QECThresholdCert
QECThresholdCert packages the assertion that five quantum error correction code families have thresholds forming a geometric sequence decaying by phi inverse. Physicists modeling QEC within Recognition Science cite it to tie the phi-ladder to thresholds such as the surface code near 1 percent. The structure is introduced directly as a definition that bundles the family cardinality, positivity of codeThreshold, and the exact decay ratio.
claimLet $Q$ be the set of five code families (repetition, surface, colour, topological, concatenated) and let threshold function $t(k) = phi^{-k}$. The certificate asserts $|Q| = 5$, $t(k) > 0$ for every natural $k$, and $t(k+1)/t(k) = phi^{-1}$ for every $k$.
background
The module introduces QECCodeFamily as an inductive type whose five constructors are repetition, surface, colour, topological and concatenated. The sibling definition codeThreshold(k) returns phi to the power minus k. This local setting follows the Recognition Science claim that adjacent code-family thresholds decay by the factor 1/phi, with the five families matching configDim D = 5. The upstream structure QECThresholdCert in QuantumErrorCorrectionThreshold supplies an analogous certificate whose fields are positivity, one-step ratio and adjacent ratio.
proof idea
The structure is defined by directly specifying its three fields. The five_families field is filled by the cardinality of the QECCodeFamily inductive type. The threshold_pos and phi_decay fields are supplied by the lemmas codeThreshold_pos and codeThreshold_decay.
why it matters in Recognition Science
This certificate is constructed by the downstream definition qecThresholdCert in the same module and is referenced by the matching structure in QuantumErrorCorrectionThreshold. It realizes the RS prediction that QEC thresholds follow the phi-ladder decay, consistent with the surface-code value near phi^{-9}. It touches the framework landmark of five families equaling configDim D = 5 and the self-similar fixed point phi.
scope and limits
- Does not compute explicit numerical thresholds beyond the phi expression.
- Does not derive the five code families from physical first principles.
- Does not address hardware-specific error rates or implementation details.
- Does not prove convergence properties of concatenated codes.
formal statement (Lean)
43structure QECThresholdCert where
44 five_families : Fintype.card QECCodeFamily = 5
45 threshold_pos : ∀ k, 0 < codeThreshold k
46 phi_decay : ∀ k, codeThreshold (k + 1) / codeThreshold k = phi⁻¹
47