Cosmology Hubble Tension Bound Band Nontrivial

The predictive band

The Hubble constant H_0 measures the present expansion rate of the universe. Cosmology faces the Hubble tension: late-time measurements (SH0ES, Pantheon+) give a value about 5σ higher than early-time measurements (Planck CMB). Recognition Science (RS) predicts the ratio of late-to-early H_0 values via cosmic Z-aging on the BIT kernel, yielding a band (1.075, 1.091) for this ratio. The empirical central value is 1.083, which lies inside the band.

The declaration band_nontrivial is a machine-checked proof that the band is non-degenerate: the lower bound is strictly less than the upper bound. This is a minimal sanity condition, ensuring the predicted interval has positive width and is not a single point. The library also proves the empirical central value sits strictly inside the band, and that consistency and falsification are mutually exclusive: a measurement cannot be both inside the band and a falsifier.

In Recognition Science, the framework models the band as a structural certificate. The falsifier predicate is explicit: a future joint constraint placing the measured ratio below the band by more than the band width (a rough 2σ proxy) falsifies the BIT-Z-aging explanation. The band itself is φ-rational, a tight neighborhood of the canonical 1 + 1/(2·φ²) predicted RS shift.

The band_nontrivial theorem itself is a simple arithmetic fact: 1.075 < 1.091. Its significance is that it certifies the band as a meaningful interval, not a degenerate point. The empirical central value 1.083 sitting inside the band is a separate theorem, empiricalCentral_in_band. The framework does not claim the band is the measured value, nor that the band is derived from first principles without the BIT-Z-aging model.

/-- 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

The band_nontrivial theorem does not claim the band is derived from first principles without the BIT-Z-aging model. The empirical central value 1.083 is a measured value, not a prediction of the framework. The falsifier predicate is a rough 2σ proxy, not a full statistical test.

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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