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

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. Within OOB the physical question, why the gauge sector has a smallest strictly positive excitation and what sets its scale, is answered: theorem-level positivity from the Gram-signature route plus this paper's scale identification. The framework's position is that the Clay formulation mis-poses the question, since continuum R^4 is pseudocontinuous and not fundamental in OOB (Papers CXXXI, CCXXV, CCCXXXI, CXXXIII); that position carries its own refutation condition, a successful Clay-standard continuum construction would refute the dissolution thesis while leaving the internal mechanism untouched. The paper offers a geometric mechanism and a scale, not the Clay-standard constructive quantum-field-theory proof. 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 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, 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 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. 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), 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)). 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. No new prediction is registered; the entropy floor remains CCCXXXI's registered prediction, and the glueball comparison is an identification against published lattice values. Section 8 states 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 in the BraneBulk Omnibus, concept DOI 10.5281/zenodo.19185171 (edition 0-CCCL, version DOI 10.5281/zenodo.21971669). Version 4 (2026-09-07): physics-only register; files: PDF, verification script with output, figure script and figure.