canonicalThreshold_pos
plain-language theorem explainer
The canonical threshold built from the golden ratio is strictly positive. Cosmology structural arguments that need a positive cost cutoff cite this fact. The proof is a one-line wrapper: unfold the definition and finish by linear arithmetic from the bound φ > 1.5.
Claim. The canonical threshold is strictly positive: $0 < T_{\mathrm{can}}$, where $T_{\mathrm{can}}$ is the real threshold constant defined from the golden ratio $\varphi=(1+\sqrt{5})/2$.
background
Module RS_COS_Structural_007 packages structural facts about Recognition Science J-cost symmetry: $J(x)=J(1/x)$, so recognition cost is ratio-symmetric. The module is marked as a structural theorem block (no sorry, no axioms).
The golden ratio $\varphi=(1+\sqrt{5})/2$ is the self-similar fixed point forced at step T6 of the Recognition forcing chain. Constants supplies a tighter elementary lower bound: $\varphi>1.5$, proved from $\sqrt{5}>2$. The canonical threshold is a named real in this module (sibling of the domain-cost definitions), arranged so that positivity is an immediate numerical consequence of that bound.
proof idea
One-line wrapper. Unfold the definition of the canonical threshold, then discharge the resulting real inequality by linarith using the upstream lemma that $\varphi>1.5$.
why it matters
Lives in the cosmology structural layer that records J-cost ratio symmetry for later certificates (sibling certificate bundle RSCOSStructural007Cert). Positivity of the threshold is the elementary gate any cost-cutoff comparison needs before domain-cost inequalities can fire. No downstream consumers are wired yet; the lemma closes a local positivity obligation inside the structural-007 package. It inherits the T6 status of $\varphi$ via the Constants import and does not itself touch the eight-tick, dimension, or fine-structure landmarks.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.