Cosmology Eta Binterval Cert

The baryon-to-photon ratio certificate

The baryon-to-photon ratio, written η_B, is a number that describes the matter content of the cosmos: it is the count of protons and neutrons (baryons) for every photon in the observable universe. The Planck 2018 measurement gives η_B = (6.10 ± 0.04) × 10⁻¹⁰, a value that has puzzled cosmologists because it sits far below what simple particle physics would predict. The cosmology eta binterval cert (the interval certificate for the baryon-to-photon ratio) proves, in a machine-checked library of formal theorems, that the golden ratio φ raised to the power −44 falls inside the range (5.5, 7.5) × 10⁻¹⁰, which brackets the observed value.

The proof is a piece of exact arithmetic, not a numerical simulation. It uses the Fibonacci identity φ⁴⁴ = F(44) × φ + F(43), where F(44) = 701408733 and F(43) = 433494437. With φ known to lie between 1.61 and 1.62, the computation gives φ⁴⁴ between 1.5 × 10⁹ and 1.6 × 10⁹, and therefore φ⁻⁴⁴ lies between 5.5 × 10⁻¹⁰ and 7.5 × 10⁻¹⁰. The theorem eta_B_interval states this bound, and a companion theorem observed_eta_in_interval confirms that the measured 6.1 × 10⁻¹⁰ sits inside it.

The number 44 is not arbitrary. The proof establishes a structural identity: 44 = 4 × 11, where 4 is the flip count of a preferred axis in a Gray code (a measure of chirality asymmetry) and 11 is 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). The claim is that both the electromagnetic coupling and the baryon asymmetry trace back to the same structural integer, a connection stated as a theorem but not itself explained physically.

In Recognition Science, this certificate is part of a larger chain that derives constants from the golden ratio. The contribution is narrow but concrete: it proves, with the precision of formal logic, that the golden ratio's 44th power lands within the measured window for η_B. It does not prove that η_B must equal φ⁻⁴⁴ exactly, nor does it derive the value from first principles; it certifies an interval and notes that the measurement falls inside. The physical bridge between the structural integer 44 and the actual baryon asymmetry of the universe remains an open question, one that is not attempted to be closed here.

/-- 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

/-- 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

This module does not prove that η_B equals φ⁻⁴⁴ exactly. It does not derive the baryon-to-photon ratio from first principles. It does not explain the physical mechanism connecting the integer 44 to baryon asymmetry.

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

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