canonicalThreshold_pos

This module develops a structural RS reading of Euler's number $e$ in terms of powers of the golden ratio $\varphi = (1+\sqrt{5})/2$, with status marked as a fully proved structural theorem (no sorry, no extra axioms). Nearby siblings introduce a domain cost, the canonical threshold itself, and an Euler-number certificate object.

The only upstream fact used here is the tighter lower bound $\varphi > 1.5$, which follows from $\sqrt{5} > 2$ and the closed form of $\varphi$. That bound is strong enough that any simple positive linear expression in $\varphi$ that appears after unfolding the threshold definition is immediately positive by linear arithmetic.

In the broader RS foundation, $\varphi$ is the self-similar fixed point forced at T6 of the unified forcing chain; positivity lemmas of this kind keep later cost and certificate constructions on the correct side of zero.

One-line wrapper. Unfold the definition of the canonical threshold so the goal is an explicit inequality in $\varphi$, then discharge it by linarith using the lemma $\varphi > 1.5$. No further case splits or RS-specific identities are required.

Gives a reusable positivity anchor for the RS Euler-number development in Foundation.Euler_Number_e_RS. Sibling material (domain cost nonnegativity, the Euler-number certificate, and its inhabitedness) sits in the same file and needs the threshold on the positive side of zero when comparing costs or assembling the certificate.

Framework-wise it is a small but load-bearing constant fact: $\varphi$ is the T6 fixed point, and many RS yardsticks and thresholds (Berry scale $\varphi^{-1}$, mass ladder rungs, etc.) inherit sign and ordering from elementary inequalities like this one. No downstream edges are recorded yet; the lemma is local scaffolding for the module's own certificate construction rather than a cross-module export.