Gravity Hubble Tension Chi2 Improvement Significant

The chi-squared improvement

The Hubble tension is the disagreement between two ways of measuring the expansion rate of the universe, H0. Measurements of the early universe, imprinted in the cosmic microwave background, give about 68.8 km/s/Mpc. Measurements of the late universe, from supernovae and other local distance indicators, give a higher value, around 73 km/s/Mpc. The gap is about 4 to 5 standard deviations, large enough that cosmologists treat it as a genuine crisis for the standard model of cosmology, ΛCDM.

Recognition Science (RS) proposes a modification called the ILG kernel that shifts the late-time inference of H0 without touching early-universe physics. Within this framework, the late-time H0 becomes 71.8 ± 1.2 km/s/Mpc, while the early-universe value stays at 68.8 ± 1.1. The tension metric, |ΔH0|/σ, drops from roughly 4σ to about 1σ. The framework's machine-checked library of formal theorems proves that this shift is positive and that the tension is reduced: the difference between the two values is less than twice the combined uncertainty.

The declaration chi2_improvement_significant establishes one specific numerical fact: the improvement in the chi-squared statistic, delta_chi2_improvement, is greater than 10. In plain terms, the ILG model fits the combined data set better than standard ΛCDM by more than 10 units of chi-squared. A chi-squared improvement of this size is conventionally regarded as statistically significant, often cited as strong evidence in favor of the modified model. The theorem is proved by direct computation from the definition, which sets delta_chi2_improvement to 13.58.

What the declaration does not claim is just as important. It does not claim that the ILG model is correct, that the chi-squared improvement arises from a physically motivated mechanism, or that the improvement is significant in any statistical sense beyond the raw threshold of 10. The theorem only asserts the arithmetic fact that 13.58 is greater than 10. It says nothing about whether the data used to compute that number are real, whether the model's assumptions are justified, or whether the improvement would survive a rigorous statistical test with a proper accounting of systematic errors.

The declaration also does not claim that the ILG model resolves the Hubble tension in any observational sense. The framework's library separately proves that the sound horizon is preserved, meaning the early-universe physics is untouched, and that the effective lensing amplitude A_L is close to unity. But those are separate theorems. The chi2_improvement_significant declaration is a narrow, formal statement about a single number being above a threshold. It is a building block in a larger argument, not the argument itself.

theorem chi2_improvement_significant : delta_chi2_improvement > 10 := by
  unfold delta_chi2_improvement; norm_num

/-- The chi-squared improvement from ILG over standard ΛCDM. -/
def delta_chi2_improvement : ℝ := 13.58

/-- The chi-squared improvement from ILG over standard ΛCDM. -/
def delta_chi2_improvement : ℝ := 13.58

What this page does not claim

The ILG model is observationally correct or physically motivated. The chi-squared improvement is statistically significant in a rigorous sense with proper error accounting. The declaration alone resolves the Hubble tension; it is one building block among several theorems.

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/Gravity/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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