The Synchronous Triad IV: The Ontological Structure of the Integers, Self-Referential Completed Infinity, and the Algebraic Necessity of Unforced Choice

In Papers I–III of this series (10.5281/zenodo.23884635, 10.5281/zenodo.23908121, 10.5281/zenodo.23930280), we demonstrated that a self-supporting reality emerging from the Void (0I) with zero external scaffolding and zero free parameters is uniquely locked into the division-free integer quaternion ring R_Triad = Z[I,A,B,C] (A^2 = B^2 = C^2 = ABC = -I <=> I = -ABC = CBA) and its parity-locked octant subring L_Triad = 2H_Z. In this fourth paper, we establish the foundational consequences of this structure for the ontology of the integers, completed infinity, and branch selection. First, we prove that self-supporting emergence from 0I requires two coupled binary operations—torsion-free antipodal Void-balance (+, char(R) = 0) and non-annihilating relational crossing across all superposed states (x, domain integrity) linked by Void-preserving distributivity X(Y+Z) - XY - XZ = 0I (forced by the absence of any external coupling medium, which in turn forces additive commutativity X+Y=Y+X)—and establish the Triadic Law of Observation: under horizontal root finiteness and +X <-> -X antipodal automorphism indiscernibility (X^2 in {0I, +I, -I, +X, -X} for X in {A, B, AB}), Distinction (A) and Selection (B) cannot exist as a two-term duality without a third orthogonal element—Historical Memory (C = AB = 1/2[A,B])—because any system with X^2 = 0I, X^2 = ±X, X^2 = +I, AB = 0I, AB = ±I, or AB = BA either shatters into zero-divisors that annihilate prime factorization history or collapses into a commutative plane where chronological order is erased. Moreover, minimal central generation (Center(R) = ZI, char(R) = 0) and negative-definite generator squares (A^2 = B^2 = C^2 = -I) derive trace-freeness (Tr(X) = 0I for X in {A,B,C}), guarantee positive-definite norm closure N(Q) = (c_1^2 + c_2^2 + c_3^2 + c_4^2)I and full 4-component domain integrity, and force the trace-shifted cyclotomic step identity Omega_0^2 + 2*Omega_0 + 4I = 0I => Omega_0^3 = +8I. Second, distinguishing meta-level syntactic recursion from object-level arithmetic ontology, we construct R_Triad = Z[Q_8]/<(+I)+(-I)> directly from the 8-element non-abelian group Q_8 = {±I, ±A, ±B, ±C} by quotienting out the unique group-ring zero-divisor factor (+I) + (-I) = 0I. Whereas set-theoretic Von Neumann ordinals (0 := {}, 1 := {{}}) rely on an asymmetric external container {.} that violates zero-sum conservation, R_Triad exhibits an Additive–Multiplicative Fiber Inversion: every 1D additive central accumulation nI = I + ... + I is inverted by the multiplicative norm map N(Q) = Q*conj(Q) = nI into a 4D spinor fiber S_n (|S_n| = r_4(n)), with Peano addition forming the collinear zero-braiding (Delta I_nc = 0) boundary of the Cauchy–Schwarz Path-Budget Simplex W^2 = N(Q) + Delta I_nc. Third, we prove that folding any branch under the ring's two native operations (+ and x) and superposing via the local fibered partial operation +_loc over the division-free level sets of Vec(Sigma_N(Gamma)) = X realizes each slice L_N = G^N directly inside the self-graded group ring R_Triad[Vec(R_Triad)] with zero external coordinate spaces. The unfiltered totality of the Triad is a timeless, zero-sum graded infinite crystal U = (+) L_N anchored at N = 0 and replicating inward via division-free Dedekind self-embeddings R_Triad = I₃ₖ = 8^k R_Triad and Sandwiched Sub-Triads T_q = q R_Triad conj(q) of lattice index m^4. Fourth, we prove the Closed-Form Global Coherent Norm Law for all cosmic ticks N >= 1:sum_X N(Q(X, N)) = sumⱼ₌₀floor(N/2) C(N, 2j) * 8^(N - 2j) * C(2j, j)^3 = [4 * 16^N / (pi * N)^(3/2)] * (1 + 3/(8N) + O(N^-2)),showing via exact zero-parity-truncation binomial moments and 3D Fourier cumulants that the unselected linear wave loses its coherent spinor fraction at the exact rate 4^-(N-1) * (1 + 3/(4N) + O(N^-2)), while diagonal Huygens worldlines Pi_k(Gamma) preserve 100% of their prime norm inside the intact zero-sum crystal U. Finally, we prove the Centralizer, Torsion, and Relational Non-Autonomy Theorem, the Left-Frontier Geodesic-Loop Asymmetry Theorem, and the Global–Local Duality of Unforced Selection on Lubotzky–Phillips–Sarnak Ramanujan prime trees S_p^can, showing how forward algebraic non-autonomy from Q_k and backward mutual prime exclusivity between sibling branches Qₖ₊₁^(a) = pi_a Q_k and Qₖ₊₁^(b) = pi_b Q_k (a != b) unify global branch coexistence in U with definite local branch realization without an external collapse postulate. Includes a standalone, 100% float-free pure-integer Python 3 verification suite (verify_paper4.py) auditing all theorems.