currentStatus
plain-language theorem explainer
The scaffolding record marks the current status of the P versus NP resolution in Recognition Science. Dissolution via linear R̂ recognition time for SAT instances is flagged proved while the conditional path remains unavailable due to an open gap on uniform circuit lower bounds. It is referenced when assembling the master certificate for the complexity claim. The definition is a direct structure construction containing one open gap description.
Claim. The status of the P versus NP resolution is the record asserting conditional proof available equals false, dissolution proved equals true, open gap equals the requirement that for some fixed positive integer $k$ every unsatisfiable CNF formula on $n$ variables has any deciding circuit of size at least $2^{n/k}$, and sorry count in chain equals 1.
background
The P vs NP Assembly module considers two resolution paths using Recognition Science tools. Path A is conditional on UniformTopologicalObstructionHyp, which requires a fixed $k$ such that every unsatisfiable formula on $n$ variables demands circuit size at least $2^{n/k}$. Path B is the unconditional dissolution via R̂ recognition time at most $n$ for satisfiable formulas together with J-cost obstruction at least 1 for unsatisfiable formulas. The PvsNPResolutionStatus structure packages the boolean flags, open gap string, and sorry count. Upstream results include the UniformTopologicalObstructionHyp definition and the dissolution components from the sibling PvsNPDissolution structure.
proof idea
This declaration is a direct record construction that populates the four fields of PvsNPResolutionStatus with the listed boolean values, string description of the open gap, and natural number. No lemmas are applied; it is a scaffolding stub.
why it matters
This status record establishes the current position in the Recognition Science treatment of P versus NP, asserting that the dissolution argument holds while the conditional proof on topological obstruction is not yet available. It connects to the master certificate declarations in the same module. The open gap points to the need for a uniform exponential lower bound on circuit size for UNSAT instances, which would complete Path A. In the broader framework this sits alongside the J-frustration and spectral emergence results that motivate the complexity measures.
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