Cosmology Baryon Asymmetry Exact Eta B Phi Scale Lt One

The baryon asymmetry rung

The baryon asymmetry, denoted η_B, is the tiny excess of matter over antimatter left over from the early universe, measured as about 6.12 × 10⁻¹⁰ baryons per photon. Cosmologists have long sought a first-principles explanation for this number. Within the Recognition Science framework, a machine-checked library of formal theorems establishes a specific structural claim: η_B sits on the −44 rung of a ladder built from powers of the golden ratio φ ≈ 1.618.

The key declaration, eta_B_phi_scale_lt_one, proves that this scale, φ⁻⁴⁴, is less than 1. This is a formal theorem, not an empirical fit. The proof shows that since φ is greater than 1, raising it to a negative power yields a positive number smaller than 1. The framework's library also proves this scale is positive, and that the rung structure derives from a product: 44 equals 4 times 11, where 4 is a chirality flip count and 11 is a generation torsion gap.

This rung assignment connects to a broader claim: the framework defines a quantity Θ_crit as φ⁴⁵, called the light field saturation threshold. A theorem proves that η_B_phi_scale times Θ_crit equals φ. This is a formal identity within the framework's definitions. The framework interprets this as a link between matter content and a concept it calls consciousness, but this interpretation is a definitional choice, not a derived physical law.

The declaration does not claim that the measured value of η_B is exactly φ⁻⁴⁴. The framework defines a separate observed central value of 6.12 × 10⁻¹⁰ and notes a discrepancy of 4.5 percent between this and the phi-scale value. The theorem establishes a structural relationship within the framework's own definitions, not a numerical match to experiment. The physical bridge from recognition events to baryogenesis remains an open target, not a proved result.

/-- 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)

/-- 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)

/-- **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

/-- **THE MATTER-CONSCIOUSNESS LINK THEOREM**:

    η_B × Θ_crit = φ

    The baryon-to-photon ratio times the light field saturation
    threshold equals the golden ratio.

    Physical interpretation: the universe creates φ times as many
    baryons per photon as would be needed to exactly fill the light
    field to capacity. The factor of φ is the self-similar overshoot —
    the same golden ratio that governs the cost function J, the mass
    law, and the 8-tick period also mediates between the matter content
    and the consciousness capacity.

    This is a THEOREM about φ-powers (pure algebra), not a hypothesis
    requiring empirical confirmation. The empirical content is in the
    rung assignment η_B ≈ φ⁻⁴⁴ (within 4.5% of observed). -/
theorem matter_consciousness_link :
    eta_B_phi_scale * theta_crit = phi := by
  unfold eta_B_phi_scale theta_crit
  exact phi_neg44_times_phi45_eq_phi'

What this page does not claim

The framework does not claim the measured baryon asymmetry equals φ⁻⁴⁴ exactly. The framework does not claim the matter-consciousness link is a derived physical law. The framework does not claim the physical bridge from recognition events to baryogenesis is proved.

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)

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Derived articles

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