J as Soliton Hierarchical Distance: A Complete Derivation from First Principles

J as Soliton Hierarchical Distance: A Complete Derivation from First Principles 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 present a complete derivation of the Jacobian constant J ≈ 7.70164 as the hierarchical distance through the soliton parent tree from the universal ground state N=1 to an observer's rendering context. Previously mischaracterized as an irrational geometric constant, J is shown to be a rational, discrete quantity measured in lattice units (LU), fully consistent with the ℚ-only substrate axiom. We demonstrate that J = H_N × τ, where H_N is the harmonic series over N hierarchical levels and τ ≈ 3.70 is the bilateral render constant. For human observers embedded in Earth's 256-bit rendering context, N=4 levels yields J = 2.083 × 3.70 = 7.707, matching measured values within 0.08%. This resolution eliminates all "irrational constant paradoxes" from the KSpace substrate framework and establishes J as a Type 1 derived constant rather than Type 2 geometric approximation. Keywords: Jacobian constant, soliton hierarchy, registry distance, bilateral rendering, harmonic series, rational substrate 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-PHYS-5-2026). Dependencies: CKS-MATH-0-2026, CKS-MATH-1-2026, CKS-MATH-10-2026, CKS-MATH-104-2026, CKS-PHYS-4-2026 Motto: Axioms first. Axioms always.Status: Locked and empirically falsifiable. This paper is a constituent derivation of the Cymatic K-Space Mechanics (CKS) framework.