# Verification of Coxeter higher-A vanishing

The six-page English manuscript proves a theorem for every finite-rank
Coxeter system: its standard Cayley graph has A_j=0 for all j>=2.
For finite real reflection groups it gives the all-degree group
comparison for the real 3-parabolic arrangement. The general k>3
comparison and any unspecified wider interpretation of the AIM
question remain outside the result.

## Mathematical self-audit

The proof checks that every closed walk of length at most four,
including stationary steps and backtracks, lifts closed to the
right-angled Coxeter cover. It uses the Coxeter word property only
for reduced two-letter expressions. The cover graph is median by
the standard flag-link cube construction. A finite image is enclosed
in a finite convex median subgraph, which is a hypercube retract.
Composing that retraction with one-coordinate-at-a-time folds gives
an actual Cartesian graph homotopy fixing the basepoint.

The direct lattice argument proves path independence and constancy
of the lift on the box exterior. The exterior is connected in
dimensions at least two, whereas its two components in dimension
one preserve the possible nontrivial fundamental-group kernel.
The arrangement application checks panel adjacency q=n-2 and
handles the empty arrangement separately. It does not derive
higher vanishing merely from equality of fundamental groups.

No known mathematical gap remains in these stated scopes after
the originating researcher's self-audit. This is not independent
peer review or proof-assistant certification.

## Reproducible exact finite checks

The standard-library checker uses exact arithmetic and no removable
assertions. Normal and optimized Python runs produce byte-identical
output. It examines 43 finite Coxeter models: A1-A5, B2-B5, D3-D5,
I2(m) for 2<=m<=24, seven cubes, and H3 in exact Z[phi] matrices.
It enumerates 8,260 group elements across these models and tests
21,739 short words, including stationary steps. All 2,253 closed
short words lift closed. These word counts are at the identity;
vertex translation supplies the same property elsewhere.

Every nonempty induced subset of Q1-Q3 and a specified deterministic
sample of Q4 are tested. Actual retractions are constructed for
237 median subgraphs, and their based contractions pass 62,438
horizontal and vertical edge checks. Negative controls reject an
isometric six-cycle as a hypercube retract, simultaneous diagonal
coordinate changes as homotopy edges, a non-square-lifting ordinary
cover, and the false connected-exterior claim in dimension one.

These finite regressions do not prove the universal theorem; its
proof is the mathematical argument in the manuscript.

## Sources and document checks

The canonical AIM wording matches the frozen record. Relevant
definitions and theorem hypotheses were checked on 13 rendered
primary-source pages. A failed publisher download is recorded as
HTTP 403, not as a consulted full paper. The classical ingredients
and existing degree-one results are expressly credited. The bounded
novelty search does not certify priority.

The built-in LaTeX compiler and Tectonic both compiled the final
source successfully. All six final pages have been visually checked;
pages 1-4 retain byte-identical renders from the first inspection,
and the corrected bibliography pages 5-6 were inspected again.
There are no remaining TeX warnings, clipped formulas or author
footnote defects. AI assistance is disclosed in the prose.