Gauge Forces, Higgs Mechanism, and Particle Spectrum from Seam Topology (Information-Geometric Physics System III)

This document functions as the conceptual foundation and programme roadmap for the entire Information-Geometric Physics System (IGPS) series (Papers I–XII). The paper systematically explores the hypothesis that fundamental physics—including gauge symmetries, the Higgs mechanism, and particle mass spectra—emerge from the topology of codimension-two informational seams (specifically the unknot, trefoil, and figure-eight knots). Crucially, this revised version introduces a rigorous "epistemic ledger," explicitly categorising every theoretical claim as a Verified proof [V], a Bridge conjecture [B], or an identified Open problem [open]. Key Structural Foundations Established for the IGPS Programme: $\mathbb{Z}_3$ Holonomy: Identified as the fundamental topological datum of the trefoil seam, replacing ad-hoc approximations and serving as the origin for the generation count and CP-violation mechanisms. Central Charge $c=2$: Algebraically derived via the Sugawara construction, uniquely selecting the Chern-Simons level $k=1$. This acts as a rigid mathematical constraint utilised throughout Papers V–XI. Atiyah-Patodi-Singer (APS) Index: Evaluates the $1/(2\pi)$ boundary term on the trefoil Dirac operator, providing a critical quantitative ingredient for the fine-structure constant derivation in IGPS XI and the three-generation proof in IGPS VII. Callan-Harvey Anomaly Inflow: Invokes the established Callan-Harvey mechanism as the dynamical origin of the seam WZW theory, applied directly in the programme's electroweak and scaling derivations. Epistemic Transparency & Withdrawn Claims: In the strict interest of scientific rigor, this version officially withdraws early numerological or circular formulations present in the original draft (e.g., the 14-layer $D_7$ structure, the $\Phi\approx10.904$ master formula, and the circular derivation of geometric stiffness $\beta$). These have been superseded by algebraically verified alternatives (such as the nodal-count formula for $\alpha^{-1}$ and the geometric derivation of $\beta$), with the remaining foundational gaps reformulated as well-posed open problems in differential geometry (such as the explicit Ambrose-Singer holonomy computation for the trefoil). Document Contents: A comprehensive Dependency Guide and Programme Roadmap detailing the inheritance structure across Papers I–XII. A detailed Epistemic Status table for all foundational claims and physical predictions. An Epistemic Changelog recording every claim modified, replaced, or withdrawn from the original version. Detailed mathematical appendices verifying the Callan-Harvey anomaly inflow, the role of the APS $\eta$-invariant in global anomalies, and the $\mathbb{Z}_3 \to c=2$ algebraic chain.