Gravity Rsbaryogenesis Eta B Within 20 Percent

The asymmetry bound

The universe holds more matter than antimatter. In the standard cosmological picture, this asymmetry is quantified by η_B, the ratio of baryons (protons and neutrons) to photons. The Planck satellite and Big Bang nucleosynthesis measurements put the observed value at about 6.1 × 10⁻¹⁰: for every billion photons, roughly one baryon survived annihilation. The Recognition Science framework, which models physical constants as consequences of a discrete recognition ledger, produces a prediction of 5.1 × 10⁻¹⁰ from the golden ratio φ alone, with no free parameters.

The declaration eta_B_within_20_percent is a formal theorem in the framework's machine-checked library of formal theorems. It states that the fractional offset between the prediction and the observed value is less than 0.20, meaning the prediction lies within 20 percent of the measurement. The theorem is proved by direct computation from the definitions of the prediction and the observation, so it is a statement about arithmetic, not a physical derivation. The framework's library also contains a certificate structure, BaryogenesisCert, that bundles this bound together with positivity and ordering conditions on the coupling constants and a spectral index prediction, all derived from φ.

What the theorem does not claim is as important as what it proves. It does not derive the observed value 6.1 × 10⁻¹⁰ from the framework's axioms; it compares a pre-defined prediction to a pre-defined observation and certifies their proximity. The prediction itself, 5.1 × 10⁻¹⁰, is a definition in the library, not a theorem. The framework does not prove that baryogenesis must proceed through its specific CP-odd channel, nor that the recognition mass scale M_rec = 2√(2π) M_Pl is the correct physical scale. These are modeling choices, not forced results.

The practical consequence is modest but real. The theorem establishes that the framework's parameter-free prediction is not wildly wrong: it is within the same order of magnitude as observation, and the fractional offset is under 20 percent. This is a consistency check, not a derivation. A reader should take it as evidence that the framework's machinery can produce numbers in the right ballpark, while remaining clear that the physical mechanism linking the recognition ledger to baryogenesis is a hypothesis, not a proved fact.

theorem eta_B_within_20_percent :
    eta_B_fractional_offset < 0.20 := by
  unfold eta_B_fractional_offset eta_B_prediction eta_B_observed
  norm_num

/-- The CMB/BBN observed value for comparison. -/
def eta_B_observed : ℝ := 6.1e-10

/-- The RS prediction for the baryon-to-photon ratio η_B.
    This is derived from the CP-odd couplings, the inflaton dynamics,
    and the freeze-out temperature, with zero free parameters.

    η_B ≈ 5.1 × 10⁻¹⁰ (16% below CMB/BBN central value ~6 × 10⁻¹⁰). -/
def eta_B_prediction : ℝ := 5.1e-10

structure BaryogenesisCert where
  lambda_from_phi : 0 < lambda_CP ∧ lambda_CP < 1
  kappa_from_phi : 0 < kappa_CP ∧ kappa_CP < 1
  grav_stronger : kappa_CP < lambda_CP
  eta_B_ok : eta_B_fractional_offset < 0.20
  spectral_ok : 0.96 < n_s_prediction 55 ∧ n_s_prediction 55 < 0.97

What this page does not claim

The theorem does not derive the observed asymmetry from first principles. The framework does not prove that its baryogenesis mechanism is the actual physical one. The prediction 5.1 × 10⁻¹⁰ is a definition, not a theorem.

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/Gravity/RSBaryogenesis.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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