canonicalThreshold
plain-language theorem explainer
The canonical error-correction threshold is the real constant φ − 3/2. Recognition-Science information arguments cite it as the native code rate R = J(φ) ≈ 0.118 that enters the Hamming-bound capacity claim. The declaration is a one-line closed-form definition in terms of the golden ratio.
Claim. The canonical threshold is the real number $\varphi - 3/2$, where $\varphi$ denotes the golden-ratio fixed point.
background
The module derives a structural Hamming-bound capacity statement from the Recognition J-cost. In RS units the self-similar scale is the golden ratio $\varphi$ (forced at T6), and the cost functional is $J(x)=(x+x^{-1})/2-1$. Direct evaluation gives $J(\varphi)=\varphi-3/2\approx 0.118$.
Module context: at rate $R=J(\varphi)$ a code is claimed to support error fraction $t/n=J(\varphi)/2\approx 0.059$, via the classical volume bound $2^{n-k}\ge V_H(n,t)$. The constant defined here is exactly that rate written in closed form, without invoking $J$ by name.
Sibling facts in the same file establish non-negativity of a domain cost and positivity of this threshold, then package them into an error-correction certificate.
proof idea
Pure definition: the real is introduced by the arithmetic expression $\varphi-3/2$. No lemma applications or tactics.
why it matters
Supplies the numeric rate that the module’s error-correction certificate and Hamming-bound argument consume. In the broader forcing chain it sits at the T5–T6 junction: J-uniqueness plus the self-similar fixed point $\varphi$ produce the unique dimensionless cost $J(\varphi)$, rewritten here as $\varphi-3/2$. That value is the RS-native code rate $R\approx 0.118$ and half of it is the supported error fraction $0.059$. The declaration is structural scaffolding for the zero-sorry capacity claim of the module; it does not itself prove the bound.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.