Materials Wettability Angle2

Wettability and the cost function

Wettability describes how a liquid spreads on a solid. The contact angle is the angle where the liquid-vapor interface meets the solid surface. A droplet that beads up has a high contact angle, above 90 degrees. A droplet that spreads flat has a low one. Superhydrophobic surfaces, like lotus leaves, push this angle past 150 degrees, so water rolls off almost without touching.

The classical theory of wetting goes back to Thomas Young in 1805. Young's equation balances the surface tensions at the three-phase contact line: solid-liquid, solid-vapor, and liquid-vapor. The contact angle falls out of that balance. Modern measurements use goniometry, where a camera photographs a sessile drop and software fits the angle. Values above 150 degrees define the superhydrophobic regime, a practical threshold for self-cleaning coatings and water-repellent fabrics.

In Recognition Science, the framework models the wetting cost through its forced cost function. The framework defines domainCost as Jcost of the ratio m over e, where m and e are quantities the subject would need to define. The framework proves three general facts about this cost. It vanishes when m equals e. It stays nonnegative when both inputs are positive. And the threshold phi minus 3/2 is positive, where phi is the golden ratio. That threshold, roughly 0.118, is the framework's canonical marker for a regime change.

The certificate bundles these three facts into one structure. The machine-checked library of formal theorems shows the certificate exists. But the framework itself proves nothing specific to wetting. The docstring records where the idea was meant to go, not a result. What would make this a theorem about superhydrophobicity is a definition of m and e in the subject's own terms, for example a surface energy or a droplet volume. Without that, the cost function is a template, not a physical law.

The honest takeaway is sharp. The framework contributes a general cost with proven properties, and a positive threshold. The leap from that threshold to 150 degrees remains a research note, not a derivation. A reader can use the framework as a scaffold: define m and e for a real surface, and the three proven facts apply. The angle itself is not yet forced.

theorem domainCost_at_eq (r : ℝ) (h : r ≠ 0) : domainCost r r = 0 := by
  unfold domainCost; rw [div_self h]; exact Jcost_unit0

theorem domainCost_nonneg (m e : ℝ) (hm : 0 < m) (he : 0 < e) : 0 ≤ domainCost m e := by
  unfold domainCost; exact Jcost_nonneg (div_pos hm he)

theorem canonicalThreshold_pos : 0 < canonicalThreshold := by
  unfold canonicalThreshold; linarith [phi_gt_onePointFive]

theorem cert_inhabited : Nonempty SuperhydrophobicityCert := ⟨cert⟩

What this page does not claim

No derivation that the contact angle equals 150 degrees or any other specific value. No claim that the framework proves anything about real materials without a definition of m and e. No assertion that the framework supersedes classical wetting theory.

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/Materials/Wettability_Angle2.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.

Derived articles

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: