canonicalThreshold_pos
plain-language theorem explainer
The canonical materials threshold is a strictly positive real. Anyone comparing the φ-ladder Mohs cutoff (diamond near φ^5 ≈ 11) to a positive gate cites this fact. The proof is a one-line wrapper: unfold the threshold and finish by linear arithmetic from φ > 1.5.
Claim. The canonical materials threshold $T$ (the real constant named in this module) satisfies $0 < T$.
background
Materials RS Module 4 is a structural (zero-sorry) fragment tying diamond on the Mohs scale to the φ-ladder: φ^5 ≈ 11.09 is read as the Mohs-10 gate. Recognition Science forces φ = (1+√5)/2 as the unique self-similar fixed point (forcing step T6); the same constant seeds mass rungs and several material cutoffs.
The only upstream fact used here is the tighter lower bound φ > 1.5, proved from √5 > 2. The canonical threshold is a real built from φ inside this module (siblings also introduce a nonnegative domain cost). Positivity of that threshold is the elementary gate needed before any comparison to Mohs integers.
proof idea
One-line wrapper. Unfold the definition of the canonical threshold, exposing an inequality linear in φ. Discharge it by linarith from the lemma φ > 1.5. No further algebraic identities or cost-function properties are required.
why it matters
Keeps the Mohs-diamond structural certificate honest: any later claim that the φ^5 gate sits above a positive materials threshold needs 0 < T first. The module status line marks the whole file as a structural theorem (0 sorry, 0 axiom). Downstream use list is presently empty, so this is a leaf positivity lemma inside the materials layer rather than a step on the T0–T8 forcing chain. It still sits on the same φ that T6 forces and that appears in the RS-native constants (ħ = φ^{-5}, G = φ^5/π).
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