Cosmology Eta Binterval Cert Phi Pow Neg44 Upper

A narrow bound on a cosmic number

The baryon-to-photon ratio, written η_B, is a number that describes the imbalance between matter and light in the universe. It counts, roughly, how many protons and neutrons exist for every billion photons left over from the Big Bang. The Planck 2018 measurement gives η_B = (6.10 ± 0.04) × 10⁻¹⁰, a value that cosmologists use to test theories of the early universe.

Within the Recognition Science framework, a specific mathematical object is claimed to relate to this ratio: the number φ⁻⁴⁴, where φ is the golden ratio (about 1.618). The framework's machine-checked library of formal theorems proves a precise bound on this number. The theorem phi_pow_neg44_upper establishes that φ⁻⁴⁴ is less than 7.5 × 10⁻¹⁰. A companion theorem proves it is greater than 5.5 × 10⁻¹⁰. Together, these form an interval certificate: φ⁻⁴⁴ lies strictly between 5.5 and 7.5 times 10⁻¹⁰.

The proof is a matter of arithmetic. The framework first proves that φ⁴⁴ lies between 1.5 × 10⁹ and 1.6 × 10⁹. This uses the Fibonacci identity φ⁴⁴ = F(44) × φ + F(43), with F(44) = 701408733 and F(43) = 433494437, and the fact that φ lies between 1.61 and 1.62. Taking the reciprocal of these bounds gives the interval for φ⁻⁴⁴. The observed value 6.1 × 10⁻¹⁰ falls comfortably inside this range.

The framework also identifies the exponent 44 as a product: 44 = 4 × 11, which it interprets as the flip count of a preferred axis in a Gray code times a torsion gap between generations. This same integer 44 appears in the framework's expression for the fine-structure constant. The framework presents this as a structural connection, not as a derivation.

What the declaration does not claim matters as much as what it proves. The theorem establishes only the numerical interval for φ⁻⁴⁴. It does not prove that the baryon-to-photon ratio itself equals φ⁻⁴⁴. The measured value falling inside the interval is an empirical check, recorded as a separate theorem that simply notes 5.5 × 10⁻¹⁰ < 6.1 × 10⁻¹⁰ < 7.5 × 10⁻¹⁰. The framework also does not claim to derive the measured value from first principles; the connection between the structural integer 44 and the cosmic ratio is asserted, not proven.

/-- φ^(-44) < 7.5 × 10⁻¹⁰. -/
theorem phi_pow_neg44_upper : phi ^ (-(44 : ℝ)) < 7.5e-10 := by
  rw [Real.rpow_neg phi_pos.le, phi_rpow_44]
  have hlower : phi ^ (44 : ℕ) > 1.5e9 := phi_pow_44_lower
  have hpos : (0 : ℝ) < phi ^ (44 : ℕ) := pow_pos phi_pos 44
  have h1 : (phi ^ (44 : ℕ))⁻¹ < (1.5e9 : ℝ)⁻¹ := by
    rw [inv_lt_inv₀ hpos (by norm_num : (0:ℝ) < 1.5e9)]
    exact hlower
  have h2 : (1.5e9 : ℝ)⁻¹ ≤ 7.5e-10 := by norm_num
  linarith

/-- φ^(-44) > 5.5 × 10⁻¹⁰. -/
theorem phi_pow_neg44_lower : phi ^ (-(44 : ℝ)) > 5.5e-10 := by
  rw [Real.rpow_neg phi_pos.le, phi_rpow_44]
  have hupper : phi ^ (44 : ℕ) < 1.6e9 := phi_pow_44_upper
  have hpos : (0 : ℝ) < phi ^ (44 : ℕ) := pow_pos phi_pos 44
  have h1 : (phi ^ (44 : ℕ))⁻¹ > (1.6e9 : ℝ)⁻¹ := by
    rw [gt_iff_lt, inv_lt_inv₀ (by norm_num : (0:ℝ) < 1.6e9) hpos]
    exact hupper
  have h2 : (1.6e9 : ℝ)⁻¹ ≥ 5.5e-10 := by norm_num
  linarith

/-- φ^44 > 1.5 × 10⁹ (uses φ > 1.61). -/
theorem phi_pow_44_lower : phi ^ (44 : ℕ) > 1.5e9 := by
  rw [phi_pow_44_fib]
  have hphi_gt : phi > 1.61 := phi_gt_onePointSixOne
  nlinarith

/-- φ^44 < 1.6 × 10⁹ (uses φ < 1.62). -/
theorem phi_pow_44_upper : phi ^ (44 : ℕ) < 1.6e9 := by
  rw [phi_pow_44_fib]
  have hphi_lt : phi < 1.62 := phi_lt_onePointSixTwo
  nlinarith

/-- 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

/-- The key integer: 44 = 4 × 11. -/
theorem forty_four_factorization : (44 : ℕ) = 4 * 11 := by norm_num

What this page does not claim

The theorem does not prove that the baryon-to-photon ratio equals φ⁻⁴⁴. The framework does not derive the measured value of η_B from first principles. The structural connection between 44 and the cosmic ratio is asserted, not proven.

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.

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