Cosmology Reionization History From Rs Reionization Epoch

The five epochs

Reionization is the cosmic era when ultraviolet light from the first stars and galaxies split neutral hydrogen into protons and electrons, ending the universe's dark ages. Astronomers divide this era into recognizable stages: the dark ages before any stars (redshift z greater than about 20), the ignition of the first stars (z near 20), the assembly of galaxies (z near 15), the bulk reionization of intergalactic hydrogen (z roughly 7 to 10), and the saturation when ionized bubbles fill the cosmos (z below 6). These five stages are the standard narrative of how the universe became transparent to starlight.

In Recognition Science, the machine-checked library of formal theorems defines these five stages as an inductive type called ReionizationEpoch, with one constructor for each stage: darkAges, firstStars, galaxyFormation, bulkReionization, and saturation. The library proves that there are exactly five such epochs, a statement that holds by direct computation. It also defines a boundary redshift for each epoch as a power of the golden ratio phi, so that the ratio of consecutive boundary redshifts is exactly phi. The library packages these facts into a certificate structure: five epochs, the phi ratio, and the positivity of every boundary redshift.

What the framework does not claim is that these five stages are the only possible division of reionization, or that the phi ladder is derived from physics rather than chosen as a model. The declaration is a formalization of a conventional five-stage history, with the geometric spacing as a definitional choice. The proof that the ratio is phi is a theorem about the definition, not a derivation of the epoch boundaries from first principles. The framework's library establishes the internal consistency of this five-epoch picture; it does not establish that the picture matches observations, nor does it claim the redshift values are predictions.

theorem reionizationEpoch_count :
    Fintype.card ReionizationEpoch = 5 := by decide

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

What this page does not claim

The five epochs are the only possible division of reionization history. The phi spacing is derived from physics rather than chosen as a model. The redshift values are predictions that match observations.

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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Derived articles

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