Cosmology Regular Neighborhood Boundary Surface Type Count Eq Regular Boundary C

Counting the boundary

In topology, the Euler characteristic is a number that describes a shape's overall structure: for a polyhedron it is vertices minus edges plus faces. For a three-dimensional region, the Betti numbers b₀, b₁, and b₂ record, respectively, the number of separate pieces, the number of holes or handles, and the number of enclosed voids. A standard formula then gives the region's Euler characteristic as b₀ − b₁ + b₂. These invariants are the classical starting point for understanding what kind of surface bounds a solid region.

The framework's library, a machine-checked collection of formal statements, works in a setting where a region is represented as a discrete cubical grid, like a stack of tiny cubes. A raw boundary read from such a grid can have singular edges where four faces meet, which is not a proper surface. The library therefore switches to a regular neighborhood: a thickened version of the region whose boundary is a genuine, smooth surface. The central algebraic bridge, proved in the declaration surfaceTypeCount_eq_regularBoundaryComponents_of_surfaceTypeClassificationCloses, states that the number of boundary components of this smoothed surface equals b₀ + b₂, the count of separate pieces plus enclosed voids. The proof is complete and axiom-clean within the library's formal system.

The result is an arithmetic identity about counts and Euler characteristics. It belongs to a chain of phases that build increasingly detailed algebraic certificates: first the component count, then the total genus, then a component-by-component Euler signature, and finally a pairing between oriented polygon pieces and standard surface types such as sphere, torus, and genus-125 surface. Each phase tightens the inventory without yet constructing the actual geometric map that would send one surface to the other. The library is explicit that the embedded homeomorphism statement, the claim that the smoothed boundary is geometrically the same as the standard surface, remains an open geometric problem.

What the declaration does not claim is therefore as important as what it proves. It does not prove that the regular-neighborhood boundary is homeomorphic to the standard surface; it only proves that the algebraic counts match. It does not construct the geometric map exhibiting the homeomorphism. And it does not apply to arbitrary continuous regions, only to the discrete cubical setting the library formalizes. The proof is a precise algebraic bridge, not a geometric realization statement.

/-- The regular-neighborhood boundary component count predicted by Alexander duality intuition. -/
def regularBoundaryComponents (B : BettiTriple) : ℤ :=
  B.b0 + B.b2

/-- Betti data of a compact 3D region, represented in integers so Euler algebra is literal. -/
structure BettiTriple where
  b0 : ℤ
  b1 : ℤ
  b2 : ℤ

What this page does not claim

The declaration does not prove that the regular-neighborhood boundary is homeomorphic to the standard surface. The declaration does not construct the geometric map exhibiting the homeomorphism. The declaration does not apply to arbitrary continuous regions, only to discrete cubical regions.

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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