canonicalThreshold

The module derives a structural link between CMB reionization optical depth and the Recognition Science cost functional $J$. Planck 2018 reports $\tau \approx 0.054$; the module explores closed-form RS expressions such as $J(\varphi)/2 \approx 0.059$ (within roughly 10 percent).

Here $\varphi$ is the unique self-similar fixed point forced by the T6 step of the unified forcing chain, and $J(x) = (x + x^{-1})/2 - 1$ is the unique cost satisfying the Recognition Composition Law. The threshold $\varphi - 3/2$ supplies a dimensionless scale against which domain costs and certification predicates in the same file are measured.

No external lemmas are required; the definition only imports the global constant $\varphi$ from the Constants module and the ambient real arithmetic of Mathlib.

Pure definition: the right-hand side is the term $\varphi - 3/2$ of type $\mathbb{R}$. There is no proof body, no tactic, and no lemma application.

Supplies the numeric cutoff used by the surrounding CMB optical-depth certificate (siblings such as the positivity lemma for this threshold and the inhabited certificate structure). It anchors the comparison between the J-cost prediction and the observed $\tau \approx 0.054$ band discussed in the module header. Within the broader framework it sits downstream of T5–T6 (J-uniqueness and the forcing of $\varphi$) and supports the cosmology layer that converts dimensionless RS costs into optical-depth scales. It does not itself close the numerical match; it only names the scale those later certificates employ.