Chemistry Polymer Chain Length From Phi Ladder Persistence Length Ratio

The persistence length ratio

A polymer chain is a long molecule that bends and coils. Its persistence length is the distance over which the chain remembers its direction before thermal motion bends it elsewhere. A stiff chain, like double-stranded DNA, has a long persistence length; a floppy one, like a loosely coiled synthetic rubber, has a short one. The persistence length is a standard quantity in polymer science, often written Lp, and it sets the scale for how the chain explores space.

In the Recognition Science framework, the persistence length is modeled as a simple ladder of discrete levels. The framework's declaration persistenceLengthRatio proves a single clean fact: the ratio of one level to the next is exactly the golden ratio, about 1.618. In symbols, if L(k) is the persistence length at level k, then L(k+1)/L(k) = φ. This is a theorem in the framework's machine-checked library of formal theorems, proved with no axioms beyond the standard logical ones. The declaration does not claim that real polymer chains have persistence lengths that are exact powers of the golden ratio. It establishes a structural relationship within the framework's model, not a measurement of any specific material.

The framework also counts five canonical polymer chain regimes: rigid rod, worm-like chain, ideal chain, excluded-volume, and collapsed. That count of five is a theorem in the same library. The framework connects this to a broader claim that the number of regimes is a structural dimension, but the declaration itself only establishes the ratio and the count. It does not derive the Flory exponent ν = 3/5 from first principles; the framework notes its own value ν ≈ 0.603 is consistent with the measured Flory value of 0.588, but that consistency is an empirical check, not a theorem.

What the declaration changes is simple: it gives a clean, checkable statement about how stiffness levels are spaced in the framework's model. A reader can see the exact ratio, verify it is the golden ratio, and know precisely what is and is not being claimed. The framework's contribution here is a structural observation, not a new law of polymer physics.

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

What this page does not claim

Real polymer persistence lengths are measured to be exact powers of the golden ratio. The Flory exponent ν = 3/5 is derived from the framework's axioms. The five-regime count is a physical law rather than a classification within the framework.

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)

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