HARD PROBLEMS · 2026 · PARTIAL COMPUTATION · OFFICIAL SOLUTIONS 0
An independently written internal checker accepted one 13-element construction. That does not prove the global optimum is 13. Closing the 14-element case still requires either a complete independent replay of every normalized branch or a proof object that an independent checker can verify. Neither gate is complete.
This note omits the coordinates and unpublished proof payload. Its subject is the boundary between a checked lower-bound witness and an unfinished global upper-bound argument.
13-element object
checked internally
Global upper bound
certificate incomplete
Outside replay
not complete
Official solutions
0
Checking one object is not an optimality proof
The direct checker establishes that a particular 13-element object satisfies the required constraints. It supports an existence statement at that size. It says nothing by itself about whether a larger object exists.
On the upper-bound side, the work produced a symmetry-reduced branch map and audited the consistency of stored search records. Missing and duplicate descriptors were checked, selected branches were replayed separately, and the encoders received targeted tests. These are useful internal controls, but they are not a complete independently checkable global certificate.
A timeout and an unchecked UNSAT line close nothing
Computational searches readily accumulate logs saying that no object was found. A time limit expiring does not prove nonexistence. Nor does a solver’s UNSAT status become a proof merely because it appears in a log. The exact input and proof output must be retained and accepted by an independent checker.
The audit package defines that next run, but the decisive proof object is absent. It therefore does not authorize “M=13 proved,” “solved,” “formally verified” or “independently reproduced.”
Between a checked example and proof that no larger example exists lies the global computation that has not yet closed.
The condition for reopening the claim
The next gate is a proof-producing run on the normalized global instance, with the input, proof and hashes preserved and independently checked. If the single instance is impractical, the branch family must be complete and disjoint, and every terminal branch must carry a checked proof.
External reproduction and official record changes would still be later, third-party decisions. The present status is narrower: one 13-element construction checked internally; global optimality and official resolution unfinished.
Project record
- Period: 2026
- Activity: witness checking, normalization and branch-map audit, search review and proof-certificate planning
- Internal evidence: witness checker, branch map, partial replay logs, tests and audit report
- External validation: no complete independent replay or global certificate
- Final outcome: one 13-element construction checked internally; no global optimality proof and zero official solutions
- Not established: global nonexistence at size 14, complete outside reproduction, peer review or official recognition