Materials Wetting Contact Angle From Jcost

The contact angle from cost

Wetting describes how a liquid droplet spreads on a solid surface. The contact angle is the angle the droplet's edge makes with the surface: a small angle means the liquid spreads easily, a large angle means it beads up. The classical relation is Young's equation, cos θ = (γ_SV - γ_SL) / γ_LV, which ties that angle to the three surface energies at the solid, liquid, and vapor interfaces. A surface is hydrophobic when the angle is large, typically above 90 degrees.

The story of this angle began in 1805 with Thomas Young, who reasoned that a droplet's shape balances the pull of the surface energies. The equation is simple, but it leaves a question open: where do the surface energies themselves come from? Recognition Science offers one answer by deriving a single cost function from five plain conditions. The framework's central theorem proves that any cost function satisfying those conditions must equal J(x) = (x + 1/x)/2 - 1. That function, called the recognition cost, measures the price of a mismatch between two quantities, and it vanishes when the quantities match.

In Recognition Science, the module for wetting applies this cost to the ratio of two surface energies. It defines a domain cost as J(m/e), where m and e are two energy values. The module proves three general facts about this cost: it is zero when m equals e, it is never negative when both inputs are positive, and a threshold value related to the golden ratio is positive. These are theorems in the machine-checked library of formal theorems. They are general facts about the cost function, not yet a full theory of wetting.

The research note attached to the module records where the idea was meant to go. At the recognition angle, the cost function yields a specific contact angle: θ_RS = arccos(φ - 2), where φ is the golden ratio, approximately 1.618. That expression evaluates to about 112.5 degrees. This is the canonical Recognition Science contact angle for hydrophobic surfaces. The number is a prediction from the cost function, not a fitted parameter.

What the module actually establishes in plain language is more modest than the note's ambition. It proves that the cost function has the right basic shape for a wetting model: it rewards matching energies with zero cost and penalizes mismatches. The specific angle of 112.5 degrees remains a research note, an identification yet to be connected to a definition of m and e in the wetting subject's own terms. The gap is explicit in the module itself: the cost is defined without reference to any particular physical quantity, so the theorems prove facts about the cost, not about wetting.

The consequence is a clear target. The framework has the cost function; what it lacks is a definition of the two energies in terms of the solid, liquid, and vapor interfaces. Once that definition is supplied, the 112.5 degree angle would become a theorem rather than a note. Until then, the module is a scaffold: a proof that the cost function behaves as a wetting model should, waiting for the physical bridge.

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]

def canonicalThreshold : ℝ := phi - 3 / 2

What this page does not claim

The module does not prove that 112.5 degrees is the measured contact angle of any real surface. The module does not define the energies m and e in terms of solid, liquid, and vapor interfaces. The module does not derive Young's equation from the recognition cost.

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/WettingContactAngleFromJCost.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.

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