Cosmology Baryon Higher Order Correction Factor In Interval

A small correction, precisely bounded

The baryon asymmetry of the universe is the observed excess of matter over antimatter. In the Recognition Science framework, the leading prediction for this asymmetry is the number phi raised to the power -44, about 6.376 times 10 to the -10. This is a prediction with a named falsifier: if precise measurements place the true value outside the range 5.5 to 7.5 times 10 to the -10 at more than 5 sigma, the leading prediction is falsified.

The declaration correction_factor_in_interval establishes a precise bound on a proposed first-order correction. The correction factor is defined as 1 minus phi to the power -8, where phi is the golden ratio. The theorem proves that this factor is strictly between 0 and 1. In plain terms, the correction is positive and it reduces the leading prediction, never increasing it or flipping its sign. The corrected prediction is phi to the -44 multiplied by this factor, which evaluates to about 6.28 times 10 to the -10.

The framework models the correction as a washout effect during the electroweak phase transition. The physical picture is that each of about 47 cycles of an eight-tick recognition process reduces a defect by a factor of phi to the -8. The net washout factor is 1 minus phi to the -8, about 0.9853. This is a HYPOTHESIS, not a theorem. The mechanism is not derived from first principles; it is a proposed model. The theorem only certifies the arithmetic: given the definition of the correction factor, it lies in the interval (0, 1).

What the declaration does not claim is just as important. It does not prove that the corrected prediction matches the measured Planck 2018 value of (6.104 ± 0.058) × 10⁻¹⁰. The correction roughly halves the gap between the leading prediction and the measurement, but it does not close it. The framework's own epistemic status labels the corrected prediction as HYPOTHESIS with a falsifier: if the true value is outside 6.0 to 6.5 times 10 to the -10 at more than 3 sigma, the corrected prediction is falsified. The theorem correction_factor_in_interval is a small, exact, machine-checked piece of arithmetic, not a physical victory.

The practical consequence is a clean separation of concerns. A reader can trust the interval (0, 1) as a proved fact about the defined quantity. The physical relevance of that quantity remains an open, testable hypothesis. This is how the framework earns trust: exact provenance for the formal claim, and an explicit list of what remains unproved.

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

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

/-- 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 declaration does not prove that the corrected prediction matches the measured Planck 2018 value. The declaration does not prove that the 8-tick washout mechanism is physically real. The declaration does not prove that the leading prediction phi^-44 is correct.

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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