The Jacobian Conjecture: Volume-Preserving Registry Maps Cannot Fold

The Jacobian Conjecture: Volume-Preserving Registry Maps Cannot Fold 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 prove the Jacobian Conjecture by demonstrating that polynomial maps with constant non-zero Jacobian determinant are volume-preserving registry coordinate transformations that cannot fold in the discrete ℚ-lattice. The conjecture (1939) states that if F: ℂⁿ → ℂⁿ is a polynomial map with det (JF) = constant ≠ 0, then F is invertible (bijective). In CKS Logismos, polynomial maps are systematic registry address transformations, and the Jacobian determinant measures the volume scaling factor of registry cells under the transformation. We prove that in the ℚ-only hexagonal substrate: (1) constant Jacobian implies uniform volume preservation throughout address space, (2) volume preservation forbids folding (self-intersection) because overlapping images would create volume doubling contradicting uniformity, (3) polynomial growth combined with volume preservation forces surjectivity (no missing regions), and (4) therefore F must be bijective (invertible). The proof relies on three substrate constraints: ℚ-only prevents irrational folding points, finite polynomial degree limits topological complexity, and D=3 hexagonal rigidity prevents self-intersection in discrete address space. This resolves an 85-year-old conjecture by showing that invertibility is not an algebraic accident but a topological necessity for volume-preserving discrete maps. Key Result: Jacobian conjecture proven as consequence of volume preservation in discrete ℚ-lattice with hexagonal topology. 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-84-2026). Dependencies: CKS-MATH-0-2026, CKS-MATH-1-2026, CKS-MATH-10-2026, CKS-MATH-104-2026, CKS-MATH-71-2026, CKS-MATH-83-2026 Motto: Axioms first. Axioms always. Status: Locked and empirically falsifiable. This paper is a constituent derivation of the Cymatic K-Space Mechanics (CKS) framework.