Cosmology Regular Neighborhood Boundary Oriented Polygon Euler List Eq Surface T

The Euler signature bridge

In topology, a surface is a two-dimensional shape like a sphere, a torus (the surface of a donut), or a higher-genus object with many holes. A key invariant of any surface is its Euler characteristic, a number computed from vertices, edges, and faces that does not change under continuous deformation. For a sphere it is 2, for a torus it is 0, and for a surface with g holes it is 2 minus 2g.

This declaration establishes an algebraic bridge in a specific computational setting. The setting starts with a compact three-dimensional region in a cubical grid, and considers the boundary of a regular neighborhood of that region: a smoothed, thickened version of the boundary that avoids the singular edges and vertices found in the raw cubical surface. The region's topology is summarized by its Betti numbers (b0, b1, b2), which count connected pieces, holes, and voids. From these, one can compute the expected number of boundary components, the total boundary Euler characteristic, and the total genus (number of holes) of the boundary.

The theorem in question concerns two lists. The first list comes from a finite polygon gluing: a set of oriented polygons with edges paired and glued together, which is a combinatorial way to build a surface. The second list is a list of standard surfaces (sphere, torus, genus-125 surface, and so on) assigned to each polygon component. The declaration proves that, component by component, the Euler characteristic of each oriented polygon component equals the Euler characteristic of its assigned standard surface, provided certain conditions hold: the lists have the same length, the total Euler characteristics match, and the classification of closed surfaces is taken as a hypothesis.

The result is a precise inventory matching: it verifies that the combinatorial polygon data and the abstract standard-surface data agree on their Euler signatures, one component at a time. This is a necessary condition for the polygon gluing to be topologically equivalent to the standard surface, and it closes the algebraic part of the argument. The theorem is machine-checked, meaning a computer program has verified the logical steps from the definitions to the conclusion, with no gaps in the reasoning.

What this declaration does not claim is equally important. It does not construct an actual geometric map between the polygon gluing and the standard surface. It does not prove that the polygon components are homeomorphic to the standard surfaces, only that their Euler characteristics match. The embedded homeomorphism, the actual continuous deformation from one shape to the other, remains an open geometric theorem. The classification of closed surfaces is assumed as a hypothesis, not proved here. This is a bridge over the algebraic gap, not a proof that the two sides are the same shape.

/-- The standard surface target is forced by the Euler characteristic: two standard surfaces with the
same Euler characteristic are equal. The classification target is therefore determined, not chosen. -/
theorem standardSurfaceType_unique_of_euler (S S' : StandardSurfaceType)
    (h : standardSurfaceEuler S = standardSurfaceEuler S') : S = S' := by
  have hgen : S.genus = S'.genus := by
    have h' := h
    unfold standardSurfaceEuler at h'
    omega
  cases S
  cases S'
  simp_all

/-- The standard surface target is forced by the Euler characteristic: two standard surfaces with the
same Euler characteristic are equal. The classification target is therefore determined, not chosen. -/
theorem standardSurfaceType_unique_of_euler (S S' : StandardSurfaceType)
    (h : standardSurfaceEuler S = standardSurfaceEuler S') : S = S' := by
  have hgen : S.genus = S'.genus := by
    have h' := h
    unfold standardSurfaceEuler at h'
    omega
  cases S
  cases S'
  simp_all

What this page does not claim

This does not claim the polygon components are homeomorphic to the standard surfaces. This does not claim the classification of closed surfaces is proved within the framework. This does not claim the embedded geometric map between the polygon gluing and the standard surface has been constructed.

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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Derived articles

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