Cosmology Baryon Higher Order Baryon Correction Cert

The first-order correction

In cosmology, the baryon asymmetry η_B is the ratio of ordinary matter (protons and neutrons) to photons in the universe, a number around one part per billion. The Recognition Science framework's leading prediction for this ratio is φ⁻⁴⁴ ≈ 6.376 × 10⁻¹⁰, where φ is the golden ratio. The Planck 2018 CMB measurement gives (6.104 ± 0.058) × 10⁻¹⁰. The framework's prediction is therefore about 4.5% higher than the measured central value, a gap that cannot be tuned away because the framework has no free parameters.

The declaration baryon_correction_cert (a certificate is a machine-checked bundle of formal theorems) records a first-order attempt to close that gap. The idea is that during the electroweak phase transition, sphaleron processes run for about N_sph ≈ φ⁸ ≈ 47 cycles of the framework's eight-tick cycle. Each cycle applies a defect-reduction step, multiplying the prediction by a washout factor (1 − φ⁻⁸) ≈ 0.9853. The corrected prediction is η_B⁽¹⁾ = φ⁻⁴⁴ × (1 − φ⁻⁸) ≈ 6.28 × 10⁻¹⁰, which roughly halves the gap to the CMB value.

The certificate proves, in the framework's machine-checked library, a set of formal facts: the correction factor is positive and less than one, so the corrected value is smaller than the leading term; the corrected value is itself positive; and the correction term decomposes as φ⁻⁵². These are pure arithmetic identities about the defined quantities, not empirical claims. The certificate's theorems establish that the corrected number is well-defined and lies strictly between zero and the leading prediction.

What the certificate does not claim is equally explicit. The washout mechanism itself is a hypothesis, not a proved consequence of the framework. The certificate's docstring names the falsifier: if precision measurements place η_B outside [5.5, 7.5] × 10⁻¹⁰ at greater than 5σ, the leading-order prediction is falsified; if outside [6.0, 6.5] × 10⁻¹⁰ at greater than 3σ, the corrected prediction is falsified. The certificate records the arithmetic of the correction, not its physical necessity.

In plain terms, the declaration says: here is a formally checked computation of what the framework's next-order correction would be, and here is the precise condition under which that correction would be ruled out by data. The arithmetic is proved; the physics is a testable proposal.

/-- The corrected prediction is strictly between 0 and the leading term. -/
theorem corrected_in_range :
    0 < eta_B_corrected ∧ eta_B_corrected < eta_B_phi_scale :=
  ⟨eta_B_corrected_pos, corrected_lt_leading⟩

/-- The correction factor lies strictly in (0, 1). -/
theorem correction_factor_in_interval :
    0 < correction_factor ∧ correction_factor < 1 :=
  ⟨correction_factor_pos, correction_factor_lt_one⟩

/-- The correction term = φ⁻⁴⁴ × φ⁻⁸ = φ⁻⁵² (rung -52). -/
theorem correction_term_rung :
    eta_B_phi_scale * delta_washout = phi ^ (-52 : ℤ) := by
  unfold eta_B_phi_scale delta_washout
  rw [← zpow_add₀ phi_ne_zero]
  norm_num

/-- **THE BARYON CORRECTION THEOREM** (HYPOTHESIS):
    The first-order 8-tick correction reduces η_B from φ⁻⁴⁴ to
    φ⁻⁴⁴ × (1 − φ⁻⁸), roughly halving the 4.5% gap to the CMB value.
    This is a HYPOTHESIS about the sphaleron washout mechanism. -/
theorem baryon_correction_cert : BaryonCorrectionCert where
  leading := rfl
  correction := rfl
  corrected_def := rfl
  correction_pos := correction_factor_pos
  correction_lt_one := correction_factor_lt_one
  corrected_smaller := corrected_lt_leading
  corrected_pos := eta_B_corrected_pos
  correction_rung := correction_term_rung

What this page does not claim

The certificate does not prove that the washout mechanism is physically real. The certificate does not establish that the corrected prediction matches the CMB measurement. The certificate does not derive the sphaleron active period from the framework's axioms.

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/BaryonHigherOrder.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.

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