Cosmology Helium Abundance3 From Jcost

Primordial helium abundance

Primordial helium is the helium produced in the first minutes after the Big Bang, before any stars existed. Big Bang nucleosynthesis, the standard model of that era, predicts a primordial helium mass fraction of about 0.245, meaning roughly 24.5 percent of the ordinary matter in the early universe was helium-4. Observations of low-metallicity galaxies, which contain gas nearly untouched by stellar processing, consistently measure values in this range, making the primordial helium abundance one of the sharpest quantitative tests of early-universe cosmology.

The Recognition Science framework derives a nearby value from its own cost function. The framework's central object is the cost, a forced measure of recognition effort defined as J(x) = (x + 1/x)/2 - 1, where x is the ratio of two quantities being compared. For the helium calculation, the framework sets x = phi, the golden ratio, and multiplies the resulting cost by 2, giving Y_p = 2 * J(phi) = 0.236. This framework value of 0.236 lands within about 3.7 percent of the measured 0.245, with nothing fitted.

In the framework's machine-checked library of formal theorems, the module named HeliumAbundance3_FromJCost proves three general facts about the cost function: it vanishes when its two inputs are equal, it is nonnegative for positive inputs, and phi - 3/2 is positive. These are all properties of the cost function itself, universally quantified, and they hold for any positive real inputs. The module does not prove that the helium abundance equals 0.236, because the identification of the helium mass fraction with J(phi) * 2 is a research note, not a theorem.

The gap between the framework's 0.236 and the measured 0.245 is real and unexplained within the framework. The framework's value is a derivation from its cost function, while the measured value comes from astronomical observations; the two numbers are close but not identical. The module itself is honest about this: its docstring states that the Lean proofs establish nothing specific to helium, because the cost function is defined without reference to any physical quantity. What would make the module a theorem about its subject is a definition of m and e in the subject's own terms.

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]

What this page does not claim

The module proves the helium abundance equals 0.236; it only proves general properties of the cost function. The framework's value of 0.236 is a measurement; it is a derivation from a model, not an observation. The framework explains why the primordial helium fraction is 0.245; the gap between the two values is unexplained.

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/HeliumAbundance3_FromJCost.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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