Cosmology Hubble Tension Pipeline From Zaging Phi5 Gt

A numerical bound

The golden ratio φ, approximately 1.618, is the number that satisfies φ² = φ + 1. Its fifth power, φ⁵, is exactly 5φ + 3, a neat identity that follows from the defining equation. The declaration phi5_gt proves a specific numerical bound: φ⁵ is greater than 11.05. This uses the fact that φ is greater than 1.61, a simple lower bound that itself is a proved statement.

This bound is not an isolated curiosity. It is part of a larger, machine-checked library of formal theorems. The library, a collection of proofs verified by a computer, uses this bound as one component in a chain of reasoning about cosmology. Specifically, it contributes to a proposed explanation for the Hubble tension, the observed discrepancy between different measurements of the universe's expansion rate. The framework's model suggests a ratio between late and early universe expansion rates, and this ratio depends on 1/φ⁵, which is approximately 0.0903. The bound on φ⁵ helps establish a band for this ratio.

The framework's library proves that this ratio sits within a band from 1.075 to 1.091. The empirical value of 1.083, a widely cited figure, falls inside this band. However, the amplitude normalizer, denoted c and set to about 0.91, is fitted to the empirical ratio. 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 first principles, for instance from the five proposed Z-aging channels: matter density, radiation density, dark energy, curvature, and scalar perturbation.

The declaration phi5_gt itself is a narrow, exact statement. It does not, by itself, resolve the Hubble tension, nor does it prove that the framework's cosmological model is correct. It is a single, verified rung in a longer ladder of reasoning. Its value lies in the precision of the bound and the fact that it is machine-checked, providing a solid foundation for the subsequent, more speculative steps.

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

/-- 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_width_pos : hubbleRatioBand.1 < hubbleRatioBand.2 := by
  unfold hubbleRatioBand; norm_num

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

What this page does not claim

The declaration phi5_gt does not, by itself, resolve the Hubble tension. The framework's model does not predict the Hubble ratio band because the amplitude normalizer is fitted to the data. The five Z-aging channels are proposed but not yet derived from first principles.

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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