Geometric Origin of Five Bosons: Two-Route Classification, w-Root Liberation in β-Decay, and the Higgs VEV as Ground-State Stacking Density

The Standard Model classifies five bosons — photon (γ), W, Z, gluon (g), and Higgs (H) — as force carriers and a symmetry-breaking scalar, but does not address the ontological question of whether these bosons are independently existing substances. This paper — PP-08 in the META Physics preprint series — derives the geometric ORIGIN of all five bosons from the uniform rotation of the four-dimensional ball B⁴(R) and the geometric structure of its three-dimensional boundary S³(R). Seven principal results are established with zero free parameters. (I) Bosons separate into two structurally independent routes: Route 1 (lepton-series: photon, Category A — individual zero-dimensional point states on S³(R)) and Route 2 (quark-series: W, Z, gluon, Category B — collective state transformations of three-quark systems). Treating all bosons as a single ontological category is a category error. (II) Neutron instability has a geometric ORIGIN: in the axis recombination of the neutron's three quarks (udd), the w-coordinate — aligned with the ISM-T radial direction and eliminated by DRAIN — occupies one of the three (x,y,z) spatial slots, creating a structural tension that makes the neutron unstable. The proton (uud) satisfies (x,y,z) exactly and is stable. (III) β-decay (n → p + e⁻ + ν̄ₑ) is a three-stage geometric process: w-root liberation → transient state (observed in WS₃ as "W⁻ boson") → reconstruction into electron (n = −1) + anti-neutrino (n = 0). The W charge sign is determined by the direction of w-root liberation: −w direction → n = −1 → Q = −1 (W⁻); +w direction → n = +1 → Q = +1 (W⁺). (IV) The Z boson preserves Hopf winding (Q = 0) and rearranges only stacking density (m ≠ 0). mZ > m_W because preserving the winding structure while rearranging stacking density incurs an additional Hopf-protection energy cost. Z and photon are orthogonal projections of the same S³(R) structure. (V) The gluon is a Jα assignment transformation between three ISM-T structures within a baryon — Category B (inter-point relation). Its mass-zero, charge-zero, color-carrying, self-interacting, confined nature follows without additional input. (VI) All five bosons (photon, W, Z, gluon, Higgs) are non-substance — none appears in the PGS list {B⁰,...,B⁵}. Each is a distinct mode by which ES structural changes are observed in WS₃. (VII) The Higgs vacuum expectation value v ≈ 246 GeV corresponds to the ground-state stacking density N̄ = v/ε₀ ≈ 1.4 × 10⁴⁴ — the uniform baseline stacking across all zero-dimensional points on S³(R). The Higgs boson (m_H ≈ 125 GeV ≈ v/2) is a fluctuation of N̄ itself, not an independent substance. Twenty-five established results are derived across 13 sections and collected in Appendix A. The complete non-substance classification of all Standard Model bosons is established, neutron instability and β-decay are derived from axis recombination geometry, and the Higgs VEV is identified as a geometric quantity rooted in the ES stacking structure. Mathematical formalization and documentation: Claude (Anthropic, Claude Opus 4.6). The core insights, physical intuitions, and axiom system originate from the author's 30 years of independent research.