IndisputableMonolith.Materials.Wettability_Angle2
Materials module that packages a domain-wall cost, a positive canonical threshold, and an inhabited superhydrophobicity certificate for contact-angle / wettability claims in RS units. A materials theorist cites it when linking surface energy to the J-cost ladder rather than empirical Young angles. The file is mostly definitions plus nonnegativity and positivity lemmas; the certificate is a Prop bundle with an explicit witness.
claimDefine a domain cost $C_{\mathrm{dom}}$ on surface configurations, prove $C_{\mathrm{dom}}\ge 0$ and an evaluation identity, fix a canonical threshold $\theta_*>0$, and package a superhydrophobicity certificate asserting that the domain cost lies above $\theta_*$ (with an inhabited witness).
background
Recognition Science materials work treats wettability as a cost comparison on the same J-functional that forces the forcing chain, not as a free Young contact angle. The Cost import supplies the nonnegative J-cost $J(x)=(x+x^{-1})/2-1$; Constants supplies the RS tick and $\varphi$-ladder units in which surface energies are expressed.
This module introduces a domain cost $C_{\mathrm{dom}}$ (sibling domainCost) meant to score interfacial mismatch or pinning on that ladder, together with a canonical positive threshold that marks the superhydrophobic regime. The certificate type bundles the inequality $C_{\mathrm{dom}}\ge\theta_*$ so downstream materials lemmas can demand a single Prop rather than ad-hoc inequalities.
No continuum fluid PDE is assumed: the setting is discrete RS cost on configurations, with nonnegativity inherited from J.
proof idea
Definition-heavy module. domainCost and canonicalThreshold are defs; domainCost_nonneg and canonicalThreshold_pos are short positivity/nonnegativity arguments from the Cost layer. domainCost_at_eq is an evaluation identity. SuperhydrophobicityCert is a structure/Prop packaging the threshold inequality; cert and cert_inhabited supply a concrete witness so the certificate type is nonempty. No deep tactic proof; the argument is assembly of cost nonnegativity plus a fixed positive cutoff.
why it matters in Recognition Science
Places superhydrophobicity inside the RS materials stack: contact-angle phenomenology is reduced to a J-cost domain score crossing a canonical threshold, consistent with the same cost that yields T5 J-uniqueness and the $\varphi$ ladder. Downstream used-by edges are empty in the current graph, so this file is a leaf certificate API for later coatings, lotus-effect, or droplet lemmas rather than a step in the T0–T8 forcing chain. It closes a materials-side interface: once cert is inhabited, any lemma needing “superhydrophobic in RS units” can take SuperhydrophobicityCert as a hypothesis instead of re-deriving the threshold comparison.
scope and limits
- Does not derive Young’s law or continuum Navier–Stokes contact-line dynamics.
- Does not compute numerical contact angles in degrees from experiment.
- Does not prove uniqueness of the canonical threshold from first principles here.
- Does not connect wettability to the eight-tick octave or D=3 forcing steps.
- Does not supply a molecular Hamiltonian; only a cost-threshold certificate API.