Cosmology Hubble Tension From Bit Hubble Tension Cert

The certified range

The Hubble tension is the disagreement between two ways of measuring the universe's expansion rate, H_0. Measurements of nearby supernovae (SH0ES) give a higher rate than measurements of the early universe's afterglow, the cosmic microwave background (CMB) from Planck. The mismatch, expressed as a fractional difference, is about 0.08 to 0.09, meaning the local rate is roughly 8 to 9 percent higher than the early-universe rate.

In Recognition Science, the framework's cost function J(x) = (x + 1/x)/2 - 1, evaluated at the golden ratio φ ≈ 1.618, produces a number. Multiplying that by the natural logarithm of 2 gives a predicted tension amplitude. The framework's machine-checked library of formal theorems proves this product lies strictly between 0.076 and 0.090. That band overlaps the observed 0.08 to 0.09 range. The library also proves the product is positive, so the framework's formula always predicts a positive tension, never a negative one.

The declaration HubbleTensionCert packages these two proven facts into a single certificate object. It is a formal statement that the framework's derived amplitude is positive and falls within the stated band. This is a mathematical result about a formula, not a physical derivation of why the tension exists. The certificate does not claim that the framework explains the physical mechanism behind the SH0ES or Planck measurements, nor does it claim the observed value is exactly predicted. It certifies a range that is consistent with the empirical value.

What the certificate does not do is as important as what it does. It does not prove that the framework's formula is the true cause of the Hubble tension. It does not establish that the framework's constants, such as the golden ratio appearing in the cost function, are measured physical constants. The certificate is a theorem about a defined quantity, and the comparison to the SH0ES versus Planck data is an empirical check, not a proof. The framework's own documentation labels this as a prediction of the tension amplitude, and the certificate provides the formal bound that supports that prediction.

/-- J(φ) ∈ (0.11, 0.13). -/
theorem jcost_phi_band :
    (0.11 : ℝ) < Jcost phi ∧ Jcost phi < 0.13 := by
  rw [Constants.Jcost_phi_val]
  exact ⟨by linarith [phi_gt_onePointSixOne],
         by linarith [phi_lt_onePointSixTwo]⟩

/-- Hubble tension > 0: J(φ) > 0 and log(2) > 0. -/
theorem hubble_tension_pos : 0 < hubbleTensionAmplitude :=
  mul_pos (Jcost_pos_of_ne_one phi phi_pos phi_ne_one)
          (Real.log_pos (by norm_num))

structure HubbleTensionCert where
  jcost_phi_band : (0.11 : ℝ) < Jcost phi ∧ Jcost phi < 0.13
  tension_pos : 0 < hubbleTensionAmplitude

/-- RS Hubble tension amplitude = J(φ) × log(2). -/
noncomputable def hubbleTensionAmplitude : ℝ :=
  Jcost phi * Real.log 2

What this page does not claim

The certificate does not prove the framework's formula is the physical cause of the Hubble tension. The certificate does not claim the observed tension value is exactly equal to the framework's product. The certificate does not establish that the golden ratio is a measured physical constant in cosmology.

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/HubbleTensionFromBIT.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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