Cosmology Void Topology From Config Dim

Cosmic voids and their five classes

Cosmic voids are the vast, nearly empty regions of space that sit between the filaments and walls of the cosmic web. They are not truly empty; they contain a few galaxies and a diffuse plasma, but their density is far below the cosmic average. Astronomers have developed several methods to identify and classify these structures, depending on the data and the physical question at hand.

Five canonical classes cover the standard approaches. The first, VIDE/ZOBOV voids, come from a watershed algorithm applied to galaxy density fields. Watershed voids are defined by the same topological technique, segmenting the density field into basins. Underdensity voids are simply regions where the galaxy count falls below a threshold. Dynamical voids are identified by their gravitational influence on surrounding matter, while supervoids are the largest, exceeding 100 megaparsecs across.

In Recognition Science, the framework models this taxonomy through a single configuration dimension, configDim, a discrete parameter that indexes the five void classes. The framework's machine-checked library of formal theorems proves that exactly five such classes exist, no more and no fewer. This is not an empirical survey; it is a structural result about the classification scheme itself.

The proof is a simple counting argument. The library defines a type for void classes with five constructors, one for each canonical type, and then proves that the cardinality of this type is exactly five. The certificate of this fact is a structure that holds the count as its only field. The entire development is axiom-clean, meaning it rests only on the standard logical postulates, with no additional assumptions.

What this establishes in plain language is that the five-fold classification is complete and consistent within the framework. If a new void-finding method were proposed, it would have to be shown to fit one of these five classes, or the framework would need to be extended. The result does not say which method is best for a given survey; it only fixes the vocabulary.

theorem voidClass_count : Fintype.card VoidClass = 5 := by decide

theorem voidClass_count : Fintype.card VoidClass = 5 := by decide

structure VoidTopologyCert where
  five_classes : Fintype.card VoidClass = 5

What this page does not claim

No claim that these five classes are the only possible classification in all of cosmology. No claim that the framework derives the existence of cosmic voids from first principles. No claim that the counting proof says anything about the physical size or density of any real void.

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/VoidTopologyFromConfigDim.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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