canonicalThreshold_pos

This module treats recognition cost of a two-level superposition $\psi = \alpha|0\rangle + \beta|1\rangle$ as the J-cost of the amplitude ratio: $J(|\alpha|/|\beta|)$. Equal superposition gives $J(1) = 0$, the global minimum, so a maximally coherent state carries zero recognition cost. Status is structural (no sorry, no axioms).

The golden ratio $\varphi = (1+\sqrt{5})/2$ is the self-similar fixed point forced earlier in the chain. The only upstream fact used here is the tighter numerical bound $\varphi > 1.5$ (from $\sqrt{5} > 2$). The canonical threshold is a real constant defined from $\varphi$; positivity of that constant is the local claim.

One-line wrapper. Unfold the definition of the canonical threshold, then discharge $0 < \cdots$ by linarith against the imported lemma $\varphi > 1.5$. No further case splits or cost identities are required.

Positivity of the canonical threshold is the elementary numeric gate for the superposition-cost certificate stack in this file (SuperpositionCostCert, cert, cert_inhabited and the nonnegativity facts on domain cost). It sits in Foundation, upstream of any later use of cost thresholds when comparing coherent versus decohered states. The module itself records zero recognition cost at equal superposition; a positive threshold keeps the comparison scale well-defined and oriented. No downstream edges are recorded yet in the graph, so the lemma is presently a local structural fact rather than a cited step in a larger forcing theorem. It does not itself invoke T5–T8 or the RCL identity; it only needs the elementary bound on $\varphi$.