Chemistry Solubility Product From Jcost

Solubility and the cost of mismatch

In chemistry, the solubility product (Ksp) is the equilibrium constant for a solid dissolving into its ions. It is the ratio of ion concentrations at saturation, the point where dissolved and precipitated forms balance. Above that ratio, the solution is supersaturated and solid forms; below it, more solid dissolves. The value of Ksp is specific to each salt and depends on temperature.

The Recognition Science framework models this balance with a cost function. The framework's central object is a ledger, a discrete record of recognition events, and a cost that the ledger must pay. The proved cost function is J(x) = (x + 1/x)/2 - 1, which is zero when its input equals 1 and positive otherwise. The chemistry module applies this by setting the input to the ratio of measured ion activity to the expected value at equilibrium.

In Recognition Science, the module proves three general facts about this cost. First, the cost is exactly zero when the measured value equals the expected value, which models saturation. Second, the cost is never negative for positive inputs, so supersaturation carries a positive cost. Third, a threshold constant involving the golden ratio is positive. These are formal theorems in the framework's machine-checked library of formal theorems.

What the module does not do is define the measured and expected quantities in chemical terms. The cost function is applied to a ratio without specifying which ions, which salt, or how concentrations convert to activities. The docstring states this plainly: the theorems prove facts about J applied to a ratio, not about any particular solubility equilibrium. The chemical interpretation is a research note, not a derived result.

The practical consequence is modest but real. The framework offers a template for how equilibrium might be recognized as a zero-cost state, and supersaturation as a positive-cost state. Turning that template into a theorem about Ksp requires defining the measured and expected quantities from chemistry itself. Until then, the module is a formal shell waiting for its subject.

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]

What this page does not claim

The module does not derive a numerical value for any specific solubility product. The module does not define which ions or salt the measured and expected quantities refer to. The chemical interpretation is a research note, not a proved result about Ksp.

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)

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