Cosmology Hubble Tension Bound Empirical Central In Band

The predictive band

The Hubble tension is the roughly five-sigma disagreement between two ways of measuring the present expansion rate of the universe, the Hubble constant H_0. Late-time measurements, from supernovae and other nearby distance indicators, give a higher value. Early-time measurements, from the cosmic microwave background observed by the Planck satellite, give a lower one. The ratio of the late-time value to the early-time value is about 1.083, and the disagreement is one of the most active problems in cosmology.

In the Recognition Science framework, the late-to-early ratio is not a free parameter. The framework models the ratio through a process it calls cosmic Z-aging on a discrete recognition kernel, and it derives a predicted band for the ratio: from 1.075 to 1.091. This band is a tight neighborhood of the canonical shift 1 + 1/(2·φ²), where φ is the golden ratio. The framework's library, a machine-checked collection of formal theorems, records the band as a structural certificate, not as a numerical coincidence.

The declaration empiricalCentral_in_band establishes a precise, formal fact: the empirical central value of 1.083 lies strictly inside the predicted band. This is a theorem in the library, proved by unfolding the definitions of the lower bound, the upper bound, and the central value, and then checking the arithmetic. The theorem is part of a larger certificate that also states the band is non-degenerate and that a value cannot be both consistent with the prediction and a falsifier of it.

What the declaration does not claim is equally important. It does not claim that the empirical value 1.083 was predicted before the measurement, nor does it prove that the framework's model is the correct explanation of the Hubble tension. The theorem only records that the chosen central value, taken as an input, falls within the derived band. The framework does provide a falsifier predicate: a future joint constraint that places the measured ratio outside the band at more than two sigma would falsify the cosmic Z-aging explanation. But the current theorem is a consistency check, not a confirmation.

For a reader, the practical meaning is simple. The framework's account of the Hubble tension makes a specific, testable claim about a ratio of two measured quantities. The central value of that ratio, as currently known, sits inside the claimed band. The claim is narrow, the check is formal, and the door is left open for future measurements to move the ratio outside the band and thereby falsify the explanation.

/-- 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

/-- 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

What this page does not claim

The declaration does not claim that the empirical value was predicted before it was measured. The declaration does not prove that the Recognition Science model is the correct explanation of the Hubble tension. The declaration does not establish that the band itself is derived from first principles within the framework.

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.

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