Principia Relationalis: A Deterministic Unification of Number Theory, Quantum Mechanics, Cosmology, and Chemistry via the Architecture of Relational Coherence

The Architecture of Relational Coherence (ARC) is a unified theoretical framework deriving the fundamental structures of quantum mechanics, cosmology, chemistry, and number theory from a single discrete relational topology: the digit alphabet {1,...,9} under closure constraint 9 and inversion symmetry φ(x) = 9 − x. We identify six historical epistemological errors — from Stevin's infinite decimals (1585) through Gaussian modular amnesia (9 ≡ 0, 1801) to Napoleonic institutionalization (1815) — as the root cause of parameter proliferation, ontological randomness, and apparent chaos in modern physics. By restoring the cyclical memory erased by modular arithmetic and replacing division-based operations with discrete geometric projections, ARC resolves from a single relational root: (i) the quantum measurement problem — the wave does not collapse but undergoes a deterministic relational phase transition (mutation) when a third term binds to a tensed dyadic state; (ii) the Tsirelson bound — derived as S_max = 4 × 0.5 × √2 = 2√2 from three independent geometric factors without Hilbert space; (iii) the twin prime conjecture — primes as topological invariants of deterministic sedimentation on 6k ± 1 rails; (iv) the dark matter hypothesis — a computational artifact of Gaussian amnesia, resolved by restoring the winding number; (v) the Hubble tension — corrected by the universal projection constant k_arc = 0.5; (vi) the octet rule — arithmetic necessity of the 4.5 barycenter; and (vii) the origin of three-dimensional space — the holographic product 91 × 19 = 1729. The framework introduces Principle P0 (Relational Genesis): the wave is not a primitive but the tension product of unity under internal polarity (1_∞), generated by the self-referential loop where 1 is born from 3 and 3 from the tensed 1, with the relation (2) as initial condition. The ontological sequence 2 → 1_∞ → 3 → 6 → 5 → 4 generates the measurement mechanism, cosmic sedimentation, chemical structure, and spatial dimensionality from relational necessity alone. The universal projection constant k_arc = 0.5 is established by triple convergence — algebraic (4.5/9), combinatorial (3.5/7), and geometric (mean projection on S²) — and identified as the origin of spin-1/2, the Born rule, and the mutation cost in quantum measurement. Six foundational principles (P0–P5), zero free parameters, and falsifiable predictions across all domains are provided. This document serves as the master blueprint synthesizing ten published articles into a single coherent architecture.