IndisputableMonolith.Physics.QuantumErrorCorrectionFromJCost
Module linking quantum error correction thresholds to the Recognition J-cost. It packages code-type counts, an eight-mode DFT identity, and a certificate that logical error vanishes whenever the cost sits strictly below J(φ). Physicists working the RS error-correction story cite the below-threshold lemma and the QECCert bundle. Argument is mostly definitional packaging plus elementary inequalities on J.
claimWhen the J-cost of a noise channel satisfies $J < J(\varphi)$, a quantum error-correcting code operates below threshold and corrects; the module records code-type enumerations, the identity that the 8-mode DFT has exactly eight modes, and a certificate bundle asserting positive logical error above threshold and vanishing logical error below $J(\varphi)$.
background
Recognition Science measures deviation from perfect reciprocity by the cost $J(x)=(x+x^{-1})/2-1$, uniquely fixed by the Recognition Composition Law. The self-similar fixed point $\varphi$ (golden ratio) sets the natural scale: $J(\varphi)$ is the critical cost separating regimes. The module imports only the Cost layer, so all statements are phrased directly in terms of $J$ and $\varphi$.
Quantum error correction enters as a discrete coding problem on an eight-tick (DFT-8) mode space, matching the T7 octave period $2^3$. Code types and mode counts are recorded as plain data; the physically loaded claim is the threshold statement that error rates with $J<J(\varphi)$ remain correctable.
proof idea
Definition-heavy module. Code types, counts, and the DFT-8 cardinality identity are closed by direct evaluation or rfl-style equalities. The below-threshold correctness lemma and the dual positive-logical-error statement are elementary comparisons against $J(\varphi)$. The certificate record simply bundles those facts into a single Prop-valued witness; no deep analytic estimates appear.
why it matters in Recognition Science
Places the RS cost threshold $J(\varphi)$ in the same role that the hashing bound or hashing threshold plays in ordinary QEC: a sharp, parameter-free divider between correctable and uncorrectable noise. Ties the eight-mode structure to the T7 eight-tick octave already forced in the UnifiedForcingChain. Downstream consumers (none yet recorded in the graph) would use the QECCert bundle when assembling fault-tolerance or decoherence arguments inside the larger RS physics stack. Closes a physics-side interface that had been only informal: error correction is not an extra postulate but a corollary of operating below the J-cost of $\varphi$.
scope and limits
- Does not derive a concrete code family or distance bound from first principles.
- Does not prove fault-tolerant gate sets or threshold theorems for concatenated codes.
- Does not relate J-cost to physical noise channels beyond the abstract threshold comparison.
- Does not address continuous-variable or infinite-dimensional QEC.
- Does not claim uniqueness of the eight-mode DFT among all possible mode counts.