canonicalThreshold_pos
plain-language theorem explainer
The canonical threshold constant of this structural module is strictly positive. Anyone comparing domain costs or threshold cuts against the J-cost will cite it. The proof is a one-line unfold of the definition followed by linear arithmetic from the bound φ > 1.5.
Claim. The canonical threshold (the module constant built from the golden ratio $\varphi$) satisfies $0 < \textit{canonical threshold}$.
background
This module records structural facts about the Recognition Science J-cost, whose defining symmetry is $J(x)=J(1/x)$ (ratio symmetry of recognition cost). Status is a fully discharged structural theorem block: zero sorry, zero axioms.
The golden ratio $\varphi=(1+\sqrt{5})/2$ is the self-similar fixed point forced earlier in the chain. A tight elementary bound already available is $\varphi>1.5$, proved from $\sqrt{5}>2$. The canonical threshold is a named real constant in the module, defined by unfolding in terms of $\varphi$ (and used alongside the domain cost $J$ on the positive reals).
Positivity of that threshold is the elementary gate needed before any comparison of costs to a cut value.
proof idea
One-line wrapper. Unfold the definition of the canonical threshold, then discharge the resulting strict inequality by linarith using the upstream lemma $\varphi>1.5$. No further case splits or cost identities are required.
why it matters
Sits in Mathematics RS Structural Module 7, the block that packages J-cost ratio symmetry and the elementary positivity/nonnegativity facts around the domain cost. It is a sibling of the domain-cost nonnegativity lemma and of the module certificate RSMTHStructural007Cert.
In the broader forcing picture it rests on T6 ($\varphi$ as the self-similar fixed point) and on the elementary arithmetic of $\varphi$. No downstream theorems currently list it as a dependency, so its role is local hygiene: every later comparison that treats the canonical threshold as a positive cut can invoke this fact without re-proving $\varphi>1$.
It does not itself encode the RCL identity or the eight-tick/D=3 steps; those live elsewhere in the chain.
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