Cosmology Bbnheliium Exact3 From Jcost Bbnheliium Exact3 Cert

The certificate's scope

In the Recognition Science framework, the declaration BBNHeliiumExact3Cert is a machine-checked certificate: a bundle of three formal theorems about a function called domainCost, which is defined as Jcost applied to the ratio of two real numbers, m and e. The three theorems are general facts about this cost function, not about any specific physical system. They state that the cost is zero when the two inputs are equal, that the cost is never negative for positive inputs, and that a certain threshold value, phi minus 1.5, is positive. These are the only facts the certificate establishes.

The name of the file suggests a connection to primordial helium-4 from Big Bang nucleosynthesis, and a research note in the file records an intended match: a measured helium mass fraction of 0.2454 against a computed value of 0.245. But the formal content of the certificate itself does not make that connection. The definition of domainCost uses the ratio m / e without ever defining what m or e mean in physical terms. To turn the certificate into a statement about helium, one would need a separate definition of these variables in the language of cosmology, which the file does not provide.

What the certificate does prove is reusable. The three facts are stated once, universally quantified, in a shared template module, and this file instantiates them. The certificate shows that the cost function behaves sensibly: it vanishes at equality, stays nonnegative for positive inputs, and has a positive threshold. This is a small but solid piece of mathematical infrastructure, ready to be used if a future definition connects it to a physical quantity. As it stands, the certificate is a proof about a function, not a proof about the universe.

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)

What this page does not claim

The certificate does not prove that the primordial helium abundance is 0.245. The certificate does not define the physical meaning of m or e. The certificate does not establish any connection between the cost function and cosmology.

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/BBNHeliiumExact3FromJCost.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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