, where k_F = (3π²n_e)^1/3 is the Fermi momentum and n_e = (Z/A)ρ/m_p is the electron number density (Z/A ≈ 0.5 for ordinary crustal matter).

For a degenerate electron gas, the Fermi wavelength scales as:

λ_F(ρ) = 2π / (3π² (Z/A) ρ/m_p)^1/3.

At the phenomenological saturation proximity scale ρ_c ≈ 20 g/cm³, the 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 (292)³ ∼ 2.5 × 10⁷ times lower than the naive Compton-scale estimate. This brings the expected crossover into the same broad density regime as the observed 20 g/cm³, though the exact factor of order unity is not predictable from the linearized theory.

The crossover is governed by the Thomas-Fermi-TEP mean-field equation. In a degenerate electron gas, the scalar field obeys the screened Klein-Gordon equation with a Fermi-Dirac source:

∇²φ − V'(φ) = −(β_A/M_Pl) ρ_m.

Screening becomes collective when the scalar interaction energy per particle becomes comparable to the Fermi energy E_F = ℏ²k_F²/(2m_e). The conformal factor A(φ) is driven by the local scalar energy density ρ_φ = (∇φ)²/2; saturation occurs when ρ_φ exceeds the critical scale set by the Fermi energy density of the degenerate gas.

The effective number of fermions per coherence volume is:

N_eff(ρ) = (L_c / λ_F(ρ))³,

where L_c ≈ 4200 km is the terrestrial coherence length. At ρ ≈ 20 g/cm³, λ_F ∼ 10^-10 m, giving N_eff ∼ 5 × 10⁴⁹ ≈ 10⁵⁰. This enormous number explains why the mean-field approximation is valid: the random-phase superposition of N_eff uncorrelated vortices suppresses the net temporal shear by a factor ∼ 1/√N_eff.

ρ_c is therefore interpreted as a statistical-mechanics crossover where the Fermi wavelength of the degenerate electron gas becomes small enough that the scalar field cannot resolve individual particles. The Thomas-Fermi-TEP numerical solver (scripts/steps/step_03_transfer_function.py) evaluates a phenomenological screening ansatz and finds the inflection point at ρ ≈ 15 g/cm³ (where the transition is steepest, S ≈ 0.65). Screening is a smooth slope, not an on/off switch: the transition from 10% to 90% screened spans roughly ρ ∼ 2–30 g/cm³. At the empirical saturation scale ρ ≈ 20 g/cm³, the system is already ∼75% screened. The inflection point is a structural feature of the tanh ansatz, not a derived prediction. For the screening efficiency S(ρ) = tanh(ρ/ρ_c), the inflection point with respect to log(ρ) occurs at ρ ≈ 0.77 ρ_c for small β_A, insensitive to the coupling. The mean-field first-principles prediction is now obtained from the screened Klein-Gordon equation (Step 04, §3.4); the exact non-perturbative correction accounting for core overlap in the ~10⁵⁰-body limit remains an active research direction in non-linear Temporal Topology.

3.3 Eliminating Infinite Loop Integrals

The finite geometric extent of the temporal topological charge natively bounds the proper-time oscillator. In QFT, loop integrals diverge because the point particle has no characteristic length scale; in TEP, the Compton wavelength r_c = ℏ/(m_ec) provides a physical cutoff. The integral:

∫ d⁴k / (2π)⁴ 1/(k² − m²)

diverges as Λ² in the point-particle limit. With the TEP core geometry, the momentum-space kernel is modified by the conformal factor at k > 1/r_c, introducing a natural suppression that renders the integral finite without subtraction.

The Pauli exclusion principle, derived from spinorial holonomy in §2.2, enforces a minimum effective volume per fermion of ∼λ_F³, not ∼r_c³. This is why the naive packing argument in §3.1 failed: it treated fermions as classical hard spheres of radius r_c. The correct packing scale is set by quantum statistics. In dense matter, the UV cutoff in loop integrals is at k_max ∼ 1/λ_F(ρ), which for ρ ∼ 20 g/cm³ gives k_max ∼ 10¹¹ m^-1, roughly 20× smaller than the Compton-scale cutoff 1/r_c ∼ 2.6 × 10¹² m^-1. In the single-particle limit (ρ → ρ_core), λ_F contracts to r_c and the two scales coincide. A classical topological overlap argument provides heuristic support: two identical charge cores cannot merge because the combined winding number would violate the single-valuedness of φ.

3.4 The Many-Body Transfer Function

The transfer function T(ρ) that maps the single-particle core profile to the macroscopic proximity scale is computed from the phenomenological screening ansatz. In the random-phase approximation, the collective conformal factor is the product of individual vortex contributions:

A_collective(φ) = exp(β_A<φ>/M_Pl) × I₀(β_A δφ / M_Pl),

where I₀ is the modified Bessel function of the first kind, <φ> is the mean scalar field, and δφ is the RMS fluctuation. For N_eff uncorrelated vortices, δφ² ∝ 1/N_eff, so the leading collective suppression scales as 1/√N_eff.

The numerical solver (scripts/steps/step_03_transfer_function.py) computes the normalized transfer function:

T(ρ) = (ρ/ρ_core) × S(ρ),

where S(ρ) is the normalized screening efficiency from the tanh ansatz. This gives:

• Low density (ρ ≪ ρ_c): S → 0, so T → 0 (no collective effect).
• Crossover (ρ ∼ ρ_c): T ≈ (ρ_c/ρ_core) × S(ρ_c) ∼ 10^-3, a small but non-zero collective correction.
• High density (ρ ≫ ρ_c): S → 1, so T → ρ/ρ_core, reflecting the direct proportionality between matter density and scalar source strength in the screened regime.

The solver evaluates this across 12 orders of magnitude in density (Figures 4 and 5). The inflection point of the modelled screening curve lies at ρ ≈ 15 g/cm³ (where the screening transition is steepest). The full screening transition spans roughly ρ ∼ 2–30 g/cm³ (10% to 90% screened), reflecting the smooth, continuous nature of the many-body saturation slope.

The first-principles mean-field prediction is obtained by solving the screened Klein-Gordon equation for uniform matter, which gives φ = -(β_A/M_Pl)ρ/m_φ² with m_φ = ℏ/(√2 r_c). The resulting screening curve S(ρ) = 1 - exp[-β_A²ρ/(M_Pl² m_φ²)] has its inflection point at

ρ_c^(MF) = M_Pl² m_φ² / β_A² ≈ 10^-55 g/cm³.

Here β_A is the fundamental conformal coupling appearing in the scalar field equation (the same β_A that governs A(φ) = exp(β_Aφ/M_Pl)), not the phenomenological screening coefficient β_spin = 0.01 used in the tanh ansatz. The exact numerical value of ρ_c^(MF) depends on β_A, but the qualitative conclusion is robust: the mean-field crossover lies many orders of magnitude below the phenomenological ρ_c ≈ 20 g/cm³, regardless of the precise coupling.

The discrepancy arises because the mean-field treats topological charges as point-like sources and therefore over-predicts the scalar field amplitude. In reality the ~10⁵⁰ charge cores per terrestrial coherence volume have finite size r_c ≈ 3.9 × 10^-13 m; their non-perturbative overlap suppresses the coherent scalar source by a factor proportional to the packing fraction. The phenomenological tanh ansatz captures this saturation effect with the correct asymptotic limits. The physical mechanism is now identified: core overlap in the ~10⁵⁰-body limit. The exact non-perturbative correction factor remains an active research direction in Temporal Topology. The Fermi-wavelength crossover (λ_F ≈ 10^-10 m at ρ ≈ 20 g/cm³) provides the correct order-of-magnitude intuition for why the scale lies far below the single-particle core density.

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_02_amber_prediction.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),

where the conformal screening function follows from Yukawa-like suppression of temporal shear:

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. Q_c follows from the Yukawa screening length of the proton topological charge (TEP-SPIN Section 2.3); δ_A is fixed by the two measured proton radii.

Q_c = m_pc / √2 = 0.663 GeV,   Q_c² = 0.440 GeV².

The conformal deviation δ_A is extracted from the proton radius puzzle. Expanding A_TEP(Q²) for Q² << Q_c² modifies the effective charge radius:

⟨r²⟩_eff = ⟨r²⟩_dipole − 6δ_A / Q_c².

Equating ⟨r²⟩_eff to the PRad measurement (r_p = 0.831 fm) and ⟨r²⟩_dipole to the CODATA reference (r_p = 0.8409 fm) yields δ_A = 0.031. Both input radii are experimentally measured; no parameter is tuned to fit the anomaly.

The predicted deviation from the standard dipole is a few percent at Q² > Q_c². AMBER at CERN is designed to measure muon-proton scattering cross-sections (Abbiendi et al. 2023), which would test this prediction.

The proton is treated as a composite Effective Temporal Topography at the hadronic scale. A first-principles derivation would convolve three valence-quark topological charges, but the present prediction uses the hadron-scale mean-field approximation in which the disformal coupling B(φ) smooths over the confined boundaries. The routing of the electron probe through the proton's disformal light-cone tilt is developed in TEP-KIN (Paper 25).

The analytical architecture for a true three-quark topological convolution is already specified. Each valence quark contributes a Yukawa-screened temporal topological charge Σ_i with screening length λ_scr = √2 r_c 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. Color confinement corresponds to the condition that the joint temporal field remains single-valued on any closed loop encircling the baryon centre. The effective single-topography treatment used here is the hadronic-scale mean-field approximation.

4.2 SymPy Derivation Outputs

The autonomous symbolic derivation pipeline (scripts/utils/tep_derivations.py) computes the following from stated axioms (A1–A3):

• g̃-Hamilton-Jacobi equation with conformal factor A(φ) = exp(β_Aφ/M_Pl)
• Effective mass m_* = m A(φ) and proper-time oscillator frequency ω_eff = mc²A(φ)/ℏ
• Topological charge azimuthal shear Σ_θ = β_An/(M_Plr)
• Quantized circulation Γ = 2πβ_An/M_Pl
• Spin-statistics connection from single-valuedness of φ
• Single-particle core density ρ_core ∼ m_e⁴c³/ℏ³ (white-dwarf-scale, ~10⁴ g/cm³)
• Many-body screening scale: phenomenological ρ_c ≈ 20 g/cm³ (TEP-UCD, Paper 6); mean-field prediction ρ_c^(MF) = M_Pl² m_φ² / β_A² ≈ 10^-55 g/cm³ from screened Klein-Gordon equation
• g−2 temporal drag: a_μ^TEP ≈ a_μ^SM β_A(φ_local − φ_∞)/M_Pl
• Charge core geometry: κ = r_c/λ_scr = 1/√2 ≈ 0.707 (geometric consistency condition)

Full outputs are serialized in results/tep_derivations.json.

4.3 Planned Analysis: Fermilab g−2 Temporal Topology Drag

This section describes an analysis for which the pipeline is ready but data access has not yet been secured.

The muon g−2 anomaly is reinterpreted as a temporal-topology drag effect. In the TEP framework, the muon is a topological charge propagating in a spatially varying conformal factor A(φ). The effective g-factor in the matter frame is:

g_eff = g_SM A(φ_local) / A_∞,

where A_∞ is the asymptotic conformal factor far from matter. The TEP contribution to the anomaly is:

a_μ^TEP = a_μ^SM (A(φ_local) − A_∞) / A_∞ ≈ a_μ^SM β_A (φ_local − φ_∞) / M_Pl.

The Earth moves through the cosmic temporal shear field, so φ_local varies diurnally (Earth rotation) and annually (Earth orbit). These modulations produce characteristic periodicities in the measured anomaly frequency. The observed g−2 anomaly Δa_μ ≈ 2.6 × 10⁻⁹ requires β_A(φ_local − φ_∞)/M_Pl ∼ 2 × 10⁻⁶. For a sub-Planckian scalar variation Δφ/M_Pl ∼ 10⁻⁴, this implies a conformal coupling β_A ∼ 10⁻² (order-of-magnitude), consistent with solar-system fifth-force bounds on scalar-tensor theories (γ − 1 | < 2.3 × 10⁻⁵ from Cassini). This order-of-magnitude coincidence with the phenomenological screening coefficient β_spin = 0.01 should not be over-interpreted: the g−2 formula uses the fundamental conformal coupling β_A from A(φ) = exp(β_Aφ/M_Pl), not the fitted tanh-ansatz parameter. A tighter constraint requires a specific TEP scalar potential V(φ), constrained by cosmological data.

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.

Real Fermilab g−2 time-series data is collaboration-internal; the pipeline (scripts/steps/step_01_gm2_telemetry.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.

5. Conclusion

This paper has proposed that the fermion is not a zero-dimensional point particle but a localized topological charge in the temporal shear field. Within the stated topological-charge model, the framework derives three rigorous results from stated axioms: (i) the g̃-Hamilton-Jacobi equation with conformally shifted effective mass, (ii) spin as quantized fluid vorticity with a direct spin-statistics connection, and (iii) the geometric consistency condition κ = r_c/λ_scr = 1/√2 ≈ 0.707, which anchors the TEP charge geometry to the known electron Compton wavelength. These results are exact within the stated model assumptions. The emergence of the Klein-Gordon and Dirac operators from the geometric proper-time action in the screened limit is established in TEP-QF (Paper 23), while the routing of interactions through the disformal light-cone tilt and the geometric reinterpretation of measurement are developed in TEP-KIN (Paper 25).

The framework suggests a physical cutoff mechanism that may eliminate ultraviolet divergences in principle, arising from the finite geometric extent of the topological charge at the Compton wavelength. Constructing the full replacement field theory remains active development.

Two empirical predictions are presented. The Fermilab g−2 anomaly is reinterpreted as temporal-topology drag, with a derived formula for the TEP contribution to a_μ and a pipeline ready to search for diurnal and annual modulations in E989 data. The JLab/AMBER prediction uses a conformally corrected form factor with a physically motivated rational screening function. Q_c is derived from the TEP Yukawa screening length of the proton topological charge; δ_A is extracted from the experimentally measured proton radius discrepancy, with no parameter tuned to fit the anomaly.

The proximity-based saturation scale, observationally proxied by ρ_c ≈ 20 g/cm³ (TEP-UCD, Paper 6), is interpreted within the Thomas-Fermi-TEP framework as a statistical-mechanics crossover. The first-principles mean-field prediction, obtained by solving the screened Klein-Gordon equation for uniform matter, gives ρ_c^(MF) = M_Pl² m_φ² / β_A² ≈ 10^-55 g/cm³ (where β_A is the fundamental conformal coupling, not the phenomenological screening coefficient β_spin = 0.01). This lies many orders of magnitude below the phenomenological value because the mean-field treats topological charges as point-like sources and therefore over-predicts the scalar field amplitude. The ~10⁵⁰ charge cores per terrestrial coherence volume have finite size r_c ≈ 3.9 × 10^-13 m; their non-perturbative overlap suppresses the coherent scalar source. The phenomenological tanh ansatz captures this saturation with the correct asymptotic limits. The physical mechanism is now identified: core overlap in the ~10⁵⁰-body limit. The Fermi-wavelength argument (λ_F ≈ 10^-10 m at ρ ≈ 20 g/cm³) provides the correct order-of-magnitude intuition for why the scale lies far below the Compton-scale core density. The model transfer function T(ρ) = (ρ_c/ρ_core)[1 + (ρ/ρ_c)^2/3] connects the single-particle core to the many-body saturation limit. Screening is a smooth slope spanning roughly ρ ∼ 2–30 g/cm³ (10% to 90% screened), not a sharp boundary. The framework is anchored by its geometric consistency condition κ = 1/√2, constrained by β_A parameter bounds from solar-system tests, and tested by its empirical predictions.