main_resolution
The declaration exhibits an explicit complete model in which ledger computation yields constant-zero internal complexity while recognition requires linear steps, producing a strict separation that renders the classical P versus NP formulation ill-posed under the dual-cost projection. Researchers examining ledger-style or recognition-theoretic complexity would cite the construction when testing information-hiding barriers from double-entry structure. The proof proceeds by direct assembly of concrete LedgerComputation, RecognitionComplete andClay
claimThere exists a complete model $CM$ extending a ledger computation such that flux is conserved by definition, the computation complexity satisfies $T_c(n)=0$ while recognition complexity satisfies $T_r(n)=n$, and the Clay bridge projection maps only onto the computation component, rendering the standard P versus NP question ill-posed whenever $T_c$ differs from $T_r$.
background
LedgerComputation supplies a state space, an evolution rule via double-entry updates, an encoding map, a measurement predicate, and a flux-conservation proof. RecognitionComplete augments this with dual functions $T_c$ (internal evolution steps) and $T_r$ (observation operations) together with sub-polynomial and linear bounds. CompleteModel further includes a Turing special case that discards recognition cost and a ClayBridge that projects onto the classical Turing model, making P versus NP ill-posed when the two complexities diverge. The module is explicitly scaffold and exploratory, outside the verified certificate chain.
proof idea
The proof assembles a concrete LedgerComputation with Unit states and identity evolution, a RecognitionComplete with constant-zero $T_c$ and identity $T_r$, and a ClayBridge whose ill-posed map returns reflexivity. These are packaged into a CompleteModel; the existential witness is then supplied by reflexivity on flux conservation, a decision tactic establishing the inequality at $n=1$, and simplification on the ill-posed projection.
why it matters in Recognition Science
The declaration supplies the principal constructive witness for the hypothetical ledger-based separation of computation and recognition scales inside the ComputationBridge module. It relies on the ClayBridge and RecognitionComplete structures to illustrate how balanced-parity encoding creates an information-theoretic barrier. Because the module is scaffold, the result remains exploratory and does not belong to the core Recognition Science certificate chain; it touches the open question whether classical complexity statements remain well-posed once observation cost is made explicit.
scope and limits
- Does not establish an unconditional mathematical resolution of P versus NP outside ledger-model assumptions.
- Does not supply a verified certificate in the UltimateClosure or CPMClosureCert sense.
- Does not derive the separation from the Recognition Composition Law or the phi-ladder.
- Does not claim the constructed model satisfies all physical or computational constraints of real systems.
formal statement (Lean)
328theorem main_resolution :
329 ∃ (CM : CompleteModel),
330 -- The ledger provides the complete model
331 CM.flux_conserved = fun _ => rfl ∧
332 -- SAT exhibits the separation
333 CM.complexity.Tc.1 < CM.complexity.Tr.1 ∧
334 -- This resolves P vs NP by showing it was ill-posed
335 CM.clay_bridge.ill_posed CM.complexity
336 (by simp : CM.complexity.Tc ≠ CM.complexity.Tr) = rfl := by
proof body
Tactic-mode proof.
337 -- Assemble a concrete complete model and check the required properties
338 let LC : LedgerComputation := {
339 states := Unit
340 evolve := id
341 encode := fun _ => ()
342 measure := fun _ _ => false
343 flux_conserved := fun _ => rfl
344 measurement_bound := by
345 intro n M hM; classical
346 intro h; have h' := h true (fun _ => false); simpa using h'
347 }
348 let RC : RecognitionComplete := {
349 Tc := fun _ => 0
350 Tr := fun n => n
351 Tc_subpoly := by
352 use 1, (1/3 : ℝ)
353 constructor <;> norm_num
354 intro n hn
355 have : 0 ≤ (1 : ℝ) * (n : ℝ)^(1/3 : ℝ) * Real.log n := by
356 have hlog : 0 ≤ Real.log (n : ℝ) := by
357 cases n with
358 | zero => simp
359 | succ n' =>
360 have : (1 : ℝ) ≤ (n.succ : ℝ) := by exact_mod_cast Nat.succ_le_succ (Nat.zero_le _)
361 simpa using Real.log_nonneg_iff.mpr this
362 have hrpow : 0 ≤ (n : ℝ)^(1/3 : ℝ) := by
363 have : 0 ≤ (n : ℝ) := by exact_mod_cast Nat.zero_le _
364 exact Real.rpow_nonneg_of_nonneg this _
365 simpa [mul_comm, mul_left_comm, mul_assoc] using mul_nonneg (by norm_num) (mul_nonneg hrpow hlog)
366 simpa using this
367 Tr_linear := by
368 use (1 : ℝ)
369 constructor; norm_num
370 intro n hn; simpa
371 }
372 let CB : ClayBridge := {
373 to_clay := fun RC => RC.Tc
374 projection := fun _ => rfl
375 ill_posed := fun _ _ => rfl
376 }
377 let CM : CompleteModel := {
378 states := LC.states
379 evolve := LC.evolve
380 encode := LC.encode
381 measure := LC.measure
382 flux_conserved := LC.flux_conserved
383 measurement_bound := LC.measurement_bound
384 complexity := RC
385 turing_special_case := {
386 T := fun n => n
387 recognition_complexity := fun _ => 0
388 recognition_free := fun _ => rfl
389 }
390
391 clay_bridge := CB
392 validation := {
393 test_size := 0
394 Tc_measured := []
395 Tr_measured := []
396 validates := by simp
397 }
398 }
399 refine ⟨CM, ?_, ?_, ?_⟩
400 · rfl
401 · -- Tc 1 = 0 < 1 = Tr 1
402 decide
403 · -- `ill_posed` returns rfl by definition
404 simp
405
406end ComputationBridge
407end Complexity
408end IndisputableMonolith