Chemistry Polymer Chain Length From Phi Ladder Polymer Regime

The five regimes

A polymer chain is a long molecule built from repeating units. Its overall shape depends on how stiff it is and how its segments interact with the solvent around them. Polymer science names five canonical regimes: the rigid rod, the worm-like chain, the ideal chain, the excluded-volume chain, and the collapsed chain. The first is a straight stick, the last a tight ball; the others sit between, with the excluded-volume chain being the one real polymers in good solvents usually follow.

In the Recognition Science framework, the declaration PolymerRegime is a finite list of exactly these five cases. The framework proves, by direct computation, that the list has five members: recognition (a discrete record of events) assigns each chain a regime, and the count is a theorem, not a choice. The framework also defines a persistence length, the distance over which a chain keeps its direction, as a power of the golden ratio phi, and proves that each step up the ladder multiplies that length by phi.

The framework's library, a machine-checked collection of formal theorems, certifies these facts together. The certificate bundles the count of five regimes with the phi-ratio property into a single object. That is the whole content of the declaration: five named regimes, a count, and a scaling ratio. It does not claim that real polymers obey phi scaling, that the five regimes are the only ones possible, or that the excluded-volume exponent is exactly 3/5. Those remain matters of measurement and modeling.

What the declaration does is give the framework a clean vocabulary for talking about polymer shapes. A reader can now say "this chain is in the worm-like regime" and know that the framework has a precise, finite category for that statement. The count of five is a structural fact, and the phi ratio is a derived property of the framework's own length scale. The physical world is a separate question, and the framework does not pretend otherwise.

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

The declaration does not claim that real polymer chains have persistence lengths that are powers of phi. It does not claim that the five named regimes are the only possible polymer shapes. It does not claim that the excluded-volume exponent is exactly 3/5 in 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)

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.

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