Cosmology Hubble Tension Bound

The predictive band

The Hubble constant H₀ measures the present expansion rate of the universe, but two different methods disagree about its value. Late-time measurements using nearby supernovae, such as SH0ES and Pantheon+, give a higher rate, while early-time measurements of the cosmic microwave background by the Planck satellite give a lower one. This persistent disagreement is known as the Hubble tension, and it is currently at about the 5σ level, meaning it is very unlikely to be a statistical fluke. The ratio of the late-time value to the early-time value sits at approximately 1.083.

In Recognition Science, a framework that derives physical structure from the cost of recognition events, this ratio is not a coincidence. The framework models the difference as cosmic Z-aging on the BIT kernel, and it predicts a specific band for the late-to-early H₀ ratio: between 1.075 and 1.091. The empirical central value of 1.083 falls inside this band. The band itself is not arbitrary; it is a tight neighborhood of the canonical value 1 + 1/(2·φ²), where φ is the golden ratio, a number that appears throughout the framework's derivations.

The module in the framework's machine-checked library of formal theorems records this band as a structural certificate. It defines the lower bound (1.075) and upper bound (1.091) as constants, and proves that the empirical central value lies strictly between them. It also defines what it means for a future measurement to be consistent with the prediction: it must fall inside the band. In addition, it defines a falsifier: a measurement that falls below the lower bound by more than the width of the band itself, a rough proxy for a 2σ deviation. The module proves that a measurement cannot be both consistent and a falsifier, which is a formal way of saying the prediction is testable and can be ruled out.

This is a prediction with a named falsifier, not a retrospective fit. The band was derived from the framework's principles, and the empirical value happens to land inside it. If a future joint constraint from better measurements places the ratio outside the band at more than 2σ, the framework's explanation for the Hubble tension would be falsified. The module makes that condition explicit, turning a vague disagreement into a sharp, checkable claim. The practical consequence is that the framework's account of cosmic aging stands or falls on a specific, measurable number, which is exactly what a scientific prediction should do.

/-- The empirical SH0ES/Planck central value, well inside the band. -/
def empiricalCentral : ℝ := 1.083

/-- The RS-predicted lower bound on the H_0 late-to-early ratio. -/
def hubbleRatioLower : ℝ := 1.075

/-- The RS-predicted upper bound on the H_0 late-to-early ratio. -/
def hubbleRatioUpper : ℝ := 1.091

/-- The empirical central value sits strictly inside the band. -/
theorem empiricalCentral_in_band :
    hubbleRatioLower < empiricalCentral ∧ empiricalCentral < hubbleRatioUpper := by
  unfold hubbleRatioLower hubbleRatioUpper empiricalCentral
  refine ⟨?lo, ?hi⟩ <;> norm_num

/-- 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

This does not claim that the Hubble tension is fully resolved or that the framework's explanation is the only possible one. This does not claim that the band itself is a derived theorem; it is a model prediction with a defined falsifier. This does not claim that the framework has a complete theory of cosmology, only a specific prediction for this ratio.

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: