Cosmology Hubble Tension Hubble Ratio From Ledger

The ledger ratio

The Hubble tension is the disagreement between two measurements of how fast the universe expands. Early-universe measurements, based on the cosmic microwave background, give about 67.4 kilometers per second per megaparsec. Late-universe measurements, using nearby supernovae and Cepheid variables, give about 73.0. The difference is far larger than the stated errors, and it has resisted conventional explanation for years.

Recognition Science offers a geometric origin for this ratio. The framework's machine-checked library of formal theorems contains a declaration, hubble_ratio_from_ledger, that proves the ratio equals (12 + 1) / 12, or 13/12. The number 13 comes from a ledger with 12 edges plus one time dimension; the number 12 is the static ledger with 12 edges. This is a theorem in the library, meaning it is a proved statement about the framework's own definitions, not an empirical claim.

The prediction from this ratio is 1.0833. The observed ratio, using the values above, is 73.04 / 67.4, which is approximately 1.0837. The match is within 0.03 percent. The library also proves that the predicted late-universe value, computed as the early value times 13/12, lies between 73.01 and 73.02, and that the relative difference from the measured 73.04 is less than 0.0005, or 0.05 percent.

What the declaration does not claim is important. The theorem proves an identity about the framework's own definitions: it shows that the ratio 13/12 is equal to (12 + 1) / 12. It does not prove that the physical universe must obey this ratio. The match with observation is a separate, weaker claim, also in the library, that the predicted value falls within a stated tolerance of the measured value. The measured values themselves are entered as definitions, not derived from the framework.

Within the framework, this is one step in a larger story. The same module also derives a dark energy density from geometry, landing at 0.6852 against a Planck measurement of 0.6847 with an error of 0.0073, a match within one sigma. These are presented as empirical checks of a geometric model, not as proofs about the cosmos.

/-- The Hubble ratio 13/12 derives from ledger edge count (12) + time dimension (1). -/
theorem hubble_ratio_from_ledger :
    hubble_ratio_topo = (12 + 1) / 12 := by
  simp only [hubble_ratio_topo]
  norm_num

/-- 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

/-- 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

/-- 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

def H_late_exp : ℝ := 73.04

def H_early_exp : ℝ := 67.4

What this page does not claim

The theorem does not prove the physical universe obeys the 13/12 ratio. The match with observation is not a proof that the framework's model is correct. The measured Hubble values are not derived from the framework; they are entered as external inputs.

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.

Derived articles

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