Chemistry Polyolefins From Jcost

The module's actual scope

Polyolefins are the workhorse plastics of everyday life: polyethylene in bags and bottles, polypropylene in ropes and containers. Their usefulness depends on crystallinity, the fraction of the material where the long molecular chains fold into orderly, repeating patterns. A highly crystalline plastic is stiff and strong; a less crystalline one is flexible and tough. For isotactic polypropylene, a common form, measured crystallinity typically falls between 65 and 75 percent.

In Recognition Science, the framework models recognition as a ledger, a discrete record of events, and assigns each event a cost, a number measuring how expensive that recognition is. The framework's central theorem forces the cost function to be J(x) = (x + 1/x)/2 - 1. A research note in the module suggests using this J to predict polymer crystallinity, with a formula that lands near 76 percent, close to the measured 75. That idea is recorded as a research note, not a result.

The module itself, named PolyolefinsFromJCost, proves three general facts about J applied to a ratio m/e. It proves that the cost vanishes when m equals e, that the cost is never negative for positive inputs, and that the constant phi - 3/2 is positive. These are properties of the cost function alone; the module never defines what m and e mean for a polymer. The definition of domainCost simply sets it to J(m/e) with no reference to polyolefins, so the chemistry is absent from the formal content.

In plain language: the machine-checked library of formal theorems shows that J behaves well, but it does not connect J to crystallinity. The gap is a missing definition of m and e in polymer terms. Until that definition exists, the module is a template shared with 2383 siblings, and the polyolefin claim remains a target, not a theorem.

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 module does not prove any statement about actual polyolefins or crystallinity. The 76 percent crystallinity figure is a research note, not a theorem or a measurement.

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.

Derived articles

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