Chemistry Solubility Rule From Jcost

Solubility and the cost ledger

In chemistry, a salt dissolves when the energy cost of pulling its ions apart into solution is smaller than the energy released when the ions are surrounded by water. The classic solubility rule in Recognition Science expresses this as a comparison of two costs: a compound dissolves when J(ΔG_solvation/ΔG_lattice) < J(φ), and is insoluble when J exceeds J(φ). Here J is the cost function, a measure of how expensive a recognition event is, and φ is the golden ratio. The claim is that the narrow range around the balance point, within ±J(φ), predicts the slightly soluble category.

The machine-checked library of formal theorems proves three general facts about this cost function. It proves that J(m/e) vanishes when m equals e, that J is nonnegative for positive inputs, and that φ − 3/2 is positive. These are real theorems, checked by the kernel. But the library proves nothing specific to solubility, because its definition of domainCost is just J(m/e) without any reference to hydration energy or lattice energy. The definition sets m and e as arbitrary real numbers, not as solvation and lattice energies.

What this means in plain language: the framework has a general cost function with a proved zero point and nonnegativity, and a proved threshold value. What it lacks is the bridge that makes those facts about solubility. The research note records where the idea was meant to go, but the formal content stops at the general properties. The library is a template shared with many sibling modules, not a theorem about salts.

In Recognition Science, this is an honest boundary. The cost function J(x) = (x + 1/x)/2 − 1 is proved to be the unique function satisfying five plain conditions. Its zero at x = 1 and its nonnegativity are consequences of that proof. The threshold φ − 3/2 is positive, which is also proved. But connecting those facts to chemistry requires defining m and e in the subject's own terms, and that definition is not present in this library.

theorem domainCost_at_eq (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]

def domainCost (m e : ℝ) : ℝ := Jcost (m / e)

What this page does not claim

The solubility rule itself is not proved; it is a research note attached to a template library. No specific salt has been shown to dissolve or not dissolve by this rule. The library does not derive the golden ratio threshold from solubility data.

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

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