Cosmology Baryon Higher Order Correction Factor Pos

The correction factor

The baryon asymmetry is the observed imbalance between matter and antimatter in the universe. In the Recognition Science framework, the leading prediction for this asymmetry is eta_B_leading = φ⁻⁴⁴, roughly 6.376 × 10⁻¹⁰. The Planck 2018 CMB measurement gives (6.104 ± 0.058) × 10⁻¹⁰, a gap of about 4.5 percent. Since the framework has no free parameters, this gap cannot be tuned away; it can only be addressed by computing a next-order correction.

The correction factor is defined as correction_factor = 1 − φ⁻⁸, where φ is the golden ratio. In plain terms, this is a number slightly less than one: φ⁻⁸ is about 0.0147, so the factor is about 0.9853. The framework's physical picture is that during the electroweak phase transition, sphaleron processes run for approximately φ⁸ cycles of the framework's eight-tick cycle, and each cycle reduces the defect by one rung, giving a net washout factor of (1 − φ⁻⁸). The corrected prediction is then η_B⁽¹⁾ = φ⁻⁴⁴ × (1 − φ⁻⁸), which comes to about 6.28 × 10⁻¹⁰, roughly halving the gap to the CMB value.

What the declaration correction_factor_pos establishes, in the machine-checked library of formal theorems, is a single, narrow fact: the correction factor is positive. It proves 0 < correction_factor. This is not a claim about the physics of the early universe, nor about whether the corrected prediction matches observation. It is a purely arithmetic statement: because φ⁻⁸ is positive and less than one, 1 − φ⁻⁸ is positive. The same library also proves the factor is less than one, that the corrected value is smaller than the leading value, and that the corrected value is positive, but those are separate declarations.

The epistemic status is explicit. The 8-tick washout mechanism is a HYPOTHESIS with a named falsifier: if precision measurements establish η_B outside [5.5, 7.5] × 10⁻¹⁰ at > 5σ, the leading-order prediction is falsified; if outside [6.0, 6.5] × 10⁻¹⁰ at > 3σ, the corrected prediction is falsified. The positivity theorem itself is proved with zero axioms beyond the standard three, but it does not validate the physical mechanism. It only certifies that the arithmetic of the correction factor is sound.

In short, the declaration tells a reader one thing with certainty: the number 1 − φ⁻⁸ is greater than zero. It does not tell the reader that the baryon asymmetry prediction is correct, that the washout mechanism is real, or that the corrected value is the final answer. Those are separate, weaker claims, and the framework's own honesty apparatus keeps them apart.

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

/-- The correction factor is positive (since δ < 1). -/
theorem correction_factor_pos : 0 < correction_factor := by
  unfold correction_factor
  linarith [delta_lt_one]

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

/-- 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

What this page does not claim

The correction factor positivity theorem does not validate the washout mechanism as physical reality. The corrected prediction is not asserted to match the CMB measurement within error bars. The framework does not claim that the baryon asymmetry prediction is complete at this order.

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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