Cosmology Omega Baryon3 From Jcost

Baryon fraction and the cost function

In cosmology, the baryon fraction is the share of the universe's matter that is made of ordinary atoms, protons, and neutrons. The Planck 2018 satellite data give a baryon density Omega_b h^2 of 0.0224 and a total matter density Omega_m h^2 of 0.143, so the ratio Omega_b/Omega_m is 0.157. That number, about 15.7 percent, is what astronomers measure when they count the ordinary matter in galaxies and gas clouds against the dark matter that outweighs it.

Recognition Science proposes a different route to a similar figure. Its core object is a cost function, a discrete record of how expensive it is for the universe to recognize one thing as another. The framework proves that any cost function satisfying five plain conditions must equal J(x) = (x + 1/x)/2 - 1. Evaluating this J at the golden ratio phi, about 1.618, gives J(phi) = 0.118. That is the framework's candidate baryon fraction: 11.8 percent, a close order of magnitude to the measured 15.7 percent.

The machine-checked library of formal theorems proves three general facts about this cost function. First, it vanishes when the two inputs are equal: J(m/e) = 0 when m = e. Second, it is never negative for positive inputs. Third, the threshold phi - 3/2 is positive. These are all true for any positive m and e. The library proves nothing specific to cosmology, because the module defines its cost as J(m/e) without ever defining what m and e mean in this subject.

In Recognition Science, the module is a placeholder. The research note records where the idea was meant to go: a derivation of the baryon fraction from the cost function alone. What would make it a theorem about cosmology is a definition of m and e in terms of baryon density and total matter density, drawn from the framework's own principles. That definition does not exist in the module. The three proved facts are general properties of J, not statements about the universe.

The plain-language upshot: the framework has a striking numerical coincidence, 0.118 against a measured 0.157, but the bridge from cost to cosmos is not built. The library establishes the cost function's basic behavior, and the cosmology claim remains a target for future work, not a result.

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)

def canonicalThreshold : ℝ := phi - 3 / 2

What this page does not claim

The module does not prove that the baryon fraction equals 0.118. The measured Planck ratio 0.157 is not derived from the framework. No definition of m and e in cosmological terms exists in the 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/Cosmology/OmegaBaryon3_FromJCost.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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