Cosmology Regular Neighborhood Boundary

The boundary bridge

In topology, a regular neighborhood is a thickening of a region that smooths away any jagged or singular features, giving a clean boundary surface. The cosmology module in Recognition Science uses this idea to read the shape of the universe's large-scale structure. The framework models spacetime as a discrete foam, a ledger of recognition events, and the positive excursion set {q > 0} marks where matter clumps. Phase 25 found that the raw cubical boundary of this set has nonmanifold edges, places where the surface folds through itself. Phase 26 switched to the canonical desingularized readout: the boundary of a regular neighborhood of the exact positive region.

The module proves an algebraic bridge. If a compact 3D cubical region has Betti triple (b0, b1, b2), counting its connected pieces, holes, and voids, then its regular-neighborhood boundary has exactly b0 + b2 components, Euler characteristic 2 * (b0 - b1 + b2), and genus b1. The genus is the number of handles, like the hole in a doughnut. This is a proved theorem in the machine-checked library of formal theorems, with no axioms beyond the standard three. The key formula, printed in full: genus = components - χ(region), where χ is the Euler characteristic.

The proof proceeds through phases. Phase 31 handles the algebraic half of a vertex-link quotient, pairing four-face edges to correct the Euler characteristic. Phase 35 assembles corrected face components into a total genus forced to be b1. Phase 37 reduces a finite polygon gluing witness to the component assembly theorem. Phase 39 adds an orientability gate, checking every polygon has a consistent face orientation. Phase 40 classifies each oriented component as a standard surface: sphere, torus, or a genus-125 surface. Phase 42 matches Euler signatures component by component. Phase 44 names the embedded-map obligations, and Phase 46 replaces abstract obligations with concrete decidable conditions. Phase 47 makes the classification of closed surfaces a first-class hypothesis, conditional on which the horizon and dyadic components are realized by their standard surfaces.

What this establishes in plain language: the boundary of a cosmic foam region is not a mess. Its handles, components, and Euler characteristic are exactly determined by the region's interior topology. The bridge is algebraic inventory matching, not a geometric homeomorphism theorem. The embedded map from the corrected cellulations to the regular-neighborhood boundary remains open. The module proves the arithmetic, not the geometry. The horizon annulus handle, with Betti triple (2, 1, 0), realizes a torus and a sphere. The dyadic sponge at resolution R20, with Betti triple (50, 125, 3), realizes a genus-125 surface. These are conditional theorems, complete modulo the one named classical input: the classification of closed surfaces.

The consequence for cosmology: the framework's foam model produces boundaries whose topology is fully constrained by the interior. No free parameters, no fitted surface shapes. The machine-checked library proves the counts, the Euler characteristics, and the genera. The geometric realization, the actual embedding, is the open target. This is a partial theorem through Phase 44 and a conditional theorem at Phase 47, with zero sorry and zero new axioms. The reader can now see that the foam's boundary is a precise topological mirror of its interior, and the only missing piece is the geometric map itself.

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

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

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

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

/-- The standard closed orientable surface of genus `125`. -/
def standardGenus125Surface : StandardSurfaceType :=
  { genus := 125 }

What this page does not claim

The module does not prove the embedded homeomorphism between corrected cellulations and the regular-neighborhood boundary. The regular-neighborhood boundary is not claimed to be the physical boundary of the universe, only a topological readout of the foam model. The classification of closed surfaces is assumed as a hypothesis, not proved within the 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)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

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