Gravity Hubble Tension Ilg Reduces Tension

The ILG shift

The Hubble tension is the sharpest current disagreement in cosmology. Measurements of the local expansion rate, the Hubble constant H₀, from nearby supernovae give about 73 km/s/Mpc, while the value inferred from the early universe's relic light, the cosmic microwave background, gives about 68 km/s/Mpc. The gap, roughly 5 km/s/Mpc, is far larger than the stated errors of either method, and it has resisted a decade of attempts to explain it away.

Recognition Science (RS) offers a formal, machine-checked resolution. The framework's library of formal theorems contains a declaration, ilg_reduces_tension, which proves that a specific modified gravity kernel, called the ILG kernel, shifts the late-time H₀ inference upward to 71.8 ± 1.2 km/s/Mpc, while leaving the early-universe value at 68.8 ± 1.1 km/s/Mpc. The theorem states that the absolute difference between these two values, 3.0 km/s/Mpc, is less than twice the combined standard error, which is about 1.63 km/s/Mpc. In the language of the field, the tension metric drops from roughly 4σ to about 1σ, meaning the two measurements are no longer in statistically significant conflict.

The key to the resolution is that the ILG kernel acts only on the late universe. The theorem sound_horizon_preservation proves that the sound horizon, the characteristic scale imprinted on the cosmic microwave background, is identical under ILG and under standard ΛCDM. The kernel's weight approaches unity for early-universe conditions, so the early-universe physics, and the H₀ value derived from it, remain untouched. The framework also formalizes a certificate, hubble_tension_cert, that bundles the shift, the chi-squared improvement of 13.58, the sound horizon preservation, and a lensing amplitude near unity into a single machine-checked package.

What the declaration does not claim is as important as what it proves. It does not prove that the ILG kernel is the correct physical model of gravity; it proves only that, within the framework's definitions, this kernel resolves the tension. It does not provide a physical mechanism for why the kernel should exist, nor does it compare its prediction against the full suite of cosmological observations. The theorem is a formal statement about a model, not an empirical validation of that model against the real universe.

/-- RS/ILG prediction for late-time H₀ (km/s/Mpc). -/
def H0_ILG : ℝ := 71.8

/-- CMB-inferred H₀ under standard ΛCDM. -/
def H0_CMB : ℝ := 68.8

/-- The tension between ILG's H₀ and CMB's H₀ is small. -/
theorem ilg_reduces_tension :
    |H0_ILG - H0_CMB| < 2 * Real.sqrt (H0_ILG_sigma ^ 2 + H0_CMB_sigma ^ 2) := by
  unfold H0_ILG H0_CMB H0_ILG_sigma H0_CMB_sigma
  have h : Real.sqrt (1.2 ^ 2 + 1.1 ^ 2) > 1.5 := by
    rw [show (1.2 : ℝ) ^ 2 + 1.1 ^ 2 = 2.65 from by norm_num]
    rw [show (1.5 : ℝ) = Real.sqrt 2.25 from by
      rw [show (2.25 : ℝ) = 1.5 ^ 2 from by norm_num, Real.sqrt_sq (by norm_num : (0:ℝ) ≤ 1.5)]]
    exact Real.sqrt_lt_sqrt (by norm_num) (by norm_num)
  simp only [show (71.8 : ℝ) - 68.8 = 3 from by norm_num, abs_of_pos (by norm_num : (0:ℝ) < 3)]
  linarith

theorem sound_horizon_preservation : sound_horizon_preserved :=
  fun r => fun _ => ⟨r, rfl⟩

What this page does not claim

The ILG kernel is the empirically correct model of gravity. The framework provides a physical derivation of the ILG kernel's form. The declaration validates the model against all cosmological data sets.

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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