Cosmology Hubble Tension Bound Consistency Excludes Falsification

The consistency check

The Hubble tension is the persistent roughly five-sigma disagreement between late-time measurements of the present expansion rate of the universe, such as SH0ES and Pantheon+, and early-time measurements from the Planck satellite's cosmic microwave background data. Recognition Science addresses this by predicting the ratio of the late-time to early-time Hubble constant through a process it calls cosmic Z-aging on its BIT kernel. The predicted ratio band is (1.075, 1.091), and the empirical central value of 1.083 sits comfortably inside it.

In Recognition Science, the framework defines a measurement as consistent with its prediction if the ratio falls strictly inside the band. It defines a falsifier as a measurement that falls below the band by more than the band width, a rough proxy for a two-sigma exclusion. The theorem consistency_excludes_falsification proves that these two conditions are mutually exclusive: no single measurement can be both consistent and a falsifier. This is a logical guarantee, not an empirical claim. It ensures the falsification criterion is well-posed, so a future measurement cannot be counted as both supporting and refuting the prediction.

The theorem is part of a structural certificate that also records the band's non-triviality and the empirical value's position inside it. The certificate is a formal object in the framework's machine-checked library of formal theorems, built without any unproven assumptions. What this does not claim is that the band itself is correct or that the empirical value confirms the prediction. The theorem only states the logical relationship between the two definitions. It does not predict any future measurement, and it does not assert that the Hubble tension is resolved. The band and the empirical value are separate facts, and the theorem simply guards the logic of how they are compared.

/-- Consistency and falsification are mutually exclusive. -/
theorem consistency_excludes_falsification {h0 : ℝ} :
    ¬ (IsConsistentWithRS h0 ∧ IsFalsifier h0) := by
  rintro ⟨⟨h_lo, _⟩, h_excl⟩
  unfold IsFalsifier at h_excl
  unfold hubbleRatioLower hubbleRatioUpper at *
  linarith

What this page does not claim

The theorem does not prove the predicted band is correct or that the empirical value confirms the prediction. It does not predict any future measurement of the Hubble ratio. It does not assert that the Hubble tension is resolved.

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/HubbleTensionBound.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

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: