Astrophysics Cmbtemperature From Phi Ladder Cmbtemp Cert

The certification structure

The cosmic microwave background (CMB) is the faint glow left over from the hot early universe, measured at 2.725 kelvin. A research note in the Recognition Science library sketches an ambition: divide the Planck temperature by powers of the golden ratio, phi, and land near this value. The note writes T_CMB = T_Planck / phi^196, with the exponent chosen to match the ratio of the two temperatures. That sentence is a research note, not a result.

What the machine-checked library of formal theorems actually proves is narrower and fully general. It defines a cost function, a measure of recognition effort, as J(m/e), where J is the framework's forced cost function and m and e are positive real numbers. Three facts follow: the cost is zero when m equals e, the cost is never negative for positive inputs, and the golden-ratio threshold phi minus 3/2 is positive. The declaration CMBTempCert packages these three facts into a single structure, and the library proves that such a certificate exists.

Nothing in the certificate mentions the CMB, a temperature, or a photon. The structure's fields are universally quantified over real numbers, with no definition tying m to a mass or e to an electron charge in this subject's terms. The docstring itself says the module proves nothing specific to its title. The name CMBTempCert is an aspiration, not an achievement.

The distinction matters for reading the library honestly. A formal proof that a cost function vanishes at equality is a theorem about cost, not about cosmology. The 2.725 kelvin value is an external measurement, and the phi-power relation is an unverified identification. What the certificate establishes is a small, clean piece of mathematics; what it does not establish is a derivation of the CMB temperature.

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 CMBTempCert := ⟨cert⟩

What this page does not claim

The CMB temperature is derived from the golden ratio. The certificate itself contains any astrophysical content. The phi-power relation is a measured or verified value.

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/Astrophysics/CMBTemperatureFromPhiLadder.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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