Cosmology Large Scale Structure From Rs Lss Regime Count

Five canonical scales

Large-scale structure is the pattern of matter in the universe at the biggest observable sizes, from the faint imprint of the cosmic microwave background to the empty voids between galaxy filaments. Astronomers routinely sort these patterns into a small set of named regimes. The Recognition Science framework's machine-checked library of formal theorems takes that sorting literally: it defines an inductive type whose five constructors are CMB acoustic scale, baryon acoustic oscillation, galaxy cluster, filament, and cosmic void, then proves the number of regimes is exactly five.

The proof is a theorem named lssRegime_count, and it is not a measurement or an empirical fit. It is a counting statement: the type LSSRegime has exactly five distinct elements, and the theorem's proof is a direct computation (Fintype.card LSSRegime = 5, by decide). The library also defines a scale function on natural numbers, scale k = phi^k, and proves that each step up the ladder multiplies the comoving length by phi, the golden ratio. A certificate structure bundles these facts: five regimes, the phi ratio between consecutive scales, and the positivity of every scale.

The framework's claim is that these five regimes sit one rung apart on a phi-ladder in comoving length. That is a structural claim about how the regimes relate, not a claim about where each regime sits in physical units. The library does not assert that the CMB acoustic scale is at any particular megaparsec value, nor does it derive the baryon acoustic oscillation scale from first principles. It establishes a discrete count and a ratio relationship, and it leaves the physical calibration to measurement.

What the declaration does not claim is as important as what it proves. It does not claim that these five regimes exhaust all possible large-scale structure, nor that the phi ladder is the only way to order them. It does not claim that the framework derives the actual observed values of these scales. The theorem is a formal counting result, and the empirical check against astronomical data is a separate step that the library does not perform.

inductive LSSRegime where
  | cmbAcoustic
  | baryonAcousticOscillation
  | galaxyCluster
  | filament
  | cosmicVoid
  deriving DecidableEq, Repr, BEq, Fintype

theorem lssRegime_count : Fintype.card LSSRegime = 5 := by decide

theorem scale_ratio (k : ℕ) : scale (k + 1) / scale k = phi := by
  unfold scale
  have hpos : (0 : ℝ) < phi ^ k := pow_pos phi_pos k
  rw [div_eq_iff hpos.ne', pow_succ]
  ring

What this page does not claim

The library does not assign physical megaparsec values to any of the five regimes. The theorem does not prove these five regimes are the only possible large-scale structures. The phi ratio is a formal relationship, not an empirical derivation of observed cosmological scales.

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/LargeScaleStructureFromRS.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.

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