Cosmology Baryon Higher Order Correction Is 8tick Rung

The 8-tick correction

The baryon asymmetry is the cosmic imbalance between matter and antimatter. The Recognition Science framework predicts a leading value for this asymmetry, written as the golden ratio φ raised to the power −44, or about 6.376 × 10⁻¹⁰. The Planck 2018 satellite measurement gives (6.104 ± 0.058) × 10⁻¹⁰. The prediction overshoots by roughly 4.5%, a gap that cannot be tuned away because the framework has no free parameters.

The framework's response is to compute a first-order correction. The theorem correction_is_8tick_rung establishes a precise algebraic identity: the washout factor, the amount by which the prediction is reduced, equals φ⁻⁸. This is one rung on the golden-ratio ladder, the same scale that organizes the framework's eight-tick cycle. The corrected prediction becomes φ⁻⁴⁴ × (1 − φ⁻⁸), which evaluates to about 6.28 × 10⁻¹⁰, roughly halving the gap to the measured value.

The theorem itself is a formal identity in the framework's machine-checked library of formal theorems. It proves that the correction factor is positive, less than one, and that the corrected value is smaller than the leading term. It also proves the corrected value remains positive. These are algebraic facts about the definitions, not measurements of the cosmos.

The physical story behind the correction is a hypothesis. The framework proposes that during the electroweak phase transition, sphaleron processes run for about φ⁸ cycles, and each cycle reduces a defect by the factor φ⁻⁸. This narrative is tagged HYPOTHESIS with explicit falsifiers: if precision measurements place the baryon asymmetry outside [5.5, 7.5] × 10⁻¹⁰ at more than 5σ, the leading prediction falls; if outside [6.0, 6.5] × 10⁻¹⁰ at more than 3σ, the corrected prediction falls.

The practical consequence is a testable target. The framework now commits to a narrower window for the baryon asymmetry than its leading-order term alone. The algebraic correction is proved; the physical mechanism that motivates it is not. What the reader can take away is a clear distinction: a formal identity about powers of the golden ratio, and a physical conjecture about the early universe that future measurements will judge.

/-- The correction factor involves φ⁻⁸ = the 8-tick rung.
    This is the SAME rung-8 that appears in the 8-tick period (T7). -/
theorem correction_is_8tick_rung :
    delta_washout = phi ^ (-8 : ℤ) := rfl

/-- The first-order corrected baryon asymmetry. -/
noncomputable def eta_B_corrected : ℝ :=
  eta_B_phi_scale * correction_factor

/-- 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 correction theorem does not prove that the physical washout mechanism occurs. The corrected prediction is not a measurement and carries no experimental uncertainty. The framework does not claim to derive the fine-structure constant or any other coupling from this correction.

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)

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