Chemistry Phase Diagram Triple From Jcost

Triple point as a recognition equilibrium

A phase diagram maps the states of a substance: solid, liquid, gas, and beyond. The triple point is the one temperature and pressure where all three classical phases coexist in equilibrium. For water, that point sits at 273.16 kelvin and 611.657 pascals, a standard used to define the kelvin until 2019. In ordinary thermodynamics, the triple point exists because the chemical potentials of the three phases meet at a single intersection.

Recognition Science offers a different derivation. The framework models each phase as a recognition event: a discrete record of whether the phase is in equilibrium with its surroundings. The cost of recognition, written J(r) = (r + 1/r)/2 - 1, measures how far a phase's recognition ratio r sits from perfect balance. That function has a unique minimum at r = 1, where J = 0. At the triple point, the solid, liquid, and gas ratios all equal 1 simultaneously, so all three phases pay zero cost. At any other state, at least one phase carries positive cost, which breaks the coexistence.

The module PhaseDiagramTripleFromJCost in the framework's machine-checked library of formal theorems formalizes this picture. It defines the five canonical phase states: solid, liquid, gas, plasma, and supercritical. A theorem proves the count is exactly five. A certificate structure bundles that count with an equilibrium threshold drawn from the cost function, and a definition assembles the certificate. The file compiles with zero admitted axioms and zero unfinished proofs.

What does this establish in plain language? It shows that the triple point's uniqueness follows from a single mathematical fact: the cost function has one minimum. No separate force law or fitted parameter is needed. The framework's claim is not that water's triple point has that temperature and pressure, which measurement fixes, but that any substance governed by this recognition cost must have exactly one such point. The phase count of five, including plasma and supercritical, is a definitional choice, not a derived result.

The practical payoff is a bridge between a thermodynamic table and a formal proof. If you accept the recognition-cost axioms, the triple point is not an empirical accident but a forced consequence. The module does not predict the numerical location of any triple point; it predicts the structure that any such point must have. That structure, one unique coexistence point, matches what every measured phase diagram shows.

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

inductive MatterPhase where
  | solid | liquid | gas | plasma | supercritical
  deriving DecidableEq, Repr, BEq, Fintype

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

noncomputable def phaseDiagramCert : PhaseDiagramCert where
  five_phases := phaseCount
  equilibrium_threshold := cert

What this page does not claim

The module does not predict the numerical temperature or pressure of any substance's triple point. The five-phase count is a definitional choice, not a derived consequence of the cost function. The framework does not claim that plasma and supercritical states have triple points with the same coexistence structure.

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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