Heinlein, Kiermaier, Kurz, and Wassermann give 333 three-dimensional subspaces of F₂⁷ with minimum subspace distance 4. The whole space has distance 4 from every three-space, so adjoining it gives 334 mixed-dimension codewords. Heinlein and Ihringer prove the upper bound 388 by semidefinite programming. The current online subspace-code table still displays 334–388 for these parameters. A dated search on 2026-07-28 found no later primary result that closes either side of the interval.

The replay decodes the 103 representatives in Appendix C and generates their orbits under the printed Klein four-group. The orbit lengths are nine fixed planes, 26 pairs, and 68 four-element orbits, giving 333 distinct three-spaces. Their minimum distance is 4. Among every subspace of F₂⁷ outside this code, exactly one has distance at least 4 from all 333 planes: F₂⁷ itself. This is a maximal-extension result for this fixed construction. It supplies no global upper bound on A₂(7,4).

For every ambient subspace outside the published code, the replay records which original planes violate distance 4. The counts for blocker-set sizes 0 through 5 are 1, 12, 16, 416, 1,784, and 3,087. Closing these blocker sets under unions of size at most 5 gives 37,591 removal sets. A deletion set can be replaced by the union of the blockers of its chosen additions. This retains every addition and restores any original plane whose deletion served no compatibility constraint, so normalized blocker unions contain an optimum. Padding a normalized witness with unused deletions gives the exact-deletion maxima. At most 11 candidate additions are eligible for any one normalized set, and direct subset enumeration checks their mutual distances. The best size is 334. With exactly five deletions the best size is 333. The output records four selected optimal witnesses with canonical encodings, individual SHA-256 digests, and aggregate digest e694fe602c2f0977efa1224876e9e61e312b1b5e9f07472e79f72849bbb0eb9f.

The fixed Singer action has order 127 and partitions all 29,212 subspaces into the recorded orbit census. Exactly 72 three-space orbits and 72 four-space orbits satisfy the distance condition internally. Their compatibility graph has 144 vertices, 72 edges, and no triangles. Its edges stay within a single dimension. Adding the compatible singleton zero or whole-space orbit gives maximum total size 255. The listed bases A and B generate compatible three-space orbits, and adjoining F₂⁷ gives a checked 255-word code.

Theorem 1.1 states the binary upper bound 388. Lemma 4.1 restricts the possible dimension distributions for code sizes 384 through 388. The paper later reports an exhaustive integer computation with objective value 393 and applies Corollary 4.6 to obtain A₂(7,4) ≤ 394. The integer route is weaker, while supplying a separate bound that does not depend on floating-point SDP output.

The audit read both primary papers, checked their versioned PDFs and source licenses, inspected the current online bounds entry, and ran dated arXiv searches for later improvements. A read-only production orientation resolved the permanent target exactly and returned an empty research-memory projection. Production check-plan calls for the bounded replacement searches through radii five and six returned proceed-with-caution because no close prior record was retrieved. The checked-out local index did not yet contain this recently published target, which is an environment lag rather than a target-integrity problem.

The search first verifies the Appendix C orbit data and all pairwise distances. It then enumerates every subspace of F₂⁷ in canonical reduced row echelon form. Direct extension leaves only F₂⁷. For the replacement search, each outside subspace carries its exact set of conflicting original planes. Every union of blocker sets of size at most five is checked, followed by direct enumeration of all mutually compatible additions. The search closes this bounded neighborhood and leaves the unrestricted problem open.

The experiment fixes the irreducible action given by multiplication by x modulo x⁷+x+1. It enumerates all subspace orbits, discards every orbit with an internal distance below 4, and builds the complete compatibility graph of the survivors. The graph calculation proves a maximum invariant code size of 255. Full order-127 invariance is therefore too restrictive for improving the lower bound.

Reuse the exact Appendix C reconstruction and compute every outside subspace with at most six conflicting original planes. Group candidates by blocker set. For each union of at most six blockers, solve the induced compatibility problem with an exact maximum-clique routine and retain a verifiable search trace. Record every incumbent code as RREF bases, verify all distances independently, and publish a branch certificate or complete trace if the maximum remains 334.

Run the repository source with the stated CPython command. The script uses integer bit masks for vectors and subspace membership. Unique reduced row echelon bases enumerate all subspaces. Every distance uses exact intersection cardinalities. The Appendix C representatives and printed Klein four-group generators are embedded with attribution. The output is one canonical compact JSON object followed by LF.