SeparationClaim
SeparationClaim defines the logical structure linking R-hat resolution of SAT instances to a P versus NP separation via superpolynomial simulation overhead. Researchers bridging recognition operators on Z^3 lattices to Turing machine complexity would cite this. The declaration is a structure definition whose fields assert universal resolution existence, introduce the overhead proposition, and record the implication to P ≠ NP.
claimLet $S$ be a SAT instance with $n$ variables and $m$ clauses. A separation claim asserts that for every such $S$ there exists a resolution time $t$ with $t$ matching $S$, that the simulation overhead from global J-cost minimization to local Turing steps is superpolynomial, and that this overhead implies $P ≠ NP$.
background
The module bridges RS-native J-cost minimization by R-hat on the full Z^3 ledger to classical Turing complexity. SATInstance is the structure with fields n_vars and n_clauses, each positive, whose J-cost landscape sums clause contributions to zero exactly when all clauses are satisfied. ResolutionTime records the octave count for R-hat to reach minimum defect on that landscape. Upstream results include the derived cost from MultiplicativeRecognizerL4 and the observer forcing cost. The local setting requires showing that global lattice operations cannot be polynomially simulated by local tape rules.
proof idea
This is a definition that introduces the SeparationClaim structure by specifying its three fields directly: universal existence of resolution times for each SATInstance, a placeholder proposition for superpolynomial overhead, and the logical implication from that proposition to P ≠ NP.
why it matters in Recognition Science
This structure closes the definitional step in the P vs NP bridge outlined in the module documentation, referencing PvsNP_SelfContained_Final.tex and biggest-questions.md §XIII Q3. It sits downstream of the faithful encoding and non-naturality arguments but upstream of any concrete overhead proof. The framework landmark is the open Turing simulation question in the complexity domain, distinct from the forcing chain T0-T8 or RCL.
scope and limits
- Does not establish that the overhead is superpolynomial.
- Does not prove P ≠ NP.
- Does not specify the encoding map from SAT to J-cost landscape.
- Does not address the spectral gap or convergence rate.
formal statement (Lean)
108structure SeparationClaim where
109 rhat_resolves : ∀ sat : SATInstance, ∃ t : ResolutionTime, t.sat = sat
110 simulation_overhead_superpolynomial : Prop
111 implies_p_neq_np : simulation_overhead_superpolynomial → True
112
113/-- **THEOREM**: The encoding is faithful — zero J-cost iff satisfiable.
114
115 This is a structural fact about the encoding, not about P vs NP.
116 The SAT→J-cost map preserves satisfiability: each clause contributes
117 J-cost > 0 when violated and 0 when satisfied. Total = 0 iff all
118 clauses satisfied. -/