RecognitionComplete
RecognitionComplete defines a structure that pairs a sub-polynomial computation function Tc with an at least linear recognition function Tr, each equipped with explicit growth witnesses. Complexity researchers modeling ledger-induced information barriers would cite this structure when separating internal evolution costs from observation costs. The declaration is a plain structure definition that directly encodes the dual-cost axioms with no proof steps or tactics.
claimA structure consisting of functions $T_c, T_r : ℕ → ℕ$ such that there exist $c, k ∈ ℝ$ with $0 < k < 1$ satisfying $T_c(n) ≤ c n^k log n$ for all $n > 0$, together with a constant $c > 0$ satisfying $T_r(n) ≥ c n$ for all $n > 0$.
background
This structure sits inside the scaffold module ComputationBridge, which examines hypothetical ledger-based separations between computation and recognition. The module treats the ledger's double-entry structure as the source of an information-theoretic barrier that hides recognition cost from standard Turing models. Upstream results supply the geometric and physical setting: the CPM2D model constructs physical models from hypothesis bundles, while the simplicial ledger continuum bridge identifies Laplacian actions with defect sums via the relation (1/2) Σ w_ij (ε_i − ε_j)² = (1/κ) Σ δ_h A_h.
proof idea
The declaration is a structure definition with no proof body. It simply records the two functions together with the existential witnesses for the sub-polynomial and linear bounds.
why it matters in Recognition Science
RecognitionComplete supplies the central interface for the hypothetical P versus NP story inside the ledger framework and is used by ClayBridge, clay_bridge_theorem, CompleteModel, and main_resolution. It encodes the claim that balanced-parity encoding forces Tc to remain sub-polynomial while Tr grows at least linearly, rendering Clay's Turing formulation incomplete. The module doc explicitly labels the entire construction as exploratory and outside the verified certificate chain, leaving open the question of discharging the ledger assumptions into the core forcing chain.
scope and limits
- Does not prove an unconditional P versus NP separation.
- Does not belong to the verified certificate chain containing UltimateClosure.
- Does not supply explicit numerical bounds beyond the stated existential witnesses.
- Does not claim that the separation holds in standard Turing models without ledger assumptions.
formal statement (Lean)
56structure RecognitionComplete where
57 /-- Computation complexity: internal evolution steps -/
58 Tc : ℕ → ℕ
59 /-- Recognition complexity: observation operations -/
60 Tr : ℕ → ℕ
61 /-- Computation is sub-polynomial -/
62 Tc_subpoly : ∃ (c : ℝ) (k : ℝ), 0 < k ∧ k < 1 ∧ ∀ n, n > 0 → Tc n ≤ c * n^k * Real.log n
63 /-- Recognition is at least linear -/
64 Tr_linear : ∃ (c : ℝ), c > 0 ∧ ∀ n, n > 0 → Tr n ≥ c * n
65
66/-- The Turing model as a special case with zero recognition complexity. -/