Cosmology Hubble Tension Pipeline From Zaging

The Hubble ratio band

The Hubble tension is the disagreement between two ways of measuring the universe's expansion rate, H_0. Measurements of nearby supernovae (SH0ES) give a higher value than measurements of the early universe's cosmic microwave background (Planck). The ratio of these two values is about 1.083.

In Recognition Science, the framework models this ratio using the golden ratio, phi, which is about 1.618. The proposed form is r_H = 1 + (1/phi^5) * c, where c is an amplitude normalizer. A known Fibonacci identity gives phi^5 = 5*phi + 3, which evaluates to a number between 11.05 and 11.11, so 1/phi^5 is between 0.0900 and 0.0906.

With c set to about 0.91, the formula produces a ratio between 1.075 and 1.091. This band contains the empirical value 1.083. The framework's library of formal theorems proves this containment, along with the bounds on phi^5 and the count of five Z-aging channels: matter density, radiation density, dark energy, curvature, and scalar perturbation.

An honest note: c is fit to the empirical ratio, so the band is a consistency check of the functional form, not a prediction. The word "predicts" does not apply. The framework's own documentation states this plainly. What would close the gap is deriving c from the five channels, each contributing its own phi-ladder rung to the total aging correction.

In plain language, the framework shows that a golden-ratio-based formula, with one fitted number, can land close to the observed Hubble tension. It does not explain why c has that value. The library proves the arithmetic and the band's containment, but the physics of c remains an open target.

/-- Hubble ratio band: (1.075, 1.091) contains empirical 1.083. -/
noncomputable def hubbleRatioBand : ℝ × ℝ := (1.075, 1.091)

/-- φ⁵ = 5φ + 3 Fibonacci identity. -/
theorem phi5_eq : phi ^ 5 = 5 * phi + 3 := by
  have h2 := phi_sq_eq
  have h3 : phi ^ 3 = 2 * phi + 1 := by nlinarith
  have h4 : phi ^ 4 = 3 * phi + 2 := by nlinarith
  nlinarith

theorem hubbleBand_contains_empirical : hubbleRatioBand.1 < 1.083 ∧ 1.083 < hubbleRatioBand.2 := by
  unfold hubbleRatioBand
  constructor <;> norm_num

noncomputable def hubbleTensionCert : HubbleTensionCert where
  phi5_fibonacci := phi5_eq
  phi5_lower := phi5_gt
  phi5_upper := phi5_lt
  five_channels := zAgingChannel_count
  band_contains := hubbleBand_contains_empirical

What this page does not claim

The module does not predict the Hubble tension; it only checks consistency of a fitted form. The value of c is not derived from first principles. The framework's library does not prove that the Hubble tension itself is resolved.

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/HubbleTensionPipelineFromZAging.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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Derived articles

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