the_open_gap
plain-language theorem explainer
the_open_gap instantiates OpenGap to record that the overhead of simulating R-hat J-cost minimization by a Turing machine is unknown. Complexity theorists examining the Recognition Science approach to P versus NP would reference this marker. The definition supplies trivial values for the two structure fields.
Claim. Define the open gap as the structure recording that the simulation overhead between the recognition operator on a J-cost landscape and Turing machine operations is unknown, together with the requirement for a spectral-to-Turing translation, both witnessed trivially.
background
The module addresses the remaining gap in connecting the Recognition Science result that the recognition operator separates satisfiable from unsatisfiable instances on a J-cost landscape to a classical Turing machine separation of P and NP. OpenGap captures the unresolved simulation overhead: proving that the simulation overhead from R-hat (global lattice minimization) to Turing machine (local tape operations) is superpolynomial. This would require showing that no polynomial-time TM can simulate the convergence of degree-normalized SpMV on an n-variable J-cost landscape. The spectral gap argument yields convergence in O(1/λ₂) octaves.
proof idea
The definition constructs the OpenGap record directly. It assigns trivial to simulation_cost_unknown and to needs_spectral_to_turing_translation. No lemmas are applied; the construction is a one-line record literal.
why it matters
This declaration supplies the gap_identified field in turingBridgeCert, which in turn supports circuitSeparation. It marks the precise location of the open question on whether recognition time grows polynomially or superpolynomially, as referenced in the paper PvsNP_SelfContained_Final.tex and biggest-questions.md §XIII Q3. The module document states that encoding faithfulness and non-naturality are established, leaving the Turing simulation as the open piece.
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