Cosmology Regular Neighborhood Boundary Componentwise Surface Inventory Matches

The componentwise surface inventory

A compact three-dimensional region has a boundary, and that boundary can be decomposed into separate pieces. In algebraic topology, the shape of each piece is captured by its Euler characteristic, a single integer that distinguishes a sphere from a torus from a higher-genus surface. The declaration componentwiseSurfaceInventory_matches_regularBoundary_of_surfaceTypeClassificationCloses proves an algebraic bridge: if a list of standard surface types is assigned to the boundary components, and each entry has the correct Euler characteristic, then the list matches the boundary's invariants component by component, not merely in aggregate.

The proof works through a chain of definitions. A Betti triple, three integers (b0, b1, b2) recording the number of connected pieces, holes, and voids in the region, determines the expected boundary data: the number of components is b0 + b2, the Euler characteristic is 2(b0 - b1 + b2), and the genus is components minus the region's Euler characteristic. A standard surface type is just a genus integer, with Euler characteristic 2 - 2g. The theorem standardSurfaceType_unique_of_euler shows that a genus is uniquely determined by its Euler characteristic, so matching Euler numbers forces matching surface types. The componentwise inventory then checks each oriented polygon component against its assigned standard surface, verifying orientation success and Euler equality for every pair.

The library applies this to two concrete cases. The horizon annulus handle, with Betti triple (2, 1, 0), is assigned a torus and a sphere, matching its two boundary components. The dyadic sponge at resolution 20, with Betti triple (50, 125, 3), is assigned 53 components whose genera sum to 125, including a genus-125 surface. These assignments are conditional theorems: they hold provided a classical classification of closed surfaces is assumed as an explicit hypothesis, never as an axiom.

What the declaration does not claim is the geometric realization. It proves the arithmetic bridge: the counts, Euler characteristics, and genera match. It does not construct a homeomorphism between the actual boundary components and the standard surfaces. The embedded map, preserving incidence, vertex links, and orientation, remains an open geometric theorem. The algebraic inventory is complete modulo one named classical input, but the geometric realization is not part of this result.

/-- 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

/-- Standard surface types for the horizon-annulus polygon components. -/
def horizonAnnulusHandleSurfaceTypes : List StandardSurfaceType :=
  [standardTorus, standardSphere]

/-- Phase-26 dyadic sponge probe at `R = 20`: `(b₀,b₁,b₂) = (50,125,3)`. -/
def dyadicSpongeR20Betti : BettiTriple :=
  { b0 := 50, b1 := 125, b2 := 3 }

/-- Standard surface types for the horizon-annulus polygon components. -/
def horizonAnnulusHandleSurfaceTypes : List StandardSurfaceType :=
  [standardTorus, standardSphere]

What this page does not claim

The declaration does not construct a homeomorphism between the actual boundary components and the standard surfaces. The declaration does not prove the embedded map is incidence-preserving, vertex-link preserving, or orientation-preserving; it only records these as obligations. The declaration does not prove the geometric realization of the regular-neighborhood boundary; that remains an open theorem.

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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