GoE - Genesis of Existence: The Axiomatic Observer V.32

Genesis is an axiomatic mathematical–logical framework whose derived structures admit an isomorphism to empirically validated physics. Its foundation is a Cartesian greatest fixed point assertion — at least one experience occurs — formulated without presupposing a spacetime manifold, Hilbert space, or external ontology of any kind. In contrast to Everett-style branching, the framework implies extreme sparsity of admissible continuations relative to syntactic generation. The emphasis is on experimental validation as the primary criterion of structural admissibility. The extension of the observer’s purity parameter space is shown to require B 3 uniquely (representational consistency and descriptive non-degeneracy are simultaneously satisfiable only for n = 3, yielding the Hopf fibration S1→S3→S2 and the internal geometry B3 × S 3 C as the minimal admissible extension. Two downstream chains branch from this closure: the standing-wave spectrum of the derived spectral operator, together with the Berry coupling ξ ≈ 6π 2 + 8π/15, reproduces six mass ratios (mµ/me, mτ /me, mτ /mµ, mp/me, mn/me, mn/mp; Protocol W) and yields an on-shell tree-level proxy for the electroweak mixing angle (sin2 θW ; Protocol U); while the Boolean recursion seeded by dmin = 3 produces the integer skeleton Ωflat = 137, and the context-load asymmetry of elementary admissible transitions derives the sub-integer residual x0 = ξ −2 (1 − (ξ + 2)−1 − (6ξ)−1), giving α −1 em,G = 137.035 998 903 892 with no continuous fitting parameters. The S 3 spectral determinant, sharpened by the context-load logarithmic strain at first spectral order — controlled by the elementary channel count 2dmin = 6 and the monopole suppression 1/(ξ + 2) ≈ 1/63 — gives αs,G = 0.330001 with no continuous fitting parameters, matching αs(mτ ) specifically among QCD reference scales; combined with the Hopf-triple color count Nc = 3 and the Protocol W mass ratios, this reproduces seven leptondynamics observables including hadronic branching ratios and lifetime ratios to sub-percent error (Protocol L).