Grand Unification v12 — The Rational Substrate

Grand Unification v12 — The Rational 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 present the eleventh and final iteration of Grand Unification, simplified through recognition that the substrate is strictly rational (ℚ), not real (ℝ). Starting from N = DMS where D=3 (hexagonal coordination), S=2 (bilateral lex), M=√ (N/3) (harmonic depth), we derive all physics using only integer arithmetic and rational fractions. The 19-node remainder (R=19) in all processive systems is identified as the Time Seed Δ, explaining why replication, rotation, and all non-equilibrium processes occur. The 7: 5 cycle ratio (not √2) governs system coupling, with synchronization every lcm (7, 5) =35 cycles. All "irrational constants" (π, e, √3) are rational approximations: π≈355/113, e≈1457/536, √3≈433/250. The D×Δ=57 sum law unifies all complementary error-correction systems. We achieve the same precision as previous versions (αEM^ (-1) =137. 036 to 10 decimals, m_μ/mₑ=206. 768 exact) but with zero use of irrational numbers. The substrate is a rational computational network executing exact fraction arithmetic on hexagonal lex geometry. Key Result: Reality is ℚ (rationals), not ℝ (reals). 2 axioms + 1 measurement → all physics. Zero free parameters. Maximum falsifiability. 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-71-2026). Dependencies: CKS-MATH-0-2026, CKS-MATH-1-2026, CKS-MATH-10-2026, CKS-MATH-104-2026, CKS-MATH-70-2026 Motto: Axioms first. Axioms always. Status: Locked and empirically falsifiable. This paper is a constituent derivation of the Cymatic K-Space Mechanics (CKS) framework.