Chemistry Polymer Chain Length From Phi Ladder Persistence Length

Persistence length in the phi ladder

In polymer physics, the persistence length measures how far a chain stays roughly straight before thermal bending makes it wander. It is the distance over which the direction of the chain loses memory of its starting orientation. A stiff rod has a long persistence length; a floppy string has a short one. The classical definition ties it to bending stiffness and thermal energy: Lp = kT/κ, where κ is the bending modulus and kT is the thermal energy scale.

The Recognition Science framework models this length scale on a discrete ladder. Its declaration persistenceLength (k) is simply phi raised to the k-th power, where phi is the golden ratio, about 1.618. The framework's machine-checked library of formal theorems proves that the ratio of consecutive rungs is exactly phi: persistenceLength (k+1) divided by persistenceLength (k) equals phi. That is, each step up the ladder multiplies the length by the golden ratio. The proof is a short algebraic identity, not an empirical fit.

The same library also names five canonical polymer regimes: rigid rod, worm-like chain, ideal chain, excluded-volume, and collapsed. It proves that there are exactly five such regimes. In the framework's account, this count matches a structural dimension of five, and the excluded-volume exponent ν is modeled as 1/φ^(1/3), about 0.603, which is close to the classical Flory value of 0.588. The framework does not claim to derive the Flory value from first principles; it offers a nearby number from its own ladder.

What the declaration does not claim is important. It does not claim that real polymer persistence lengths are exactly powers of phi. It does not claim that the golden ratio is the only possible scaling. It does not claim that the five regimes are physically exhaustive or that the framework's exponent replaces measured polymer data. The declaration is a definitional model, not a measurement or a derivation of polymer physics from first principles.

theorem persistenceLengthRatio (k : ℕ) :
    persistenceLength (k + 1) / persistenceLength k = phi := by
  unfold persistenceLength
  have hpos := pow_pos phi_pos k
  rw [pow_succ, div_eq_iff hpos.ne']
  ring

theorem polymerRegimeCount : Fintype.card PolymerRegime = 5 := by decide

noncomputable def persistenceLength (k : ℕ) : ℝ := phi ^ k

What this page does not claim

Real polymer persistence lengths are exactly powers of phi. The framework derives the Flory exponent 0.588 from first principles. The five named regimes are physically exhaustive.

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/PolymerChainLengthFromPhiLadder.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

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: