Chemistry Polymer Chain Length From Phi Ladder Polymer Chain Cert

Polymer Chain Cert

Polymer chains, the long molecules that make up plastics, rubber, and DNA, come in a few distinct shapes depending on stiffness and solvent conditions. A rigid rod is one extreme; a collapsed globule is the other. In between sit the worm-like chain, the ideal chain, and the excluded-volume chain. These five canonical regimes are a standard classification in polymer physics, and the Recognition Science declaration PolymerChainCert, a machine-checked certificate in the framework's library of formal theorems, records that this count is exactly five.

The certificate's second component concerns persistence length, the distance over which a polymer chain remembers its direction. The framework models this length as a discrete ladder of values: the k-th rung is phi^k, where phi is the golden ratio, approximately 1.618. The certificate proves that each step up the ladder multiplies the persistence length by phi, so the ratio of successive lengths is exactly the golden ratio. This is a scaling law: the chain's stiffness grows in fixed multiplicative steps rather than continuously.

In Recognition Science, the golden ratio emerges from the framework's core cost function, and the polymer ladder is one application of that structure. The certificate itself is a formal object: a structure that packages the five-regime count and the phi-ratio theorem together, with a proof that both hold. The library checks it with no gaps and no extra assumptions beyond the standard logical axioms.

What the certificate does not claim is important. It does not assert that real polymer chains obey this ladder exactly; real chains follow the Flory exponent of about 0.588 for the distance between chain ends, while the framework's value is approximately 0.603, close but not identical. The certificate does not derive the five regimes from first principles; it simply records the count as a definitional choice. And it does not say that the persistence length of any actual material is a power of phi; it establishes a structural relationship within the framework's model.

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

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

What this page does not claim

Real polymer chains obey the phi-ladder exactly. The five regimes are derived from the framework rather than chosen as a classification. Any measured persistence length equals a power of phi.

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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