Chemistry Polyolefins From Jcost Polyolefin Cert

A certificate with no subject

In the Recognition Science framework, a ledger (a discrete record of events) assigns a cost to every recognition event. The declaration PolyolefinCert is a machine-checked certificate that bundles three proven facts about that cost. First, the cost is zero when the two quantities being compared are equal. Second, the cost is never negative when both inputs are positive. Third, a specific constant, the golden ratio minus 1.5, is greater than zero. These are the complete contents of the certificate.

The certificate's name is misleading. It contains no definition of a polyolefin, no measure of crystallinity, and no chemical parameter. The cost function it uses is defined generically as J(m/e), where m and e are real numbers, with no interpretation attached to them. The three facts it proves are true for any positive real numbers, not for any particular material. The library's own documentation states this plainly: the proof establishes general facts about the cost function and nothing specific to polyolefins.

A research note attached to the library sketches an intended application: estimating isotactic polypropylene crystallinity from the cost function. It suggests a formula, crystallinity = 1 − 2×J(φ), which yields about 76.4 percent, close to a measured range of 65 to 75 percent. But this note is explicitly marked as a plan, not a result. No theorem in the library connects the cost function to crystallinity, and no definition links m and e to any chemical quantity. The numerical agreement is an unverified suggestion, not a derived conclusion.

What the certificate does establish is a small, clean piece of mathematics. It shows that the framework's cost function behaves sensibly at equality, stays nonnegative, and has a positive threshold constant. Those facts are proved in the machine-checked library of formal theorems and are available for any future work that actually defines m and e in chemical terms. The certificate is a foundation stone, not a building.

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 any relationship between the cost function and polyolefin crystallinity. The numerical agreement with isotactic polypropylene crystallinity is an unverified research note, not a theorem. No claim is made that the golden ratio constant has any chemical significance beyond being a positive number.

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/PolyolefinsFromJCost.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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