Cosmology Regular Neighborhood Boundary Oriented Polygon Euler Total Eq Surface

The Euler bridge

In topology, the Euler characteristic is a number that describes a surface's shape: a sphere has 2, a torus has 0, and a surface with 125 holes has -248. The framework's machine-checked library of formal theorems proves that when you have a list of standard surface types (each identified by its genus, or number of holes), the total Euler characteristic is simply the sum of the individual Euler characteristics. This is the definitional identity that the declaration orientedPolygonEulerTotal_eq_surfaceTypeEulerTotal_of_surfaceTypeClassificationCloses uses as its final step.

The theorem itself is a bridge between two inventories. On one side, you have a list of oriented polygon components, each a finite combinatorial description of a surface built by gluing polygon edges. On the other side, you have a list of standard surface types, each a sphere, torus, or higher-genus surface. The declaration establishes that if the two lists have the same number of components, and if the classification wrapper has already matched them component-by-component, then the total Euler characteristic of the polygon list equals the total Euler characteristic of the standard surface list. In plain terms: the two ways of counting holes agree in the aggregate.

The theorem is part of a larger chain in the framework's cosmology module, which studies the boundary of a regular neighborhood of a positive region in a cubical foam. The chain builds algebraic bridges: first from Betti numbers to boundary invariants, then through polygon gluing and orientability witnesses, then to the standard surface classification. This particular declaration is the final algebraic step that closes the Euler-total identity, conditional on the classification wrapper having already matched each component.

What the theorem does not claim is the geometric realization. It does not prove that the oriented polygon components are homeomorphic to the standard surfaces, nor that the embedded map exists. The classification of closed surfaces is named as an explicit hypothesis, not proved. The embedded homeomorphism theorem remains open, and this declaration is explicitly a finite combinatorial certificate, not a geometric realization theorem.

/-- Total Euler characteristic across the assigned standard surface types. -/
def surfaceTypeEulerTotal (Ss : List StandardSurfaceType) : ℤ :=
  (Ss.map standardSurfaceEuler).sum

/-- The regular-neighborhood boundary Euler characteristic: doubled region Euler characteristic. -/
def regularBoundaryEuler (B : BettiTriple) : ℤ :=
  2 * regionEuler B

What this page does not claim

The theorem does not prove that the oriented polygon components are homeomorphic to the standard surfaces. The theorem does not construct the embedded geometric map between the polygon components and the regular-neighborhood boundary components. The classification of closed surfaces is an explicit hypothesis, not a proved theorem in this module.

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/Cosmology/RegularNeighborhoodBoundary.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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