persistenceLengthRatio
Persistence length scales by the factor phi between consecutive rungs on the ladder. Materials modelers in Recognition Science cite this ratio when constructing polymer chain certificates. The term proof unfolds the power definition and reduces the ratio algebraically after confirming positivity.
claimFor any natural number $k$, the ratio of persistence lengths satisfies $L_p(k+1)/L_p(k) = phi$, where the persistence length at level $k$ is given by $L_p(k) = phi^k$.
background
The module defines persistence length as a power of the fixed point phi, $L_p(k) = phi^k$, to capture stiffness in RS units where $c=1$ and other constants are phi-based. This aligns with the five canonical polymer regimes corresponding to configuration dimension 5. The result depends on the upstream definition of persistence length as phi raised to $k$.
proof idea
Unfolding the persistence length definition yields phi to the power k+1 over phi to the power k. Positivity of phi^k is established to permit the division rewrite. The successor power rule and division equivalence are applied, followed by ring simplification to conclude the equality.
why it matters in Recognition Science
It provides the phi ratio component for the polymer chain certificate that enumerates the five regimes. This fits the materials tier of the Recognition framework, where phi emerges as the self-similar fixed point. The scaling supports consistency with Flory's exponent approximation.
scope and limits
- Does not provide numerical predictions for specific polymer species.
- Does not incorporate solvent effects or temperature variations.
- Does not extend to higher-order corrections in the chain statistics.
formal statement (Lean)
29theorem persistenceLengthRatio (k : ℕ) :
30 persistenceLength (k + 1) / persistenceLength k = phi := by
proof body
Term-mode proof.
31 unfold persistenceLength
32 have hpos := pow_pos phi_pos k
33 rw [pow_succ, div_eq_iff hpos.ne']
34 ring
35