Chemistry Polymer Chain Statistics From Jcost

Polymer chain statistics

Polymer chain statistics describes the shape of a long molecule made of repeating units. The simplest model, the freely jointed chain, treats each unit as a rigid rod that can point in any direction. The mean squared distance between the two ends of such a chain is ⟨r²⟩ = Nl², where N is the number of monomers and l is the length of each one. This is the classic random-walk result.

Real polymers are stiffer than this model suggests. Their persistence length l_p, the distance over which the chain direction is remembered, is longer than a single monomer. In the framework of Recognition Science, the persistence length is related to the monomer length by l_p = l_0 × φ^k, where φ is the golden ratio and k depends on the polymer class. For DNA, l_p ≈ 50 nm, which is close to φ⁸ × 1 nm, since φ⁸ ≈ 47.

In Recognition Science, the cost function J(x) = (x + 1/x)/2 - 1 measures the price of a mismatch between two quantities. The module under discussion defines a domain cost as J(m/e), where m and e are two positive real numbers. The Lean code proves three general facts about this cost: it is zero when m equals e, it is never negative for positive inputs, and the threshold φ - 3/2 is positive. These are properties of the cost function itself, not of any specific polymer.

The module does not yet connect these general facts to polymer physics. To do so, one would need to define m and e in terms of polymer properties, such as the persistence length and the monomer length. The current module is a template, shared with many other subjects, that records where the idea was meant to go. The proof that the cost vanishes at equality and is nonnegative is a foundation, not a finished theory.

What the module does establish is a small, machine-checked certificate: the cost function behaves sensibly at equality and stays nonnegative. This is a necessary step, but the step from these general facts to a statement about polymer chains remains open. The next move is to define m and e in the language of polymer physics, which would turn the template into a theorem about the subject.

def domainCost (m e : ℝ) : ℝ := Jcost (m / e)

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 module proves no specific polymer fact; it proves general properties of the cost function. The freely jointed chain formula is a classical model, not a result of the framework. The persistence length scaling is a research note, 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/PolymerChainStatisticsFromJCost.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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