is_complete_correction
A correction protocol C for a state map X is complete when every initial state reaches a zero J-cost ground state after finitely many applications of the correction map. Researchers analyzing fault-tolerant ledger dynamics in Recognition Science thermodynamics cite this definition when establishing stability of physical laws. The definition is a direct predicate that encodes finite-time convergence via iteration of the protocol's correction function.
claimA correction protocol $C$ for a function $X:Ω→ℝ$ is complete if for every state $ω$ there exists a natural number $n$ such that the $J$-cost of $X$ at the $n$-fold iterate of the correction map on $ω$ equals zero.
background
In the error-correction formulation of Recognition Science thermodynamics, physical stability arises because ledger dynamics implements fault tolerance. Recognition defects are deviations from the ground state where the functional J vanishes; the J-cost measures the associated energy, with the upstream defect functional defined as J itself for positive arguments. Density scales as powers of phi on the ladder. The CorrectionProtocol structure supplies a correction map that is cost-decreasing and fixes all ground states. This module identifies the ledger with a stabilizer code, defects with errors, the eight-tick cycle with the correction period, and the phi-ladder with code distance.
proof idea
This is a direct definition that encodes the completeness predicate using the iterated application of the correction map from the CorrectionProtocol structure together with the J-cost functional. No lemmas are invoked; the body is the explicit universal quantifier over states and the existential quantifier over iteration count.
why it matters in Recognition Science
This definition supplies the completeness notion required for the fault-tolerance threshold and is_fault_tolerant predicate in the same module. It fills the error-correction step in Recognition Science thermodynamics, linking directly to the eight-tick octave (T7) and the phi-ladder structure for code distance. It supports the claim that ledger dynamics realizes fault-tolerant computation, though no downstream theorems yet apply it.
scope and limits
- Does not bound the number of iterations uniformly across initial states.
- Does not construct an explicit correction map from ledger rules.
- Does not quantify defect accumulation rate during correction.
formal statement (Lean)
84def is_complete_correction {X : Ω → ℝ} (C : CorrectionProtocol X) : Prop :=
proof body
Definition body.
85 ∀ ω, ∃ n : ℕ, Jcost (X (C.correct^[n] ω)) = 0
86
87/-! ## Fault Tolerance Threshold -/
88
89/-- The fault tolerance threshold: below this defect density,
90 errors can be corrected faster than they accumulate. -/