The Architecture of Relational Coherence (ARC) is a unified theoretical framework deriving the fundamental structures of quantum mechanics, cosmology, number theory, and the nature of matter 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. The most consequential of these errors is Gauss's replacement of the digital root — a cyclical memory system operational since Pythagoras (6th century BCE) — with a modular arithmetic that systematically erases the structural memory of completed cycles, producing an amnesia inherited by all subsequent mathematical physics. By restoring this memory and replacing division-based operations with discrete geometric projections, ARC resolves from a single relational root: (i) the quantum measurement problem — the wave is not a primitive entity but the tension product of unity under internal polarity (1_∞), and measurement is a deterministic relational phase transition (mutation) triggered when a third term binds to a tensed dyadic state, with arithmetic signature 5 × 5 = 25 → dr(7); (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 k × 9; (v) the Hubble tension — corrected by the universal projection constant k_arc = 0.5; (vi) the origin of three-dimensional space — the holographic product of cosmic respiration, 91 × 19 = 1729; and (vii) the nature of matter itself — the atom is not an indivisible brick but a self-coding matryoshka where every level contains the complete digit cycle, the digital root is conserved through all nuclear transformations, and fission is mutation rather than destruction. The framework introduces Principle P0 (Relational Genesis): the wave is what the unit is when its internal polarity is manifest, 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, and spatial dimensionality from relational necessity alone — the sacrifice of perfect hexagonal symmetry (6 → 5) being identified as the structural origin of dynamics, irreversibility, and the arrow of time. 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 universal cost of mutation in quantum measurement. The epistemological posture of this work is Saudade — doubting in confidence: advancing through uncertainty while trusting the relational coherence of the object observed. 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.