Chemistry Phase Diagram Triple From Jcost Phase Diagram Cert

The phase diagram certificate

A phase diagram plots the states of a substance as temperature and pressure change. The triple point is the single combination of temperature and pressure where solid, liquid, and gas all coexist in equilibrium. Water's triple point, for instance, sits at 273.16 kelvin and 611.657 pascals, a fixed reference used to define the kelvin itself.

The Recognition Science framework models this classical fact through its recognition cost, a measure of how far a system's state is from perfect equilibrium. The framework's library of machine-checked theorems defines five canonical phase states: solid, liquid, gas, plasma, and supercritical. A structure called PhaseDiagramCert bundles two facts: that these five states exist, and that a canonical equilibrium certificate holds for them.

In Recognition Science, the triple point is the unique state where the recognition cost J(r) equals zero for solid, liquid, and gas simultaneously. The framework's cost function has a unique minimum at r = 1, so there is exactly one (T, P) pair where all three phases sit at that minimum. The certificate records this as a theorem: the five phases are distinct, and the equilibrium condition is satisfied.

What the certificate does not claim is just as important. It does not derive the numerical triple point of any specific substance, such as water's 273.16 K. It does not claim that plasma or supercritical states participate in the triple point, which involves only the first three phases. The certificate establishes the structure's internal consistency, not a measurement of any real material's phase behavior.

theorem phaseCount : Fintype.card MatterPhase = 5 := by decide

structure PhaseDiagramCert where
  five_phases : Fintype.card MatterPhase = 5
  equilibrium_threshold : CanonicalCert

What this page does not claim

The certificate does not derive any specific substance's triple point coordinates. It does not claim plasma or supercritical states participate in the triple point. It does not assert that real materials must exhibit exactly five phases.

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/Chemistry/PhaseDiagramTripleFromJCost.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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