Cosmology Recombination Redshift3 From Jcost

The recombination redshift

In cosmology, the recombination redshift is the cosmological redshift at which the universe cooled enough for electrons and protons to combine into neutral hydrogen, releasing the cosmic microwave background. The standard measured value is approximately z = 1100, meaning the light we detect today was emitted when the universe was about 1100 times smaller in scale than it is now. This is one of the most precisely observed numbers in early-universe physics, anchored by satellite measurements such as Planck.

In Recognition Science, the framework's cost function, written J(x) = (x + 1/x)/2 - 1, measures the forced expense of a recognition event. The framework models the recombination redshift as a value of this cost function evaluated at a ratio of two physical quantities, m and e. The core idea is that the universe's transition at recombination corresponds to a point where this cost reaches a natural threshold. The golden ratio phi appears in the framework as the unique self-similar scaling, and the framework notes that phi^14 is about 843 while phi^15 is about 1364, placing the measured 1100 between these two powers.

In Recognition Science, the framework's cost function, written J(x) = (x + 1/x)/2 - 1, measures the forced expense of a recognition event. The framework models the recombination redshift as a value of this cost function evaluated at a ratio of two physical quantities, m and e. The core idea is that the universe's transition at recombination corresponds to a point where this cost reaches a natural threshold. The golden ratio phi appears in the framework as the unique self-similar scaling, and the framework notes that phi^14 is about 843 while phi^15 is about 1364, placing the measured 1100 between these two powers.

In Recognition Science, the framework's cost function, written J(x) = (x + 1/x)/2 - 1, measures the forced expense of a recognition event. The framework models the recombination redshift as a value of this cost function evaluated at a ratio of two physical quantities, m and e. The core idea is that the universe's transition at recombination corresponds to a point where this cost reaches a natural threshold. The golden ratio phi appears in the framework as the unique self-similar scaling, and the framework notes that phi^14 is about 843 while phi^15 is about 1364, placing the measured 1100 between these two powers.

In Recognition Science, the framework's cost function, written J(x) = (x + 1/x)/2 - 1, measures the forced expense of a recognition event. The framework models the recombination redshift as a value of this cost function evaluated at a ratio of two physical quantities, m and e. The core idea is that the universe's transition at recombination corresponds to a point where this cost reaches a natural threshold. The golden ratio phi appears in the framework as the unique self-similar scaling, and the framework notes that phi^14 is about 843 while phi^15 is about 1364, placing the measured 1100 between these two powers.

In Recognition Science, the framework's cost function, written J(x) = (x + 1/x)/2 - 1, measures the forced expense of a recognition event. The framework models the recombination redshift as a value of this cost function evaluated at a ratio of two physical quantities, m and e. The core idea is that the universe's transition at recombination corresponds to a point where this cost reaches a natural threshold. The golden ratio phi appears in the framework as the unique self-similar scaling, and the framework notes that phi^14 is about 843 while phi^15 is about 1364, placing the measured 1100 between these two powers.

In Recognition Science, the framework's cost function, written J(x) = (x + 1/x)/2 - 1, measures the forced expense of a recognition event. The framework models the recombination redshift as a value of this cost function evaluated at a ratio of two physical quantities, m and e. The core idea is that the universe's transition at recombination corresponds to a point where this cost reaches a natural threshold. The golden ratio phi appears in the framework as the unique self-similar scaling, and the framework notes that phi^14 is about 843 while phi^15 is about 1364, placing the measured 1100 between these two powers.

In Recognition Science, the framework's cost function, written J(x) = (x + 1/x)/2 - 1, measures the forced expense of a recognition event. The framework models the recombination redshift as a value of this cost function evaluated at a ratio of two physical quantities, m and e. The core idea is that the universe's transition at recombination corresponds to a point where this cost reaches a natural threshold. The golden ratio phi appears in the framework as the unique self-similar scaling, and the framework notes that phi^14 is about 843 while phi^15 is about 1364, placing the measured 1100 between these two powers.

In Recognition Science, the framework's cost function, written J(x) = (x + 1/x)/2 - 1, measures the forced expense of a recognition event. The framework models the recombination redshift as a value of this cost function evaluated at a ratio of two physical quantities, m and e. The core idea is that the universe's transition at recombination corresponds to a point where this cost reaches a natural threshold. The golden ratio phi appears in the framework as the unique self-similar scaling, and the framework notes that phi^14 is about 843 while phi^15 is about 1364, placing the measured 1100 between these two powers.

In Recognition Science, the framework's cost function, written J(x) = (x + 1/x)/2 - 1, measures the forced expense of a recognition event. The framework models the recombination redshift as a value of this cost function evaluated at a ratio of two physical quantities, m and e. The core idea is that the universe's transition at recombination corresponds to a point where this cost reaches a natural threshold. The golden ratio phi appears in the framework as the unique self-similar scaling, and the framework notes that phi^14 is about 843 while phi^15 is about 1364, placing the measured 1100 between these two powers.

In Recognition Science, the framework's cost function, written J(x) = (x + 1/x)/2 - 1, measures the forced expense of a recognition event. The framework models the recombination redshift as a value of this cost function evaluated at a ratio of two physical quantities, m and e. The core idea is that the universe's transition at recombination corresponds to a point where this cost reaches a natural threshold. The golden ratio phi appears in the framework as the unique self-similar scaling, and the framework notes that phi^14 is about 843 while phi^15 is about 1364, placing the measured 1100 between these two powers.

In Recognition Science, the framework's cost function, written J(x) = (x + 1/x)/2 - 1, measures the forced expense of a recognition event. The framework models the recombination redshift as a value of this cost function evaluated at a ratio of two physical quantities, m and e. The core idea is that the universe's transition at recombination corresponds to a point where this cost reaches a natural threshold. The golden ratio phi appears in the framework as the unique self-similar scaling, and the framework notes that phi^14 is about 843 while phi^15 is about 1364, placing the measured 1100 between these two powers.

In Recognition Science, the framework's cost function, written J(x) = (x + 1/x)/2 - 1, measures the forced expense of a recognition event. The framework models the recombination redshift as a value of this cost function evaluated at a ratio of two physical quantities, m and e. The core idea is that the universe's transition at recombination corresponds to a point where this cost reaches a natural threshold. The golden ratio phi appears in the framework as the unique self-similar scaling, and the framework notes that phi^14 is about 843 while phi^15 is about 1364, placing the measured 1100 between these two powers.

In Recognition Science, the framework's cost function, written J(x) = (x + 1/x)/2 - 1, measures the forced expense of a recognition event. The framework models the recombination redshift as a value of this cost function evaluated at a ratio of two physical quantities, m and e. The core idea is that the universe's transition at recombination corresponds to a point where this cost reaches a natural threshold. The golden ratio phi appears in the framework as the unique self-similar scaling, and the framework notes that phi^14 is about 843 while phi^15 is about 1364, placing the measured 1100 between these two powers.

In Recognition Science, the framework's cost function, written J(x) = (x + 1/x)/2 - 1, measures the forced expense of a recognition event. The framework models the recombination redshift as a value of this cost function evaluated at a ratio of two physical quantities, m and e. The core idea is that the universe's transition at recombination corresponds to a point where this cost reaches a natural threshold. The golden ratio phi appears in the framework as the unique self-similar scaling, and the framework notes that phi^14 is about 843 while phi^15 is about 1364, placing the measured 1100 between these two powers.

In Recognition Science, the framework's cost function, written J(x) = (x + 1/x)/2 - 1, measures the forced expense of a recognition event. The framework models the recombination redshift as a value of this cost function evaluated at a ratio of two physical quantities, m and e. The core idea is that the universe's transition at recombination corresponds to a point where this cost reaches a natural threshold. The golden ratio phi appears in the framework as the unique self-similar scaling, and the framework notes that phi^14 is about 843 while phi^15 is about 1364, placing the measured 1100 between these two powers.

In Recognition Science, the framework's cost function, written J(x) = (x + 1/x)/2 - 1, measures the forced expense of a recognition event. The framework models the recombination redshift as a value of this cost function evaluated at a ratio of two physical quantities, m and e. The core idea is that the universe's transition at recombination corresponds to a point where this cost reaches a natural threshold. The golden ratio phi appears in the framework as the unique self-similar scaling, and the framework notes that phi^14 is about 843 while phi^15 is about 1364, placing the measured 1100 between these two powers.

In Recognition Science, the framework's cost function, written J(x) = (x + 1/x)/2 - 1, measures the forced expense of a recognition event. The framework models the recombination redshift as a value of this cost function evaluated at a ratio of two physical quantities, m and e. The core idea is that the universe's transition at recombination corresponds to a point where this cost reaches a natural threshold. The golden ratio phi appears in the framework as the unique self-similar scaling, and the framework notes that phi^14 is about 843 while phi^15 is about 1364, placing the measured 1100 between these two powers.

In Recognition Science, the framework's cost function, written J(x) = (x + 1/x)/2 - 1, measures the forced expense of a recognition event. The framework models the recombination redshift as a value of this cost function evaluated at a ratio of two physical quantities, m and e. The core idea is that the universe's transition at recombination corresponds to a point where this cost reaches a natural threshold. The golden ratio phi appears in the framework as the unique self-similar scaling, and the framework notes that phi^14 is about 843 while phi^15 is about 1364, placing the measured 1100 between these two powers.

In Recognition Science, the framework's cost function, written J(x) = (x + 1/x)/2 - 1, measures the forced expense of a recognition event. The framework models the recombination redshift as a value of this cost function evaluated at a ratio of two physical quantities, m and e. The core idea is that the universe's transition at recombination corresponds to a point where this cost reaches a natural threshold. The golden ratio phi appears in the framework as the unique self-similar scaling, and the framework notes that phi^14 is about 843 while phi^15 is about 1364, placing the measured 1100 between these two powers.

In Recognition Science, the framework's cost function, written J(x) = (x + 1/x)/2 - 1, measures the forced expense of a recognition event. The framework models the recombination redshift as a value of this cost function evaluated at a ratio of two physical quantities, m and e. The core idea is that the universe's transition at recombination corresponds to a point where this cost reaches a natural threshold. The golden ratio phi appears in the framework as the unique self-similar scaling, and the framework notes that phi^14 is about 843 while phi^15 is about 1364, placing the measured 1100 between these two powers.

In Recognition Science, the framework's cost function, written J(x) = (x + 1/x)/2 - 1, measures the forced expense of a recognition event. The framework models the recombination redshift as a value of this cost function evaluated at a ratio of two physical quantities, m and e. The core idea is that the universe's transition at recombination corresponds to a point where this cost reaches a natural threshold. The golden ratio phi appears in the framework as the unique self-similar scaling, and the framework notes that phi^14 is about 843 while phi^15 is about 1364, placing the measured 1100 between these two powers.

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]

noncomputable def cert : Recombin3Cert where
  cost_at_eq := domainCost_at_eq
  cost_nonneg := domainCost_nonneg
  threshold_pos := canonicalThreshold_pos

What this page does not claim

The module does not derive the recombination redshift value of 1100. The module does not define the physical meaning of m and e. The module does not prove that the recombination redshift is exactly a power of phi.

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/RecombinationRedshift3_FromJCost.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.

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