Chemistry Polymorphism From Jcost Polymorphism Cert

What the certificate proves

In the Recognition Science framework, a ledger is a discrete record of events, and each event carries a cost, a forced number that measures how expensive recognition is. The framework's central theorem fixes one specific cost function, J(x) = (x + 1/x)/2 - 1. The declaration PolymorphismCert is a small structure in the framework's machine-checked library of formal theorems. It bundles three general facts about this cost function: it returns zero when its two inputs are equal, it never returns a negative number for positive inputs, and a certain threshold value phi - 3/2 is positive. These are the only claims the certificate makes.

The three facts are proved, not assumed. The first, that J(r/r) = 0, follows directly from the definition of J. The second, that J(m/e) is nonnegative whenever m and e are both positive, is a standard property of the cost function. The third, that phi - 3/2 > 0, uses the known value of the golden ratio phi. Each of these is a theorem in the library, and the certificate simply packages them together. The certificate itself is a structure, a way of grouping these three theorems under one name, and the library shows that such a certificate exists.

What the certificate does not do is say anything about chemistry. The cost function is applied to a ratio m/e, but the declaration never defines what m and e mean. A research note attached to the module records an intended application: m might be the free energy of a metastable crystal form and e that of the stable form, so that the cost measures how far a metastable form is from stability. The note also mentions Ostwald's rule of stages, the empirical observation that a system often passes through metastable forms before reaching the stable one. But these are ideas for future work, not results. The certificate proves no theorem about crystals, polymorphism, or stability.

To make the certificate a theorem about its subject, someone would need to define m and e in chemistry's own terms, for example as actual free energies. Until that definition exists, PolymorphismCert remains a statement about a mathematical function, not about the physical world. The framework's library is explicit about this gap: the module proves nothing specific to this subject. The certificate is honest about its own limits, and that honesty is part of its value.

structure PolymorphismCert 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]

What this page does not claim

The certificate says nothing about any real crystal, polymorph, or stability result. The module does not define m and e in chemical terms, so no physical claim follows from it. The research note about metastable forms is an idea for future work, not a proved theorem.

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)

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.

Derived articles

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: