IndisputableMonolith.QuantumComputing.ErrorCorrectionThresholdFromJCost
This module defines the recognition quantum J(φ) and derives the surface code error correction threshold together with an error rate cost function from the J-cost in Recognition Science. Researchers working on quantum error correction inside the RS framework would cite it when linking the phi-ladder to fault-tolerant thresholds. The module is built from a short chain of definitions and positivity lemmas that rest directly on the imported Cost and Constants modules.
claimLet $J$ be the recognition cost function. The module introduces the recognition quantum $J_0 = J(φ)$ and defines the surface-code threshold $p_{th}$ together with the error-rate cost function $C(p)$ such that $C(p_{th}) = 0$ and $C(p) > 0$ for $p > p_{th}$, with an inhabited certificate $E$ witnessing that error rates below threshold are admissible under the J-cost.
background
The module sits inside the QuantumComputing domain and imports the RS-native time quantum τ₀ = 1 tick from Constants together with the J-cost machinery from Cost. It introduces the auxiliary definitions recognitionQuantum (equal to J(φ) by the sibling lemma recognitionQuantum_eq_Jph), surfaceCodeThreshold, errorRateCost, and the certificate type ErrorCorrectionCert. These objects are placed on the phi-ladder so that the Berry creation threshold φ^{-1} and the dream fraction φ^{-3} control the admissible error rates.
proof idea
This is a definition module, no proofs. The structure consists of successive definitions (recognitionQuantum, surfaceCodeThreshold, errorRateCost, ErrorCorrectionCert) followed by three short lemmas (recognitionQuantum_pos, surfaceCodeThreshold_pos, errorRateCost_nonneg) that establish non-negativity and the equality recognitionQuantum = J(φ).
why it matters in Recognition Science
The module supplies the ErrorCorrectionCert and the threshold relation that later quantum-computing results in the Recognition framework depend on. It directly implements the connection between the J-cost and fault-tolerant quantum computation, closing the step from the eight-tick octave and D = 3 to concrete error-rate bounds inside the phi-ladder.
scope and limits
- Does not derive the full quantum error correction threshold from first principles outside the J-cost.
- Does not compute a numerical value for p_th without fixing φ to its RS value.
- Does not address non-surface-code architectures.
- Does not prove stability under arbitrary noise models beyond the J-cost.