RecognitionDefect
plain-language theorem explainer
A recognition defect packages a state whose J-cost is strictly positive. Researchers formalizing error correction and fault tolerance in Recognition Science thermodynamics would cite this structure when defining defect energy and correction protocols. The definition is a direct packaging of the state together with the positivity condition on its J-cost.
Claim. A recognition defect for a map $X:Ω→ℝ$ consists of a state $s∈Ω$ together with the assertion that the J-cost of $X(s)$ is positive.
background
In Recognition Science the J-cost function quantifies deviation from the ground state where J=0. Upstream results define the defect functional as J applied to positive arguments and derive costs from multiplicative recognizers or recognition events. The module develops the error-correction viewpoint of RS thermodynamics, in which ledger dynamics implements fault tolerance so that physical laws remain stable. Recognition defects are deviations from J=0; their energy is the J-cost itself. The module links this picture to quantum error correction by treating the ledger as a stabilizer code, defects as errors, and the 8-tick cycle as the correction period.
proof idea
This is a structure definition that directly encodes the state and the positivity of its J-cost. No lemmas are applied; the construction serves as a type for the downstream definitions defect_energy and defect_energy_pos.
why it matters
The structure supplies the basic object for the error-correction treatment of RS thermodynamics. It is used directly by defect_energy to extract the J-cost and by defect_energy_pos to establish positivity. It fills the role of formalizing defects within the module's connection to the 8-tick octave and φ-ladder code distance, supporting the claim that ledger dynamics implements fault tolerance.
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