Chemistry Phase Coexistence From Jcost Phase Coexistence Topology

Five shapes of coexistence

In physical chemistry, a phase diagram maps which state of matter is stable at a given temperature, pressure, and composition. Where two phases can exist together, the diagram shows a boundary; where three or more meet, it shows a point or a line. Across thousands of systems, five shapes recur: the simple two-phase binodal curve, the three-phase eutectic point, the four-phase peritectic point, the azeotrope where liquid and vapor share composition, and the tricritical point where three phases become identical. These are the standard topologies a chemist learns when reading a diagram.

The Recognition Science framework, a machine-checked library of formal theorems, takes these five shapes as a single object. Its declaration PhaseCoexistenceTopology lists exactly those five, and its proof phaseTopology_count shows the list has five members and no more. The framework then connects one of them to its central cost function: the curvature of the binodal curve is gated by a canonical band on the chemical-potential ratio, a band that comes from the same cost function that forces the golden ratio and three spatial dimensions elsewhere in the framework. In plain terms, the framework claims that the shape of a two-phase boundary is not arbitrary; it is constrained by a universal cost of recognition.

The declaration itself is small. It is an inductive type with five constructors, a theorem that the count is five, and a certificate structure that bundles that count. The proof is a direct computation, with no axioms beyond the kernel's standard three. That means the claim "there are five topologies" is proved in the framework's logic. But the framework does not prove that real chemical systems must show only these five shapes, nor that every observed phase diagram falls into one of them. It defines a classification and proves the count of that classification.

The link to the binodal curvature is stated as a docstring, not as a proved theorem in this declaration. The framework's own library marks the physical recognition-to-linking bridge as open. So a careful reader should treat the five-topology count as a proved structural fact, and the binodal curvature claim as a stated direction, not a finished result. The page that matters for a chemist is the one that tests the curvature prediction against measured phase diagrams; that page does not exist yet.

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

This declaration does not prove that all real chemical systems show only these five topologies. The binodal curvature claim is a stated direction, not a proved theorem in this declaration. The physical recognition-to-linking bridge remains open.

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: