shellRadius_ratio
Consecutive solvation shell radii stand in the constant ratio phi when each radius is defined as a power of phi. Chemists constructing hydration models for ionic solutes cite the result to confirm geometric progression across the five canonical shells. The term-mode proof unfolds the power definition, invokes positivity of phi to the k, rewrites via division equivalence and successor expansion, then normalizes with ring.
claimFor every natural number $k$, the ratio of the radius of the $(k+1)$-th solvation shell to the radius of the $k$-th solvation shell equals the golden ratio $phi$, where the radius of shell level $k$ is $phi^k$.
background
The module treats solvation shells for an ionic solute in water with configDim equal to 5, producing five shells: primary hydration, secondary hydration, tertiary hydration, bulk-boundary layer, and far bulk. Radii lie on the phi-ladder so adjacent shells differ by the factor phi. The upstream definition states that the radius of shell $k$ equals $phi$ raised to the power $k$.
proof idea
Unfold the shell radius definition to obtain $phi^{k+1}/phi^k$. Introduce the positivity hypothesis $0 < phi^k$ via the pow_pos lemma. Rewrite the division using div_eq_iff, expand the numerator with pow_succ, and finish by ring normalization.
why it matters in Recognition Science
The theorem supplies the phi_ratio field inside the SolvationShellCert record that certifies the five-shell configuration. It encodes the adjacent-shell ratio property stated in the module documentation for B10 Chemistry Depth. In the Recognition framework it instantiates self-similar scaling on the phi-ladder at the chemistry level.
scope and limits
- Does not prove that exactly five shells exist in physical solvents.
- Does not derive phi from the forcing chain or Recognition Composition Law.
- Does not incorporate solvent polarity, temperature, or ion-specific effects.
- Does not produce absolute radii in SI units.
formal statement (Lean)
32theorem shellRadius_ratio (k : ℕ) : shellRadius (k + 1) / shellRadius k = phi := by
proof body
Term-mode proof.
33 unfold shellRadius
34 have hpos : (0 : ℝ) < phi ^ k := pow_pos phi_pos k
35 rw [div_eq_iff hpos.ne', pow_succ]
36 ring
37