Black Hole Formation as G₂ Permutohedron Collapse: Comonoid Pinch Maps, Eilenberg–Zilber Failure, and the Three-Stage Topological Transition to the Fano (7,3) Bulk

We develop a topological and algebraic description of black hole formation within the One- Octonion Brane-Bulk framework. As a massive object accumulates sufficient mass, the geometry undergoes a three-stage transition detected by two algebraic structures: the comonoid pinch map on the suspended event-horizon 2-sphere, and the Eilenberg–Zilber (EZ) chain morphism on the product decomposition of the surrounding spacetime. Stage I (exterior, r > r_S): the full G₂ permutohedron with 14 generators governs the brane dynamics, the EZ morphism : C_*(t,r) C_*(S²) → C_*(M⁴) is a chain equivalence, and the T2* sump boundary ∇ ⊗ lies well outside the horizon for all known black holes. Stage II (horizon, r = r_S): the EZ/Alexander–Whitney (AW) chain homotopy fails because the radial coordinate r becomes null, preventing a consistent front/back face decomposition of simplices; simultaneously the comonoid pinch map Δ: ΣS² → ΣS² ΣS² becomes non-trivial, splitting the bifurcation sphere ∨ into inequivalent future and past sheets. The intermediate geometry at the horizon is the A₂ sub- root-system of G₂ — the six short roots forming a regular hexagon in root space — which is the maximal sub-algebra surviving on the null surface. Stage III (bulk interior, r < r_S): r becomes timelike, the suspension becomes temporal, all 14 G₂ generators are active, and the full (7,3) Fano incidence geometry of PG(2,2) re-emerges with the zero eigenmode driving the temporal collapse toward r = 0. A critical mass M_crit = c⁴/(4Ga₀) ≈ 1.5×10²³ M marks the threshold ☉ above which the Schwarzschild radius engulfs the T2* boundary; no known structure reaches this threshold, confirming that T2* boundary emission (511 keV, 20 GeV) always escapes to exterior observers. Part of the One-Octonion Brane-Bulk Framework series. Anchor DOI: 10.5281/zenodo.19120873. Community: one-octonion-brane-bulk. Author: Bharathi Dasan Jagadeesan, M.D., University of Minnesota. ORCID: 0000-0002-1143-941X.