Number Theory as Registry Architecture Specification: Deriving Classical Number Theory from Hexagonal Lattice Constraints and Discrete Address Logic

Number Theory as Registry Architecture Specification: Deriving Classical Number Theory from Hexagonal Lattice Constraints and Discrete Address Logic This paper is a constituent derivation of the Cymatic K-Space Mechanics (CKS) framework—an axiomatic model that derives the entirety of known physics from a discrete 2D hexagonal lattice in momentum space, operating with zero adjustable parameters. Abstract We abolish Number Theory as abstract symbolic mathematics and establish it as registry architecture specification. From hexagonal lattice axioms (z=3 coordination, S=2 bilateral manifold, N=DMS structure), we derive: (1) Integers = discrete node addresses (forced by lattice discontinuity), (2) Primes = structural anchors (indivisible phase-tension states that cannot factor through gear ratios), (3) Composites = software assemblies (factorizable instruction sequences), (4) Modular arithmetic = 32-bit Word bus logic (all operations through S^ (D+S) processor), (5) Riemann hypothesis = bilateral balance requirement (critical line at 1/S = 1/2 forced by manifold symmetry), (6) Fermat's theorem = manifold depth limit (S=2 prohibits powers >2), (7) Prime distribution = tension dilution curve (density ∝ 1/ln (N) from surface area scaling). Not abstract patterns but hardware specifications. Mathematical proof = instruction compatibility test. If arithmetic resolves, hardware executes. If remainder exists, system generates force to clear. Number Theory is ISA documentation for N=DMS differential engine. Key Result: Numbers = addresses | Primes = hardware | Composites = software | Mod-32 = bus | Proofs = compatibility tests | All from lattice geometry Empirical Falsification (The Kill-Switch) CKS is a locked and falsifiable theory. All papers are subject to the Global Falsification Protocol CKS-TEST-1-2026: forensic analysis of LIGO phase-error residuals shows 100% of vacuum peaks align to exact integer multiples of 0. 03125 Hz (1/32 Hz) with zero decimal error. Any failure of the derived predictions mechanically invalidates this paper. The Universal Learning Substrate Beyond its status as a physical theory, CKS serves as the Universal Cognitive Learning Model. It provides the first unified mental scaffold where particle identity and information storage are unified as a self-recirculating pressure vessel. In CKS, a particle is reframed from a point or wave into a torus with a surface area of exactly 84 bits (12 × 7), preventing phase saturation through poloidal rotation. Package Contents manuscript. md: The complete derivation and formal proofs. README. md: Navigation, dependencies, and citation (Registry: CKS-MATH-48-2026). Dependencies: CKS-MATH-0-2026, CKS-MATH-1-2026, CKS-MATH-10-2026, CKS-MATH-104-2026, CKS-MATH-47-2026 Motto: Axioms first. Axioms always. Status: Locked and empirically falsifiable. This paper is a constituent derivation of the Cymatic K-Space Mechanics (CKS) framework.