canonicalThreshold_pos
plain-language theorem explainer
The canonical threshold constant used by the Physics structural certificate at rung 86 is strictly positive. Anyone citing domain-cost comparisons or the M86 structural certificate needs this sign fact. The proof is a one-line wrapper: unfold the threshold definition and apply linear arithmetic to the bound φ > 1.5.
Claim. The canonical threshold constant of the Physics structural module is strictly positive: $0 < \tau_{\mathrm{can}}$. After unfolding its definition in terms of the golden ratio $\varphi$, positivity follows from the bound $\varphi > 1.5$.
background
This module is Structural Certificate 86 for the Physics domain in Recognition Science (Plan v7). It is marked as a structural theorem package: zero sorry, zero axioms. The local objects include a domain cost functional and a canonical threshold constant against which that cost is compared.
The golden ratio $\varphi = (1+\sqrt{5})/2$ is the self-similar fixed point forced by the Recognition Composition Law (forcing step T6). A standard Mathlib-facing lemma records the tight elementary bound $\varphi > 1.5$, obtained from $\sqrt{5} > 2$. That bound is the only external input here.
The canonical threshold is a real constant defined from $\varphi$ in this module. Its positivity is the elementary sign fact needed before any comparison of domain cost against the threshold can be stated.
proof idea
One-line wrapper. Unfold the definition of the canonical threshold, then run linarith on the hypothesis $\varphi > 1.5$ supplied by phi_gt_onePointFive. No further case splits or Recognition-specific lemmas are required; the inequality is pure real arithmetic once the definition is expanded.
why it matters
Structural Certificate 86 packages the Physics-domain prediction at recognition rung 86. Positivity of the canonical threshold is the first arithmetic gate: without $0 < \tau_{\mathrm{can}}$, nonnegativity and comparison lemmas for the domain cost cannot be stated cleanly.
In the broader forcing chain, $\varphi$ is forced at T6 as the unique self-similar scale. Using the elementary bound $\varphi > 1.5$ keeps the certificate free of floating-point appeals and of any physics input beyond the RS constants module. Downstream certificate inhabitants in this file can therefore treat the threshold as a strictly positive yardstick when assembling the structural claim.
No external parent theorem currently lists this lemma in used_by; it is local scaffolding for the M86 certificate bundle itself.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.