# Verification report

## Result and source scope

For the finite Coxeter group H3, the identity-deleted absolute-order complex is homotopy equivalent to a wedge of four circles and 872 two-spheres. Deleting the central involution as well gives a shellable wedge of 867 two-spheres. The classical and dual Coxeter-generated ideals, with the identity deleted, are shellable wedges of 163 and 367 two-spheres.

The central involution has a link consisting of five disjoint hexagons. This violates the Cohen-Macaulay link condition over every field in H3; the same interval is an edge link in H4 and gives the corresponding obstruction there.

The uniform finite-Coxeter-group source question, type-A/type-B EL-labelings and the full H4 homotopy type are not resolved. The frozen source record remains PARTIALLY_SOLVED. Classical and dual conventions are explicitly distinguished; their different sphere counts are intentional.

## Written argument

The proof uses the exact model A5 x C2, reconstructs the reflection set, describes the central interval through five Klein four subgroups, and checks three complete shellings. A shelling attachment argument establishes wedges of two-spheres, not merely the ranks of homology groups. Reattaching the central vertex is a mapping-cone calculation: the nullhomotopic inclusion of five circles into the simply connected deleted complex contributes the unreduced suspension, hence four circles and five additional two-spheres.

The H4 argument uses preservation of absolute intervals in finite parabolic subgroups and Carter's reflection-length criterion. Existing Coxeter theory, the classical/dual distinction and related interval results are credited. The bounded literature review does not certify absolute priority.

## Exact certificates

The standalone `verify_h3.py` constructs the 120-element group by generator closure, the fifteen reflections by conjugation, reflection lengths by breadth-first search, and absolute order by directed reachability. It compares reachability with the defining length identity on all 14,400 ordered pairs.

The two certificate files supply three complete ordered facet lists:

| Complex | Vertices | Edges | Triangles | Two-spheres |
| --- | ---: | ---: | ---: | ---: |
| Classical Coxeter ideal, identity deleted | 74 | 510 | 600 | 163 |
| Dual Coxeter ideal, identity deleted | 98 | 930 | 1200 | 367 |
| Full order, identity and central involution deleted | 118 | 1530 | 2280 | 867 |

For all 4080 facets, the checker verifies the complete facet set, uniqueness, purity and the shelling intersection criterion. It also checks the central-link graph, the full f-vector (119, 1560, 2310) and the cone decomposition. Eighteen corrupt certificates, including correct facet sets in invalid orders, are rejected.

Normal and optimized Python runs produce identical JSON. All operations are exact and use only the standard library. Tests use explicit exceptions, not assertions removed by optimization. The checker does not import the exploratory or shelling-search programs, and the reported success flags in certificate files are not trusted.

## Review and package status

The proof is AI-assisted and self-audited. “Separate checker” refers to an independent implementation path, not review by another researcher. No independent expert review or proof-assistant verification has been completed.

The package build record documents clean-directory source compilation, exact checker replays and page-by-page visual inspection. The source archive redistributes the author's manuscript, audit notes, code and certificate data, not third-party source PDFs or extracted full texts.