Binary Cuntz Bridge Framework v1.0: Corrected Scale Relation and Conditional Spin(10) Phenomenology

Abstract

This is version 1.0 of the Binary Cuntz Bridge Framework, released as a preliminary, non-peer-reviewed working paper. This version corrects and strengthens the scale relation by introducing a parametrized total suppression S_tot = S_bridge + Delta S. For the preferred branch sigma = 1/7, the bridge suppression factor is exp[-31.486]. Matching E_0 = 2.25 meV to M_* = 2.2e16 GeV requires an additional suppression Delta S = 32.964, whose microscopic origin is not derived in this paper. Therefore, the identification M_* = M_GUT is stated as a provisional phenomenological target, not as a derived prediction. The operator-algebraic core remains proven, while the seven-bridge geometry and Spin(10) phenomenology remain conditional. ## Changes in v3.0 (Revised and Integrated Edition) This is a revised and integrated edition of the original framework. Main updates include: 1. **Corrections**: - Bridge torus fundamental group corrected: π₁(T_bridge) ≅ ℤ⁷ - Bit–action convention clarified with explicit ln 2 factor - Kirchberg absorption statement restricted to safe form - Original scale branch (σ = 1/7) numerically re-evaluated 2. **Mathematical Upgrades**: - Explicit G_bridge-stable embedding K_P ≅ spin(7) ↪ A₃^ℝ - Proven decomposition A₃^ℝ = K_P ⊕ B₂₈ ⊕ F₇₉ - 107 = 28 + 79 realized as vector-space decomposition - Degree-count no-go theorem for 79 - Missing-partner Schur complement theorem 3. **Phenomenological Extensions**: - Extended Spin(10) Higgs sector: 210_H + 126_H + 126̄_H + 10_H + 120_H - Explicit Yukawa symmetry structure - Clebsch–Gordan coefficient framework 4. **Status Classification**: All statements now tagged as [Proven], [Conditional], [Corrected], [Upgraded], or [Open].