IndisputableMonolith.Information.QuantumErrorCorrectionThreshold
The module defines the quantum error correction threshold at each rung k of the phi-ladder and proves the threshold lies strictly below unity while decreasing with higher rungs. Quantum information researchers working inside Recognition Science would cite these bounds when estimating error tolerance in self-similar physical systems. The module consists of a core definition plus positivity and ratio lemmas that follow directly from the ladder's recursive structure.
claimLet $T(k)$ be the quantum error correction threshold at phi-ladder rung $k$. Then $T(k) < 1$ for every positive integer $k$, and the sequence is strictly decreasing: $T(k+1) < T(k)$.
background
This module belongs to the Information domain and imports only the Constants module, whose sole content is the RS-native time quantum $τ_0 = 1$ tick. The phi-ladder is the discrete scaling hierarchy generated by the golden-ratio fixed point, with rungs indexed by positive integers $k$. The QEC threshold is defined relative to this ladder so that error rates in quantum systems inherit the self-similar phi scaling. The module documentation states the central claim: the threshold at rung $k$ lies below unity and falls as the rung index rises.
proof idea
This is a definition module. It introduces the threshold function qecThresholdAt together with four supporting declarations: a positivity lemma, two ratio lemmas that encode the strict decrease via the ladder's recursion, and a packaged certificate QECThresholdCert that bundles the bounds for later use. No further proof tactics appear inside the module itself.
why it matters in Recognition Science
The module supplies the concrete information-theoretic bound needed to apply quantum error correction inside phi-scaled Recognition Science models. It directly realizes the module-level claim that the threshold at rung $k$ is below unity with higher rungs giving lower thresholds. No parent theorems or downstream usages are recorded yet, leaving the result as an interface definition awaiting integration into larger information-processing arguments.
scope and limits
- Does not evaluate the threshold at any concrete numerical rung.
- Does not extend the definition beyond the phi-ladder.
- Does not incorporate specific decoherence channels or hardware noise models.
- Does not address correlated multi-qubit errors.