Cosmology Regular Neighborhood Boundary Surface Type Euler Total Eq Regular Boun

The genus bridge

In topology, a compact three-dimensional region has Betti numbers that count its basic features: b0 counts connected pieces, b1 counts tunnels or handles, and b2 counts enclosed voids. The boundary of such a region is a two-dimensional surface, and its shape is summarized by its Euler characteristic, a single integer that combines vertices, edges, and faces. For a smooth boundary made of standard pieces, the total Euler characteristic of all boundary components follows a fixed formula: twice the region's own Euler characteristic, which is b0 minus b1 plus b2.

The Recognition Science framework's machine-checked library of formal theorems proves an algebraic bridge. If a compact 3D cubical region has Betti triple (b0, b1, b2), and its regular-neighborhood boundary has boundary components b0 + b2, boundary Euler characteristic 2 * (b0 - b1 + b2), and genus computed by components minus region Euler, then the total desingularized boundary genus is exactly b1. This means the number of handles in the smoothed boundary equals the first Betti number of the original region.

The declaration surfaceTypeEulerTotal_eq_regularBoundaryEuler_of_surfaceTypeClassificationCloses establishes the final algebraic step: when the assigned standard surface list has the same component count as the oriented polygon list, and its total Euler characteristic is derived from the standard formula, then the resulting count, Euler, and genus match the regular-neighborhood boundary invariants. This is a theorem in the framework's library, proved with no axioms beyond the standard three, and it closes the arithmetic inventory.

In Recognition Science, this result matters for cosmology. The framework models the early universe as a discrete recognition foam, and its horizon annulus handle has Betti triple (2, 1, 0), meaning two components, one handle, no voids. The theorem confirms that its smoothed boundary has genus exactly 1, matching a torus and a sphere. Similarly, a dyadic sponge with Betti triple (50, 125, 3) gets a boundary with total genus 125, a standard genus-125 surface.

The theorem does not claim that the abstract standard surface is geometrically embedded in the original space as the actual boundary. That homeomorphism remains an open geometric problem. The proof establishes the numbers match, not that a specific map exists realizing the standard surface inside the cubical complex. The classification of closed surfaces is taken as a hypothesis, not proved within the framework.

/-- Total genus of the regular-neighborhood boundary, using `χ(boundary) = 2χ(region)`. -/
def regularBoundaryGenus (B : BettiTriple) : ℤ :=
  regularBoundaryComponents B - regionEuler B

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

/-- Phase-26 horizon-annulus handle at `R = 20` and `R = 32`: two components, one tunnel, no void. -/
def horizonAnnulusHandleBetti : BettiTriple :=
  { b0 := 2, b1 := 1, b2 := 0 }

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

What this page does not claim

The standard surface is geometrically homeomorphic to the actual regular-neighborhood boundary. The classification of closed surfaces is proved within the framework. The theorem constructs the embedded map between the polygon gluing and the standard surface.

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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