Chemistry Polymer Crystal3 From Jcost Poly Crystal3 Cert

The certificate's scope

PolyCrystal3Cert is a formal certificate, a small bundle of three proved facts about a mathematical function called cost. In plain terms, the cost function measures the price of a ratio of two numbers, and the certificate records that this price is zero when the two numbers are equal, that it is never negative when both numbers are positive, and that a certain constant, phi minus 1.5, is greater than zero. These are general properties of the cost function itself, proved for any positive real numbers. The certificate does not mention polymers, crystals, or temperatures anywhere in its statements.

The certificate lives in a machine-checked library of formal theorems, a collection where every proof is verified by a computer. The three facts are proved from the definition of the cost function alone, with no extra assumptions about chemistry. The certificate is inhabited, meaning the three facts are actually proved and assembled into one object. That is the whole content: three general inequalities and an equality, nothing more.

In Recognition Science, the cost function is the framework's central object, and its unique form is forced by five plain conditions. The certificate reuses that function but does not connect it to any polymer-specific quantity. A research note in the same file suggests a possible application: crystallization rate might peak when the ratio of temperature to melting temperature equals the reciprocal of the golden ratio, about 0.618, which matches an empirical range of roughly 0.6 to 0.7 times the melting temperature. That note is a hypothesis, not a proved result, and the certificate itself proves none of it.

What the certificate does establish is a reusable template. Because the three facts hold for any positive inputs, any future work that defines the polymer's recognition variables, the discrete record of events the framework tracks, in terms of measurable quantities like temperature and melting temperature can inherit these proofs. Until such a definition exists, the certificate remains a general lemma about the cost function, not a theorem about polymer crystallization.

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 PolyCrystal3Cert := ⟨cert⟩

structure PolyCrystal3Cert 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

What this page does not claim

This certificate does not prove that polymer crystallization rate peaks at any specific temperature ratio. This certificate does not define what the numbers m and e mean for a polymer. This certificate does not establish the empirical claim about the crystallization peak near 0.6 to 0.7 times the melting temperature.

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/Polymer_Crystal3_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.

Derived articles

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