is_complete_correction
plain-language theorem explainer
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.
Claim. A 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
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.
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