turingBridgeCert
plain-language theorem explainer
This definition assembles the TuringBridgeCert record that packages the faithful SAT-to-J-cost encoding, the non-natural character of the R-hat certificate, linear landscape scaling with instance size, and identification of the remaining simulation gap. Researchers working on the Recognition Science treatment of P versus NP would cite it to record the current bridge status. The construction is a direct record instantiation that references the established module components without further reduction.
Claim. The Turing bridge certificate consists of: the equivalence that a J-cost landscape has minimum cost zero if and only if the encoded SAT instance is satisfiable; the R-hat certificate is non-natural; for any SAT instance the number of variables plus clauses is at least the number of variables; and the open gap in the simulation from global lattice minimization to Turing-machine tape operations is identified.
background
In the Recognition Science setting a J-cost landscape is obtained by mapping each clause of a SAT instance to a positive defect contribution under the J-cost functional, with total cost zero precisely when the instance is satisfiable. The R-hat operator performs global minimization over the full Z^3 ledger. The module document states that the encoding step and the non-naturality of R-hat are already established, while the Turing-machine simulation step remains open. Upstream results supply the faithful-encoding theorem (an identity on the min-cost-zero condition) and the non-naturality theorem (R-hat evaluates the entire assignment at once rather than via local polynomial circuits).
proof idea
The definition is a direct structure constructor that assigns the four fields of TuringBridgeCert to the corresponding sibling definitions: encoding_faithful supplies the equivalence, rhat_is_non_natural supplies the non-natural property, landscape_linear supplies the linear size bound, and the_open_gap supplies the identified simulation question. No tactics or lemmas are applied beyond the record construction itself.
why it matters
This definition supplies the assembled certificate for the P versus NP bridge in the Recognition Science framework, feeding the open question of whether R-hat's global J-cost minimization can be simulated by a polynomial-time Turing machine. It incorporates the established faithful encoding and non-naturality results while marking the remaining gap whose closure would require a superpolynomial overhead argument from the spectral-gap contraction to Turing steps. The module references the paper PvsNP_SelfContained_Final.tex and question Q3 on convergence rate.
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