Gravity Rsbaryogenesis

The asymmetry from gravity

Baryogenesis is the process that produced the observed excess of matter over antimatter in the early universe. In standard cosmology, the asymmetry is quantified by η_B, the ratio of baryons to photons, measured at about 6.1 × 10⁻¹⁰. The standard model of particle physics cannot explain this number; it requires new physics beyond what experiments have found.

Recognition Science offers a candidate mechanism. The framework's central claim is that a discrete record of events, called a ledger, forces a specific cost function J(x) = (x + 1/x)/2 - 1. From that cost function, the framework derives the golden ratio φ ≈ 1.618 as the unique self-similar scaling. The baryogenesis module uses only φ to set the couplings that break the symmetry between matter and antimatter.

In Recognition Science, the asymmetry comes from gravity. The module defines two coupling constants from φ alone: λ_CP = φ⁻⁷ and κ_CP = φ⁻⁹. These are CP-odd couplings, meaning they treat matter and antimatter differently. The module proves these couplings are positive and less than one, and that κ_CP is weaker than λ_CP. The recognition mass scale is set to M_rec = 2√(2π) M_Pl, where M_Pl is the Planck mass.

The model then predicts η_B ≈ 5.1 × 10⁻¹⁰, which is within 20 percent of the measured value. The library proves this agreement in a theorem called eta_B_within_20_percent. It also predicts the spectral index of primordial fluctuations, n_s, to lie between 0.96 and 0.97 at 55 e-folds of inflation, matching observations. The module packages these results into a single certificate structure, BaryogenesisCert, which asserts all five conditions: the couplings come from φ, gravity is stronger than the other coupling, the asymmetry is within 20 percent, and the spectral index is in the observed range.

What this establishes is a parameter-free derivation. The asymmetry is not fitted; it is computed from φ, which itself comes from the framework's forcing chain. The agreement with observation is an empirical check, not a theorem. The module shows that a single number, the golden ratio, can account for the matter-antimatter asymmetry and the tilt of the cosmic microwave background spectrum. This is a concrete, testable consequence of the framework's core structure.

/-- The CP-odd gravitational coupling: λ_CP = φ⁻⁷.
    This determines the strength of the χRR̃ term in the
    CP-violating Lagrangian. -/
noncomputable def lambda_CP : ℝ := phi ^ (-(7 : ℝ))

/-- The CP-odd electromagnetic coupling: κ_CP = φ⁻⁹.
    This determines the strength of the χFF̃ term. -/
noncomputable def kappa_CP : ℝ := phi ^ (-(9 : ℝ))

theorem lambda_CP_bounds : 0 < lambda_CP ∧ lambda_CP < 1 :=
  ⟨lambda_CP_pos, lambda_CP_lt_one⟩

theorem kappa_CP_bounds : 0 < kappa_CP ∧ kappa_CP < 1 :=
  ⟨kappa_CP_pos, kappa_CP_lt_one⟩

/-- phi^(-7) > phi^(-9) because -7 > -9 and phi > 1. -/
theorem lambda_gt_kappa : kappa_CP < lambda_CP := by
  unfold lambda_CP kappa_CP
  exact Real.rpow_lt_rpow_of_exponent_lt one_lt_phi (by norm_num : (-(9 : ℝ)) < -(7 : ℝ))

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

theorem n_s_at_55 : 0.96 < n_s_prediction 55 ∧ n_s_prediction 55 < 0.97 := by
  unfold n_s_prediction; constructor <;> norm_num

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 framework does not prove that baryogenesis actually occurred; it derives a value for the asymmetry under its model assumptions. The agreement with the observed η_B is an empirical check, not a theorem; the measured value is an input, not an output. The module does not derive the fine-structure constant or any other Standard Model coupling.

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)

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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