canonicalThreshold_pos
plain-language theorem explainer
The canonical threshold constant in the electrolyte-solution activity model is strictly positive. Citation target for anyone assembling the ESSolution5 structural certificate or checking domain-cost positivity. Proof is a one-line unfold of the definition, closed by linear arithmetic from φ > 1.5.
Claim. The canonical threshold (the real constant used as the reference scale in the electrolyte-solution $J$-cost model) is strictly positive: $0 < t_{\mathrm{can}}$.
background
Module ESSolution5 treats electrolyte-solution activity coefficients from the Recognition Science $J$-cost. The structural claim is that $\gamma \to 1$ at infinite dilution, with the RS form $\log(\gamma) = J(\varphi)\cdot C_{\mathrm{DH}}\cdot I^{1/2}$. At ionic strength $I = J(\varphi)^2$ one obtains the pure number $\log(\gamma) = -J(\varphi)^3 \approx -0.00164$.
Here $J$ is the unique nonnegative cost $J(x)=(x+x^{-1})/2-1$ forced by the Recognition Composition Law (T5), and $\varphi=(1+\sqrt{5})/2$ is the self-similar fixed point (T6). The sibling definition canonicalThreshold is the real scale constant against which domain costs in this module are compared.
The sole upstream fact is the tight lower bound $\varphi > 1.5$, obtained from $\sqrt{5}>2$.
proof idea
One-line wrapper. Unfold the definition of the canonical threshold, then close with linarith using the lemma $\varphi > 1.5$. No further algebraic identities or chemistry lemmas are required; positivity is pure arithmetic once the definition is expanded.
why it matters
Local positivity lemma inside the Chemistry.ESSolution5 package (Plan v7, structural theorem, 0 sorry / 0 axiom). It underwrites the nonnegativity and certificate siblings (domainCost_nonneg, ESSolution5Cert, cert, cert_inhabited) even though no downstream edge is recorded yet.
In the broader RS frame it sits on T5–T6: $J$-uniqueness and $\varphi$ as the forced fixed point. The module’s numerical anchor $\log(\gamma)=-J(\varphi)^3\approx-0.00164$ is only meaningful once the reference threshold is known to be positive. No open scaffold remains; the claim is fully proved.
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