1 open statement

• The conjectured separation of P and NP, left as an open question.

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1 open statement

• Theorem 3.2. A function between types is rational if and only if it is the denotation of some rational term.

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1 open statement

• A black-box word-RAM program achieves the refined crossing bound within near-linear time, with constant success probability.

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1 open statement

• Gaifman's normal-form theorem. There is a uniform transformation of every first-order formula over $σ$ into an equivalent Gaifman normal form. For an input of quantifier rank $q$ with $k$ free variables, its common radius is at most $7 ^ q$, each basic local sentence has at most $q + k$ witnesses, and a sentence is translated using basic local sentences only. Equivalence holds in every finite $σ$-structure.

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1 of 2 statements open

• Two false claims admit derivations from each other and from themselves. None has a proof grounded in the background axioms. This additional control tests the least-fixed-point interpretation of the proof network.

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1 open statement

• Descent from stage 300 to stage 290.

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