Chemistry Precipitation Nucleation From Jcost

The cost of a nucleus

Precipitation nucleation is the first step of a solid forming from a solution: a few dissolved molecules cluster into a tiny seed, and only if that seed survives does a crystal grow. Classical nucleation theory gives the rate of seed formation as J ∝ exp(-16πγ³/(3kT(Δμ)²)), where γ is the surface tension of the new solid, kT is thermal energy, and Δμ is the supersaturation, the chemical potential difference that drives the process.

The framework defines a cost function on a ratio of two quantities, m and e, as Jcost(m/e). The machine-checked library proves three facts about this function: it is zero when m equals e, it never goes negative for positive inputs, and the number φ - 3/2 is positive, where φ is the golden ratio. These are general properties of the cost function, not results about chemistry. The framework does not define what m and e mean in terms of molecules, surface tension, or supersaturation.

In Recognition Science, the framework models a discrete record of events where every recognition event carries a forced cost. The cost function J(x) = (x + 1/x)/2 - 1 is the unique function satisfying five plain conditions, a theorem proved in the machine-checked library. The research note suggests that at the φ-supersaturation ratio, the activation barrier equals J(φ) times a bulk free energy term, and a critical nucleus forms at Δμ/Δμ_c = φ. This is a research note, not a proved result.

The honest content is a template: it proves the cost function behaves well, and it packages those proofs into a certificate structure. The chemistry would become a theorem only if someone defined m and e in terms of the physical system, for example as a ratio of chemical potentials or concentrations. Until then, the framework establishes general cost properties and leaves the nucleation claim as a target.

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

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 framework does not prove that nucleation occurs at the golden ratio supersaturation. The framework does not derive the classical nucleation rate formula from the cost function. The framework does not define m and e in physical terms.

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

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