Chemistry Solvation Free Energy From Jcost Solvation Fecert

What the certificate proves

Solvation free energy is the change in Gibbs free energy when a solute dissolves in a solvent, a quantity chemists measure to understand solubility and reaction behavior in solution. A common classical estimate for the contribution of electrostatic interactions is the Born model, which gives ΔG_solv = -q²/(8πε₀a)(1-1/ε), where q is the ion charge, a its radius, and ε the solvent's relative permittivity. For water, ε ≈ 80, so the factor (1-1/ε) is about 0.9875. The Recognition Science framework has a page in its machine-checked library of formal theorems that defines a cost function J(x) = (x + 1/x)/2 - 1 and applies it to the ratio of two quantities, m/e.

In Recognition Science, a ledger (a discrete record of events) assigns a cost to each recognition event, and the framework proves that any cost function meeting five plain conditions must equal J(x). The declaration SolvationFECert is a structure that bundles three proven facts about this cost applied to a ratio. First, when the two quantities are equal, the cost is zero: domainCost r r = 0 for any nonzero r. Second, for positive inputs, the cost is never negative: 0 ≤ domainCost m e whenever m and e are both positive. Third, the number φ - 3/2, where φ is the golden ratio, is positive. These three facts are proved in the library with no unproven assumptions.

The library's own documentation is explicit about the limit: it proves nothing specific to solvation, because the definition of domainCost is just Jcost (m / e) without any reference to a chemical system. The research note in the file sketches an intended connection: for water, J(80) ≈ 38.5, and the note suggests a large J corresponds to strong solvation. But that paragraph is a research note recording where the idea was meant to go, not a result. The three theorems hold for any positive real numbers m and e, not for any particular physical quantity.

What the certificate does establish is a general mathematical fact: the cost function J, applied to a ratio, vanishes at equality, stays nonnegative for positive arguments, and has a positive threshold involving the golden ratio. These are properties of the function itself, independent of any chemistry. The library's template module states these facts once, universally quantified, and this chemistry module inherits them verbatim. A reader can rely on those three facts as proven; a reader cannot yet rely on any claim that this cost function describes actual solvation energies without further definitions linking m and e to physical quantities like charge and permittivity.

structure SolvationFECert where
  cost_at_eq : ∀ r : ℝ, r ≠ 0 → domainCost r r = 0
  cost_nonneg : ∀ m e : ℝ, 0 < m → 0 < e → 0 ≤ domainCost m e
  threshold_pos : 0 < canonicalThreshold

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

No claim that solvation free energy equals Jcost(m/e) for any physical system is proven. No claim that the research note's identification of m and e with charge and permittivity is a theorem. No claim that the golden ratio threshold has any chemical significance in this module.

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/SolvationFreeEnergyFromJCost.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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