Cosmology Baryon Higher Order Corrected Lt Leading

The corrected baryon asymmetry

The baryon asymmetry of the universe is the tiny surplus of matter over antimatter left over from the early cosmos. In the Recognition Science framework, the leading prediction for this asymmetry is η_B = φ⁻⁴⁴, a number built purely from the golden ratio. That prediction, at about 6.376 × 10⁻¹⁰, sits roughly 4.5 percent above the value measured by the Planck 2018 CMB observation, (6.104 ± 0.058) × 10⁻¹⁰.

The declaration corrected_lt_leading is a theorem in the framework's machine-checked library of formal theorems. It proves that the corrected value, obtained by multiplying the leading term by a correction factor of 1 − φ⁻⁸, is strictly smaller than the leading term. In plain terms, the theorem establishes that the correction moves the prediction downward, from about 6.376 × 10⁻¹⁰ to about 6.28 × 10⁻¹⁰, roughly halving the gap to the measured value. The theorem also proves the corrected value remains positive, so the prediction stays physically sensible.

The correction factor itself is not arbitrary. It comes from a proposed mechanism: during the electroweak phase transition, sphaleron processes are active for about φ⁸ cycles of an eight-tick recognition cycle. Each cycle is said to reduce a defect by a factor of φ⁻⁸, so after roughly 47 cycles the net washout is (1 − φ⁻⁸), about 0.9853. This physical story is a hypothesis, not a proved result. The theorem only certifies the arithmetic: if the mechanism were real, this is the number it would produce.

The theorem's reach is deliberately narrow. It does not prove that the washout mechanism actually occurs in nature, nor does it prove that the corrected value matches the Planck measurement. The framework's own certificate labels the mechanism as HYPOTHESIS with an explicit falsifier: if precision measurements establish η_B outside [6.0, 6.5] × 10⁻¹⁰ at more than 3σ, the corrected prediction is falsified. The theorem's contribution is to show that the correction is mathematically consistent, positive, and smaller than the leading term, nothing more.

/-- The corrected prediction is less than the leading term.
    The 8-tick washout reduces η_B. -/
theorem corrected_lt_leading : eta_B_corrected < eta_B_phi_scale := by
  unfold eta_B_corrected
  have h1 : 0 < eta_B_phi_scale := eta_B_leading_pos
  have h2 : correction_factor < 1 := correction_factor_lt_one
  calc eta_B_phi_scale * correction_factor
      < eta_B_phi_scale * 1 := by
        apply mul_lt_mul_of_pos_left h2 h1
    _ = eta_B_phi_scale := mul_one _

/-- The corrected prediction is positive. -/
theorem eta_B_corrected_pos : 0 < eta_B_corrected :=
  mul_pos eta_B_leading_pos correction_factor_pos

/-- The first-order correction factor: 1 − φ⁻⁸. -/
noncomputable def correction_factor : ℝ := 1 - delta_washout

/-- HYPOTHESIS: The 8-tick washout mechanism.
    Physical basis: during the EW phase transition, sphalerons are active
    for N_sph ≈ φ⁸ recognition cycles. Each cycle, R̂ reduces the baryon
    excess by a factor δ = φ⁻⁸.
    Epistemic status: HYPOTHESIS with falsifier
    (η_B outside [6.0, 6.5] × 10⁻¹⁰ at > 3σ would falsify this). -/
structure BaryonCorrectionCert where
  /-- Leading term -/
  leading : eta_B_phi_scale = phi ^ (-44 : ℤ)
  /-- Correction factor -/
  correction : correction_factor = 1 - phi ^ (-8 : ℤ)
  /-- Corrected prediction -/
  corrected_def : eta_B_corrected = eta_B_phi_scale * correction_factor
  /-- Correction is positive -/
  correction_pos : 0 < correction_factor
  /-- Correction is less than 1 -/
  correction_lt_one : correction_factor < 1
  /-- Corrected prediction is smaller -/
  corrected_smaller : eta_B_corrected < eta_B_phi_scale
  /-- Corrected prediction is positive -/
  corrected_pos : 0 < eta_B_corrected
  /-- The correction term is at rung -52 -/
  correction_rung : eta_B_phi_scale * delta_washout = phi ^ (-52 : ℤ)

What this page does not claim

The theorem does not prove that the washout mechanism occurs in nature. The theorem does not prove that the corrected value matches the Planck measurement. The theorem does not establish that the leading-order prediction φ⁻⁴⁴ is itself derived from first principles.

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.

Derived articles

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: