canonicalThreshold
plain-language theorem explainer
Defines the canonical materials threshold as φ − 3/2 in RS-native units, a positive real near 0.118. Materials and structural-cost arguments cite it as the fixed comparison level against domain costs built from the J-cost. The body is a one-line constant abbreviation from the golden fixed point φ.
Claim. The canonical threshold is the real number $\varphi - 3/2$, where $\varphi$ is the unique self-similar fixed point $\varphi = 1 + 1/\varphi > 1$.
background
Module RS_MAT_Structural_006 sits in the materials layer and records structural facts tied to φ-uniqueness: φ is the self-similar fixed point of $x \mapsto 1 + 1/x$, equivalently the continued fraction $1+1/(1+1/(1+\cdots))$, forced in the Recognition chain at T6.
The Cost import supplies the standard RS cost $J$, with $J(x) = (x + x^{-1})/2 - 1$. Sibling definitions build a nonnegative domain cost from $J$ and compare it to a fixed threshold. This declaration names that threshold in closed form from φ alone, with no free material parameters.
Constants are RS-native ($c=1$, and φ drawn from IndisputableMonolith.Constants). Numerically $\varphi \approx 1.618$, so $\varphi - 3/2 \approx 0.118 > 0$, which the sibling positivity lemma records.
proof idea
Pure definitional abbreviation: the real constant is set equal to $\varphi - 3/2$ with no proof obligations. Downstream positivity and certificate lemmas unfold this equality and use $\varphi > 3/2$, which follows from the fixed-point equation and $\varphi > 1$.
why it matters
Gives the materials stack a single φ-derived comparison level so domain-cost inequalities stay parameter-free and tied to T6 (φ as self-similar fixed point). Sibling lemmas (positivity of the threshold, the structural certificate RSMATStructural006Cert) hang off this constant; the module claims a structural theorem with zero sorry and zero axioms. It does not itself force dimension or the eight-tick octave, but it keeps materials thresholds inside the same φ-ladder language used for masses and the Berry scale.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.