Chemistry Polymerization Kinetic From Jcost Polymerization Cert

The certificate's scope

In chemistry, radical polymerization builds long chains from monomers, and the degree of conversion measures how much monomer has been consumed. A common empirical observation is that most radical polymerizations reach roughly 85 to 95 percent conversion before the reaction effectively stops. A research note in the Recognition Science library suggests that at the gel point, conversion equals 1 minus J(φ), about 88.2 percent, which would sit inside that empirical band.

The machine-checked declaration PolymerizationCert (a formal certificate, a bundle of proved facts) does not establish that chemical claim. What it proves is three general facts about a cost function, a measure of recognition effort, applied to a ratio of two real numbers m and e. First, when m equals e, the cost is zero. Second, for positive inputs, the cost is never negative. Third, the quantity phi minus 3/2, a threshold, is positive. These are facts about the cost function itself, universally quantified over all real numbers.

The certificate's definition of domainCost is simply Jcost (m / e), with no reference to monomers, chains, or gel points. Nothing in the formal proof connects m and e to any chemical quantity. The 88.2 percent figure lives in a research note above the code, not in the theorems. The certificate is shared verbatim across 2383 sibling modules, each with its own subject heading, which is why the body text warns that it proves nothing specific to this subject.

What would make the certificate a theorem about polymerization is a definition of m and e in chemical terms: for instance, m as monomer concentration and e as some equilibrium or endpoint value. Until such a definition exists, the certificate remains a template, a reusable shell of general facts. The empirical consistency of 88.2 percent with the observed 85 to 95 percent band is a research aspiration, not a proved result.

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 certificate does not prove that polymerization reaches 88.2 percent conversion. The certificate does not define m or e in any chemical terms. The certificate does not establish the gel point conversion as a derived result.

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/PolymerizationKineticFromJCost.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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