Cosmology Baryon Higher Order Corrected Decomposition

The corrected decomposition

The baryon asymmetry of the universe is the observed excess of matter over antimatter, usually written as the ratio η_B of baryon number density to entropy density. The Recognition Science framework derives a leading prediction for this ratio: η_B = φ⁻⁴⁴, where φ is the golden ratio. That number evaluates to about 6.376 × 10⁻¹⁰, which is about 4.5% higher than the Planck 2018 CMB measurement of (6.104 ± 0.058) × 10⁻¹⁰. Because the framework has no free parameters, this gap cannot be tuned away; the only way to address it is to compute a correction from the framework's own structure.

The correction comes from a washout mechanism. During the electroweak phase transition, sphaleron processes are active for about φ⁸ ≈ 47 cycles of the framework's eight-tick recognition cycle. Each cycle, the ledger, a discrete record of recognition events, reduces the defect by a factor δ = φ⁻⁸. Over all cycles, the net washout factor is (1 − δ)^N_sph, which the framework approximates as 1 − φ⁻⁸ ≈ 0.9853. The corrected prediction is therefore η_B^(1) = φ⁻⁴⁴ × (1 − φ⁻⁸), which evaluates to about 6.28 × 10⁻¹⁰, roughly halving the gap to the measured value.

The declaration corrected_decomposition is a theorem in the framework's machine-checked library of formal theorems. It states that the corrected value equals the leading value minus the leading value times the washout factor: η_B_corrected = η_B_phi_scale − η_B_phi_scale × δ_washout. The proof is a direct algebraic expansion, and the library also proves that the correction factor is positive and less than one, that the corrected value is positive, and that it is strictly smaller than the leading value. These are all formal facts about the defined quantities, not empirical claims.

What the declaration does not claim is equally important. The washout mechanism itself is a hypothesis, not a theorem. The framework's own epistemic status labels it as HYPOTHESIS with an explicit falsifier: if precision measurements establish η_B outside [5.5, 7.5] × 10⁻¹⁰ at more than 5σ, the leading prediction is falsified; if η_B is outside [6.0, 6.5] × 10⁻¹⁰ at more than 3σ, the corrected prediction is falsified. The decomposition theorem does not prove that the washout mechanism is physically real, nor that the corrected value matches any measurement. It only establishes the algebraic relationship between the defined quantities, leaving the physical interpretation and empirical status to the hypothesis and its falsifier.

/-- The corrected η_B = φ⁻⁴⁴ × (1 − φ⁻⁸) = φ⁻⁴⁴ − φ⁻⁵². -/
theorem corrected_decomposition :
    eta_B_corrected = eta_B_phi_scale - eta_B_phi_scale * delta_washout := by
  unfold eta_B_corrected correction_factor
  ring

/-- 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 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 _

/-- 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 corrected value matches the Planck 2018 measurement within its error bars. The washout mechanism is a proven physical process; it is only a hypothesis. The framework derives the fine-structure constant or any other coupling constant.

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: