One-Octonion Brane-Bulk Framework - Paper CCCLII: The S4 Permutohedron as a Positive Geometry — Exact Canonical Form, Facet Factorization, the Torsor Labeling of the Klein Quartic's 24 Heptagons, and a Candidate Scattering Polytope for the Matter Tetrahedron

Abstract

The series identifies every Standard-Model interaction vertex with a Reidemeister move on knotted brane loops, and the path integral with a sum over Reidemeister sequences (Papers VIII, XII); the standing open item is the measure. This paper establishes the geometric half of a candidate answer; the physical half is stated as open. Three structural results are proved by exact computation. (1) The Ursa Major S4, the index-7 maximal subgroup of PSL(2,7) permuting the matter basis H+ (Paper CXXI), acts simply transitively on the 24 heptagons of the {7,3} tiling of the Klein quartic: the heptagons form an S4-torsor. This replaces a wrong "duality" reading of the 24-to-24 coincidence: the cell-complex dual of the heptagonal tiling is the {3,7} triangle tiling. (2) The S4 permutohedron (24 vertices, 36 edges, 14 facets) is a smooth positive geometry; unimodularity forces unit numerators, so the canonical function is the flag sum Omega = sum 1/(s1 s2 s3), equal to 16/3 at the Weyl center. (3) Omega factorizes geometrically on its boundary: the 14 facets are the 14 nonempty proper subsets of H+; residues on the six square facets are A1xA1 canonical forms (the three class-2A pairings, candidate Yukawa pair channels), and on the eight hexagonal facets A2 canonical forms; the centroid residues 6 and 16 are computed by hand in the text and independently verified. The walk operator on the permutohedron edge graph decomposes into the same abstract S4 irreducibles the series assigns to up-type (3), down-type (3'), and charged-lepton (1+2) Yukawa structure, with exact spectrum +-3, +-sqrt(3), +-{1+sqrt(2), 1, sqrt(2)-1}, an abstract-type match rather than a dynamical statement. The positive-geometry calculus of Arkani-Hamed, Bai, and Lam is imported as mathematics; the N=4 amplituhedron object is not used. One statement is registered as a conjecture: under a derived kinematic embedding, the tree-level Reidemeister-sequence sum equals the canonical function of the permutohedron. Three missing bridges are named, K1 the kinematic embedding (including the discrete-sum versus differential-form mode of correspondence), K2 the strain bridge to S = U_trail (Paper LXXVIII), K3 the move-to-edge map with vertex-factor weights, with kill conditions D1/D2 stated as necessary, not sufficient. No experimental prediction is registered. Strictly contingent on the established One-Octonion Brane-Bulk framework (Omnibus, concept DOI 10.5281/zenodo.19185171; edition cited: 0-CCCL, version DOI 10.5281/zenodo.21971669). Files: PDF, the verification script, and the figure. Version 2 (2026-09-08): physics-only register; footer constant corrected to f_bulk = 3.6919%.