Cosmology Baryon Asymmetry Exact Eta B Eq Phi Over Theta Crit

The baryon asymmetry relation

The baryon asymmetry is the observed imbalance between matter and antimatter in the universe. For every billion or so antiparticles, there are roughly a billion plus one particles, and this tiny surplus is why any matter exists at all. Cosmologists measure it as η_B, the ratio of baryons (protons and neutrons) to photons, with a central observed value near 6.12 × 10⁻¹⁰. The standard explanation, electroweak baryogenesis, produces this number from three ingredients: the amount of charge-parity violation, the number of particle species, and a washout factor that suppresses the surplus.

Recognition Science (RS) derives each of those ingredients from its own framework and then assembles them into a single formal statement. The declaration eta_B_eq_phi_over_theta_crit proves that η_B equals φ divided by Θ_crit, where φ is the golden ratio (about 1.618) and Θ_crit is defined as φ⁴⁵. Since φ⁻⁴⁴ times φ⁴⁵ equals φ, this is equivalent to saying η_B sits on the φ⁻⁴⁴ rung of a ladder of powers. The number 44 itself is not arbitrary: the framework proves it is the product of a chirality flip count (4) and a generation torsion gap (11). The theorem also establishes that η_B is positive and less than one, and that Θ_crit is greater than one, so the ratio is a small positive number consistent with observation.

In plain terms, the framework claims the baryon asymmetry is not a free parameter but a forced consequence of its structure: the same product 4 × 11 that appears in its fine-structure constant expression also fixes the matter-antimatter surplus. The golden ratio mediates between the matter content and a quantity the framework calls the consciousness capacity, defined as the light field saturation threshold. The full derivation chain is machine-checked in the framework's library of formal theorems, with zero unproved axioms in its audit.

What this declaration does not claim is equally important. It does not claim to derive the fine-structure constant itself, whose exact value remains an open target. It does not claim the observed value 6.12 × 10⁻¹⁰ is derived from first principles; that number is an input from measurements, and the framework's φ⁻⁴⁴ value differs from it by about 4.5 percent. It does not claim the physical bridge from recognition events to baryon production is complete; that link remains open. The theorem establishes an exact algebraic relation within the framework, not a complete physical derivation.

/-- The link expressed as a ratio: η_B = φ / Θ_crit. -/
theorem eta_B_eq_phi_over_theta_crit :
    eta_B_phi_scale = phi / theta_crit := by
  have h_tc_pos : 0 < theta_crit := lt_trans (by norm_num : (0 : ℝ) < 1) theta_crit_gt_one
  rw [eq_div_iff (ne_of_gt h_tc_pos)]
  exact matter_consciousness_link

/-- **THEOREM**: The baryon asymmetry rung 44 is the product of the
    chirality flip count and the torsion gap.

    This is not a coincidence — it reflects the deep connection between
    CP violation (from the chirality of the Gray code) and the mass
    hierarchy (from the torsion spectrum). -/
theorem rung_44_is_product :
    flip_count_gen0 * torsion_gap_01 = 44 := by
  native_decide

/-- η_B scale is positive. -/
theorem eta_B_phi_scale_pos : 0 < eta_B_phi_scale := by
  unfold eta_B_phi_scale
  exact zpow_pos phi_pos (-44)

/-- η_B scale is between 0 and 1.
    φ⁻⁴⁴ is positive because φ > 0, and it is < 1 because φ > 1. -/
theorem eta_B_phi_scale_lt_one : eta_B_phi_scale < 1 := by
  unfold eta_B_phi_scale
  have h : phi ^ (-44 : ℤ) = 1 / phi ^ (44 : ℤ) := by
    rw [zpow_neg, one_div]
  rw [h]
  rw [div_lt_one (zpow_pos phi_pos (44 : ℤ))]
  exact one_lt_zpow₀ one_lt_phi (show (0 : ℤ) < 44 by norm_num)

/-- Θ_crit is large (φ⁴⁵ >> 1). -/
theorem theta_crit_gt_one : 1 < theta_crit := by
  unfold theta_crit
  exact one_lt_zpow₀ one_lt_phi (show (0 : ℤ) < 45 by norm_num)

What this page does not claim

The fine-structure constant α is not derived by this declaration; its exact value remains an open target. The observed baryon asymmetry value 6.12 × 10⁻¹⁰ is not derived from first principles; it is an input from measurements. The physical bridge from recognition events to baryon production is not established by this theorem; that link remains open.

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/BaryonAsymmetryExact.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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