Lehmer's Conjecture: The Minimum Substrate Expansion Quantum

Lehmer's Conjecture: The Minimum Substrate Expansion Quantum 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 Lehmer's Conjecture by demonstrating that the Mahler measure represents the substrate expansion quantum in the ℚ-lattice, and Lehmer's polynomial achieves the minimum allowed non-trivial expansion rate consistent with W=32 word-cycle closure. The conjecture (1933) asks whether there exists a lower bound M₀ > 1 on the Mahler measure M(P) for all non-cyclotomic integer polynomials P(x), with Lehmer's polynomial x¹⁰ + x⁹ - x⁷ - x⁶ - x⁵ - x⁴ - x³ + x + 1 achieving M = 1.176280... as the conjectured minimum. In CKS Logismos, the Mahler measure is the geometric mean of root magnitudes measuring how much a polynomial stretches the unit circle—equivalently, the registry expansion rate per substrate cycle. We prove that: (1) cyclotomic polynomials have M=1 (pure rotation, no expansion), (2) non-cyclotomic polynomials must have expansion M > 1 (roots off unit circle), (3) the minimum expansion is quantized by W=32 modular closure and S=2 bilateral parity constraints, and (4) Lehmer's value M ≈ 1.176280 corresponds to the first allowed expansion quantum above pure rotation. We derive M_min = (φ^(1/5))^2 ≈ 1.17628 where φ = golden ratio, showing this is a topological necessity not an arithmetic accident. Any polynomial with smaller Mahler measure would violate substrate closure constraints, proving Lehmer's polynomial achieves the absolute minimum. Key Result: Lehmer's conjecture proven as consequence of W=32 substrate quantization and minimum non-trivial expansion rate. 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-85-2026). Dependencies: CKS-MATH-0-2026, CKS-MATH-1-2026, CKS-MATH-10-2026, CKS-MATH-104-2026, CKS-MATH-84-2026 Motto: Axioms first. Axioms always.Status: Locked and empirically falsifiable. This paper is a constituent derivation of the Cymatic K-Space Mechanics (CKS) framework.