QECCert
plain-language theorem explainer
QECCert bundles four certification properties for the RS quantum error correction protocol into a single structure. Researchers studying phi-harmonic scheduling cite it to record the five code families, the vanishing of J-cost at unity, positivity for all other positive rates, and the eight DFT modes. The structure is populated directly by four sibling lemmas with no additional reasoning steps.
Claim. A structure $QECCert$ is defined by the conjunction of four assertions: the set of quantum error correction code types has cardinality 5, $J(1)=0$, $J(r)>0$ for every positive real $r≠1$, and the discrete Fourier transform of order 8 has exactly 8 modes.
background
In the Recognition Science framework the J-cost function measures deviation from self-similarity and serves as the error-rate metric for quantum error correction. The module implements RS patent 015, which schedules harmonic pulses via an 8-mode DFT at 5φ frequency. The inductive type QECCodeType enumerates the five families (repetition, surface, colour, topological, flagCode) whose count equals the configuration dimension D=5. Upstream, dft8ModeCount is defined as 2^3 in the Fourier analysis module, confirming the eight-tick octave required by T7 of the forcing chain.
proof idea
The declaration is a structure definition that packages four independent properties. Each field is supplied by a sibling definition or lemma: the code count from qecCodeCount, the zero at unity from below_threshold_correct, the positivity from logical_error_positive, and the mode count from dft8_eq_8. No tactics or reductions are required beyond the direct assignment.
why it matters
QECCert supplies the certificate object consumed by qecCert, which in turn witnesses the RS QEC protocol of patent 015. It closes the B16 QEC Depth claim by confirming the five-code configuration, the J-cost threshold at r=1, strict positivity elsewhere, and the DFT-8 schedule demanded by the eight-tick octave (T7) and D=3. The construction directly supports the phi-harmonic scheduling rule that error correction succeeds precisely when the error rate lies below the J(φ) band.
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