Gravity Rsbaryogenesis Kappa Cp Bounds

The CP-odd coupling bounds

In particle physics, a coupling constant is a number that sets the strength of an interaction. The symbol kappa_CP names one such constant in the Recognition Science framework, and it is the strength of a particular kind of interaction that treats matter and antimatter differently. The framework's library of machine-checked formal theorems proves that this constant is always greater than zero and always less than one. That is the entire content of the declaration kappa_CP_bounds: it establishes the two inequalities 0 < kappa_CP < 1.

The value itself comes from the golden ratio phi, the number approximately equal to 1.618 that solves the equation r² = r + 1. The framework defines kappa_CP as phi raised to the power -9, a number near 0.013. The proof that this value lies between zero and one is not a numerical accident but a formal theorem: because phi is greater than one, any negative power of phi is less than one, and because phi is positive, every power of phi is positive. The same reasoning applies to a second coupling, lambda_CP, defined as phi to the -7, and the framework also proves that kappa_CP is smaller than lambda_CP.

These bounds matter because the framework uses them in a proposed explanation for baryogenesis, the process that produced the observed excess of matter over antimatter in the early universe. The framework's model combines nine ledger parities, where a ledger is a discrete record of recognition events, to select a unique CP-odd channel, meaning a channel that behaves differently under charge conjugation and parity reversal. From that channel it derives a prediction for the baryon asymmetry, the ratio of matter to photons, of about 5.1 × 10⁻¹⁰, compared with the observed value of about 6.1 × 10⁻¹⁰. The framework's library proves that this prediction is within 20 percent of the observed value, and that the predicted spectral index of primordial fluctuations, a measure of how density variations vary with scale, falls between 0.96 and 0.97 for 55 to 60 e-folds of inflation.

What the declaration does not claim is just as precise. It does not prove that the golden ratio is the correct source of the coupling, nor that this particular baryogenesis mechanism is the one nature uses. The bounds are a formal statement about a defined quantity, not an empirical measurement. The comparison to the observed baryon asymmetry is a separate theorem that checks a numerical offset, and the mechanism itself remains a model, a definitional choice within the framework, not a forced consequence of the axioms. The declaration also does not claim that the framework derives the fine-structure constant or any other standard-model coupling from first principles.

theorem kappa_CP_bounds : 0 < kappa_CP ∧ kappa_CP < 1 :=
  ⟨kappa_CP_pos, kappa_CP_lt_one⟩

/-- The CP-odd electromagnetic coupling: κ_CP = φ⁻⁹.
    This determines the strength of the χFF̃ term. -/
noncomputable def kappa_CP : ℝ := phi ^ (-(9 : ℝ))

/-- phi^(-7) > phi^(-9) because -7 > -9 and phi > 1. -/
theorem lambda_gt_kappa : kappa_CP < lambda_CP := by
  unfold lambda_CP kappa_CP
  exact Real.rpow_lt_rpow_of_exponent_lt one_lt_phi (by norm_num : (-(9 : ℝ)) < -(7 : ℝ))

theorem eta_B_within_20_percent :
    eta_B_fractional_offset < 0.20 := by
  unfold eta_B_fractional_offset eta_B_prediction eta_B_observed
  norm_num

What this page does not claim

No claim that the golden ratio is the empirically correct source of the CP-odd coupling. No claim that this baryogenesis mechanism is the one nature uses. No claim that the framework derives the fine-structure constant 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/Gravity/RSBaryogenesis.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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