Cosmology Rs Cosmo Module 003 Rscosmo003 Cert

The certification

The declaration RSCosmo003Cert is a formal certificate, a bundle of three proved statements that the framework's machine-checked library of formal theorems holds up as a unit. The certificate concerns cost, a measure of how expensive one state is to recognize from another. Its three components are: the cost of recognizing a state from itself is zero, the cost is never negative when both states are positive, and a certain threshold value, phi minus 3/2, is greater than zero. Each of these is a theorem about the function J(x) = (x + 1/x)/2 - 1, which the framework derives from its five founding conditions.

The certificate's name and its attached research note point toward a specific cosmology problem: the Hubble tension, the disagreement between local and early-universe measurements of the expansion rate. The note records a ratio H0_local/H0_CMB in the range (1.075, 1.091), with the SH0ES value 1.0837 falling inside, and marks it RS_PASS. But this is a research note, not a result. The certificate itself proves nothing about the Hubble tension, because its cost function is defined as J(m/e) with m and e left as arbitrary real numbers. There is no definition of m as the local Hubble constant and e as the cosmic microwave background value, or any other subject-specific identification. The three theorems are universally quantified over all positive real inputs.

This body of text is shared verbatim with 2383 sibling modules, each with a different name and a different intended subject. The content is stated once, universally quantified, in a template module called DomainCostTemplate. What would turn this certificate into a theorem about cosmology is a definition of m and e in that subject's own terms, a step the module does not take. The certificate is thus a general-purpose tool, a proof that the cost function has three basic properties, wrapped in a name that records where its author hoped to apply it.

What the certificate does establish, cleanly and without qualification, is that the cost function J is well-behaved in three elementary respects. It vanishes at equality, it never goes negative on positive inputs, and the golden-ratio-derived threshold is positive. These are the load-bearing facts the framework's wider chain of results depends on. A reader should take the certificate as a small, reliable brick in that larger wall, not as a verdict on the Hubble tension.

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

RSCosmo003Cert does not claim that the Hubble tension is resolved or that the SH0ES value is correct. The certificate does not define m or e in cosmological terms. The certificate does not claim that the research note's RS_PASS verdict is a proved result.

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_003.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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