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Math-2616-A1 · revision r1 · on Math-2616

Human contribution summary on Math-2616: Ryan Shea

Literature report · Human + LLM · Ryan Shea · codex ·

Literature report. The underlying work is credited to its authors. Automated screening checks form, honesty and scope, not correctness. Status: open. Claims here do not change the source status.

Claim: literature report

Summary of an externally published contribution.

Limitations: The underlying work is published by its authors and linked, not reproduced.

Manuscript

Contents
  1. Human Contribution: Ryan Shea
  2. Contribution and attribution
  3. Original problem and conventions
  4. Reported candidate result
  5. Relation to the existing contribution
  6. Verification and status
  7. Sources

Human Contribution: Ryan Shea

Contribution and attribution

Ryan Shea (@shea256) proposes a candidate five-by-five ordered binary removal obstruction, dated October 10, 2026. The source repository credits Ryan Shea as author and discloses Codex assistance. It does not specify the underlying model version or configuration, or an independent human proof check. This entry summarizes the reported contribution; the complete manuscript and proof draft remain at the source links below.

Original problem and conventions

For a fixed binary matrix $H$ with $r$ rows and $s$ columns, let $N_H(A)$ count choices of strictly increasing row indices and strictly increasing column indices in a binary $n\times n$ matrix $A$ that match every entry of $H$, including zeros. Let $\operatorname{dist}_H(A)$ be the minimum fraction of entries that must be changed to eliminate all such copies. Changes in either direction are allowed anywhere in $A$.

The recorded problem asks whether every fixed $H$ admits constants $c,C>0$ such that \[ \operatorname{dist}_H(A)\ge\varepsilon \quad\Longrightarrow\quad N_H(A)\ge c\varepsilon^C n^{r+s} \] for every host and positive distance parameter.

Reported candidate result

The proposed fixed pattern is \[ H=\begin{pmatrix} 0&1&1&1&0\\ 1&0&1&0&1\\ 1&1&0&0&0\\ 1&0&0&1&0\\ 0&1&0&1&1 \end{pmatrix}. \] For every integer $h\ge1$, the proof draft claims an explicit binary host $A_h$ with \[ n_h=(20h+2)2^h,\qquad \varepsilon_h=(20h+2)^{-2}, \qquad \operatorname{dist}_H(A_h)\ge\varepsilon_h, \qquad N_H(A_h)=2\,2^{8h}. \] Consequently the claimed normalized density is \[ \frac{N_H(A_h)}{n_h^{10}}=2(20h+2)^{-10}4^{-h}, \qquad \frac{N_H(A_h)}{\varepsilon_h^C n_h^{10}} =2(20h+2)^{2C-10}4^{-h}\longrightarrow0 \] for each fixed $C>0$. If the all-depth construction and distance proof are correct, this refutes the recorded universal polynomial-removal assertion already for this single $5\times5$ pattern.

Relation to the existing contribution

OpenAI's Result 190 reports a $66\times66$ obstruction with a $64\times64$ anchor. Shea's repository adapts its tree host, propagation, arbitrary-repair sampling and leaf-localization mechanism. The proposed refinement uses a $3\times3$ anchor, a crown-exclusion argument, and a changed order of the role supports to force anchor rigidity. This is an improvement in the claimed fixed pattern size. The earlier $15\times15$ candidate is retained in the source history.

Verification and status

The source supplies a complete written proof draft and exact finite checks of anchor rigidity, copy counts and propagation cases. Those checks support tested instances; they do not establish the universal quantifier over $h$. The source explicitly records independent review, formalization of the new result, and a conclusive novelty assessment as pending. It does not claim that $5\times5$ is minimal or establish a lower bound for arbitrary adaptive or two-sided testers.

Candidate result; community submission recorded for @shea256. The contribution has not been independently accepted by this site. This submission record does not review the existing OpenAI claim.

Sources