Constructive Integer Physics from the Synchronous Triad

Abstract: 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 $\mathbb{Z}$, $\mathbb{H}_{\mathbb{Z}}$ (integer-scaled Hurwitz quaternions), and $\mathbb{O}_{\mathbb{Z}}$ (integer octonions), with zero continuum limits, zero irrational square roots, and zero adjustable parameters. Starting from the closed non-commutative triad ($A^2 = B^2 = C^2 = ABC = -I$) updating synchronously via $\Omega(\mathbf{s}_t) = s_0 I + \mathbf{v}(\mathbf{s}_t)$ with $\mathbf{s}_t \in \{+1, -1\}^3$ and $s_0 = -s_x s_y s_z$ on the body-centered cubic lattice $\Lambda_{\text{BCC}}$, twelve self-contained integer theorems establish: (1) 4D Hadamard orthogonality ($HH^T = 4I_4$), the $2T / Q_8 \cong \mathbb{Z}_3$ coset sweep ($C_1 \to 2C_2 \to 8Q_8$), and $16\text{D}$ sedenion zero-divisor termination at $N = 3$; (2) exact step-reversal quadrance relativity ($3N^2 = Q_{\mathbf{X}} + T_\tau$) and proper clock quadrance $Q_0 = N^2 - Q_{\mathbf{X}}$; (3) single-tick octonion conjugation, the quark/lepton confinement split, and the charge-color parabola $\vert{}\mathbf{s} \times \mathbf{c}\vert{}^2 = 4k(3 - k) \in \{0, 8, 8, 0\}$; (4) the unscaled electroweak identity $\hat{Q} = \hat{T}_3 + \hat{Y}_L$, the $72 : 120 : 192$ trace ratios ($S_W^{(0)} = 3/8$), and exact anomaly cancellation; (5) zero-sum Hydrogen baryogenesis ($p + n + e^- + \nu_L = 0$), Deuteron six-quark frustration stability, and the 3D nucleon magnetic relation $3\hat{\mu}_n + 2\hat{\mu}_p = 0$; (6) $N = 2$ vacuum scalar cancellation ($-48I + 48I = 0I$), discrete exterior calculus ($d^2 \equiv 0$), 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 $\text{lcm}(2, 3) = 6$ fermion knot closure, universal loop parity $(-1)^m$, $\theta_{\text{QCD}} \equiv 0$, the $\text{lcm}(3, 4) = 12$ associator braid, and the $(6, 3, 2)$ commutator overlap matrix; (9) the $N = 4$ return-loop partition ($216 = 24 + 144 + 48$), $120^\circ$ cyclic decoherence shift, Wigner's Friend turnstile, and Tsirelson bound ($\hat{C}^2 = 8I$); (10) the Six $24$s uniqueness of $d = 3$ and the $24m^2 + 2$ Euler causal shell identity; (11) the exact integer Euler–Schwarzschild quadrance equation, telescoping $4M^3 + 6M^2 + 3M$ horizon evaporation, and the Baryonic Tully–Fisher rotation trough $N\,\Delta Q_\perp^2 \approx 4M\,\Delta N^4$ without dark matter; and (12) 3D tetrahedral rotor aperiodicity ($U^2 - 2U + 9I = 0$).