SolventExtractionCert
plain-language theorem explainer
Packages the three algebraic side-conditions needed to treat liquid-liquid extraction as a J-cost problem: diagonal vanishing of the domain cost, non-negativity off the diagonal, and positivity of the canonical extraction threshold. Chemists and RS auditors cite it when wiring distribution-ratio optimality at φ. It is a pure structure definition; inhabitants are assembled from sibling lemmas.
Claim. A solvent-extraction certificate is a triple of facts: (i) for every nonzero real $r$, the domain cost of the pair $(r,r)$ vanishes; (ii) for all positive reals $m,e$, the domain cost of $(m,e)$ is nonnegative; (iii) the canonical extraction threshold is strictly positive.
background
The module derives the liquid-liquid extraction distribution ratio $D = [\mathrm{analyte}]{\mathrm{org}}/[\mathrm{analyte}]{\mathrm{aq}}$ from the Recognition Science J-cost. In RS units the efficient operating point is $D_{\mathrm{opt}} = \varphi$, where $J(\varphi) \approx 0.118$, read as roughly 11.8% analyte remaining aqueous (88.2% extracted).
The domain cost is the local cost functional on positive concentration (or mass/energy) pairs that the module uses in place of raw $J$. Its diagonal vanishing and off-diagonal non-negativity mirror the global fact that recognition cost is nonnegative and minimized on the identity ray. The canonical threshold is the positive cutoff against which extraction efficiency is scored; positivity is required so the comparison is well-posed.
Upstream, ObserverForcing already records that every recognition-event cost is nonnegative via $J$-cost non-negativity on positive states. The present certificate simply specializes that discipline to the extraction domain.
proof idea
No proof body: this is a structure declaration. The three fields are Prop-valued requirements (diagonal cost zero, cost nonnegative for positive arguments, threshold positive). Concrete inhabitants are built later by assigning the sibling lemmas domainCost_at_eq, domainCost_nonneg, and canonicalThreshold_pos into those fields.
why it matters
Gives the module a single named interface for "extraction is a well-posed J-cost problem." Downstream, cert is the canonical noncomputable inhabitant, and cert_inhabited records Nonempty SolventExtractionCert, closing the structural theorem (0 sorry, 0 axiom) advertised in the module header.
In the broader RS chain this sits under chemistry applications of T5 J-uniqueness and the self-similar fixed point $\varphi$ (T6): the claim that $D_{\mathrm{opt}} = \varphi$ is only meaningful once cost is nonnegative, vanishes on matched phases, and is compared to a positive threshold. The certificate is the bookkeeping object that makes those three facts available as one hypothesis package.
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