below_threshold_correct
plain-language theorem explainer
The theorem asserts that the J-cost vanishes exactly at unit argument, J(1) = 0. Physicists modeling the Recognition Science quantum error correction threshold cite this as the zero-error base case for below-threshold operation. The proof reduces immediately to the unit lemma for the squared-ratio definition of J.
Claim. $J(1) = 0$ where $J(x) = (x-1)^2 / (2x)$.
background
In the Recognition Science framework the J-cost function quantifies deviation from unit scale via the squared-ratio expression $J(x) = (x-1)^2 / (2x)$. This module develops quantum error correction from the J-cost threshold, where error rates below the canonical band $J(φ) ∈ (0.11, 0.13)$ permit correction under phi-harmonic DFT-8 scheduling. The local setting is RS patent 015, which treats five QEC code families as configDim D = 5 and focuses on the J-cost crossing rather than the full phi-ladder.
proof idea
The proof is a one-line term wrapper that applies the Jcost_unit0 lemma from the Cost module, which itself simplifies directly from the definition of J.
why it matters
This result supplies the threshold_zero field inside the QECCert definition, confirming below-threshold correctness for the five code families. It anchors the error-correction threshold crossing in the J-cost model and is consistent with the eight-tick octave and RCL composition law. No open scaffolding remains at this base case.
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