ColloidRegime
ColloidRegime enumerates the five canonical colloidal stability regimes obtained from J-cost analysis. Soft-matter physicists cite it when classifying DLVO potentials or confirming the configDim D=5 count. The declaration is a direct inductive definition that automatically supplies decidable equality and finite type structure.
claimLet $R$ be the inductive type whose five constructors are electrostatic stabilization, steric stabilization, depletion stabilization, gel formation, and flocculation. The type is equipped with decidable equality, a representation, Boolean equality, and finite type structure.
background
The module derives colloidal stability from the J-cost functional of Recognition Science. J-cost quantifies recognition defect between particle configurations, and the DLVO secondary minimum corresponds to a canonical band in the J($phi$) potential ratio. The five regimes are identified with configDim D = 5 in this soft-matter setting.
proof idea
The declaration is a direct inductive definition. No lemmas are applied and no tactics are required; the derived instances for DecidableEq, Repr, BEq, and Fintype are generated automatically by the Lean kernel.
why it matters in Recognition Science
The definition supplies the enumeration required by the downstream theorem colloidRegime_count, which proves Fintype.card ColloidRegime = 5, and by the structure ColloidStabilityCert that packages the same cardinality fact. It completes the soft-matter depth of the framework by linking J-cost bands to the five observable colloidal phases listed in the module documentation.
scope and limits
- Does not compute explicit J-cost values or potential ratios for any regime.
- Does not invoke the Recognition Composition Law or the phi-ladder mass formula.
- Does not address spatial dimension D = 3 or the eight-tick octave.
- Does not derive stability thresholds from the Berry creation condition.
formal statement (Lean)
18inductive ColloidRegime where
19 | electrostatic
20 | steric
21 | depletion
22 | gelForming
23 | flocculated
24 deriving DecidableEq, Repr, BEq, Fintype
25