Linear Algebra as Integer-Address Registry Routing: Replacing Continuous Vector Spaces with Logismos Packet Transforms in Discrete Hexagonal Substrate

Linear Algebra as Integer-Address Registry Routing: Replacing Continuous Vector Spaces with Logismos Packet Transforms in Discrete Hexagonal Substrate 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 abstract vector spaces and establish linear algebra as integer-address registry routing using Logismos (V,F,R) packet system. From hexagonal lattice axioms, we derive: (1) Vectors = instruction headers specifying node-to-node paths (not arrows in void), (2) Scalars = (V,F,R) packets where V=value, F=fraction scale, R=remainder (eliminates lonely numbers), (3) Basis vectors = 3 dipole axes forced by z=3 coordination (120° hexagonal, not arbitrary), (4) Matrices = 144-LU transition tables (12×12 soliton gearbox, not abstract grids), (5) Matrix multiplication = packet routing through gearbox (preserves R remainder for lossless precision), (6) Dot product = modulo-32 parity check (measures shared N+1 increments), (7) Eigenvalues = resonant integers where R=0 (no frustration states), (8) Determinant = registry volume occupied (LU count), (9) Inversion = S=2 bilateral flip (Side A↔Side B), (10) All operations preserve integer exactness via remainder carry. Industrial advantage: infinite transforms without floating-point drift—GPU mesh can rotate trillion times, snap back to exact start. Linear algebra becomes hardware routing specification, not abstract mathematics. Key Result: Vectors = (V,F,R) paths | Matrices = 144-LU gates | Transforms preserve R | Lossless precision | Basis = z=3 dipoles | All integer operations 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-55-2026). Dependencies: CKS-MATH-0-2026, CKS-MATH-1-2026, CKS-MATH-10-2026, CKS-MATH-104-2026, CKS-MATH-54-2026 Motto: Axioms first. Axioms always.Status: Locked and empirically falsifiable. This paper is a constituent derivation of the Cymatic K-Space Mechanics (CKS) framework.