currentStatus
The definition currentStatus assembles the resolution status for P versus NP inside Recognition Science. It flags the dissolution path as proved while marking the conditional P ≠ NP path unavailable, with a single open gap tied to the uniform topological obstruction hypothesis. This record acts as the central status tracker referenced by downstream master certificates in the complexity module. The construction is a direct sorry stub that populates the four fields of the PvsNPResolutionStatus structure without invoking lemmas.
claimThe current status of the P versus NP resolution in Recognition Science is the record with conditional proof available set to false, dissolution proved set to true, open gap equal to the statement that there exists fixed $k > 0$ such that every unsatisfiable CNF formula on $n$ variables requires any deciding circuit of size at least $2^{n/k}$, and sorry count in chain equal to 1.
background
The PvsNPResolutionStatus structure records two resolution paths for P versus NP. Path A is conditional on the UniformTopologicalObstructionHyp, which asserts a uniform exponential circuit-size lower bound for all UNSAT formulas. Path B is the unconditional dissolution argument that recognition time R̂ for SAT instances is at most linear in the number of variables while Turing-machine simulation incurs structural overhead, so the classical P versus NP question conflates distinct complexity measures. Upstream results include the definition of UniformTopologicalObstructionHyp (a strengthened hypothesis with fixed k) and the J-cost and spectral-gap machinery imported from JCostLaplacian and SpectralEmergence that underwrite the frustration and circuit bounds.
proof idea
The declaration is a sorry stub that directly constructs the PvsNPResolutionStatus record by assigning the four fields: conditional_proof_available to false, dissolution_proved to true, open_gap to the quoted description of the uniform bound, and sorry_count_in_chain to 1. No tactics or lemmas are applied; the body is a literal field assignment terminated by the sorry marker.
why it matters in Recognition Science
This declaration supplies the master status record that downstream certificates such as PvsNPMasterCert consult. It marks the dissolution argument as proved while isolating the remaining gap in the conditional path, thereby closing the local sorry count in the PvsNPAssembly module. The open gap points directly to the uniform topological obstruction hypothesis whose proof would complete the conditional P ≠ NP route via J-frustration and exponential circuit lower bounds, linking to the Recognition Science treatment of complexity through J-cost and the phi-ladder.
scope and limits
- Does not prove the uniform topological obstruction hypothesis.
- Does not establish P ≠ NP unconditionally.
- Does not claim a complete resolution of P versus NP.
- Does not discharge the open gap string into a formal theorem.
closing path
The scaffold path is to replace the sorry by proving the UniformTopologicalObstructionHyp structure, i.e., exhibiting a fixed positive integer k such that every unsatisfiable formula on n variables requires circuits of size at least 2^{n/k}.
formal statement (Lean)
91def currentStatus : PvsNPResolutionStatus where
92 conditional_proof_available := false
proof body
Body contains sorry. Scaffold only; not proved.
93 dissolution_proved := true
94 open_gap := "UniformTopologicalObstructionHyp: prove that for some fixed k, " ++
95 "every UNSAT formula on n variables requires circuits of size >= 2^(n/k)."
96 sorry_count_in_chain := 1
97
98/-! ## Master Certificate -/
99