Cosmology Baryon Asymmetry Derivation Baryon Asymmetry Cert

The asymmetry certificate

The baryon asymmetry is the observed fact that the universe contains far more matter than antimatter. The standard measure is the baryon-to-photon ratio η_B, the number of protons and neutrons left over for every photon. Planck 2018 measured η_B = (6.12 ± 0.04) × 10⁻¹⁰, and Big Bang Nucleosynthesis gives (6.1 ± 0.3) × 10⁻¹⁰. Cosmology needs an explanation for why this number is positive, tiny, and not zero.

The Recognition Science framework's ledger, a discrete record of recognition events, supplies a derivation chain that ends in a certificate named BaryonAsymmetryCert. The certificate packages a proof that the asymmetry's sign is positive: matter dominates. The chain runs from the framework's forced cost function through the golden ratio, an eight-tick cycle, three spatial dimensions, the gauge group Q₃, a chiral Gray code, and the CKM matrix, arriving at a positive Jarlskog invariant J_CP, the standard measure of CP violation. Combined with the Sakharov conditions, the three requirements for generating a matter-antimatter asymmetry, this proves η_B > 0. The sign is the derived content; the framework proves that matter exists, not how much.

The certificate also contains a structural scaffold, a textbook proportionality skeleton η_B ∝ J_CP / g_★, where g_★ = 106.75 is the number of relativistic degrees of freedom at the electroweak scale. This scaffold is explicitly not the prediction. With the framework's own J_CP ≈ 3 × 10⁻⁵, it gives η_B ≈ 3 × 10⁻⁷, about 500 times larger than observed. The missing factor is the washout constant c, which requires the detailed dynamics of the electroweak phase transition, a piece of physics that remains OPEN. The certificate proves only that this structural value is positive and less than one.

In Recognition Science, the observed magnitude is a separate object: a hypothesis that η_B ≈ φ⁻⁴⁴·(1−φ⁻⁸)², where φ is the golden ratio. This expression lands within the Planck band, but it is tagged HYPOTHESIS, not theorem. The exponent −44 is an empirical rung assignment, supported by the numerical match itself. The certificate includes the exact integer arithmetic that η_B's rung (−44) plus a saturation exponent (45) equals 1, but this proves only a relation between two defined rung assignments, not a physical mechanism. The falsifier is named: a tightened CMB measurement of η_B outside the φ⁻⁴⁴·(1−φ⁻⁸)² band would refute the hypothesis.

What the certificate changes is the epistemic status of the sign. Before it, matter dominance was an observed fact awaiting a derivation. After it, within the framework, the sign is a proved consequence of the same forcing that yields the golden ratio and three dimensions. The magnitude remains a separate, falsifiable guess. The reader can now see the exact boundary: the framework derives that matter exists, and it does not derive how much.

/-- 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 certificate does not predict the observed magnitude of the baryon asymmetry. The rung arithmetic does not establish a physical link between matter content and the Light Field. The framework does not derive the fine-structure constant or the exact value of the CP-violating phase.

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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