Cosmology Reionization History From Rs Redshift Ratio

The redshift ladder

In cosmology, reionization is the era when the first stars and galaxies ionized the neutral hydrogen left over from the Big Bang, ending the cosmic dark ages. Standard accounts divide this process into a sequence of stages: the dark ages before any stars, the appearance of the first stars, the growth of galaxies, the main bulk reionization, and finally the saturated state of the modern universe. The Recognition Science framework models these five stages as a discrete ladder of boundary redshifts, meaning the redshift values at the transitions between stages.

The framework's ledger, a discrete record of events, places the boundary of each epoch at a redshift given by phi raised to an integer power, where phi is the golden ratio, approximately 1.618. The central theorem, called redshift_ratio, proves that the ratio of any boundary redshift to the one before it is exactly phi. In symbols, if boundaryRedshift(k) is the k-th boundary, then boundaryRedshift(k+1) / boundaryRedshift(k) = phi. This is a machine-checked proof in the framework's library of formal theorems, with zero unproved assumptions. The proof does not derive the redshift values from physics; it starts from the definition that the boundaries lie on a geometric ladder with ratio phi.

The theorem also certifies that every boundary redshift is a positive number, which is physically sensible since redshift is always positive. Together with the five-epoch structure, this forms a certificate object that packages the claims: exactly five epochs, the phi ratio between consecutive boundaries, and positivity. The framework uses this to describe a reionization history where the cosmic dark ages end above redshift 20, the first stars appear around redshift 20, galaxy formation occurs near redshift 15, bulk reionization spans redshift 7 to 10, and saturation is complete below redshift 6.

What the theorem does not claim is that real reionization actually followed this exact ladder. It establishes a consistency condition within the Recognition Science model: if the boundaries are defined this way, then the ratio follows necessarily. It does not predict measured redshifts, does not assert that the golden ratio is a fundamental constant of cosmology, and does not claim that the five epochs are the only possible division. The framework's own scaling law, which forces the golden ratio as a self-similar scaling, is a separate result; this theorem merely applies that scaling to a five-step ladder.

The practical consequence is that within the Recognition Science account, reionization is not an arbitrary sequence of events but a structured progression where each stage boundary is linked to the next by a fixed multiplicative factor. This gives cosmologists a precise, checkable statement about the framework's model of reionization, even though it leaves open whether the real universe obeys this particular spacing.

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

theorem redshift_pos (k : ℕ) : 0 < boundaryRedshift k :=
  pow_pos phi_pos k

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

What this page does not claim

The theorem does not claim that real reionization followed the phi-spaced ladder. It does not derive the redshift values from physical laws. It does not assert the golden ratio is a fundamental constant of cosmology.

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

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