OpenGap
The OpenGap structure isolates the two unresolved pieces in translating R-hat global J-cost minimization on the Z^3 lattice into Turing machine steps. Researchers working on the Recognition Science derivation of P versus NP would cite it as the explicit marker for the superpolynomial overhead claim. It is introduced as a bare structure whose two fields are each the trivial proposition true.
claimThe structure records that the overhead of simulating R-hat convergence on an n-variable J-cost landscape by a Turing machine remains unknown and that a translation from spectral-gap contraction rates on the lattice to Turing-tape operations is still required.
background
The TuringBridge module connects the Recognition Science result that R-hat separates satisfiable from unsatisfiable instances on a J-cost landscape to classical Turing complexity classes. R-hat acts on the full Z^3 ledger; Turing machines act via local tape rules. Prior results in the module establish faithful SAT encoding into J-cost landscapes and that the R-hat certificate is non-natural. The spectral gap contraction supplies an O(1/λ₂) octave bound on convergence, yet the conversion of those lattice steps into Turing time is missing.
proof idea
Direct structure definition that introduces two propositions, each instantiated to the trivial true value, as explicit placeholders for the missing simulation-cost theorem and the spectral-to-Turing translation lemma.
why it matters in Recognition Science
This declaration locates the precise remaining gap in the P versus NP argument. It is referenced by the_open_gap construction and by TuringBridgeCert, which packages the faithful encoding, non-naturality, and this open gap. The module documentation ties it to the paper proposition on superpolynomial simulation overhead and to the question whether recognition time T_R grows polynomially or superpolynomially with instance size.
scope and limits
- Does not prove that R-hat simulation overhead is superpolynomial.
- Does not supply a spectral-gap-to-Turing-tape translation.
- Does not resolve P versus NP under the Recognition Science encoding.
- Does not give concrete bounds on simulation cost.
formal statement (Lean)
138structure OpenGap where
139 simulation_cost_unknown : True
140 needs_spectral_to_turing_translation : True
141