Cosmology Cmbtemp3 From Jcost

The cosmic microwave background

The cosmic microwave background (CMB) is the oldest light in the universe, a faint glow that fills all of space. It was discovered in 1965 by Arno Penzias and Robert Wilson, who found a persistent hiss in their radio antenna that they could not explain. That hiss turned out to be radiation left over from the hot, dense state of the early universe, emitted about 380,000 years after the Big Bang. As the universe expanded, this light stretched and cooled. Today it has a nearly perfect blackbody spectrum with a measured temperature of 2.725 kelvin, a value established by the COBE satellite in the 1990s and refined by later missions such as WMAP and Planck.

The standard cosmological story explains this temperature through the expansion of the universe and the physics of the early plasma. In Recognition Science, the framework's library contains a module that attempts to connect this measured temperature to a different kind of structure: a discrete record of recognition events, called a ledger. The module, named CMBTemp3_FromJCost, defines a cost function on ratios of two quantities, m and e, as J(m/e), where J(x) = (x + 1/x)/2 - 1. This is the same cost function that the framework proves is forced by five plain conditions, a result checked in the machine-checked library of formal theorems.

What the module actually proves is three general facts about this cost function. First, the cost vanishes when the two quantities are equal: J(1) = 0, meaning there is no recognition cost when the ratio is unity. Second, the cost is never negative for positive inputs, so it is a meaningful measure of difference. Third, the golden ratio phi minus 3/2 is positive, a small technical fact about the constant phi. These three facts are packaged into a structure called a certificate, and the module proves that this certificate exists. The prose above the code records a research note: the intended application was to set m and e to temperatures, with T_CMB = T_vac * phi^(-196), giving 2.73 kelvin when T_vac is the Planck temperature. But the note itself states clearly that this is a research note, not a result.

The module does not establish that the CMB temperature is 2.725 kelvin, nor that the phi-ladder relation is correct. What it establishes is a small, clean piece of mathematics about the cost function, and it does so with the same rigor as the rest of the framework's library. The research note records an aspiration, not a derivation. The actual derivation of the CMB temperature from the framework's principles remains open.

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

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)

What this page does not claim

The module does not prove that the CMB temperature is 2.725 kelvin. The module does not derive the phi-ladder relation from the framework's axioms. The research note is not a theorem; it is an aspiration.

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/CMBTemp3_FromJCost.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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