ErrorSyndrome
plain-language theorem explainer
ErrorSyndrome packages a detectable signature for recognition defects in the RS ledger. Thermodynamic modelers and fault-tolerance researchers cite it when deriving correction protocols from the J-cost. The declaration is a structure definition that supplies a syndrome map together with the single axiom that the map vanishes exactly on ground states.
Claim. Let $X:Ω→ℝ$. An error syndrome for $X$ is a map $s:Ω→ℝ$ such that $s(ω)=0$ if and only if $J(X(ω))=0$ for all $ω∈Ω$, where $J$ denotes the J-cost.
background
The module develops the error-correction viewpoint of RS thermodynamics: physical laws remain stable because ledger dynamics implements fault tolerance. Recognition defects are deviations from the ground state J=0; the J-cost supplies the associated energy. The eight-tick cycle acts as the error-correction period and the φ-ladder supplies code distance.
proof idea
This is a structure definition. It directly encodes the interface by declaring the syndrome function and the biconditional axiom that links zero syndrome to zero J-cost. No lemmas or tactics are invoked; the declaration itself provides the type used by the downstream instantiation jcost_syndrome.
why it matters
ErrorSyndrome supplies the detection interface for the thermodynamics module and is instantiated by jcost_syndrome to recover the natural syndrome given by the J-cost. The construction links RS to quantum error correction, with the ledger as stabilizer code, defects as errors, and the eight-tick cycle (T7) as correction period. It supports later development of correction dynamics and fault-tolerance thresholds.
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