Cosmology Reionization Endpoint3 From Jcost

The reionization endpoint

Reionization is the epoch when ultraviolet light from the first stars and galaxies split neutral hydrogen into protons and electrons, ending the cosmic dark ages. Observations of the Lyman-alpha forest and the cosmic microwave background place this transition at a redshift z around 5.5 to 6, meaning the light we see left those sources when the universe was roughly a billion years old.

The module ReionizationEndpoint3_FromJCost in the Recognition Science library was planned to derive that endpoint from a cost function. The cost function, written J(m/e), is a measure of recognition cost: the price a ledger pays to register a comparison between two quantities m and e. The module defines this cost as J(m/e), where J is the unique cost function forced by the framework's five axioms.

What the machine-checked code actually proves is three general facts about this cost function, not a statement about reionization. First, the cost vanishes when the two quantities are equal: J(r/r) = 0 for any nonzero r. Second, the cost is never negative for positive inputs: J(m/e) ≥ 0 whenever m and e are both positive. Third, the golden-ratio constant phi satisfies phi - 3/2 > 0, a small numerical threshold. These three facts are packaged into a structure called ReionEnd3Cert, and the library proves that this structure is inhabited, meaning the certificate exists.

The docstring records the intended application: a research note suggests z_end = phi^4 ≈ 6.85, which is consistent with the observed 5.5 to 6 range. But the code itself contains no definition of m and e in cosmological terms. The module proves nothing about reionization because the quantities m and e are left as arbitrary real numbers. To make this a theorem about the reionization endpoint, one would need to define m and e in terms of, say, ionizing photon production and hydrogen density, and then prove that J(m/e) reaches a threshold at the observed redshift.

In plain language: the module is a template with a name. It proves that the cost function has three basic properties that any such cost should have. The reionization claim is a hope recorded in a comment, not a result established by the proof. The library's own documentation states this clearly: the paragraph above is a research note recording where the idea was meant to go, not a result.

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]

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

What this page does not claim

This module derives the reionization redshift z ≈ 6 from the cost function. The research note's value phi^4 ≈ 6.85 is a proved consequence of the framework. The observed reionization endpoint is evidence for the Recognition Science framework.

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/ReionizationEndpoint3_FromJCost.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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