A Phenomenological Approximation of the Inverse Fine-Structure Constant via Z/6Z Topology and Informational Entropy

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)