IndisputableMonolith.Chemistry.SolventExtractionFromJCost
Defines solvent-extraction energetics from the RS cost J, with a domain cost, a positive canonical threshold, and an inhabited certificate type. RS chemistry work cites it when tying partition or extraction steps to the unique J-functional. The module is mostly definitions plus elementary nonnegativity and positivity lemmas over Constants and Cost.
claimA chemistry module that introduces a domain cost built from the RS cost $J(x)=(x+x^{-1})/2-1$, proves that cost is nonnegative and agrees on equal arguments, fixes a strictly positive canonical threshold, and packages these facts in an inhabited solvent-extraction certificate.
background
Recognition Science forces a unique nonnegative cost $J$ on ratios (T5): $J(x)=\cosh(\log x)-1$, equivalently $(x+x^{-1})/2-1$, obeying the Recognition Composition Law. The Cost import supplies that functional and its basic calculus; Constants supplies the RS-native tick and related units.
This module sits in the Chemistry domain and treats solvent extraction as a cost comparison on concentration or partition ratios. Sibling definitions introduce a domain cost (evaluation of $J$ on the relevant ratio), record that equal inputs give equal cost, and prove nonnegativity. A canonical threshold is fixed as a positive real cutoff against which extraction is judged favorable or not.
The certificate type bundles those facts so downstream chemistry lemmas can assume a single inhabited witness rather than re-proving positivity and threshold properties at each use site.
proof idea
Definition-heavy module, not a single deep theorem. Domain cost is defined by applying $J$ (from Cost) to the extraction ratio; equality-on-equals and nonnegativity are short wrappers on the corresponding $J$ lemmas. The canonical threshold is a concrete positive constant; positivity is a one-line arithmetic check. The certificate structure assembles these fields, and inhabitation is by exhibiting the default threshold and the nonnegativity proof.
why it matters in Recognition Science
Gives Chemistry a first-class hook from macroscopic solvent extraction into the forced cost $J$, so partition energetics inherit T5 uniqueness rather than an ad hoc free energy. No downstream used_by edges are recorded yet; the natural parents are later extraction, solubility, or phase-equilibrium theorems that need a nonnegative domain cost and a positive cutoff. In the broader chain it is scaffolding for RS-native chemistry constants (phi-ladder mass and coupling scales remain upstream). Closes the gap between pure Cost and applied chemical inequalities without introducing new free parameters.
scope and limits
- Does not derive experimental partition coefficients or real-solvent free energies.
- Does not prove uniqueness of the canonical threshold beyond positivity.
- Does not connect extraction cost to the phi-ladder mass formula or alpha band.
- Does not model kinetics, temperature dependence, or multi-component activity.
- Does not supply numerical fits to laboratory extraction data.