Fractal Coherent Logic (LFC/FCL): A Unified Arithmetic Framework — Complete Series (Papers 1–16)

A series of sixteen papers presenting Fractal Coherent Logic (LFC/FCL), a mathematical framework founded on the multiplicative group (ℤ/60ℤ)*, whose φ(60) = 16 coprime elements form 8 symmetric pairs (a, 60−a) with axis at 30. The modulus 60 = 2²·3·5 emerges from the Fibonacci entry ranks: lcm(α(2), α(3), α(5)) = lcm(3,4,5) = 60. Each result is classified: [P] proved, [D] derived, [C] calculated, [H] hypothesis. Paper 1 — FCL: The Universal Arithmetic Foundation. Introduces the progression 0→1→2→3→4, the mod 60 substrate derived by exhaustive elimination, the 16 coprime slots, Klein V₄, Binary Collapse (p² mod 60 ∈ {1,49}), the fractal ladder L0–L9, and the triangle of α⁻¹ (gnomon 137, shadow π, hypotenuse α⁻¹). Establishes q = 60 as the substrate identity via CRT and Siegel-Walfisz. Paper 2 — The Algebra of the Map: V₄, Hadamard, and Conductors. Proves 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. Grand Simplicity mod 60 via Strong Multiplicity One. Paper 3 — The Closed Circuit and the Riemann Hypothesis. Three theorems: T1 (Re[∂ξ/∂σ] = 0 ⟺ σ = 1/2, from functional equation), T2 (Grand Simplicity via SMO), T3 (W_im = g² > 0 → σ = 1/2, conditional). Two independent paths: Path A (algebraic identity ḟ = −g) and Path B (Wronskian-ODE, Im(V) antisymmetry). Paper 4 — The Invariant Template: Expansion, Retraction, and Anti-correlation. Proves the template is invariant (16 slots conserved, axis 30 invariant, anti-correlation ρ = −1/15 as structural constant). Retraction: Var/Mean → 1/φ refuted (2026-03-24); replaced by algebraic chain ρ → 1/φ via continued fraction. Paper 5 — The Riemann Hypothesis as a Consequence of the Substrate. Synthesises Papers 1–4, adds spectral verification (FFT of prime oscillation = zeros of ζ, 50/50 match, two orthogonal signals), Sub-Poisson CRT Theorem, and Path D (positional argument): the 16 angular positions are fixed properties of the group, axis 30 is forbidden (gcd(30,60)≠1), g(σ,t) cannot change sign because the geometry does not change. 10 steps, all [P], zero [H]. The simplicity of zeros follows from the fixed substrate geometry, not as an external hypothesis. Paper 6 — P ≠ NP: Structural Asymmetry of the Binary Model. The P/NP gap is the lossless/lossy gap of {0,1}: directional information is lost at zero crossings in binary but preserved in {−1,−0,+0,+1}. Paper 7 — Yang-Mills: Spectral Gap 16/15. The covariance matrix of (ℤ/60ℤ)* has eigenvalues {0, 16/15}, providing spectral gap Δ = 16/15 > 0 from arithmetic structure. Structural analogy [A] with Yang-Mills mass gap. Paper 8 — Navier-Stokes: Regularity via Directional Preservation. Blow-up requires directional loss at vorticity zero crossings. In the quaternary model, −0 ≠ +0 preserves direction, yielding Lipschitz continuity and no blow-up via BKM/CFM criteria. Conditional [H] on the physical model. Paper 9 — BSD: Local-Global via Cosets. Reorganises the Euler product of L(E,s) over the 4 cosets of (ℤ/60ℤ)*/V₄ with Hadamard decomposition. Conjectures rank r counts simultaneous collapses at s = 1. Paper 10 — Ramanujan: 23 Results as Projections of (ℤ/60ℤ)*. Shows that 23 classical results of Ramanujan — mock theta functions, partition congruences, τ-function, 1729, Rogers-Ramanujan, formulae for π — are projections of the mod 60 structure. Paper 11 — Poincaré: Discrete-Continuous Convergence. The LFC discrete construction (CRT, V₄, cosets) and Perelman's continuous proof (Ricci flow) converge on the same topological object. Structural analogy [A]. Paper 12 — α⁻¹ = 137.036: The Portal. Four independent derivations of the integer 137 [P]. Complete formula α⁻¹ = 137 + 9/250 − 224/5¹² yields 137.035999082496, within 0.07σ of CODATA. Integer [P], fractional form [H]. Paper 13 — Hodge: The Codec as Algebraic Class Detector. The Codec Hodge compression operator (byte >> 2 + byte & 3) separates algebraic (harmonic) from residual components. Shannon entropy reduced by 53%. Structural analogy [A] with the Hodge conjecture. Paper 14 — Singularity of n = 60. Proves C(φ(n),2) = π_Pisano(n) is satisfied uniquely by n = 60 (verified for n ≤ 10,000). Four independent routes to the singularity. Diophantine uniqueness: n = (φ(n)−1)×λ(n) has unique solution n = 60. Paper 15 — The Hierarchy L0→L9. The factorisation 60 = 2²·3·5 forces a 10-level hierarchy (L0–L9) where each level emerges from the previous by the same funnel 0→∞→1. L7–L9 correspondence with replication/transcription is [A]. Paper 16 — Perception as Casimir Effect. Proposes T5b = 16/π² as the observer-field coupling constant. Testable predictions [H] in four domains: colour opponency (ρ = −1/15), musical scale (12 = τ(60)), binocular fusion, phonemic classes (φ(60) = 16). All papers in three languages (PT-BR, EN, ES). Computational verification scripts included (Python). The framework is self-contained: each paper builds strictly on the preceding ones. Version 3.0 (2026-03-30) includes Path D resolving the simple-zeros gap.