persistenceLength
Persistence length for polymer chains is assigned the value phi raised to the power k, with k a natural number indexing the rung on the phi-ladder. Materials modelers connecting stiffness scales to Recognition Science constants would cite this when building chain length derivations. The definition is a direct power assignment in the reals requiring no lemmas or tactics.
claimThe persistence length at rung $k$ equals $L_p(k) = phi^k$, where $phi$ is the golden ratio fixed point forced by the self-similar condition in the unified chain.
background
The module sets polymer length scales in Recognition Science units, with persistence length quantifying stiffness through the relation $L_p = kT/κ$ and scaling as $phi^k$. Five regimes (rigid rod, worm-like chain, ideal chain, excluded-volume, collapsed) are identified, matching configDim $D=5$, and the end-to-end exponent is given as $1/phi^{1/3} ≈ 0.603$ for consistency with Flory $ν ≈ 0.588$. This rests on the phi self-similar fixed point from T6 of the forcing chain and the eight-tick octave structure.
proof idea
The declaration is a direct definition that sets the persistence length at integer rung $k$ to the power $phi^k$. No lemmas are applied and no tactics are used; the body consists solely of the power expression.
why it matters in Recognition Science
This supplies the base length scale referenced by the persistence length ratio theorem, which proves consecutive ratios equal phi, and by the PolymerChainCert structure that records the five regimes together with the phi ratio property. It occupies the materials tier C slot, linking polymer stiffness directly to the phi fixed point at T6 and the spatial dimension $D=3$ from T8. No open questions are closed.
scope and limits
- Does not derive the rung index $k$ from microscopic parameters.
- Does not convert results to SI units such as nanometers.
- Does not assign numerical $k$ values for specific polymers.
- Does not incorporate temperature dependence beyond the RS scaling.
formal statement (Lean)
27noncomputable def persistenceLength (k : ℕ) : ℝ := phi ^ k
proof body
Definition body.
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