Cosmology Reionization Redshift Rs

Reionization redshift

Reionization redshift is the cosmic epoch, conventionally written z_reion, when the first stars and galaxies emitted enough ultraviolet light to strip electrons from the neutral hydrogen that filled the early universe. Astronomers usually place this era at a redshift between about 7 and 10, meaning the light we see from that time has been stretched to roughly eight to eleven times its original wavelength. The name records the transition: the universe had cooled enough for protons and electrons to combine into neutral hydrogen, and then the first luminous sources re-ionized that gas, making the universe transparent to ultraviolet radiation.

In Recognition Science (RS), the framework models this epoch with its cost function J(x) = (x + 1/x)/2 - 1, which measures the forced price of a recognition event. The module defines domainCost(m, e) as J(m/e), a dimensionless cost comparing two quantities m and e. The framework notes that the golden ratio powers phi^4 ≈ 6.85 and phi^5 ≈ 11.09 bracket the observed reionization range, and calls this consistency. That is a research note, not a derived result: the module proves three general facts about the cost function, namely that it vanishes when m equals e, that it is nonnegative for positive inputs, and that phi - 3/2 is positive. It proves nothing specific to reionization, because the definition of domainCost does not connect m and e to any physical quantity like ionizing photon production or hydrogen density.

The machine-checked library of formal theorems does establish a certificate structure, ReionizationCert, which packages the three general facts into a single object. The certificate exists, and it is inhabited, meaning the three properties hold. But the certificate would only become a theorem about reionization if the module defined m and e in terms of the epoch's own physics. That definition is absent. The same three general facts appear verbatim in 2383 sibling modules, each sharing the template without subject-specific content.

What the module genuinely establishes, in plain language, is that the cost function has three useful properties: it costs nothing when the two inputs are equal, it never returns a negative cost for positive inputs, and there is a positive threshold value related to the golden ratio. These are properties of the cost function itself, not of reionization. The physical bridge, the definition of m and e as quantities that actually describe the reionization era, remains open.

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

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]

What this page does not claim

The module does not prove that reionization occurred at redshift phi^4 or phi^5. The framework does not derive the reionization redshift from first principles. The bracket of z_reion by phi^4 and phi^5 is a stated consistency note, not a theorem.

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/Reionization_Redshift_RS.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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