Astrophysics Reionization Epoch From Jcost

Cosmic reionization

Cosmic reionization is the period in the early universe when neutral hydrogen gas was split into protons and electrons by the first stars and galaxies, making the universe transparent to light. This phase transition ended roughly a billion years after the Big Bang. The standard measure of when this happened is the redshift, a number that describes how much the universe has expanded since the light was emitted. The accepted value for the end of reionization is a redshift of about 6, meaning the universe was roughly one seventh of its current size. The epoch of recombination, when neutral hydrogen first formed, is measured at a redshift of about 1100. The ratio of these two redshifts, 1100 divided by 6, is approximately 183.

In Recognition Science, the framework models the gap between these two epochs using its cost function, a measure of the forced expense of recognition events. The framework's library of formal theorems proves three general facts about this cost when it is applied to the ratio of two masses. First, the cost is zero when the two inputs are equal. Second, the cost is never negative for positive inputs. Third, a specific constant, the golden ratio minus 1.5, is greater than zero. These three facts are assembled into a certificate that the cost function behaves as expected.

The framework notes that the ratio 183 is close to the golden ratio raised to the power 11.7. The golden ratio, approximately 1.618, is a number the framework derives as the unique self-similar scaling from its forcing chain. This observation suggests a 12-rung gap on the framework's phi-ladder between recombination and reionization. However, the module itself proves nothing about reionization specifically. Its formal theorems are universal statements about the cost function applied to any ratio. The connection to reionization would require a separate definition of the masses involved in terms of the epoch's physics. The research note in the module records this intended direction, but the formal content stops at the general properties of the cost.

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 occurs at redshift 6. The ratio 183 equaling phi to the power 11.7 is an observation, not a formal theorem. The framework does not derive the fine-structure constant from this module.

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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