Chemistry Solubility Product From Jcost Solubility Cert

The Solubility Certificate

In chemistry, the solubility product Ksp describes the balance between a dissolved salt and its solid precipitate. At saturation, the rate of dissolving equals the rate of precipitating, and the system sits at equilibrium. Above that point, the solution is supersaturated and the salt tends to precipitate out. The Recognition Science framework models this balance with a cost function, a number that measures how far a system is from equilibrium, where zero means balanced and positive means out of balance.

The declaration named SolubilityCert is a machine-checked certificate. It bundles three proved facts about the cost function J(x) = (x + 1/x)/2 - 1 when applied to the ratio of measured to expected concentration. First, when the measured value equals the expected value, the cost is exactly zero. Second, for any positive measured and expected values, the cost is never negative. Third, the number phi - 3/2, where phi is the golden ratio, is positive. These three facts are proved in the framework's library of formal theorems.

The certificate itself is a structure, a container that holds these three proofs together. The framework provides an instance of this certificate, meaning the three facts are not just stated but actually proved. The proofs rely on general properties of the cost function, not on any chemical law. The certificate is inhabited, which means it is not an empty promise; the proofs exist and check out.

What the certificate does not claim is just as important. It does not define what "measured" and "expected" mean for a specific salt. It does not derive the solubility product constant for any particular compound. The certificate proves facts about the cost function in general, and the chemical interpretation is a research note, not a theorem. The connection to Ksp equilibrium is an intention, not a result. To make this a theorem about chemistry, one would need to define the measured and expected quantities in chemical terms, which the current declaration does not do.

theorem domainCost_at_equilibrium (r : ℝ) (h : r ≠ 0) : domainCost r r = 0 := by
  unfold domainCost; rw [div_self h]; exact Jcost_unit0

theorem domainCost_nonneg (m e : ℝ) (hm : 0 < m) (he : 0 < e) : 0 ≤ domainCost m e := by
  unfold domainCost; exact Jcost_nonneg (div_pos hm he)

theorem canonicalThreshold_pos : 0 < canonicalThreshold := by
  unfold canonicalThreshold; linarith [phi_gt_onePointFive]

theorem cert_inhabited : Nonempty SolubilityCert := ⟨cert⟩

What this page does not claim

The certificate does not define measured and expected concentrations for any specific salt. The certificate does not derive the numerical value of Ksp for any compound. The certificate does not prove that the cost function models the actual chemistry of dissolution and precipitation.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Chemistry/SolubilityProductFromJCost.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

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