Cosmology Baryon Asymmetry Derivation Eta B Positive

The derived sign

The baryon-to-photon ratio η_B counts how many protons and neutrons exist for every photon left over from the Big Bang. The measured value is about 6.1 × 10⁻¹⁰, a tiny surplus of matter over antimatter. This surplus is why stars, planets, and people exist at all. The Recognition Science framework, which starts from the idea that reality keeps a discrete record of events at a forced cost, contains a machine-checked theorem about this ratio.

The theorem, named eta_B_positive, proves that the structural formula for η_B is positive. That structural formula is a textbook skeleton: a CP-violating quantity divided by the number of relativistic particle species at the electroweak scale. The framework derives the CP-violating piece from a chain starting with its forced cost function, through the golden ratio, an eight-tick cycle, three dimensions, and the Gray-code chirality of the quark mixing matrix. The chain ends with the Sakharov conditions, the three requirements for generating a matter-antimatter asymmetry. The theorem's content is the sign: matter dominates antimatter, not the reverse.

What the theorem does not claim is the magnitude. The structural value it proves positive is about 3 × 10⁻⁷, roughly 500 times larger than the observed 6.1 × 10⁻¹⁰. The missing factor is the washout constant, which requires detailed transport dynamics through the electroweak phase transition; that calculation remains open. A separate hypothesis matches the observed number with a golden-ratio power law, η_B ≈ φ⁻⁴⁴·(1−φ⁻⁸)², landing within the Planck 2018 band. That match is an empirical rung assignment with a named falsifier, not a theorem.

The practical consequence is a clear separation of what is derived and what is not. The sign is a proved theorem in the machine-checked library, anchored to the framework's forced cost function. The magnitude is a hypothesis awaiting either confirmation or a tightened CMB measurement that falsifies it. A reader can trust the sign claim as kernel-checked, while treating the number as an open empirical question.

/-- η_B is positive: matter dominates over antimatter.
    This follows directly from J_CP > 0 and is the genuine derived content
    (the SIGN of the asymmetry, not its magnitude). -/
theorem eta_B_positive : eta_B_structural > 0 := by
  unfold eta_B_structural
  apply div_pos jarlskog_positive
  norm_num [g_star]

/-- The structural η_B: proportional to J_CP / g_★.
    SCAFFOLD ONLY — numerically ≈ 3×10⁻⁷, about 500× the observed value,
    because the order-one-suppressed washout constant c is not derived.
    Used solely for the sign theorem (η_B > 0) and the smallness bound. -/
noncomputable def eta_B_structural : ℝ := jarlskog_structural / g_star

/-- The φ-rung exponent for the baryon asymmetry scale.
    φ⁴⁴ ≈ 1.568 × 10⁹, so φ⁻⁴⁴ ≈ 6.376 × 10⁻¹⁰.
    The observed η_B ≈ 6.1 × 10⁻¹⁰ is within ~4.5% (before the
    (1−φ⁻⁸)² prefactor). HYPOTHESIS-grade rung assignment. -/
def eta_B_rung : ℤ := -44

What this page does not claim

The theorem does not derive the observed magnitude of the baryon asymmetry. The structural formula is not a prediction and is not used as one. The φ⁻⁴⁴ rung assignment is not a derived mechanism, only an empirical match. The framework does not claim to have computed the washout factor or the detailed phase transition dynamics.

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/BaryonAsymmetryDerivation.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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