Cosmology Reionization History From Rs Redshift Pos

The redshift ladder

In cosmology, reionization is the epoch when ultraviolet light from the first stars and galaxies ionized the neutral hydrogen that filled the early universe. Astronomers divide this history into five canonical stages: the cosmic dark ages (z > 20), the first stars (z ~ 20), galaxy formation (z ~ 15), bulk reionization (z ~ 7-10), and saturation (z < 6). The boundary redshift between each stage is a number that marks a transition in the universe's ionization state.

The Recognition Science framework models these boundary redshifts as a geometric ladder. Its machine-checked library of formal theorems defines a boundary redshift at step k as phi raised to the power k, where phi is the golden ratio, approximately 1.618. A proved theorem, redshift_pos, states that every boundary redshift is strictly positive: for any natural number k, the value phi^k is greater than zero. This is a simple but load-bearing fact, because the framework's model of the reionization ladder divides one boundary by the next to show that the ratio is always phi, and that division requires the denominator to be nonzero.

The same library proves that the ratio of consecutive boundary redshifts is exactly phi, and that there are exactly five reionization epochs. Together these theorems form a certificate, a packaged proof that the model is internally consistent. The certificate states that the five epochs exist, that the boundary redshifts form a geometric progression with ratio phi, and that all boundaries are positive. This is what the declaration redshift_pos establishes: within the framework's model, the ladder is well-defined and its rungs never collapse to zero or below.

What redshift_pos does not claim is that the absolute values of these boundary redshifts match observed cosmology. The theorem proves positivity and the geometric ratio, but it does not anchor the ladder to a specific starting redshift. The framework identifies the golden ratio as the scaling, but the first rung, the value of phi^0, is simply 1, and the framework does not derive from first principles why the dark ages should end near z = 20 rather than some other number. The match between the geometric ladder and the observed five-stage history is a modeling choice, not a derived prediction. The theorem guarantees the ladder's shape, not its position on the redshift axis.

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

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

noncomputable def reionizationCert : ReionizationCert where
  five_epochs := reionizationEpoch_count
  phi_ratio := redshift_ratio
  boundary_always_pos := redshift_pos

What this page does not claim

The theorem does not establish that the absolute redshift values match observed cosmology. The framework does not derive the first rung of the ladder from first principles. The match between the geometric ladder and the observed five-stage history is a modeling choice, not a derived prediction.

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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