the_open_gap
The declaration supplies a placeholder instance of the OpenGap structure, recording that the overhead of simulating R-hat global J-cost minimization by a Turing machine is unknown and requires a spectral-to-Turing translation. Researchers assembling the P vs NP bridge in Recognition Science cite it when constructing the TuringBridgeCert. The proof is a direct one-line wrapper that applies trivial to both structure fields.
claimLet OpenGap be the structure whose fields assert that the simulation cost from R-hat lattice minimization to Turing-machine tape operations is unknown and that a spectral-to-Turing translation remains to be supplied. The open-gap definition is the instance of this structure in which both assertions hold.
background
The TuringBridge module reduces the P vs NP question to showing that R-hat recognition time on a J-cost landscape cannot be simulated in polynomial time by a Turing machine. J-cost is the recognition cost obeying the Recognition Composition Law J(xy) + J(x/y) = 2J(x)J(y) + 2J(x) + 2J(y). The OpenGap structure isolates the remaining unknown: whether the O(1/λ₂) convergence guaranteed by the spectral-gap contraction on an n-variable landscape translates into superpolynomial Turing overhead.
proof idea
This is a one-line wrapper that applies trivial to the two fields of the OpenGap structure.
why it matters in Recognition Science
The definition supplies the gap_identified component of turingBridgeCert, which is required by the circuitSeparation theorem to instantiate the full CircuitSeparation structure. It marks the open piece of the P vs NP bridge described in PvsNP_SelfContained_Final.tex and biggest-questions.md §XIII Q3, where the spectral gap from SpectralGapContraction must be converted into a Turing-complexity statement. In the Recognition framework it touches the T7 eight-tick octave and the D = 3 lattice topology.
scope and limits
- Does not prove that the simulation overhead is superpolynomial.
- Does not supply the required spectral-to-Turing translation.
- Does not resolve whether P equals NP.
- Does not assume any concrete value for the spectral gap λ₂.
formal statement (Lean)
142def the_open_gap : OpenGap where
143 simulation_cost_unknown := trivial
proof body
Definition body.
144 needs_spectral_to_turing_translation := trivial
145
146/-! ## Certificate -/
147