One-Octonion Brane-Bulk Framework — Paper CCCLXII: The Yang–Mills Mass Gap as Finite Incidence Geometry — the Shortest Colour-Singlet Loop Floor and its Hubble-Scale Dilational Complement

Version 3 (2026-09-04): register revision, no claim-content change - prize-disclaimer and community-judgment meta removed throughout; Section 1.2 restructured to two claim levels (the framework's solution, CLAIMED conditional on the framework; the dissolution position, with its refutation condition); scope facts stated once, declaratively. The physics is presented as the physics. Version 2 (2026-09-04, same day): (a) adds Section 1.2, "What solution means here": the paper now claims the framework's SOLUTION of the mass-gap question explicitly at the conditional level (within OOB the physical question is answered - theorem-level positivity from the Gram-signature route plus this paper's scale identification), states the framework's POSITION that the Clay formulation mis-poses the question (continuum R^4 is pseudocontinuous and not fundamental in OOB: Papers CXXXI, CCXXV, CCCXXXI, CXXXIII; map-not-territory), gives that position its own refutation condition (a successful Clay-standard continuum construction would refute the dissolution thesis while leaving the internal mechanism untouched), and states plainly that the Millennium Prize as formulated is NOT claimed. (b) adds Section 2.1, "Relation to the framework's earlier mass-gap statements" — the paper is the loop-geometric leg of a published family: the Gram-matrix signature theorem of Papers CVIII/CXXIV (Delta = |mu_0| Lambda_QCD > 0, mu_0 = -1+6 gamma, confinement as rank reduction, with its own Clay-scope discussion), the least-action/pseudocontinuity route of Paper CXXXI, the C_2(G_2) = 4 Casimir route of Paper CXXXIII, the non-associativity-energy reading of Paper CLXXVII, and the Seifert-form positivity reading of Papers CXXVI-CXXVII; the charge-level closure machinery is Paper LIX's topological winding definition of colour (pi_2(S^2) = Z; singlet = zero net winding; SU(3)_C = Stab_G2(e_7)). Section 8's open program is re-scoped against these corpus anchors (what remains open is the selection of the 14/L form and the count-level reduction, not the existence of dynamical routes). v1 omitted this map; the omission was caught by the author. Mechanism paper for the gauge-sector mass gap of the One-Octonion Brane-Bulk (OOB) framework, isolated as a standalone finite-incidence statement with its scale identification and strict sector bookkeeping. The paper states at every load-bearing point that it does NOT supply the Clay-standard constructive quantum-field-theory proof: it offers a geometric mechanism and a scale, not the theorem. In the gauge sector, a lone coloured excitation is not a physical asymptotic state; the first brane-readable excitation is a closed colour-singlet loop, which must visit all three colour reservoirs of the Klein-quartic pants decomposition. The shortest such loop is taken to be the closed geodesic triangle of length L = 3 arccosh(cos(pi/3)/sin(pi/7)) = 1.6358 — a contractible piecewise-geodesic cycle with corners at the sector visits, deliberately not a smooth closed geodesic: the surface's smooth systole is 8 arccosh(1/2 + cos(2pi/7)) = 3.9359 (Katz-Schaps-Vishne, J. Diff. Geom. 76 (2007) 399, Table 7.1), and the series' systole seam is 2r = 1.0905. With the singlet-closure winding count N_w = dim G2 = 14 (not fitted) and the framework's conventional strong-scale anchor Lambda_QCD = 200 MeV (an input; the framework content is the dimensionless factor 14/L = 8.558, the energy law itself an identification rather than a result derived from Yang-Mills dynamics), the identification reads Delta_YM = 14 Lambda/L = 1711.7 MeV — 0.19 sigma below the anisotropic-lattice 0++ glueball mass 1730(50)(80) MeV of Morningstar and Peardon, with the lattice window 1700-1730 MeV corresponding to anchors in [198.6, 202.1] MeV. Positivity is tied to the framework pinning the surface at the Hurwitz-maximal point of genus-3 moduli space, where the {7,3} inradius is a constant. In the brane/dilational sector the same finite-incidence refusal of an unconstrained zero mode appears mechanically: the auxetic endpoint K = 0, mu > 0 (nu = -1 in both 2D and 3D, machine-verified), the Maxwell-Calladine count 3V - 6 - E = -6 on the complete K7 incidence truss (six self-stress states in the generic rigid realization), and the pi/21 angular tick stalling the Wick relaxation a half-tick short at curvature debt 1/21, with the exact entropy-production floor (k_B/42T) ln(11/10) whose leading order in the debt d = 1/21 is k_B/(441 T) (Papers CCCXXXI and CCCLXI). The surviving scalar breathing mode is lifted to the Hubble floor m ~ H0 (hbar H0 = 1.44e-33 eV, range c/H0 = 1.37e26 m; Paper CCCV) — forty-two decades below the gauge floor, and deliberately kept there. No new prediction is registered; the entropy floor remains CCCXXXI's registered prediction, inherited unchanged, and the glueball comparison is an identification against published lattice values. Section 8 declares the open mathematical program: the admissible singlet loop space, the minimality proof, a basis-independent representation-theoretic closure theorem for N_w = 14, the SU(3)-from-G2 reduction of the count, a dynamical derivation of the inverse-length energy law, a unified mechanical model for the dilational sector, and quantum stability of the classical lengths. Contingent on the established OOB framework (G2 = Aut(O) reduction, Klein-quartic pants decomposition and systole seam, auxetic theorem and infrared fixed point, tick superselection); full proofs and gate scripts in the BraneBulk Omnibus, concept DOI 10.5281/zenodo.19185171 (edition 0-CCCL, version DOI 10.5281/zenodo.21971669). Supplementary files: gate script + green output (43/43 checks), figure script + figure, the Fugu cross-model claim-audit report with raw reviewer outputs (fidelity, kill-argument memo and defense), and the intake/premise ledger.