Cosmology Hubble Tension Bound Is Consistent With Rs

The consistency predicate

The Hubble tension is the persistent disagreement between two ways of measuring the present expansion rate of the universe. Late-time measurements, such as those from SH0ES and Pantheon+, give a higher value for the Hubble constant H_0 than early-time measurements from the Planck satellite's study of the cosmic microwave background. The disagreement is about 5 sigma, meaning it is very unlikely to be a statistical fluke. Recognition Science (RS) offers an explanation for this gap: it predicts a specific ratio between the late-time and early-time values of H_0.

The ledger, a discrete record of events, is central to the framework. RS models the universe's expansion as a process of cosmic Z-aging on this ledger, which produces a predicted shift in the H_0 ratio. The framework's prediction is a band: the late-to-early H_0 ratio should lie between 1.075 and 1.091. This band is a tight neighborhood around the canonical value of 1 + 1/(2·φ²), where φ is the golden ratio. The empirical central value from the SH0ES and Planck measurements is 1.083, which sits comfortably inside this predicted band.

The declaration IsConsistentWithRS is a formal definition in the framework's machine-checked library of formal theorems. It states that a given H_0 ratio is consistent with the RS prediction if and only if it is strictly greater than the lower bound of 1.075 and strictly less than the upper bound of 1.091. This is not a vague notion of agreement; it is a precise, checkable condition. The library also proves that the band is non-degenerate, meaning the lower bound is indeed less than the upper bound, and that the empirical central value of 1.083 lies strictly inside the band.

The framework also defines what would falsify its explanation. A measurement is a falsifier if it falls below the predicted band by more than the band's width, a rough proxy for a 2 sigma deviation. The library proves a key structural fact: a measurement cannot be both consistent with the RS prediction and a falsifier. These two conditions are mutually exclusive by definition. This gives the framework a clear, testable prediction: if a future joint constraint places the measured ratio outside the band at more than 2 sigma, the RS explanation is falsified.

In Recognition Science, this declaration is a structural certificate. It does not prove that the RS explanation is correct, nor does it prove that the empirical value of 1.083 is the true ratio. It establishes a precise, machine-checked criterion for what would count as evidence for or against the framework's specific prediction. The value of this is in its clarity: the framework has made a falsifiable claim, and the declaration makes the conditions for that claim's success or failure unambiguous. This is a model of how a scientific theory can be stated with enough precision to be tested.

The declaration does not claim that the RS explanation is the only possible one for the Hubble tension, nor does it claim that the empirical value of 1.083 is exact. It also does not claim that the band itself is derived from first principles within the framework; the band is a prediction based on the cosmic Z-aging model. What it does claim is that a measurement falling within the band is consistent with the RS prediction, and a measurement falling below the band by more than its width would falsify it. This is the core of what IsConsistentWithRS establishes.

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

/-- A measurement is consistent with the RS prediction iff it sits in
the predicted band. -/
def IsConsistentWithRS (h0_ratio : ℝ) : Prop :=
  hubbleRatioLower < h0_ratio ∧ h0_ratio < hubbleRatioUpper

/-- The band is non-degenerate. -/
theorem band_nontrivial : hubbleRatioLower < hubbleRatioUpper := by
  unfold hubbleRatioLower hubbleRatioUpper; norm_num

/-- 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

Not a proof that the RS explanation for the Hubble tension is correct. Not a derivation of the predicted band from the framework's fundamental axioms. Not a claim that the empirical value of 1.083 is the exact true 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: