Cosmology Hubble Tension Hubble Ratio Bounds

The 13/12 ratio

The Hubble tension is the observed disagreement between two ways of measuring how fast the universe expands. Early-universe measurements, based on the cosmic microwave background, give a Hubble constant near 67.4 km/s/Mpc. Late-universe measurements, using distant supernovae and other local distance ladders, give a value near 73.0. The ratio of these two numbers is about 1.0837, and no conventional model fully explains why they differ.

In Recognition Science, the framework's machine-checked library of formal theorems contains a declaration named hubble_ratio_bounds. This theorem proves a narrow numerical fact: the framework's predicted ratio, defined as 13/12, lies strictly between 1.0833 and 1.0834. The number 13/12 comes from a geometric model: a dynamic ledger, a discrete record of events with 12 edges plus one time dimension, divided by a static ledger with 12 edges. The theorem hubble_ratio_bounds is a pure arithmetic statement, verified by direct computation, and it does not by itself assert anything about telescopes or the actual universe.

The framework then connects this ratio to observation through a separate theorem, hubble_ratio_match. That theorem states that the predicted late-universe Hubble constant, computed as 67.4 times 13/12, differs from the measured value 73.04 by less than 0.05 percent of the measured value. The predicted value falls between 73.01 and 73.02. This is a numerical agreement, not a derivation of the tension itself; the framework's model chooses the ratio 13/12 from its ledger geometry, and the match to observation is an empirical check.

What hubble_ratio_bounds does not claim is important. It does not prove that the Hubble tension is resolved, nor that 13/12 is the only possible ratio. It does not derive the measured Hubble constants from first principles; those values, 67.4 and 73.04, are entered as experimental definitions in the library. The theorem only bounds the rational number 13/12 between two decimals. The framework's broader claim, that this ratio explains the tension, rests on the separate match theorem and on the geometric interpretation, both of which are model choices rather than forced consequences.

/-- Helper: 13/12 numerical bounds. -/
theorem hubble_ratio_bounds :
    (1.0833 : ℝ) < (hubble_ratio_topo : ℝ) ∧ (hubble_ratio_topo : ℝ) < (1.0834 : ℝ) := by
  simp only [hubble_ratio_topo]
  norm_num

/-- The Hubble Ratio matches observation to within 0.05%.

    pred = 67.4 * (13/12) = 73.0166...
    obs  = 73.04
    |pred - obs| / obs = |73.0166 - 73.04| / 73.04 = 0.00032 < 0.0005 ✓

    This is now PROVEN, not axiomatized. -/
theorem hubble_ratio_match :
    abs (H_late_pred - H_late_exp) / H_late_exp < 0.0005 := by
  simp only [H_late_pred, H_late_exp, H_early_exp, hubble_ratio_topo]
  norm_num

/-- Predicted late Hubble value. H_late_pred = 67.4 * (13/12) = 73.01666... -/
theorem H_late_pred_value :
    (73.01 : ℝ) < H_late_pred ∧ H_late_pred < (73.02 : ℝ) := by
  simp only [H_late_pred, H_early_exp, hubble_ratio_topo]
  norm_num

What this page does not claim

The theorem does not prove that the Hubble tension is resolved or that 13/12 is the only possible ratio. The measured Hubble constants 67.4 and 73.04 are entered as experimental definitions, not derived from the framework. The geometric interpretation of 13/12 as a ledger ratio is a model choice, not a forced consequence of the theorem.

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/HubbleTension.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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