[Refuted Pathway] (Pillar IV Math) A Mathematical Foundation for the Canvas Model: Gauge Couplings, Flavor Structure, and CP Violation from Canvas Temporal Mathematics

This paper presents a unified mathematical foundation for the Canvas Model of fundamental physics, built on Canvas Temporal Mathematics (CTM). The foundation consists of three integrated layers: the abstract mathematical framework of CTM (amplitudes, the equality processor, S-symmetry, and steering dynamics), a formal model providing rigorous semantics, and the physical instantiation describing gauge fields, fermion generations, and threshold crossings. What this paper derives: Within this foundation, we derive the following results from the eight CTM primitives: · The gauge group SU(3)×SU(2)×U(1) from the Charge primitive· Exactly three fermion generations from the spectral theorem for the 3×3 threshold tensor· The gauge coupling ratios g₁² : g₂² : g₃² = 1 : 2/3 : 2/π and absolute values g₃² = 25π²/1024, g₂² = 25π³/3072, g₁² = 25π³/2048 at the Planck scale· The normalization-free gauge ratio g₃²/g₂² = 3/π matching data to 1.2%· The CKM Wolfenstein parameter λ = 1/5 = 0.2 from gauge subspace dimensions· The CKM hierarchy V_us ~ λ, V_cb ~ λ², V_ub ~ λ³· The V_ub suppression factor V_ub = λ³/3 ≈ 0.0036, matching the observed 0.00361 ± 0.00030· The PMNS mixing angles θ₁₂ = 1/√3 ≈ 33.1°, θ₂₃ = π/4 = 45°, and θ₁₃ = arcsin(1/(3√5)) ≈ 8.57°, matching the observed central value exactly· The CP-violating CKM phase δ_CKM = 68°, matching the observed 68° ± 4°, derived from the asymmetric threshold crossing dynamics encoded in the equality processor Why this matters: Every result traces back to the eight primitives. No experimental inputs are used for the gauge couplings, mixing angles, or CP phase. The paper establishes that the Canvas Model, grounded in Canvas Temporal Mathematics, provides a rigorous mathematical foundation for the gauge and flavor sectors of the Standard Model, including the origin of CP violation. What remains open: The remaining open problems—the Yukawa sector (factor ~25 discrepancy at M_P), the neutrino mass squared differences (factors of 3–34 off), the electroweak scale, vacuum stability, and cosmology—are identified as precisely formulated computations within the CTM framework. The mathematical structure for these computations exists; the specific instantiation for mass generation and dimensional reduction remains to be completed. Keywords: Canvas Temporal Mathematics, Canvas Model, gauge coupling unification, CKM matrix, PMNS matrix, CP violation, fermion generations, mathematical foundation, equality processor, spectral truth