QECCodeType
plain-language theorem explainer
QECCodeType enumerates five quantum error correction code families used in the Recognition Science phi-harmonic scheduling protocol of RS_PAT_015. Researchers modeling J-cost thresholds for error rates in the band (0.11, 0.13) cite this enumeration to fix the configuration dimension at five. The declaration is a direct inductive definition with five constructors and automatic derivation of finite-type and equality instances.
Claim. Let $C$ denote the set of quantum error correction code types. Then $C = $ {repetition, surface, colour, topological, flagCode}.
background
The module Quantum Error Correction from J-Cost implements RS_PAT_015, which derives QEC thresholds from the J-cost function $J(x) = (x + x^{-1})/2 - 1$. Error correction is effective when the error rate $r$ satisfies $J(r) < J(φ)$ with $J(φ) ∈ (0.11, 0.13)$. The module fixes five code families to realize configuration dimension 5, consistent with the DFT-8 harmonic pulse scheduling at $5φ$ Hz. No upstream lemmas are required; the definition stands alone and supplies the finite set used by the downstream QECCert structure.
proof idea
The declaration is an inductive type definition that introduces the five constructors repetition, surface, colour, topological, flagCode and derives the instances DecidableEq, Repr, BEq, Fintype in one step.
why it matters
This definition supplies the finite set of five code types required by the downstream QECCert structure, which asserts five_codes : Fintype.card QECCodeType = 5 together with the threshold conditions threshold_zero : Jcost 1 = 0 and error_positive. It directly realizes the configDim D = 5 stated in the module documentation for RS patent 015. The enumeration anchors the below-threshold criterion J(r_error) < J(φ) → correct without reference to the spatial dimension D = 3 of the forcing chain.
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