Chemistry Polymorphism From Jcost

Polymorphism and the cost threshold

Polymorphism is the ability of a solid material to exist in more than one crystal structure. The same chemical compound, like carbon, can arrange its atoms as graphite or diamond. These forms, called polymorphs, have different physical properties but identical chemistry. Which form appears depends on conditions such as temperature and pressure, and on the path the system takes as it solidifies.

A central puzzle is why a system often passes through a less stable form before reaching the stable one. In 1897, Wilhelm Ostwald stated a rule of stages: a system does not go directly to the most stable form, but moves stepwise through metastable forms that are closest in free energy to the starting state. A metastable form is one that is not the most stable but persists because the barrier to change is high. The stable form is the one with the lowest Gibbs free energy under given conditions.

In Recognition Science, the framework models this competition with its cost function. The cost of recognizing a state is J(x) = (x + 1/x)/2 - 1, where x is a ratio of two quantities. The framework defines a domain cost as J(m/e), where m and e are two real numbers. The module proves three general facts about this cost: it is zero when m equals e, it is never negative for positive inputs, and the threshold phi - 3/2 is positive. Here phi is the golden ratio, about 1.618.

The framework uses this threshold to state a condition for polymorph selection. A metastable form is stable when J(ΔG_metastable/ΔG_stable) is less than J(phi). In words, the cost of recognizing the metastable form relative to the stable one must stay below a fixed bound. This is a research note, not a proved result, because the module does not define what m and e mean in chemical terms. The Lean code proves the general inequalities but nothing specific to crystals.

The machine-checked library of formal theorems shows what is and is not established. What is proved: the cost vanishes when the two inputs are equal, the cost is nonnegative, and the threshold is positive. What is not proved: that any real crystal obeys this condition. The module is a template, shared with many sibling modules, waiting for a definition of m and e from chemistry itself.

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]

theorem cert_inhabited : Nonempty PolymorphismCert := ⟨cert⟩

What this page does not claim

No theorem states that any real crystal obeys the cost condition. The module does not derive the existence of polymorphism from the cost function. The threshold phi - 3/2 is not shown to match any measured free-energy ratio.

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

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