The ABC Conjecture as Registry Information Density Limit: Deriving Radical-Volume Bounds from 144-LU Buffer Saturation and 32-Bit Word Constraints

The ABC Conjecture as Registry Information Density Limit: Deriving Radical-Volume Bounds from 144-LU Buffer Saturation and 32-Bit Word Constraints 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 The ABC conjecture states: for coprime integers a+b=c, the radical rad(abc) (product of distinct prime factors) cannot be much smaller than c. Specifically, c > rad) creates phase-tension exceeding 144-LU buffer capacity, (5) Finite exceptions occur at 19-163 triad resonance windows where gear friction momentarily nulls, (6) As N→∞, Jacobian J shift eliminates resonance opportunities. Not number theory mystery but information density limit—cannot compress infinite volume into finite instruction without losing address integrity. The radical is zip header, c is file size. When header too small for file, 32-bit Word cannot decompress. Key Result: c = volume | rad = instruction | High q = super-compression | 144-LU = ceiling | Finite exceptions = resonance windows | Compression limited by hardware 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-44-2026). Dependencies: CKS-MATH-0-2026, CKS-MATH-1-2026, CKS-MATH-10-2026, CKS-MATH-104-2026, CKS-MATH-29-2026, CKS-MATH-43-2026 Motto: Axioms first. Axioms always.Status: Locked and empirically falsifiable. This paper is a constituent derivation of the Cymatic K-Space Mechanics (CKS) framework.