Independent examination of the linear-intersection bound

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grade: Separate grades for the non-2-colorable intersection theorem and the three-colorable counterexample, with the reconstruction ruling, independence evidence, exact coverage, and acceptance limits.

report: Whole-statement examination of the frozen linear-intersection argument, including finite and infinite hypergraphs, recoloring, distinct edges, the uniform constant, and the two problem formulations.

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On 14 October 2026, an independent adversarial reviewer and a separate independent grader, both using model gpt-6-astra in fresh contexts, examined the preserved argument for the second question of Problem 836. The report supplies the full proof and its attacks; the grade passes each of these distinct scoped verdicts:

• refutation-failed: for every integer $r\ge2$, every intersecting $r$-uniform hypergraph with no proper $2$-coloring has distinct edges meeting in at least $\lfloor r/2\rfloor\ge r/3$ vertices.
• refuted-as-stated: mere proper $3$-colorability does not imply any uniform positive linear intersection bound. The complete two-edge common-center family has maximum intersection ratio $1/r$.

The first conclusion includes arbitrary infinite hypergraphs in classical set theory with choice. Its compactness proof is an essential foundational expansion expressly permitted by the examination's assignment. The grade accepts that reconstruction; it does not certify a proof in the unseen source. The vertex-count question, historical attribution, external-source fidelity and all Lean artifacts are excluded. No choice-free infinite conclusion or later changed argument is covered.

The grade passes independence on the separate assignments, primary isolation records, content assessment and disclosed reading. No excluded exposure was identified. It also states the evidentiary limit: no complete historical record of every read or possible incoming communication was supplied, so it does not independently certify every historical no-read assertion. The mathematical findings and the review are credited to their respective claimants and to Plasma AI as the claim's standing records.

Exact subjects and record maintenance

The target, argument and manifest preserve the mathematical extraction frozen at 2026-10-14T11:26:16.490126+00:00. They are unchanged. The original submitted report bytes and grade bytes are also preserved exactly. Reading ranges in the grade refer to those submitted bytes.

The filed report and grade add wiki metadata and normalize their headings to their supplied titles. Their mathematical bodies, paths, dates, qualifications and verdicts are unchanged. These are the only editorial changes. No executable mathematics was supplied or is needed: the report and grade contain the complete symbolic arguments and all finite witness sets. Reproduction consists of checking them against the preserved inputs.