What does this research mean for the field?

Black hole formation undergoes a three-stage topological transition governed by G₂ permutohedron collapse, with a critical mass threshold of ~1.5×10²³ M☉ ensuring that T2* boundary emissions always escape to exterior observers. Novelty: ClaimNovelty.METHODOLOGICAL. Consensus alignment: ConsensusAlignment.NEUTRAL.

What question did this study set out to answer?

This research aims to describe black hole formation through topological and algebraic perspectives within a novel framework.

June 8, 2026Open Access

Paper CLXXXIX: 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

Key Points

This research aims to describe black hole formation through topological and algebraic perspectives within a novel framework.
Developed a topological and algebraic model of black hole formation using the One-Octonion Brane-Bulk framework.
Analyzed three stages of geometric transition at the event horizon and in the bulk interior relative to mass accumulation.
Utilized algebraic structures such as comonoid pinch maps and Eilenberg–Zilber chain morphisms to describe the transitions.
At critical mass, black holes exhibit a significant transition to G₂ permutohedron dynamics.
Eilenberg–Zilber chain morphism fails at the horizon due to null radial coordinates, complicating the simplicial decomposition.
The Fano (7,3) bulk geometry re-emerges at the interior with all dynamics active below the Schwarzschild radius.

Abstract

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 > rS): 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 = rS): 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 < rS): 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 Mcrit = 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.

Paper CLXXXIX: 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

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