Cosmology Hubble Tension Pipeline From Zaging Zaging Channel

The five channels

The Hubble tension is the disagreement between two ways of measuring how fast the universe expands. The early universe, seen in the cosmic microwave background, gives one rate; nearby supernovae give another. The ratio of the two is about 1.083, and the disagreement is real enough that cosmologists debate whether it signals new physics or a hidden error. A machine-checked catalog in the Recognition Science framework names five physical ingredients that could contribute to the aging correction that reconciles the two rates.

The catalog is a formal list of five channels: matter density, radiation density, dark energy, curvature, and scalar perturbation. These are the standard components of a cosmological model, each with its own effect on how light ages as it travels. The declaration proves that the list has exactly five entries, no more and no fewer. That count is a theorem, checked by a computer from first principles, with no unproved assumptions.

The five channels sit inside a larger pipeline that connects the golden ratio φ to the Hubble ratio. The pipeline uses the identity φ⁵ = 5φ + 3, which follows from φ² = φ + 1. The number 1/φ⁵ is about 0.0903, and the framework's formula for the late-to-early ratio is 1 + (1/φ⁵)·c, where c is an amplitude normalizer. With c near 0.91, the ratio lands near 1.083, inside the empirical band of (1.075, 1.091).

In Recognition Science, the five channels are a ledger, a discrete record of physical contributions, each assigned to a rung of the φ-ladder. The framework proves the arithmetic facts: that φ⁵ lies between 11.05 and 11.11, that the band contains the empirical value 1.083, and that the channel list has five entries. These are the kernel-checked results. What is not proved is that the amplitude c equals 0.91; that number is fitted to the empirical ratio, so the band is a consistency check of the functional form, not a prediction.

What the declaration changes is the status of the five ingredients: they are no longer an informal list but a formally enumerated set with a proved count. The open question is whether the amplitude c can be derived from the five channels themselves, each contributing its own φ-ladder correction. Until that derivation exists, the word "predicts" does not apply to this result.

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

/-- 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 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 declaration does not prove that the Hubble tension is resolved. It does not derive the amplitude c from the five channels. It does not claim that the five channels are the only possible ingredients.

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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