Turing_incomplete
Any Turing model that sets recognition complexity to zero fails to capture measurement costs for balanced-parity instances inside a ledger computation. Researchers examining dual computation-recognition frameworks would cite the result when contrasting pure evolution models against those that track observation overhead. The proof constructs an explicit ledger instance with constant-false measurement, derives a contradiction from the parity lower bound on a size-one mask, and enforces the zero-recognition property by extensionality.
claimFor any Turing model whose recognition complexity function is identically zero, there exists a problem type together with a ledger computation model such that some input size $n$ and proper-subset mask $M$ witness a measurement failure on the balanced-parity encoding, while the Turing recognition function remains the zero map.
background
The module is explicitly labeled a scaffold for hypothetical ledger-based complexity exploration and lies outside the verified certificate chain. TuringModel is the structure whose recognition complexity is axiomatically zero for every input size. LedgerComputation augments a state space and evolution rule with an encode operation and a measure operation whose measurement_bound axiom encodes lower bounds on recoverable information. The balanced-parity encoder maps a bit and a mask to either the mask or its complement, creating an information-theoretic barrier between computation and observation.
proof idea
The tactic proof first builds a concrete LedgerComputation with unit states, identity evolution, trivial encoding, and a measure that always returns false. It populates the measurement_bound field by classical reasoning, instantiating the assumed universal recovery at true with the all-false mask to reach an immediate falsehood. The existential is witnessed by a one-element problem using the empty mask of cardinality zero, applying simpa to the bound; the second conjunct follows by function extensionality from the input model's recognition-free axiom.
why it matters in Recognition Science
The declaration illustrates the Turing incompleteness hypothesis inside the scaffold module's discussion of ledger dual complexity. It shows how double-entry structure forces recognition costs absent from pure Turing evolution and supports the module's hypothetical claim that P equals NP at computation scale but not at recognition scale. No downstream theorems depend on it, and the module doc explicitly cautions against citing the result as a proven resolution of P versus NP.
scope and limits
- Does not prove an unconditional P versus NP separation in classical complexity theory.
- Does not supply explicit asymptotic bounds for any concrete decision problem.
- Does not apply to models lacking an explicit measurement protocol.
- Does not discharge any open questions inside the verified Recognition Science core.
formal statement (Lean)
161theorem Turing_incomplete (TM : TuringModel) :
162 ∃ (problem : Type) (LC : LedgerComputation),
163 -- The ledger model captures costs Turing ignores (existence of a hard measurement instance)
164 (∃ (n : ℕ) (M : Finset (Fin n)) (hM : M.card < n),
165 ¬ (∀ b R, LC.measure (LC.encode (BalancedParityHidden.enc b R).toList) M = b)) ∧
166 -- Turing counts only evolution, not measurement
167 TM.recognition_complexity = fun _ => 0 := by
proof body
Tactic-mode proof.
168 -- Witness: any problem with balanced-parity output
169 let LC : LedgerComputation := {
170 states := Unit -- placeholder
171 evolve := id
172 encode := fun _ => ()
173 measure := fun _ _ => false
174 flux_conserved := fun _ => rfl
175 measurement_bound := by
176 intro n M hM
177 -- Apply the balanced-parity lower bound
178 classical
179 intro h
180 -- Instantiate the universal claim at `b = true` with any mask `R`.
181 -- Our `measure` always returns `false`, so it cannot equal `true`.
182 have h' := h true (fun _ => false)
183 simpa using h'
184 }
185 use Unit, LC
186 -- Provide a concrete hard instance using the bound and trivial size witness.
187 refine ⟨?_, ?_⟩
188 · refine ⟨1, (∅ : Finset (Fin 1)), by decide, ?_⟩
189 -- Instantiate the universal impossibility from the `measurement_bound` field.
190 simpa using (LC.measurement_bound 1 (∅) (by decide))
191 · funext n
192 exact TM.recognition_free n
193
194
195/-- P vs NP resolution through recognition -/