OpenGap
plain-language theorem explainer
OpenGap packages the two unresolved assertions required to convert the Recognition Science R-hat separation on J-cost landscapes into a classical Turing-machine P versus NP statement. Complexity theorists studying the natural-proof barrier would cite the structure when quantifying the overhead of simulating global lattice minimization by local tape operations. The declaration is introduced as a bare structure whose two fields are each the constant True.
Claim. An open gap consists of the pair of assertions that the simulation cost of R-hat convergence on a J-cost landscape to a Turing machine remains unknown and that a translation from spectral-gap contraction on the integer lattice to Turing-tape steps is still required.
background
The module develops the remaining bridge between Recognition Science and classical complexity. R-hat is the recognition operator that minimizes J-cost on the full Z³ ledger; a J-cost landscape encodes a SAT instance so that minimum cost is zero precisely when the instance is satisfiable. The spectral gap supplies an O(1/λ₂) octave convergence rate for degree-normalized sparse matrix-vector multiplication, yet this rate lives on the lattice and must still be converted into steps on a finite Turing tape.
proof idea
Structure definition that directly records the two propositions as fields, each set to the constant True.
why it matters
OpenGap is referenced by the_open_gap (the concrete witness) and by TuringBridgeCert (the certificate that assembles encoding faithfulness, non-naturality of the R-hat certificate, landscape linearity, and the open gap). It marks the precise location of the missing superpolynomial-simulation argument in the P vs NP bridge described in PvsNP_SelfContained_Final.tex and biggest-questions.md §XIII Q3.
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