Cosmology Hubble Tension Bound Lower Pos

The lower bound

The Hubble tension is the persistent disagreement, around five standard deviations, 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 than early-time measurements from the Planck satellite's cosmic microwave background data. The ratio of the late-time to early-time Hubble constant is the quantity in dispute.

In the Recognition Science framework, this ratio is predicted to fall in a specific band, from 1.075 to 1.091. The empirical central value, 1.083, sits inside that band. The declaration lower_pos is a theorem in the framework's machine-checked library of formal theorems. It proves that the lower edge of this band, 1.075, is greater than zero. This is a basic sanity condition: a negative or zero lower bound for a ratio of expansion rates would be physically meaningless, so the theorem confirms the band is a valid interval of positive numbers.

The theorem does not claim that the band is correct, that the empirical value will remain inside it, or that the framework's explanation of the tension is true. It only establishes a formal property of a defined constant. The framework also defines a falsifier predicate: a future joint measurement placing the ratio below 1.059, which is the lower bound minus the band width, would falsify the framework's explanation. The theorem says nothing about whether such a measurement will occur.

In Recognition Science, the framework models the late-to-early ratio via a process called cosmic Z-aging on the BIT kernel. The band is described as phi-rational, a tight neighborhood of the canonical predicted shift of 1 plus 1 over twice the golden ratio squared. The lower_pos theorem is a small piece of this larger structure, ensuring the arithmetic of the prediction is coherent.

/-- The lower bound is positive. -/
theorem lower_pos : 0 < hubbleRatioLower := by
  unfold hubbleRatioLower; norm_num

/-- The lower bound is positive. -/
theorem lower_pos : 0 < hubbleRatioLower := by
  unfold hubbleRatioLower; 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

/-- A measurement is a falsifier iff it sits below the predicted band by
more than the band width (rough 2σ proxy). -/
def IsFalsifier (h0_ratio : ℝ) : Prop :=
  h0_ratio < hubbleRatioLower - (hubbleRatioUpper - hubbleRatioLower)

What this page does not claim

The theorem does not assert that the predicted band is empirically correct. The theorem does not claim that the empirical central value will remain inside the band in future measurements. The theorem does not establish that the framework's explanation of the Hubble tension is true.

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)

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