Cosmology Large Scale Structure From Rs Scale Ratio

The scale ratio

The golden ratio, often written φ, is the number that satisfies φ² = φ + 1, approximately 1.618. It appears throughout mathematics and art, from the pentagon's geometry to the Fibonacci sequence, where each term is the sum of the two before it. In the Recognition Science framework, this same number emerges from a proved chain of results about recognition cost, and it is used to define a ladder of cosmic scales.

Within the framework, the declaration scale_ratio establishes a precise relationship: for any nonnegative integer k, the scale at position k+1 divided by the scale at position k equals φ. Here scale is defined as φ raised to the power k, so the ratio of consecutive scales is exactly the golden ratio. The theorem is proved in the machine-checked library of formal theorems, with no unproved assumptions. It also proves that every scale is positive, so the division is always valid.

The framework uses this ladder to model five canonical large-scale structure regimes: the cosmic microwave background acoustic scale, baryon acoustic oscillations, galaxy clusters, filamentary structure, and cosmic voids. Each regime sits one rung up the φ-ladder in comoving length. The library contains a certificate structure that bundles these facts: five regimes, the φ ratio, and positivity of all scales.

What scale_ratio does not claim is equally important. The theorem only establishes the ratio between consecutive rungs of an abstract ladder. It does not identify which observed astronomical objects correspond to which rung, nor does it derive the absolute length of any cosmic structure. The mapping from the five named regimes to the ladder is a definitional choice, not a proved result. The framework's library proves the arithmetic of the ladder, not the physical correspondence.

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

noncomputable def scale (k : ℕ) : ℝ := phi ^ k

theorem scale_pos (k : ℕ) : 0 < scale k := pow_pos phi_pos k

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

What this page does not claim

The theorem does not identify any specific observed cosmic structure with a particular rung of the ladder. The framework does not derive the absolute length of any cosmic structure from the scale ratio alone. The correspondence between the five named regimes and the ladder is a definitional choice, not a proved result.

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)

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