canonicalThreshold_pos

This module derives an RS-native data-compression rate from the J-cost. Shannon entropy $H$ is the information-theoretic floor; Recognition Science adds a fixed per-symbol overhead $J(\varphi)$, so an RS-optimal code has average length $H + J(\varphi)$. At $H = 1$ bit/symbol the module quotes length $1.118$, i.e. overhead $\approx 0.118$.

The J-cost is the unique nonnegative cost forced by the Recognition Composition Law (forcing step T5): $J(x) = (x + x^{-1})/2 - 1$. The golden ratio $\varphi = (1+\sqrt{5})/2$ is the self-similar fixed point (T6). The canonical threshold is the concrete real constant obtained by evaluating that cost (or an equivalent closed form) at $\varphi$.

The only upstream fact used here is the elementary bound $\varphi > 1.5$, proved from $\sqrt{5} > 2$.

Pure wrapper. Unfold the definition of the canonical threshold so the goal is an explicit inequality in $\varphi$. Discharge it by linarith using the imported lemma $\varphi > 1.5$. No further lemmas or case splits.

Positivity of the threshold is the minimal sanity check that the RS compression overhead is a genuine positive cost, not zero or negative. It underwrites the module's structural claim that RS-optimal codes sit strictly above the Shannon floor by $J(\varphi)$ bits per symbol (about $0.118$ at unit entropy).

The result sits in the Information domain and inherits its meaning from J-uniqueness (T5) and the forcing of $\varphi$ (T6). No downstream consumers are wired yet in the graph; the natural parents are the compression certificate and any later theorem that needs $T > 0$ to keep code lengths strictly super-entropic. It does not touch the eight-tick octave, $D = 3$, or the $\alpha$ band.