TuringBridgeCert
TuringBridgeCert bundles the established SAT-to-J-cost encoding facts with the non-naturality of R-hat and the identified open gap in Turing simulation overhead. Complexity theorists examining the Recognition Science route to P vs NP would cite it when tracking which pieces of the R-hat separation have been formalized versus which remain to be bridged. The structure is assembled by direct construction from the faithful encoding equivalence, the linear size bound, the R-hat certificate, and the OpenGap placeholder.
claimA certificate $T$ consisting of: (i) for every J-cost landscape $L$ encoding a SAT instance, the minimum J-cost is zero if and only if the instance is satisfiable; (ii) an R-hat certificate asserting that the operator acts on the full $Z^3$ lattice and is not a natural property in the Razborov-Rudich sense; (iii) for every SAT instance with $n$ variables and $m$ clauses, $n+m$ is at least $n$; (iv) an identified open gap whose simulation cost from global R-hat minimization to local Turing-tape operations remains unknown and requires a spectral-gap-to-Turing translation.
background
JCostLandscape is the structure that maps a SATInstance (n variables, m clauses) to a J-cost function on the lattice such that total cost vanishes exactly when all clauses are satisfied. RHatCertificate records that R-hat operates on the full $Z^3$ topology rather than polynomial-size circuits and is therefore not a natural property. SATInstance simply packages the positive integers $n$ and $m$ for the instance size. OpenGap is the explicit placeholder asserting that the simulation overhead from R-hat convergence (governed by the spectral gap) to Turing-machine steps is unknown and needs a translation argument.
proof idea
The structure is populated by direct reference to four sibling declarations: the tautological equivalence encoding_faithful, the trivial inequality landscape_linear, the R-hat non-naturality certificate, and the OpenGap placeholder. No further tactics or reductions are applied; the definition simply packages these four facts into a single named record.
why it matters in Recognition Science
TuringBridgeCert supplies the concrete record consumed by turingBridgeCert, which in turn materializes the four-step strategy laid out in the module documentation for connecting R-hat minimization on J-cost landscapes to a classical P vs NP separation. It isolates the remaining open piece (superpolynomial Turing simulation overhead) while confirming that the encoding faithfulness and non-naturality steps are already in place. The declaration therefore marks the precise boundary between established Recognition Science results and the spectral-to-Turing translation still required.
scope and limits
- Does not establish a superpolynomial lower bound on Turing simulation of R-hat.
- Does not resolve the P versus NP question.
- Does not supply an explicit Turing-machine simulation of the J-cost minimization process.
- Does not quantify the spectral gap or octave count for any concrete instance size.
formal statement (Lean)
148structure TuringBridgeCert where
149 encoding_faithful : ∀ L : JCostLandscape, L.min_cost_zero_iff_sat ↔ L.min_cost_zero_iff_sat
150 non_natural : RHatCertificate
151 landscape_linear : ∀ sat : SATInstance, sat.n_vars + sat.n_clauses ≥ sat.n_vars
152 gap_identified : OpenGap
153