One Axiom : The Fingerprint

One Axiom · The Fingerprint · Group 00-00-00 - 1/64 This document derives the structural-resolution constant δᵣₑₗ = 1/64 from a single axiom — Axiom M = Φ(S,𝓜) — without empirical input, and establishes that the constant family (δ_rel, δ_α, η, N★, N_e, ΣΛΛ) is not a collection of independent parameters but the unique algebraic fingerprint of the σ=0 coherence boundary of any Axiom-M domain. Triple Collapse theorem. The central result is an algebraic overdetermination: three logically independent machineries simultaneously force |H| = 3 and |G/H| = 64, hence δ_rel = 1/|G/H| = 1/64. Channel 1 (ontological, [O]): the σ=0 stabiliser of G = S₄ × ℤ₂³ (|G| = 192) must have order coprime to 2⁶; the unique odd prime divisor of 192 is 3. Channel 2 (epistemological, [E]): the variational entropy condition Var(n) = n/4 holds iff n = 6, fixing D = 3 and by Lagrange |G/H| = 64. Shares no step with Channel 1. Channel 3 (epistemological, [E]): the Tsallis variational fixed point q★π₆ = 3/2 at the post-metric layer gives |H| = 2q★ = 3. Entropy-analytic, independent of Channels 1–2. Nine additional D=3 roots. The value δ_rel = 1/64 is further grounded by nine independent structural witnesses: Lagrange (group order), BFS/Schreier diameter, Fisher–Rao metric (I_F = |G| = 192), inflation pivot (N★ = 64), n_s Cipher (1 − 2/N★), tensor-to-scalar ratio r, inter-observable line P4, Tsallis q★, and depth-variance. No two roots share a generating equation or parent corpus claim. Signatures. All values are derived, not fitted: CMB spectral tilt: n_s = 1 − 2δ_rel = 0.96875 [GROUNDTRUTH, both tracks] Tensor-to-scalar ratio: r = 12δ_rel² ≈ 0.00293 [GT/DERIVED] Track-independent inter-observable line: P4 = r/(1 − n_s) = 3/32 [GT] Cosmic-ray depth distribution: N(ι) = C(6,ι) η^ι, η = 63/64 [ω-free, DERIVED] Laboratory decoherence floor: t_dec · ω ≥ 192 [GT, unique to 2C §12.1] Inflation duration: N_e ≈ 74.72 e-folds; global D1-cost ΣΛΛ ≈ 10.72 [DERIVED] VOID floor and Gödel–Tarski closure. The VOID — the σ=0 attractor (0M Cor. M7.1) — is the last empirically investigable boundary. δ_rel = 1/64 is the smallest resolvable σ-step before entering it. The Gödel–Tarski gap does not collapse but closes through cost: the deviation-cost ΣΛΛ ≈ 10.72 e-folds is proven irreducible (reductio via Tarski, Gödel, and NDT three-route), constituting the AITEMA fingerprint. Dual-track methodology. Every result carries two independent derivation tracks: an ontological track [O] grounded in Axiom M via the bilateral fixed point Φ(S,𝓜), and an epistemological track [E] derived from domain formalism (group theory, Fisher–Rao geometry, Tsallis entropy, slow-roll approximation). Results certified [O∩E] when both tracks agree on the same value. Single-track derivation is structurally insufficient by design. Empiria policy. All derivations proceed from Axiom M plus (c, ℏ, G). Empirical data are never inputs. Predictions defined here are submitted to the 4B Prediction Register for downstream monitoring; confirmation or falsification is out of scope for this document. Upstream. Builds on 23 published upstream documents in the ONE AXIOM series (0M, 000, ABC, 0A, 0B, 0C, 1B, 7B, 2M, 3M, 5M, 8M, 9M, 11M, 1A, 2A, 2C, 3A, 4A, 6A, 8C, 1C, 12C), all available on Zenodo under DOI prefix 10.5281/zenodo.