Fractal Coherent Logic (LFC): A Unified Arithmetic Framework — Complete Series (Papers 1–10)

A series of ten papers presenting Fractal Coherent Logic (LFC), a mathematical framework founded on the multiplicative group (ℤ/60ℤ)*, whose φ(60) = 16 coprime elements form 8 symmetric pairs (a, 60−a). The modulus 60 = 2²·3·5 emerges from the Fibonacci entry ranks: lcm(α(2), α(3), α(5)) = lcm(3,4,5) = 60. Paper 1 — Fractal Coherent Logic (FCL): The Arithmetic Substrate of Primes. Introduces the 4-state space {−1,−0,+0,+1}, the mod 60 substrate, and proves the Binary Collapse Theorem: p² mod 60 ∈ {1,49} for every prime p > 5, with exact 8/8 distribution. Establishes the local distribution rule via ord₁₅(2) = 4. Paper 2 — The Algebra of the Map: V₄, Hadamard, and the Conductors. Proves that V₄ = {1,11,49,59} is a normal subgroup isomorphic to the Klein four-group. The quotient yields 4 cosets {A,T,G,C} with Hadamard H₄ orthogonality and distinct conductors {1,5,12,60} via CRT. Paper 3 — Grand Simplicity mod 60: The Klein Kernel, the Closed Circuit, and the Riemann Hypothesis. Presents three theorems: T1 (analytic confinement via Wronskian), T2 (Grand Simplicity via SMO with distinct conductors), T3 (RH as conservation law of the closed circuit). Paper 4 — The Breathing of the Field: Invariant Template, W_im > 0, and Extension to All Zeros. Proves the invariant template theorem (slots, axis, proportion φ preserved as N → ∞) and establishes W_im > 0 globally via the functional equation identity. Paper 5 — The Riemann Hypothesis Is a Tautology. Synthesizes Papers 1–4: ζ is the observable of a circulating field with exactly one collapse axis (σ = 1/2). RH is a consequence of the object's definition, not an external restriction. Paper 6 — P ≠ NP: The Asymmetry Is in the Model, Not the Problem. Identifies the P/NP gap as the lossless/lossy gap of the binary computation model: directional information is lost at zero crossings in {0,1} but preserved in {−1,−0,+0,+1}. Paper 7 — Yang-Mills and Mass Gap: Spectral Confinement from the Arithmetic Circulant Field. The covariance matrix of (ℤ/60ℤ)* has spectral gap Δ = 16/15 > 0, providing mass gap from arithmetic structure. W_im > 0 confirms absence of zero modes. Paper 8 — Navier-Stokes Global Regularity via Directional Preservation at Zero Crossings. Blow-up requires directional loss at vorticity zero crossings. In the quaternary model, −0 ≠ +0 preserves direction, yielding Lipschitz continuity of ξ = ω/|ω| and no blow-up via Constantin-Fefferman-Majda criterion. Paper 9 — Birch and Swinnerton-Dyer: The Euler Product Decomposed by Cosets. Reorganises the Euler product of L(E,s) over the 4 cosets of (ℤ/60ℤ)*/V₄. Conjectures rank r counts simultaneous Hadamard collapses at s = 1. Paper 10 — Ramanujan's Map: 23 Results as Projections of (ℤ/60ℤ)*. Shows that 23 classical results of Ramanujan — mock theta functions, partition congruences, τ-function, 1729, Rogers-Ramanujan identities, formulae for π — simplify to arithmetic in (ℤ/60ℤ)*. All papers include computational verification scripts (Python, zero external dependencies). The framework is self-contained: each paper builds strictly on the preceding ones.