# Verification report

Verification date: 2026-08-29.

Verdict: the round-\(S^4\) construction is a complete counterexample to the
uniform-lower-bound reading of AIM Problem 4.4.

The proof was audited obligation by obligation:

- the exact quotient metric was recovered by horizontal projection;
- its full second-order jet at a fixed point was inserted into the coordinate
  curvature identity;
- the resulting tensor has the standard positive O'Neill sign on real
  planes;
- the selected complex vectors are unitary and totally isotropic;
- their singular contraction is exactly \(-ab/\varepsilon^2\);
- codimension at least four supplies two nonzero rotation blocks;
- the round \(S^4\) action supplies a closed, unit-weight example; and
- the one-block case is rank-one positive semidefinite, matching the reported
  codimension-two boundary.

Two separate exact-arithmetic programs passed. The first uses the polarized
singular tensor over Gaussian rationals and checks 1,824 coordinate identities,
7,296 curvature symmetries, the complete negative eigenspace, total isotropy,
the sphere value, and the one-block boundary. The second starts only from the
quotient metric and differentiates exact rational 2-jets; across four
parameter cases it checks 1,024 tensor entries and independently recovers the
same value. At \(\varepsilon^2=1/7\), it obtains \(-6\) on round \(S^4\).

The computations are audit aids; the manuscript contains the uniform symbolic
proof. No expert review or peer review is claimed. Public post-publication
scrutiny is expected. The eventual submitting author must review and assume
responsibility for the proof, citations, disclosure, and final PDF.

## Public release and license

The author approved public release on 2026-09-02. This paper, its source files, and this verification report are licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0): https://creativecommons.org/licenses/by/4.0/