Zitterbewegung Revisited: Insights into Spacetime

Zitterbewegung Revisited: Insights into Spacetime Damian Pikor, Paweł Kurzawski Abstract Standard physics treats Zitterbewegung - trembling motion predicted by the Dirac theory - as a mathematical artifact. This is because it suggests an internal structure for the electron, whereas experiments consistently show the electron to be a point particle. In this paper, we propose a revision of this view. We postulate that Zitterbewegung is a real phenomenon, not as a motion of the particle’s charge, but as a persistent vacuum resonance; an imprint of its creation. We argue this dynamic constitutes a stable, internal phase geometry associated with the particle. The central point of the model is the strict separation of the point-like particle from this associated geometry. We formalize this by introducing two distinct form factors. The charge form factor ( FEM ) describes the particle itself and remains operationally point-like ( FEM ≡ 1), ensuring full consistency with scattering data. Simultaneously, we define a new, internal observable – the phase form factor ( Fϕ ) – which describes the coherence of this associated phase geometry at the Compton scale ( ¯ λC ). This model is calibrated against the anomalous magnetic moment of the muon (∆ aµ ). This calibration then predicts the observed "silence" of the electron anomaly (∆ ae ) and a significant, testable anomaly for the tau lepton (∆ aτ ). This approach resolves the fundamental paradox, linking the g− 2anomaly to the Dirac equation’s kinematic substructure. The model predicts that this phase structure ( Fϕ ), while invisible to scattering, is accessible via coherence-sensitive protocols, such as interferometry, where it manifests as a measurable visibility V ( q ) ∝ |Fϕ ( q2 ) | linked to CHSH violation thresholds. Key words: Zitterbewegung, Lepton Geometry, Dirac Equation, Compton Scale, Anomalous Magnetic Moment, g-2, SMEFT. 1 1. Introduction The relativistic Dirac equation is a cornerstone of lepton dynamics, successfully describing spin and predicting antimatter [1-3]. However, it also yields one of physics’ most enduring puzzles: Zitterbewegung (ZBW), an ultra-rapid trembling of the free particle’s position operator. This motion is characterized by the ZBW frequency ωZ = 2 mc2/ℏ (which is ∼ 10 20 Hz for the electron, not ∼ 10 15 Hz) and an amplitude on the order of the reduced Compton wavelength, ¯ λC = ℏ/ ( mc ). This prediction is in direct conflict with observation. High-energy scattering experiments and precision QED tests consistently constrain the lepton’s charge radius to be much smaller than its Compton wavelength, ≪¯ λC [4, 5]. Because ZBW implies an internal structure at the ¯ λC scale, it is traditionally dismissed as an unphysical mathematical artifact [1, 3, 6]. In this work, we revise this view. We postulate that the ZBW paradox arises from a misinterpretation of the phenomenon. Our central hypothesis is that ZBW is a real, physical process, but it is not the motion of the particle’s charge. Instead, we posit that ZBW is a persistent, dynamic resonance of the vacuum itself, imprinted in the particle’s vicinity during its creation (e.g., pair production). The particle remains point-like but is inextricably "dressed" by this associated, coherent phase-geometry. The key to our model is a strict sector separation hypothesis. We formalize this by defining two distinct form factors for the lepton: 1. The Charge Form Factor ( FEM ( q2 )): This is the standard observable probed in scattering experiments. In our model, the charge density is a point-like delta function, ρEM (x) = δ(3) (x). Consequently, FEM ( q2 ) ≡ 1. This "geometric silence" in the charge sector ensures full consistency with scattering data and precision QED tests [4, 5]. 2. The Phase Form Factor ( Fϕ ( q2 )): This is a new, internal observable we introduce. It does not describe charge, but rather the spatial coherence of the associated ZBW vacuum resonance. Unlike FEM , this phase-form factor possesses a non-trivial structure, Fϕ(q2)= 1, with a characteristic scale defined by ¯ λC. This model resolves the paradox: FEM ≡ 1explains why scattering experiments see a point particle, while Fϕ = 1 allows ZBW to be a real phenomenon with its characteristic Compton scale. Most importantly, this framework is not just a conceptual model; it has concrete, falsifiable consequences. We propose that this underlying phase-geometry is the source of the anomalous magnetic moment (∆ aℓ ) of the leptons. We use the persistent, high-precision discrepancy in the muon anomalous magnetic moment, ∆ aµ , as a calibration input for our model. By anchoring our geometry to the observed ∆ aµ , the model becomes predictive. It correctly "post-dicts" the observed null result for the electron anomaly (∆ae≈0) and predicts a specific, large, and potentially measurable anomaly for the tau lepton, ∆aτ. This article is organized as follows: Section 2 details the formal, covariant theory of the phase-geometric world-surface Σ, its stability, its gauge compliance (WT/ST identities), its derivation of spin (Noether charges), and its extension to a cosmological fluid. Section 3 presents the results of our numerical analysis, showing how the model is calibrated on ∆ aµ and its predictions for ∆ ae and ∆ aτ across the UV and Planck-scale parameter space, including its projection to hadronic (CLN/BGL) sectors. Section 4 synthesizes these findings, discussing the model’s implications, its connection to the r.m.s. phase radius r(ϕ) rms , and its falsifiability via interferometry. This approach links the IR slope of Fϕ to 2 coherence visibility V ( q )and CHSH violation bounds, supported by representative ( V, S ) data points from modern Franson/emitter and bright SPDC configurations. Detailed derivations and numerical protocols, including the interferometric procedure (Appendix F.2), are provided in the Appendices. 2. Theoretical Framework 2.1. Assumptions and Sector Separation (WT/ST Compliance) We introduce a strict separation of observables into a charge sector and a phase-geometric sector. In the rest frame, the charge density is point-like, ρEM (x) = δ(3) (x), yielding FEM(q)≡1[7, 8]. This ensures operational point-likeness in scattering. This strict separation is the key to gauge compliance. By positing FEM ≡ 1, the QED vertex remains unmodified, ensuring the Ward-Takahashi (WT) identity ( qµ Γ µ = S−1 ( p′ ) −S−1 ( p )) is satisfied by construction [6]. Similarly, when extended to quarks, the q-q-g vertex remains standard, preserving QCD’s Slavnov-Taylor (ST) identities. All new physics from Fϕ is implemented via allowed EFT operators, not by modifying gauge vertices. The Fϕ field (via its IR slope) maps to a catalog of the lowest-dimension, gaugeinvariant operators that describe this coherence, selected by C, P, T and SU (2) ×U (1) symmetries. The primary operator is the dimension-6 electromagnetic dipole (see Appendix E). This effective dipole arises from the Σ-field’s interaction with the lepton, not from QED loop corrections, thus avoiding double-counting with standard renormalization. The RGE running of this operator’s coefficient from the matching scale Λto the lepton mass mℓ is required and introduces a calculable (but schema-dependent) uncertainty, ensuring the entire framework remains WT/ST-compliant [11]. The new sector possesses a radial coherence density ρϕ ( r )with a natural scale ¯ λC and an independent phase form factor Fϕ(q2). We define the phase form factor in the rest frame via the spherical transform Fϕ(q) = 4πZ∞ 0 dr r2ρϕ(r)sin(qr) qr and treat it relativistically as a function solely of the invariant q2 = qµqµ , which ensures manifest covariance. The r.m.s. phase radius is determined by the standard slope-radius relation ⟨r2⟩ϕ = − 6 d Fϕ/ d q2|q2=0 , with r(ϕ) rms = p⟨r2⟩ϕ . This formally maps the form-factor apparatus onto coherence observables. 2.2. Covariant Resonance World-Surface (Σ) The coherence carrier is defined as a smooth, two-dimensional resonance world-surface Σ ⊂R1,3 with an embedding Xµ ( σa ), induced metric γab = ηµν∂aXµ∂bXν , two normals nµ I , a second fundamental form KI ab , curvatures H and KG , and the Laplace-Beltrami operator ∆Σ. All observables are built exclusively from Poincaré scalars. The covariant phase density in spacetime is defined distributionally as ϱϕ(x;u) = NZΣ d2σ√−γ δ(4)(x−X(σ)). 3 This is projected onto the hyperplane orthogonal to the four-velocity uµvia ρϕ(x) = Zdτ ϱϕ(xµ=τuµ+xµ ⊥;u), which guarantees a frame-independent definition of the radial density ρϕ ( r ). A sketch of the proof of covariance, including the use of projectors and independence from reparameterization in the IR limit, is provided in Appendix A. 2.3. Action Functional on Σ The minimal, manifestly covariant action functional combines surface tension, curvature elasticity, and a phase field ϑ on Σ, with a "soft" resonance condition at the Compton scale ℓ∼¯ λC . The full action S [ X, ϑ ]and the derivation of the Helfrich/Willmore-type shape equation [9, 10] are detailed in Appendix B. The stability of this action is ensured by a full non-linear stability analysis. As detailed in Appendix B, the second variation δ2S must be positive definite. The resonance term µ (2 ℓH − 1) 2 provides a positive "mass" term, stabilizing the solution against normal fluctuations within a well-defined parameter space of ( T, κ, ¯κ, µ, h ). This action also defines the model’s stress-energy tensor via the standard variational principle Tµν Σ = −2 √−g δSΣ δgµν , which allows the Σ-field to be treated as a cosmological fluid (see Sec. 2.11). 2.4. Projection and Definition of ρϕ(r) The projection from ϱϕ ( x ; u )onto the rest frame using the projector Pµν = ηµν−uµuν leads to an isotropic radial density ρϕ ( r )satisfying the normalization 4 πR∞ 0 d r r2ρϕ ( r ) = 1. This defines the moments ⟨rn⟩ϕ and the spherical transform Fϕ . The procedure is reparametrization-invariant, and UV/IR regulators on Σensure a unique projection and the existence of moments, as is standard in analyses of surfaces with bending energy. This provides a direct interface to interferometric measurements via the Fϕ(q2)slope. 2.5. Geometric Ansatz: "Gabriel’s Horn" For an axially-symmetric surface of revolution with profile r ( x ) = a/x , the mean curvature H asymptotically realizes the resonance condition 2 ℓH ≃ 1and selects the scale r∼ℓ . A small-slope approximation H≃1 2 ( −r′′ +1 /r )with a self-similar ansatz r ( x ) = ax−α yields an energy minimum at α≃ 1, justifying the inverse profile. After non-dimensionalization ( ˆr = r/ℓ, ˆx = x/ℓ ), this anchors the geometry to ¯ λC . An analysis of the variational equations (Appendix B) confirms the existence and local uniqueness of this energy minimum for the "Gabriel’s Horn" profile, provided UV/IR regulators [ rmin, rmax ]are applied. The full derivation of the normalization constant C−1 is provided in Appendix C. 2.6. Spectral Ansatz: Bessel + Localization Complementarily, we employ a spectral ansatz ρϕ ( r ) = A r J2 1 ( kr ) W ( r ; rmin, rmax )with a C∞ window function W , where k = c′/ℓ sets the Compton scale. The normalization constant is A−1 = 4 πR d r r3J2 1 ( kr ) W ( r ). The J1 ( z ) ∼z/ 2behavior at r→ 0ensures UV regularity, while the asymptotic behavior necessitates the IR-damping window W . 4 The calculation of Fϕ ( q2 )is realized as a spherical Bessel transform j0 ( qr ) = sin ( qr ) / ( qr ) using methods detailed in Appendix D. 2.7. Low-q2Expansion and "Shape-Blindness" The expansion of j0(z) = 1 −z2/6+... yields the low-q2expansion for the form factor: Fϕ(q2)=1−q2 6⟨r2⟩ϕ+(q2)2 120 ⟨r4⟩ϕ+O((q2)3). This defines the slope-radius relation dFϕ/dq2|0 = −⟨r2⟩ϕ/ 6. The "shape-blindness" property implies that at O ( q2 ), Fϕ depends only on the second moment ⟨r2⟩ϕ . Any two normalized ansätze (e.g., Geometric and Spectral) with the same ⟨r2⟩ϕ will have an identical slope and r.m.s. radius, with differences only appearing at O (( q2 ) 2 )via the ⟨r4⟩ϕ moment, IR anchoring with ⟨r2⟩ϕ = 1 . 5(Compton units) and the demonstration of shape-blindness (agreement in the slope O ( q2 )between ansätze after matching ⟨r2⟩ϕ ) are illustrated through data and plots in Appendix F.2. 2.8. Parameter Estimation and Anchoring The ansatz parameters (e.g., a, k ) are determined by a unified optimization problem. The procedure minimizes the geometric action Sgeo (see Appendix B) subject to the constraints of normalization (4 πR d r r2ρϕ ( r )=1) and anchoring to the Compton scale ℓ = ¯ λC . A critical consistency test ("slope-match") ensures that both ansätze yield the same ⟨r2⟩ϕ and thus the same low-q2physics. 2.9. Ansätze Comparison: IR Equivalence and UV Divergence As established in Sec. 2.7, the Geometric and Spectral ansätze are equivalent in the lowq2 (IR) regime, provided their second moments ⟨r2⟩ϕ are matched. Their divergence occurs at highq2 (UV), which is dictated by the larger tails of their respective density profiles and the choice of regularization window W ( r ). This ensures that while the lowq2 observables ( r(ϕ) rms ,∆ aℓ ) are robust, the highq2 behavior remains distinct. A numerical verification of this IR equivalence is provided in Appendix C. 2.10. Noether Charges, Spin, and the Dirac Factor The internal phase field ϑ on Σ(Sec. 2.3) acts as a U(1) field. By imposing a spinorial holonomy (a 4 π periodicity required for a double covering), the associated Noether charge for rotations on Σyields the intrinsic angular momentum S = ℏ/ 2. This framework geometrically derives the Dirac "factor of 2" ( S = ℏ/ 2and g = − 2) from the world-surface topology, rather than postulating it. The ZBW frequency ωZ = 2 mc2/ℏ is likewise recovered from the resonance condition 2 ℓH ≃ 1when the scale is set to ℓ = ¯ λC . This approach can be generalized to spin-3/2 fields by modifying the holonomy and constraints on Σ, while respecting the Velo-Zwanziger consistency conditions [1]. 2.11. Cosmological Bridge: The Σ-Fluid The stress-energy tensor Tµν Σ derived from SΣ (Sec. 2.3) allows the model to act as a "cosmological bridge". When populated in the early universe, the ensemble of Σ5 resonances behaves as an effective fluid ρΣ ( a ). Our stability analysis (Appendix B) and long-wavelength limit (Sec. 2.7) show that the fluid is "cold" ( c2 s,Σ≈ 0) and has negligible anisotropic stress ( σΣ≈ 0). This Σ-fluid contributes to Ω m and must be consistent with cosmological data (BBN, CMB, BAO). This framework connects the lab-measured IR-slope ⟨r2⟩ϕ (via Fϕ and ∆ aℓ ) to the cosmological parameters ρΣ and wΣ . Admissibility curves for ρΣ ( a )show consistency with BBN and CMB constraints. We generate a map of ( ⟨r2⟩ϕ, Ω Σ )which shows that labbased metrology of ⟨r2⟩ϕ can strongly constrain the allowed cosmological parameter space, breaking degeneracies with other sectors (e.g., Neff or early dark energy) in global fits to Planck and DESI data [5, 11]. 2.12. Consistency with QFT Principles The model’s consistency with fundamental QFT principles is ensured by the FEM ≡ 1 separation. • Heisenberg Uncertainty Principle (HUP): The canonical commutators [ ˆx, ˆp ] are unchanged. Fϕ only modulates the coherence of a state, not the fundamental algebra, fully preserving ∆x∆p≥ℏ/2. • Larmor/Unruh Radiation: As the EM vertex is standard, the Larmor formula ( P∝a2 ) and the Unruh temperature ( TU∝a ) remain unmodified. The Σ-field does not couple directly to the photon field in this way. • EPR States (CHSH): Fϕ acts as a "phase-damping" channel. It does not violate locality but modulates the coherence (visibility V ) of the entangled state. This correctly predicts a reduction in S≈ 2 √2V . To distinguish this from instrumental noise, a full open-system model (e.g., Kraus dephasing) is needed to separate the intrinsic Fϕ effect from depolarization (Werner/singlet mixing) and multipair/detector noise ( η, κ, p2 ). The goal is to isolate Fϕ in the q2≤ 0 . 02 − 0 . 04 window, as specified in Appendix F.2. We note that representative ( V, S )data from modern Franson/emitter setups and S vs. V curves from bright SPDC sources operationally justify this phase-law approach [1, 3]. 3. Numerical Analysis and Phenomenological Projections 3.1. UV Grid and Planck-Line Scans We performed a numerical analysis based on the framework detailed in Section 2 and Appendix D. We scanned the parameter space in two primary ways: 1. UV Grid Scan (54 nodes): We performed a Cartesian scan over the regulator space (Λ , p ). This scan confirms a robust separation of roles. The muon anchor ∆ aµ = 2 . 5 × 10 −9 is held constant by construction, and the electron anomaly remains negligible, with a mean ⟨ ∆ ae⟩ ≈ 1 . 084 × 10 −17 (std. dev. 1 . 93 × 10 −19 ). In contrast, the tau anomaly exhibits a wide, stable plateau, with a mean ⟨ ∆ aτ⟩ ≈ 2 . 463 × 10 −7 (std. dev. 4.40 ×10−9), as shown in smeft_uv_summary.csv. 6 2. Planck-Line Trajectory (Λ(ℓ) = c/ℓ): We probed the scale-dependence by sampling a trajectory along the "Planck-Line." This scan ( smeft_planck_summary.csv ) confirms that the light-lepton anchors (∆ ae, ∆ aµ ) are invariant. The tau channel, however, exhibits the characteristic geometry-controlled crossover, interpolating from the classical plateau (max. ∆ aτ≈ 2 . 867 × 10 −4 ) to a deep compression window (min. ∆aτ≈5.842 ×10−9). These scans numerically validate the "IR-anchored, UV-sensitive" nature of the model. The heavy-lepton channel ( τ ) acts as a short-distance probe, while the light-lepton channels ( e, µ ) are protected by the lowq2 "shape-blindness" of the geometry. The uncertainties on these predictions (∆ ae ,∆ aτ ) include the full experimental and theoretical error budget: (1) the experimental uncertainty on the ∆ aµ calibration input, and (2) the theoretical uncertainty on the dFϕ/dq2|0 slope, estimated from its stability across the regulator grid (see Appendix D) and its dependence on the IR window choice (Appendix F.2). The predicted "plateau" value for the tau anomaly, ⟨ ∆ aτ⟩ ≈ 2 . 463 × 10 −7 , is consistent with current experimental limits [14, 15], although precision is not yet sufficient to probe this value. Table 1: Numerical stability of the IR-slope ⟨r2⟩ϕ (in Compton units) against regulator and grid variations. Metrics (e.g., AIC/BIC) confirm stability. Scan / Window W(r)Grid Density (q)⟨r2⟩ϕ(Mean) AIC/BIC (relative) UV-Grid (W1) Standard 1.5002 Ref. UV-Grid (W1) 2x Density 1.5001 +0.1 UV-Grid (W2) Standard 1.4998 +0.5 Planck-Line (W1) Standard 1.5000 Ref. 3.2. SMEFT Projection and Consistency We project these dipole predictions into the Warsaw basis of the Standard Model Effective Field Theory (SMEFT) [11]. We map the anomalies ∆ aℓ to the electromagnetic dipole operator coefficient Cℓeγ and its SU (2) L×U (1) Y decomposition ( CℓeB, CℓeW ). The full conventions and RGE procedure are detailed in Appendix E. Note on g-factor and aℓ conventions: We adhere to the standard PDG convention aℓ = ( g− 2) / 2[5]. The derivation of g = − 2in Sec. 2.10 refers to the base gyromagnetic ratio from the Dirac equation’s structure, which is the starting point before anomalies ( g≈ 2) are considered. All ∆ aℓ values reported are in the PDG convention. This mapping confirms the results of our scans at the operator level. • UV Grid: Cµeγ is constant with zero variance (mean 9 . 610 × 10 −11 , std. dev. 3 . 88 × 10 −26 ). Ceeγ is ultra-small (mean 1 . 084 × 10 −17 , std. dev. 1 . 93 × 10 −19 ). Cτeγ shows a stable plateau (mean 2.463 ×10−7, std. dev. 4.40 ×10−9). • Planck-Line: Cµeγ and Ceeγ remain fixed, while Cτeγ spans orders of magnitude, from a minimum of 5.842 ×10−9to a maximum of 2.867 ×10−4. Critically, we verify the lowq2 consistency at every node by calculating the reconstruction residuals ( εℓ ). As shown in Appendix D (and smeft_planck_summary.csv ), these residu7 als are zero (e.g., dat_resid mean 1 . 07 × 10 −18 , std. dev. 3 . 70 × 10 −18 ) for all leptons across all scans, certifying that the low-q2slope-radius relation is exactly preserved. When extending this framework to the quark sector, the same gauge-safe principle applies. The Fϕ effect is not implemented by modifying the q-q-g vertex (preserving Slavnov-Taylor identities), but via gauge-invariant operators in the relevant EFT, such as HQET or SCET [11]. This maps the Fϕ coherence parameter onto the hadronic form factors (e.g., in CLN or BGL parameterizations) used to describe decays like B→D(∗)ℓν . We performed a stability test using the same unitarity prior for both CLN and BGL parameterizations, confirming that the results are robust against parameterization bias. The resulting effect on the differential decay rate [ d Γ /dw ] EFT/ [ d Γ /dw ] SM is a smooth, O (10 −3 )correction, consistent with EFT power counting and standard PDG analysis methods [5, 11]. 1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 1.5 1.55 1.6 0.99 1 1 1.01 1.01 Recoil parameter, w Ratio [dΓ/dw]EFT/[dΓ/dw]SM Hadronic Projection Stability (CLN vs. BGL) CLN (Unitarity Prior) BGL (Unitarity Prior) 10−8 10−7 10−6 10−5 10−4 10−3 ∆aτLimit Excluded ∆aτ[14, 15] Figure 1: Placeholder: Stability of Fϕ -induced corrections in B→D(∗)ℓν decays under CLN and BGL parameterizations, using identical unitarity priors. The gray shaded region (right axis) indicates the current experimental exclusion limits for ∆aτ. 3.3. Neutrino and Z′Portal Consistency We performed two unit tests to ensure our framework is consistent with other phenomenological constraints (details in Appendix E). 1. Neutrino Unit Test: We defined an effective neutrino magnetic moment proxy, ξν , and determined the maximum allowed value consistent with experimental limits. Across the 54-node UV grid ( smeft_nu_summary.csv ), this value is uniformly small and stable: ξmax ν has a mean of 6 . 940 × 10 −11 with a standard deviation of only 6 . 347 × 10 −13 (min 6 . 531 × 10 −11 , max 6 . 959 × 10 −11 ). This confirms that our model induces no tension with neutrino constraints, demonstrating robust sector separation. 2. Leptophilic Z′ Portal: We confirmed that a modular Z′ portal can be introduced to account for the g− 2anomaly. A scan over ( mZ′, g′ )shows viable parameter 8 space (e.g., mZ′≃ 0 . 261 GeV , g′≃ 1 . 78 × 10 −3 ) that matches the ∆ aµ target while passing a proxy "trident" screen [12, 13]. This demonstrates that the Z′ can be treated as an orthogonal effector, leaving the core ZBW-driven ∆ ae/ ∆ aτ predictions intact. 4. Discussion We reinterpret Zitterbewegung (ZBW) as a phase coherence resource encoded in a gaugesafe form factor, Fϕ ( q2 ). This form factor is normalized as Fϕ (0) = 1 and defined by its infrared (IR) slope ⟨r2⟩ϕ . It is fundamentally separated from charge scattering, for which the form factor remains FEM ≡1. This separation elevates ZBW from a mathematical artifact to an operational observable. It is measurable via interferometric visibility, V ( q ) ∝ |Fϕ ( q2 ) | , and is linked to CHSH nonlocality through the approximation S≈ 2 √2V . This holds in the dominant phasedephasing regime, with a violation threshold of V> 1 /√2 . This relationship has numerous direct proxies in existing experiments. Franson/emitter systems and bright SPDC sources both demonstrate that S degrades as visibility V decreases—a process governed by parameters like pure dephasing (γd/Γ) or multipair probability. Metrology, Noise Models, and IR Stability This "lab-toFϕ " mapping is realized by a joint fit V ( q2 ) = V0 (1 + c1q2 )to extract ⟨r2⟩ϕ=−6c1. In practice, the S(q)prediction requires a hierarchical noise model: S=2√2ηV0|Fϕ(q2)| 1+κp2 Here, η (depolarization) is estimated tomographically on complementary bases, and p2 (multipair probability) is constrained by brightness and second-order coincidences. This model reproduces the observed dependencies in Franson/emitter and bright SPDC experiments, calibrating |Fϕ| as an intrinsic coherence metric independent of instrumental degradation. For metrological robustness, we define two IR windows: a primary window where q2 linear residuals are white, and a secondary window to control O ( q4 )terms. We utilize heteroscedastic weights and an optional Gaussian Process (GP) for phase drifts (calibrated by Allan variance), mandating the report of Durbin-Watson statistics or GP hyperparameters. A nominal Fisher/CRLB analysis for a q2 = { 0 , 0 . 01 , 0 . 02 } grid (IR units) at SNR=2 yields σc1≈ 3 × 10 −3 and σ ( ⟨r2⟩ϕ ) ≈ 0 . 018. This sets the baseline requirements for integration time and phase stability needed to measure ⟨r2⟩ϕ and the S(q)curve. QFT Rigor: WT/ST Compliance The model’s QFT rigor is ensured by construction. The electromagnetic ( FEM ≡ 1) and QCD ( q−q−g ) vertices remain unmodified. Consequently, the **Ward-Takahashi identity** ( qµ Γ µ = S−1 ( p′ ) −S−1 ( p )) and the non-abelian **Slavnov-Taylor identities** are preserved, maintaining renormalizability and BRST symmetry. All Fϕ effects enter exclusively as permissible operators within EFT currents, thus separating charge scattering from coherence observables. Key diagnostic risks are explicitly defined: 9 •ξmax ν(min): 6.531 ×10−11 •ξmax ν(max): 6.959 ×10−11 •ξmax ν(mean): 6.940 ×10−11 •ξmax ν(std. dev.): 6.347 ×10−13 The extreme stability and smallness of this value confirms the "neutrino silence" of the model. E.3. Z′Portal Scan The Z′ scan (Sec 3.3) was performed over a grid spanning mZ′∈ [0 . 1 , 1 . 0] GeV and g′∈ [10 −4, 10 −2 ]. The acceptance mask ok_all = True requires passing both the ∆ aµ target window (∆ atarget µ = 2 . 5 × 10 −9 , δtol = 0 . 5 × 10 −9 ) and a proxy trident production screen. Figure 3 shows the accepted parameter space corridor, which follows the expected g′∝mZ′scaling. 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 10−4 10−3 10−2 Z’ Mass, mZ′(GeV) Coupling, g′ Z′Scan Acceptance Region for ∆aµ≈2.5×10−9 Accepted Region (ok_all = True) Example Point (0.261, 1.78e-3) Figure 3: The phenomenologically accepted parameter space for the Z′ portal. The blue band represents the "narrow stripe" of ( mZ′, g′ )values that simultaneously satisfy the ∆ aµ target window and pass trident constraints [12, 13]. The red dot highlights the specific benchmark point cited in the text. E.4. Literature and Limitations The ∆ aτ observable is bounded by experimental limits [14, 15] and global SMEFT fits [11]. The Z′ constraints from trident production are taken from [12, 13]. The model’s primary limitation is the sensitivity of highq2 observables (like the ∆ aτ plateau) to the choice of UV regulator (W(r)or Λ, p), though this does not affect the low-q2r(ϕ) rms prediction. 16 F. Experimental Protocol for r(ϕ) rms Measurement The key prediction, r(ϕ) rms ≈p3/2¯ λC , is not accessible via scattering but can be measured using coherence-sensitive protocols. F.1. Interferometric Protocol A practical measurement can be achieved via electron interferometry (e.g., in a RamseyBordé or Mach-Zehnder setup) by measuring the coherence factor (fringe visibility) V as a function of momentum transfer q . The visibility is directly proportional to the modulus of the phase form factor, V(q)∝ |Fϕ(q2)|. Procedure: 1. Prepare a coherent electron beam (e.g., from a cold-atom source or a high-brightness field-emission tip). 2. Use a beam splitter (e.g., a nanofabricated grating) to separate the wave packet. 3. Impart a differential momentum kick qto one arm of the interferometer. 4. Recombine the beams and measure the fringe contrast V(q)at the detector. 5. Plot V(q)vs. q2in the low-q2regime (q≪1/¯ λC). 6. Fit the data to the low-q2expansion: V(q2)≈ V01−q2 6⟨r2⟩ϕ. 7. The linear slope c1=−V0⟨r2⟩ϕ/6directly yields ⟨r2⟩ϕ, and thus r(ϕ) rms. Systematics: The primary uncertainty budget will be dominated by the initial beam coherence V0, the precision of the momentum kick q, and decoherence from stray fields. F.2. Alternative Protocols Alternatively, the ZBW phase geometry can be probed by: • Spin-Echo Protocols: Using Ramsey-type pulse sequences (spin-echo or echoRamsey interferometry) to measure the dephasing caused by the ZBW dynamics. • Interferometric Phase Estimation: The ZBW frequency ωZ = 2 mc2/ℏ is ∼ 10 20 Hz, far beyond telecommunication or THz ranges. This eliminates "beat-note" spectroscopy as a viable direct measurement. The only viable approach, detailed in Appendix F.2, relies on interferometry in the q≪ 1 /¯ λC regime (e.g., q2≤ 0 . 02 − 0 . 04 in IR units) to estimate small, q2 -dependent phase shifts, parameterized by V ( q2 ) ≃ V0(1+c1q2), rather than demodulating ωZ. 17 Appendix G: Interferometric IR test of the phase form factor G.1. Goal and setup This appendix details a direct, lowq interferometric test of the phase form factor Fϕ that is independent of charge scattering. In the IR, the spherical transform implies Fϕ(q2)=1−q2 6⟨r2⟩ϕ+O(q4), so the slope at q2→ 0recovers ⟨r2⟩ϕ and rrms ( ϕ ) = p⟨r2⟩ϕ . We measure the visibility V ( q ) ∝ |Fϕ ( q2 ) | in a small window of q , and fit V ( q2 ) ≈V0 (1+ c1q2 ), yielding ⟨r2⟩ϕ = − 6 c1 . This approach is necessary as the ZBW frequency ( ∼ 10 20 Hz) is inaccessible to direct "beat-note" demodulation. G.2. IR anchoring and shape-blindness We enforce the IR anchor ⟨r2⟩ϕ = 1 . 5(Compton units), i.e., rrms ( ϕ ) = p3/2¯ λC . Two independent ansätze (GH and spectral) are matched to this target. After the match: • The IR slopes coincide to numerical precision, ∆ c1≃ 3 × 10 −5 at q2 = 0 . 05, demonstrating IR shape-blindness (agreement in order q2). • The normalization is correct, Fϕ (0) ≈ 1, and IR residuals r2 slope −r2 moment are numerically small across UV-grid and along the Planck-line after IR rematch, confirming stability of the IR metric. G.3. Data products and plots We provide Planck-line and UV-grid lowq visibility series V ( q )after ⟨r2⟩ϕ = 1 . 5, suitable for linear fits: •Planck-line series: zmg_cl_ir150_planck_V_series.csv (columns: idx, α,q,V). •UV-grid series: zmg_cl_ir150_uv54_V_series.csv (columns: node, q,V). • Summary table: zmg_cl_ir150_summary.csv for GHscaled and SPmatched showing r2 moment ≈1.5,r2 slope ≈1.495,F(0) −1≈0, and ∆c1in the IR window. Illustrative plots (provided in the repository) include: •V(q)vs q2with linear fit V0(1+c1q2). • " F ( q )vs q2 " in the IR, using V≈ |Fϕ| , with the reference curve 1 −q2 6⟨r2⟩ϕ at ⟨r2⟩ϕ= 1.5. G.4. Robustness across regulators To demonstrate regulator-independent IR: • UV-grid (54 nodes): after IR rematch, residuals r2 slope − 1 . 5and F (0) − 1remain numerically small for all nodes, confirming IR invariance of the observable slope. 18 • Planck-line: scanning α with rematched scale preserves the IR slope and normalization along the trajectory, as quantified by the same residual diagnostics. G.5. Practical SNR and error budget A pilot SNR scan on ideal (noise-free) synthetic series returns extremely small per-point σ (SNR ∼ 10 9 ), an artifact of dense, clean data. For realistic planning, we recommend the “SNR 2.0” budget: • Decompose σ into statistically and systematically motivated components: σstat ∼ 1/N,σsys (phase/V0stability at 10−3–10−2), and σcal (q-axis calibration). •Define the target IR window: q2≤0.02 −0.04 (in IR units). • Implement phase drift control (e.g., via Gaussian Process (GP) priors or Allan variance diagnostics, reporting GP hyperparameters or Durbin-Watson statistics). • Fit V0 jointly with c1 . The target precision, based on nominal CRLB calculations, is σ(c1)∼10−3, implying a target of σ(⟨r2⟩ϕ)∼10−2to confirm the slope. • Cross-validate fits across multiple IR windows (e.g., q2≤ 0 . 02 , 0 . 03 , 0 . 04) and quote the worst-case as the requirement. For CHSH tests (Sec. 2.12), this S≈ 2 √2V approximation must be refined using an open quantum system approach (e.g., Kraus dephasing) to distinguish the Fϕ signal from depolarization and multipair noise ( η, κ, p2 ). As an order-of-magnitude (OOM) example for CHSH: with q2 = 0 . 02 (IR units), V0 = 0 . 95, η = 0 . 9, p2 = 10 −3 , and κ≈ 1, we find S≈2√2·η·V0·|Fϕ(0.02)|/(1+κp2)≈2.2, which is clearly above the classical bound. G.6. Summary The IR interferometric protocol directly measures ⟨r2⟩ϕ from the visibility slope, recovers rrms ( ϕ ) = p3/2¯ λC , and exhibits IR shape-blindness after ⟨r2⟩ϕ matching. Stability across regulator families (UV-grid, Planck-line) and a practical SNR roadmap together provide a clear, falsifiable path to experimental validation of the phase-geometry sector without relying on charge scattering. Funding This research received no external funding. Conflicts of Interest The authors declare no conflict of interest. Data and Code Availability All analysis code and numerical artifacts supporting this study are publicly available. 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