Cosmology Rs Cosmo Module 005 Rscosmo005 Cert

The certificate

A certificate in a machine-checked library of formal theorems is a packaged set of proved statements. The one called RSCosmo005Cert collects three facts about a mathematical function called cost, a measure of recognition effort. The first fact: when the two inputs to the cost are equal, the cost is zero. The second: for positive inputs, the cost is never negative. The third: a certain threshold value, phi minus 1.5, is greater than zero. These are the entire contents of the certificate.

The three facts follow from the general theory of the cost function, not from anything specific to cosmology. The cost is defined as J(m/e), where J is a function that satisfies a forced composition law. The certificate proves properties of this J, but the module never defines what m and e mean in physical terms. The docstring records a research note: the intended application was to compare a tensor value 2/(44*phi^2) against a Planck threshold, but that comparison is not part of the proved certificate.

What the certificate does not claim is important. It does not claim that the tensor value satisfies the Planck bound. It does not claim that m and e represent any particular physical quantities. It does not claim that the threshold phi minus 1.5 has cosmological significance. The certificate is a template: the same three facts are stated once, universally, in a shared module, and this cosmology module inherits them verbatim. To make this a theorem about cosmology, one would need to define m and e in the subject's own terms, and no such definition appears here.

The honest reading is that RSCosmo005Cert is a well-formed but empty vessel. It proves real mathematics about the cost function, and that mathematics is correct. But the leap from a general cost property to a statement about the universe is not made in this file. The research note is a statement of intention, not a result. A reader who wants a cosmology theorem must look elsewhere, or supply the missing definitions themselves.

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 tensor value satisfies the Planck bound. The certificate does not establish that phi minus 1.5 has any cosmological meaning. The certificate is not specific to cosmology; it is a shared template.

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/RS_Cosmo_Module_005.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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