ResolutionTime
ResolutionTime records a SAT instance together with the octave count at which R-hat reaches its minimum defect on the associated J-cost landscape. Complexity theorists bridging recognition operators to Turing classes would cite the structure when stating resolution-time bounds. The declaration is a plain structure with three fields and carries no proof obligations.
claimA resolution time for a satisfiability instance consists of the instance (with $n$ variables and $m$ clauses), a natural number of octaves, and a proposition asserting that the minimum defect on the J-cost landscape has been attained.
background
SATInstance is the structure holding the number of variables and clauses of a Boolean satisfiability problem. The J-cost landscape encodes each clause as a local additive term; the total J-cost vanishes precisely when every clause is satisfied. ResolutionTime augments this data with the octave count at which R-hat drives the defect below threshold on that landscape.
proof idea
The declaration is a direct structure definition. No lemmas are invoked; the three fields are introduced by type alone.
why it matters in Recognition Science
ResolutionTime supplies the time parameter required by the downstream SeparationClaim, which concludes that superpolynomial simulation overhead implies P ≠ NP. It occupies the convergence-rate slot in the module's P-vs-NP bridge strategy, connecting R-hat's global minimization on the Z³ lattice to the eight-tick octave structure of the forcing chain. The remaining open piece is the demonstration that no polynomial-time Turing machine can replicate the global operation.
scope and limits
- Does not prove that the required octave count is polynomial in instance size.
- Does not establish faithfulness of the SAT-to-J-cost encoding.
- Does not exhibit a Turing-machine simulation of R-hat.
formal statement (Lean)
70structure ResolutionTime where
71 sat : SATInstance
72 octaves : ℕ
73 reaches_minimum : Prop
74
75/-! ## The Non-Naturality Argument -/
76
77/-- A natural property (Razborov-Rudich) has two characteristics:
78 1. Constructivity: computable in polynomial time from the truth table.
79 2. Largeness: satisfied by a random function with probability ≥ 1/poly(n).
80
81 R-hat's certificate is non-natural because it operates on the full Z³
82 lattice topology, not on polynomial-size truth tables. -/