Cosmology Cmboptical Depth3 From Jcost Cmbopt Depth3 Cert

What the certificate proves

The cosmic microwave background (CMB) is the oldest light in the universe, released about 380,000 years after the Big Bang. Its optical depth measures how much that light was scattered by free electrons on its way to us; a value near zero means the early universe was mostly transparent. The Planck 2018 measurement gives an optical depth of about 0.054.

The declaration CMBOptDepth3Cert (a certificate: a packaged bundle of machine-checked proofs) establishes three facts about a cost function J. The cost function measures the penalty for recognizing one quantity as another; here it is applied to the ratio m/e of two real numbers. The certificate proves that this cost is zero when the two inputs are equal, that it is never negative when both inputs are positive, and that the quantity phi minus 1.5 is positive, where phi is the golden ratio, about 1.618. These are general facts about the cost function, not specific to cosmology.

In the Recognition Science framework, this certificate is a stepping stone. The framework's own research note records an intended application: setting the optical depth tau equal to J(phi)/2 gives about 0.059, which is within 10 percent of the Planck value. That numerical agreement is a research note, not a proved result. The certificate itself proves nothing about the CMB, because the quantities m and e are not defined in terms of cosmology; they are just placeholders.

What the certificate does establish is a reusable template. The same three facts are stated once, universally, in a shared module, and this certificate instantiates them. Any future work that defines m and e in physical terms can build on these proofs. Until then, the optical depth derivation remains a target, not a theorem.

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 certificate does not derive the optical depth of the CMB. The numerical agreement with Planck 2018 is a research note, not a proved result. The certificate does not define m and e in physical terms.

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/CMBOpticalDepth3FromJCost.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.

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