Materials Polymorphism3 From Jcost

Polymorphism and its cost

Polymorphism is the ability of a solid material to exist in more than one crystal form. The same chemical compound, for example a drug molecule, can pack its molecules into different arrangements, and each arrangement has its own properties. One form might dissolve easily in the body, another might not. This matters enormously for pharmaceuticals: roughly half of active pharmaceutical ingredients show polymorphism, so drug developers must know which form they are making.

The module materials polymorphism3 from jcost is a small file in the Recognition Science framework's machine-checked library of formal theorems. The framework's central object is a cost, a number that measures how hard one recognition event is relative to another. The file defines a domain cost as the framework's cost function J applied to the ratio of two real numbers m and e. It then proves three general facts about that cost: it is zero when m equals e, it is never negative when m and e are both positive, and a certain threshold built from the golden ratio is positive.

The file's own docstring is explicit about the limits. The three proved facts are general properties of the cost function, not facts about any particular material. The file defines m and e only as real numbers, with no connection to molecules or crystals. The docstring records a research note that the intended subject was crystal polymorphism, and it mentions a suggestive numerical agreement: the framework's cost at the golden ratio, square-rooted, gives about 34.4 percent, which sits inside the 30 to 50 percent range of real-world polymorphism rates. But that note is a plan, not a proved result.

What the module does establish, in plain language, is that the cost function has the basic hygiene any sensible measure should have: matching inputs cost nothing, and positive inputs never produce a negative cost. The threshold fact, that phi minus 1.5 is positive, is a small arithmetic consequence of the golden ratio being greater than 1.5. These are the building blocks one would need if a later module ever defined m and e in terms of actual crystal properties. Until that definition exists, the file remains a template, not a theory of polymorphism.

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]

What this page does not claim

The module does not prove any fact about real crystals or pharmaceutical compounds. The 34.4 percent polymorphism rate is a research note, not a theorem. The file does not establish that the cost function is the correct measure for polymorphism.

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/Materials/Polymorphism3_FromJCost.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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