Constructive Integer Physics from the Synchronous Triad

A parameter-free, constructive mathematical and physical verification specification deriving relativistic kinematics, Standard Model quantum numbers, quantum measurement dynamics, nuclear stability, and discrete general relativity strictly within the integer rings Z, HZ (integer-scaled Hurwitz quaternions), and OZ (integer octonions), with zero continuum limits, zero irrational square roots, and zero adjustable parameters. Starting from the closed non-commutative triad (A² = B² = C² = ABC = -I) updating synchronously via Ω(sₜ) = s₀ I + v(sₜ) with sₜ ∈ \+1, -1\³ and s₀ = -sₓ sy sz on the body-centered cubic lattice ΛBCC, twelve self-contained integer theorems establish: (1) 4D Hadamard orthogonality (HHT = 4I₄), the 2T / Q₈ Z₃ coset sweep (C₁ → 2C₂ → 8Q₈), and 16D sedenion zero-divisor termination at $N = 3$; (2) exact step-reversal quadrance relativity (3N² = QX + T_τ) and proper clock quadrance Q₀ = N² - QX; (3) single-tick octonion conjugation, the quark/lepton confinement split, and the charge-color parabola s × c² = 4k(3 - k) ∈ \0, 8, 8, 0\; (4) the unscaled electroweak identity Q̂ = T̂₃ + ŶL, the $72 : 120 : 192$ trace ratios (SW⁽⁰⁾ = 3/8), and exact anomaly cancellation; (5) zero-sum Hydrogen baryogenesis (p + n + e⁻ + νL = 0), Deuteron six-quark frustration stability, and the 3D nucleon magnetic relation 3μ̂ₙ + 2μ̂ₚ = 0; (6) $N = 2$ vacuum scalar cancellation ($-48I + 48I = 0I$), discrete exterior calculus (d² ≡ 0), and orthogonal photon polarizations; (7) the $N = 3$ Hurwitz chiral lock, the $1$-to-$1$ forward-choice vs. backward-determinism bijection, and the $(3,3,5,2,5,2,5,2)$ inner-core spectrum; (8) the lcm(2, 3) = 6 fermion knot closure, universal loop parity (-1)ᵐ, θQCD ≡ 0, the lcm(3, 4) = 12 associator braid, and the $(6, 3, 2)$ commutator overlap matrix; (9) the $N = 4$ return-loop partition ($216 = 24 + 144 + 48$), 120^∘ cyclic decoherence shift, Wigner's Friend turnstile, and Tsirelson bound (Ĉ² = 8I); (10) the Six $24$s uniqueness of $d = 3$ and the 24m² + 2 Euler causal shell identity; (11) the exact integer Euler–Schwarzschild quadrance equation, telescoping 4M³ + 6M² + 3M horizon evaporation, and the Baryonic Tully–Fisher rotation trough N\,Δ Q_⊥² ≈ 4M\,Δ N⁴ without dark matter; and (12) 3D tetrahedral rotor aperiodicity (U² - 2U + 9I = 0).