MatterPhase
plain-language theorem explainer
The MatterPhase inductive enumerates the five canonical states of matter used in the Recognition Science phase diagram. Chemists modeling J-cost minimization at the triple point would cite this enumeration to fix the phase space cardinality at five. The declaration is a direct inductive definition that derives decidable equality, representation, boolean equality, and finite type instances automatically.
Claim. Let MatterPhase be the inductive type whose constructors are solid, liquid, gas, plasma, and supercritical, equipped with decidable equality, a representation instance, boolean equality, and a finite type instance.
background
The module treats the triple point as the unique state where J-cost reaches its global minimum simultaneously for the solid, liquid, and gas phases, which occurs precisely when each phase has recognition ratio equal to one. Five canonical phases are introduced to match configDim D = 5, with the triple point involving only the first three. This chemistry version adapts the upstream inductive MatterPhase from CondensedMatterPhasesFromRS, which instead lists solid, liquid, gas, plasma, and BEC.
proof idea
The declaration is an inductive definition with an empty proof body. The deriving clause for DecidableEq, Repr, BEq, and Fintype is supplied directly by Lean, generating the required instances without further proof steps.
why it matters
This enumeration is required by the downstream phaseCount theorem asserting Fintype.card MatterPhase = 5 and by PhaseDiagramCert, which pairs the five phases with an equilibrium threshold. It also feeds CondensedMatterPhaseCert and totalPhaseCount in the physics module, where total phases equal 10 = 2 × D. The definition thereby supplies the phase space dimension for the triple-point uniqueness claim that J-cost is minimized for all three phases at exactly one (T, P) pair.
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