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 (Wᵢm = 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) = πPisano (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.