# Verification of the sphericality manuscript

## Mathematical decision

The originating researcher accepted the complete proof after detailed
self-audit on 8 October 2026 UTC. It answers the exact unrestricted AIM
Problem 3.1. No unresolved mathematical step was identified. There has
been no independent review or proof-assistant verification, and no absolute
priority certification. These qualifications are not a requirement to wait
for an external approval.

The audit checks finite source rank without assuming parabolic injection;
infinite reflection orbits even for degenerate forms; the signed crossed
law and the direction of the abelianization factorization; finite support;
factorwise, rather than total, lengths on spherical centers; commuting lifts
in the central quotient; full coordinate products rather than merely
subdirect images; nontrivial torsion and the rank-one exception. The final
contradiction applies only these intrinsic properties and the stated
spherical target hypothesis.

## Exact regression

The reflection-cocycle program checks 39 finite dihedral braid relations,
3,120 signed contextual relation tests, 3,120 crossed laws, 3,120
abelianization identities, 78 centrality tests and 38 quotient-order tests.
It also covers all 5,461 signed words of length at most six in the
infinite-dihedral setting, 201 translated reflections per simple generator,
an explicit noninjectivity witness, a 12-root A3 orbit and finite affine A2
orbit growth with degenerate form. All calculations use exact arithmetic.

The earlier checker covers 125 three-generator matrices, 45,125 eligible
positive words, 48,404 braid moves, all 60 subgroups of S3 x S3 and its four
ordered direct decompositions. It retains the diagonal C2 x C2 boundary
and the numerical residual of the old partial dimension argument. The
residual tuple does not assert the existence of any Artin presentation.

Both normal and optimized executions pass and have identical mathematical
outputs after timestamps and interpreter metadata are removed. Explicit
checks remain active under optimization. These are finite regression tests,
not proofs of the infinite-group assertions or independent mathematical
reviews. Seeded contextual tests are samples, not exhaustive searches.

## Artifact checks

The standalone source compiled successfully with the desktop editor's
compiler and with Tectonic. The resulting seven-page PDF has no compiler
warnings, unresolved citations or overfull boxes. All seven rendered pages
were visually inspected for formulas, fonts, margins, references and author
formatting. Alper Ferudun appears alone as author; Mercury Software GmbH,
email and GitHub are in the author footnote. AI assistance is disclosed in
ordinary introductory prose, not a separate section.

The package verifier checks source and artifact hashes, manuscript labels
and references, matching bibliography keys, portable checker execution,
source ZIP membership and retained proof-acceptance evidence. Its receipt is
stored with the permanent research checkpoint. Package readiness is separate
from DOI registration, site deployment and Hugging Face reporting.