Cosmology Eta Binterval Cert Eta B Interval

The baryon-to-photon interval

The baryon-to-photon ratio η_B measures how many protons and neutrons exist for every photon in the observable universe. The measured value from the Planck 2018 data is (6.10 ± 0.04) × 10⁻¹⁰, a number that has puzzled cosmologists because it is not predicted by the Standard Model of particle physics.

In Recognition Science, the declaration eta_B_interval establishes a precise interval for this ratio. The framework's machine-checked library of formal theorems proves that the golden ratio φ raised to the power -44 falls strictly between 5.5 × 10⁻¹⁰ and 7.5 × 10⁻¹⁰. The proof uses the Fibonacci identity φ⁴⁴ = F(44) × φ + F(43), with F(44) = 701408733 and F(43) = 433494437, and bounds φ between 1.61 and 1.62 to derive the interval.

The framework models the number 44 as a product of two structural integers: 4 (the flip count of a preferred axis in a Gray code) times 11 (the torsion gap between two generation levels). This same 44 appears in the framework's expression for the fine-structure constant α⁻¹ = 44π × exp(-w₈ ln φ / 44π), suggesting a common structural origin for both the electromagnetic coupling and the baryon asymmetry.

The observed value 6.1 × 10⁻¹⁰ falls inside the proved interval, but this is an empirical check, not a theorem. The interval certificate does not derive the exact measured value, nor does it explain why the baryon asymmetry exists. It only establishes that the golden-ratio power φ⁻⁴⁴ lands within the stated bounds.

/-- φ^(-44) ∈ (5.5 × 10⁻¹⁰, 7.5 × 10⁻¹⁰).
    The observed η_B = (6.10 ± 0.04) × 10⁻¹⁰ falls inside this interval. -/
theorem eta_B_interval :
    phi ^ (-(44 : ℝ)) > 5.5e-10 ∧ phi ^ (-(44 : ℝ)) < 7.5e-10 :=
  ⟨phi_pow_neg44_lower, phi_pow_neg44_upper⟩

/-- The observed η_B ≈ 6.1 × 10⁻¹⁰ is inside the predicted RS interval. -/
theorem observed_eta_in_interval :
    (5.5e-10 : ℝ) < 6.1e-10 ∧ (6.1e-10 : ℝ) < 7.5e-10 := by
  norm_num

/-- **STRUCTURAL THEOREM**: 44 = flip_count(axis 0) × |Δτ₁₂|.
    The rung of the baryon asymmetry is the product of:
    - The chirality asymmetry of the Gray code (flip count of preferred axis)
    - The generation torsion gap (CW filtration level difference)

    This is the SAME "44" that appears in α⁻¹ = 44π × exp(-w₈ ln φ / 44π). -/
theorem rung_44_equals_flip_times_torsion :
    (44 : ℕ) = bitFlipCount 0 * (torsionGap 0 1).natAbs := by
  simp only [bitFlipCount, torsionGap, τ]
  native_decide

What this page does not claim

The exact measured value of η_B is not derived, only an interval. The framework does not explain why baryons outnumber antibaryons. The structural identity 44 = 4 × 11 is a proved theorem, but its physical interpretation as chirality and torsion is a model choice.

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/EtaBIntervalCert.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

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: