qecCodeCount
plain-language theorem explainer
The theorem states that the finite type of quantum error correction codes has exactly five elements. Researchers on phi-harmonic error correction would reference this cardinality to confirm the configuration dimension. The proof applies the decide tactic to compute the cardinality from the inductive definition.
Claim. The set of quantum error correction code families has cardinality five: $|$repetition, surface, colour, topological, flagCode$| = 5$.
background
Module QuantumErrorCorrectionFromJCost implements RS patent 015 on phi-harmonic QEC scheduling. The inductive type QECCodeType enumerates five families: repetition, surface, colour, topological and flagCode, each deriving Fintype. This establishes configDim D = 5 for the protocol where error correction activates below the J(phi) threshold in (0.11, 0.13). The upstream inductive definition provides the DecidableEq and Fintype instances required for the cardinality statement.
proof idea
The proof is a one-line wrapper that invokes the decide tactic. Since QECCodeType is a finite inductive type with an explicit Fintype instance, decide computes Fintype.card directly and confirms equality to 5.
why it matters
This result supplies the five_codes component to the qecCert definition, completing the certificate for the RS QEC protocol. It aligns with the five code families required for DFT-8 scheduling at 5 phi Hz and matches the configDim D = 5 in the Recognition Science framework. The declaration closes the count for the J-cost threshold analysis without addressing the ladder or explicit error rates.
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