Chemistry Solvation Shells From Config Dim Shell Radius Pos

Shell radius positivity

In the chemistry of solutions, a solvation shell is the layer of solvent molecules that surrounds a dissolved ion. The Recognition Science framework models these shells in water as five distinct regions: primary hydration, secondary hydration, tertiary hydration, a bulk-boundary layer, and far bulk. Each shell is assigned a radius on a ladder where the ratio between neighboring radii is the golden ratio, phi, approximately 1.618.

The declaration shellRadius_pos proves that for any shell index k, the radius phi^k is strictly greater than zero. This is a basic but necessary property: a radius cannot be zero or negative, and the proof guarantees the model never produces an invalid shell size. The statement follows directly from the fact that phi is positive and any positive number raised to a natural power remains positive.

In Recognition Science, this positivity result is one of three facts bundled into a certificate for the solvation shell model. The other two are that there are exactly five shells and that the adjacent-shell ratio equals phi. Together they form a compact, machine-checked summary of the model's internal consistency.

The declaration does not claim that real water has exactly five solvation shells around every ion, nor that measured shell radii follow a golden-ratio ladder. It establishes only a formal property of the model's own definition: the radius function is always positive. The physical comparison to experiment remains a separate empirical question.

theorem shellRadius_pos (k : ℕ) : 0 < shellRadius k := pow_pos phi_pos k

structure SolvationShellCert where
  five_shells : Fintype.card SolvationShell = 5
  phi_ratio : ∀ k, shellRadius (k + 1) / shellRadius k = phi
  radius_always_pos : ∀ k, 0 < shellRadius k

inductive SolvationShell where
  | primaryHydration
  | secondaryHydration
  | tertiaryHydration
  | bulkBoundary
  | farBulk
  deriving DecidableEq, Repr, BEq, Fintype

What this page does not claim

The declaration does not claim that real water has exactly five solvation shells around every ion. The declaration does not claim that measured shell radii follow a golden-ratio ladder.

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/SolvationShellsFromConfigDim.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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