canonicalThreshold
plain-language theorem explainer
The canonical recognition threshold for qubit channel capacity is the real constant φ − 3/2. Researchers bounding effective classical capacity under RS recognition overhead cite it as the fixed comparison scale. It is introduced by direct arithmetic on the golden ratio; no further proof is required.
Claim. The canonical threshold is the real number $\varphi - 3/2$, where $\varphi$ denotes the golden ratio.
background
Recognition Science forces the golden ratio $\varphi$ as the unique self-similar fixed point of the cost functional (forcing-chain step T6). In this module the classical identity “one qubit equals one bit” is adjusted by recognition overhead: effective quantum channel capacity is stated as $\varphi^{-1}$ times qubit count, or roughly 0.618 bits per qubit.
The constant $\varphi - 3/2$ compares that golden-ratio scale against the classical three-halves offset that appears in binary information bookkeeping. Sibling definitions introduce a non-negative domain cost and a positivity certificate for the same threshold; the module is marked structural (zero sorry, zero axiom).
proof idea
Definitional abbreviation only. The real is bound to the term $\varphi - 3/2$ using the imported constant $\varphi$ from Constants and ordinary Mathlib real arithmetic. No lemmas are applied and no tactic proof is present.
why it matters
Gives the numeric threshold that the module’s qubit-channel certificate and positivity lemma rely on. It sits inside the RS information story that replaces bare qubit count by $\varphi^{-1}$ times that count after recognition overhead. The value is not itself a forcing-chain landmark (T5–T8), yet it is the concrete scale against which domain cost is compared when the structural theorem of the module is assembled. Downstream positivity of the threshold underwrites the claim that the effective capacity remains strictly positive.
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