Cosmology Reionization History From Rs Reionization Epoch Count

Five epochs

Cosmic reionization is the period when ultraviolet light from the first stars and galaxies stripped electrons from neutral hydrogen throughout the universe, ending the cosmic dark ages. Astronomers commonly divide this process into phases: the dark ages (redshift z greater than about 20), the first stars (z near 20), galaxy formation (z near 15), bulk reionization (z between roughly 7 and 10), and saturation (z below 6). The Recognition Science framework encodes these five phases as a formal object called ReionizationEpoch, and its machine-checked library of formal theorems proves a counting statement: the number of distinct epochs is exactly five.

The proof is a computation. The declaration reionizationEpoch_count states that the cardinality of the type ReionizationEpoch equals 5, and the proof term is the single word decide, meaning the kernel checked the finite enumeration directly. The same file defines a boundary redshift for each step as phi raised to an integer power, where phi is the golden ratio. A companion theorem proves that the ratio of consecutive boundary redshifts is always phi, and another proves every boundary redshift is positive. These three results are bundled into a structure called ReionizationCert, assembled by the definition reionizationCert.

What the declaration does not claim is just as important. It does not assert that the five-epoch division is the empirically correct history of the real universe; that is a physical identification, not a theorem. The formal result only says that, within the Recognition Science framework, a five-epoch model with golden-ratio-spaced boundaries is internally consistent. It also does not claim that the boundary redshifts match observed values, nor that the epochs have equal duration in time. The framework derives the number five from its own structural principles, and the mapping to astronomical epochs is a modeling choice layered on top.

In practice, this means the theorem is a structural scaffold, not a measurement. It tells a cosmologist that if the framework's geometry applies, reionization should naturally split into five stages with boundaries descending by successive powers of the golden ratio. Whether the real sky follows that ladder is a question for observation, and the framework's own library does not settle it.

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

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

structure ReionizationCert where
  five_epochs : Fintype.card ReionizationEpoch = 5
  phi_ratio : ∀ k, boundaryRedshift (k + 1) / boundaryRedshift k = phi
  boundary_always_pos : ∀ k, 0 < boundaryRedshift k

What this page does not claim

The theorem does not assert that the five-epoch division matches the empirically observed history of reionization. The framework does not claim that the boundary redshifts equal measured values from astronomical surveys. The declaration does not imply the epochs have equal duration in cosmic time.

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)

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.

Derived articles

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