Cosmology Hubble Tension Pipeline From Zaging Hubble Band Width Pos

The band's width

The Hubble tension is the disagreement between two ways of measuring the universe's expansion rate, H₀. Measurements of nearby supernovae give a faster expansion than measurements of the early universe's leftover light, by roughly 8 percent. The Recognition Science framework proposes a specific ratio between these two values: r_H = 1 + (1/φ⁵) · c, where φ is the golden ratio and c is an amplitude normaliser. With c set near 0.91, this ratio lands close to 1.083, inside the empirical neighborhood of the SH0ES and Planck measurements.

The declaration hubbleBand_width_pos is a small piece of that proposal. It states, as a formally checked theorem, that the proposed band (1.075, 1.091) has positive width: the lower endpoint is strictly less than the upper endpoint. This is a sanity check on the band's definition, not a statement about the cosmos. The band could be entirely wrong and this theorem would still hold, because it only asserts that 1.075 is a smaller number than 1.091.

In the machine-checked library of formal theorems, this result is one of several that assemble into a certificate for the whole proposal. The certificate also records that φ⁵ = 5φ + 3, that φ⁵ lies between 11.05 and 11.11, that there are five aging channels, and that the empirical value 1.083 sits inside the band. The width theorem is the least substantive of these pieces; it exists to keep the band's definition well-formed.

What the declaration does not claim is more important than what it proves. The amplitude normaliser c ≈ 0.91 is fitted to the very empirical ratio the framework is compared against. The band is therefore a consistency check of the functional form, not a prediction. The word "predicts" does not apply to this framework until c is derived from the five aging channels, each contributing its own φ-ladder rung, or until the band is retired as evidence. The formal theorem hubbleBand_width_pos is silent on all of this; it only certifies that the interval has room.

theorem hubbleBand_width_pos : hubbleRatioBand.1 < hubbleRatioBand.2 := by
  unfold hubbleRatioBand; 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

/-- 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

/-- φ⁵ > 11.05 (using φ > 1.61). -/
theorem phi5_gt : (11.05 : ℝ) < phi ^ 5 := by
  rw [phi5_eq]
  linarith [phi_gt_onePointSixOne]

/-- φ⁵ < 11.11 (using φ < 1.62). -/
theorem phi5_lt : phi ^ 5 < (11.11 : ℝ) := by
  rw [phi5_eq]
  linarith [phi_lt_onePointSixTwo]

theorem zAgingChannel_count : Fintype.card ZAgingChannel = 5 := by decide

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

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

What this page does not claim

The band (1.075, 1.091) is a prediction of the Hubble ratio. The declaration hubbleBand_width_pos says anything about the physical correctness of the band. The framework derives the Hubble tension from first principles without a fitted parameter.

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)

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

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