Chemistry Phase Coexistence From Jcost Phase Coexistence Cert

The five coexistence shapes

In chemistry, when two or more phases of a substance exist together at equilibrium, the pattern they form is called a phase coexistence topology. The classical examples are familiar: a liquid and its vapor meeting along a curve, a eutectic where two solids and a liquid meet at a single point, or an azeotrope where boiling does not change the composition. A standard encyclopedia would list these as separate phenomena, each with its own name and diagram.

The Recognition Science declaration PhaseCoexistenceCert takes a different step: it defines a finite list of five such topologies and proves, by a direct check, that the list has exactly five members. The five are the two-phase binodal, the three-phase eutectic, the four-phase peritectic, the azeotrope, and the tricritical point. The declaration is a ledger, a discrete record of recognized shapes, and it certifies that the ledger contains five entries. The proof is a machine-checked computation, with no unproved assumptions and no axioms beyond the standard logical ones.

The declaration does not claim that these five are the only phase behaviors that can occur in real materials. It does not derive the shapes from the J-cost function, even though the module name mentions J-cost. The connection to the framework's cost function, which forces the golden ratio and the number three in other contexts, is not part of this declaration. The declaration only fixes a count: five canonical topologies, no more, no less.

What the declaration changes is the status of the list. Instead of a textbook enumeration that could be extended without limit, the framework treats the five as a closed set, the same way the number of spatial dimensions is forced to be three. The practical consequence is that any phase diagram, no matter how complex, is expected to decompose into these five building blocks. That expectation is a target for further work, not a result of this declaration.

theorem phaseTopology_count : Fintype.card PhaseCoexistenceTopology = 5 := by decide

inductive PhaseCoexistenceTopology where
  | binodal
  | eutectic
  | peritectic
  | azeotrope
  | tricritical
  deriving DecidableEq, Repr, BEq, Fintype

def phaseCoexistenceCert : PhaseCoexistenceCert where
  five_topologies := phaseTopology_count

What this page does not claim

The declaration does not derive the five topologies from the J-cost function. The declaration does not claim that real materials exhibit only these five phase behaviors. The declaration does not provide a classification of all possible phase diagrams.

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/PhaseCoexistenceFromJCost.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.

Derived articles

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: