Astrophysics Reionization Epoch From Jcost Reionization Cert

The Reionization Certificate

Cosmic reionization is the epoch, roughly a billion years after the Big Bang, when ultraviolet light from the first stars and galaxies stripped electrons from the neutral hydrogen that filled the early universe. Astronomers date its completion to a redshift of about z ≈ 6, meaning the light we see from that era has been stretched to about seven times its original wavelength. The standard picture places recombination, when neutral hydrogen first formed, at z ≈ 1100, so the ratio between the two redshifts is roughly 183.

In the Recognition Science framework, a machine-checked library of formal theorems, the declaration named ReionizationCert is a structure, a bundled package of three proved statements. Each statement concerns a function called domainCost, which the framework defines as Jcost(m / e), where Jcost is the framework's forced cost function and m and e are real numbers standing for two quantities. The certificate proves that domainCost equals zero when m equals e, that it is never negative when both inputs are positive, and that a certain threshold value, phi minus 3/2, is greater than zero. These are general facts about the cost function, true for any positive real numbers you might choose.

What the certificate does not do is connect those facts to reionization. The library's own documentation says so directly: it proves nothing specific to this subject, because domainCost is defined without any reference to astrophysics. The numbers m and e are not defined as, say, the mass of a hydrogen atom or the electron charge in a reionization context. The certificate is one of 2,383 sibling modules that share the same formal body, each with a different subject in its filename but the same three general theorems. A research note in the file records the intended direction, that the 12-rung gap on the phi-ladder between recombination and reionization might match the redshift ratio, but that note is not a proved result.

For the reader, the practical upshot is a clear boundary. The certificate is a legitimate piece of formal mathematics, a small proof package that the library checks and that any user can cite. What it is not is evidence about the cosmos. To make it a theorem about reionization, someone would need to define m and e in astrophysical terms and prove that those definitions satisfy the certificate's conditions. Until that definition exists, the certificate remains a template, not a discovery.

theorem domainCost_at_eq (r : ℝ) (h : r ≠ 0) : domainCost r r = 0 := by
  unfold domainCost; rw [div_self h]; exact Jcost_unit0

theorem domainCost_nonneg (m e : ℝ) (hm : 0 < m) (he : 0 < e) : 0 ≤ domainCost m e := by
  unfold domainCost; exact Jcost_nonneg (div_pos hm he)

theorem canonicalThreshold_pos : 0 < canonicalThreshold := by
  unfold canonicalThreshold; linarith [phi_gt_onePointFive]

def domainCost (m e : ℝ) : ℝ := Jcost (m / e)

What this page does not claim

ReionizationCert does not prove that cosmic reionization occurred at any specific redshift. The certificate does not define m or e in astrophysical terms. The phi-ladder ratio of 183 is a research note, not a proved result about the cosmos.

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

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