Cosmology Helium Abundance3 From Jcost Helium Abund3 Cert

The certificate's scope

Primordial helium is the helium forged in the first minutes after the Big Bang, when protons and neutrons fused into nuclei. Its mass fraction, written Y_p, is a standard output of Big Bang nucleosynthesis theory. The measured value is about 0.245, meaning roughly a quarter of the ordinary matter in the early universe ended up as helium-4. The framework's research note records a hoped-for match: applying its cost function to the golden ratio gives 0.236, within about 4 percent of the measured 0.245. That agreement is a research note, not a result.

What the machine-checked library of formal theorems actually proves is narrower. The declaration HeliumAbund3Cert assembles three general facts about a cost function, written Jcost, applied to a ratio of two positive numbers. First, when the two numbers are equal, the cost is zero. Second, for positive inputs, the cost is never negative. Third, the number phi minus 3/2 is positive, where phi is the golden ratio. Each fact is proved from the definition of Jcost; the certificate packages them into one structure and proves that structure is inhabited, meaning the three facts are consistent and available.

None of those facts mentions helium, mass fractions, nucleosynthesis, or cosmology. The module defines its cost as Jcost (m / e) for arbitrary real numbers m and e, without saying what m or e are. The document's own status line says it proves nothing specific to this subject. The helium paragraph is a research note recording where the idea was meant to go, not a result. To turn the certificate into a statement about primordial helium, one would need a definition of m and e in terms of actual physics, such as the neutron-to-proton ratio at freeze-out.

In Recognition Science, this is a normal and honest state: the general cost theorem is proved, but the bridge from cost to a specific physical quantity is open. The certificate's value is that it isolates the three general properties cleanly and checks them. What a reader can safely take away is that the cost function behaves sensibly on positive ratios. What the reader cannot take away is any established prediction of the helium abundance.

structure HeliumAbund3Cert where
  cost_at_eq : ∀ r : ℝ, r ≠ 0 → domainCost r r = 0
  cost_nonneg : ∀ m e : ℝ, 0 < m → 0 < e → 0 ≤ domainCost m e
  threshold_pos : 0 < canonicalThreshold

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]

theorem cert_inhabited : Nonempty HeliumAbund3Cert := ⟨cert⟩

What this page does not claim

The certificate does not prove any value for the primordial helium abundance. The certificate does not define m or e in physical terms. The research note's 0.236 match is not a theorem and carries no uncertainty estimate.

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)

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.

Derived articles

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