Analytical Evaluation of the Electromagnetic Coupling Constant via Modular Substrate Vacuum Invariants

Description

This deposit contains the preprint manuscript and the complete computational laboratory for the analytical evaluation of the inverse fine-structure constant ($\alpha^{-1}$) formulated as a deterministic response of the quantum vacuum substrate (Modular Substrate Theory, MST).

Conceptual Logic

Rather than utilizing empirical data-fitting, this framework treats $\alpha^{-1}$ as a renormalization flow originating from a pristine geometric manifold ($4\pi^3 + \pi^2 + \pi$). This bare coupling is sequentially screened by the vacuum's fundamental informational impedance ($R_{\text{fund}} = \ln 2 / (6 \ln 3)$), which we derive from the Standard Model $\mathbb{Z}_6^{(1)}$ global 1-form gauge symmetry and holographic Cantor string dynamics.

Key Theoretical Components

• The Master Equation: We present a non-perturbative series:

  •                                                                                      $\alpha^{-1} \approx \text{Geometry} - \Delta_{\text{Holographic}} - \Delta_{\text{Torsional}}$.

  • $$\alpha^{-1} \approx (4\pi^3 + \pi^2 + \pi) - \frac{R_{\text{fund}}^3}{4} - \left(1 + \frac{1}{4\pi}\right)R_{\text{fund}}^5$$

• Holographic Partitioning: Implements the Bekenstein-Hawking bound ($1/4$) projected onto the $\mathbb{Z}_6$ partition space.

• Topological Scattering: Incorporates the vacuum manifold's torsional cross-section ($1 + 1/4\pi$) as an infrared fixed-point correction at the fifth-order complexity scale ($R_{\text{fund}}^5$).

• Metrological Convergence: The formulation yields a result of $137.035999206...$, matching the CODATA 2022 baseline with a residual absolute deviation of $\sim 1.5 \times 10^{-14}$.

Computational Validation & Falsifiability

To demonstrate that this convergence is not a "Look-Elsewhere Effect" (LEE) artifact, this repository provides a 100-digit precision computational audit.

• Stochastic Search Audit: Monte Carlo simulations (up to $10^6$ syntactic tree structures) confirm that the master equation constitutes a unique global minimum of algorithmic complexity.

• Non-Perturbative Stability Audit: We demonstrate that the model occupies a steep "phenomenological potential well." Micro-perturbations ($\epsilon = 10^{-6}$) to the input topological invariants ($R_{\text{fund}}$ or the $\mathbb{Z}_6$ scaling) cause the predictive accuracy to collapse by $>10^4$ orders of magnitude, effectively falsifying the hypothesis of arbitrary parameter tuning.

Contents

• Analytical_Evaluation_Alpha.pdf: Full manuscript submitted to PTEP (Paper ID: T06182).

• Validation_Suite.ipynb / Validation_Suite.pdf: Interactive environment for independent replication of all invariants and stochastic audit logs.

• Algebraic_Naturalness.png: Visual audit map illustrating the structural isolation of the MST result.

Endorsement & Peer Review

For colleagues in the High Energy Physics (hep-ph) community interested in providing an arXiv endorsement to support this submission, please utilize the following direct link: https://arxiv.org/auth/endorse?x=JFXY7X

Companion Foundational Work: The derivation of the invariants is detailed in Zenodo DOI: 10.5281/zenodo.20546608.

Last Update: June 2026 | Status: Under Review (PTEP)