← P/NP Dual Rehearsal / Research Rounds / Round 8
Deliberately picks problems whose usual narrative isn't built around a classical solution closure, yet which are genuinely in P, to stress-test Round 7's bridge: general-graph maximum matching's Edmonds blossom contraction (which preserves the existence of augmenting paths, plus the Tutte matrix's algebraic escape route linking the existence of a matching to whether a determinant is the zero polynomial), maximum flow's residual network (which preserves information about future improvability, paired with a min-cut duality certificate), the determinant's Gaussian elimination (compressing n! permutation terms into polynomially many steps), shortest path's semiring aggregation (where exponentially many paths need only keep the optimal value), and treewidth dynamic programming's boundary summaries. These algorithms share no single polymorphism, yet a more general pattern of exact quotienting keeps recurring across them. Proposes the Polynomial Exact Quotient Scheme (PEQS), but the Equals Team immediately proves that if any solver's machine state is allowed to serve as the quotient state, PEQS degenerates into a tautology for “L∈P” — so six non-circular admissibility conditions must be added (solver-independence, composability, answer-blindness, exact semantic preservation, a verifiable conservation law, and resource completeness).
Relationship to other documents, stated as far as possible in the document's own words, not my interpretation.
“P problems don't necessarily share one kind of closure, but they often have some mathematical way of ‘exactly quotienting away future-equivalent states.’ The real game now is: does SAT simply lack this kind of quotient, or have we just not found it yet?” — excerpted from the closing “This Round's One-Liner” of this round's document. Tentative score: P=NP: 7, P≠NP: 7.
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