Cosmology Baryon Asymmetry Derivation Saturation Exponent

The saturation exponent

The baryon-to-photon ratio η_B measures how many more protons than antiprotons filled the early universe, about 6.1 × 10⁻¹⁰ according to Big Bang Nucleosynthesis and Planck 2018 CMB measurements. In the Recognition Science framework, this number is approached from two separate directions. One is a structural scaffold, the textbook proportionality η_B ∝ J_CP / g_★, which the framework's own library shows is about 500 times too large to match observation; only its positivity, meaning matter dominates over antimatter, is proved as a theorem. The other is an empirical match: the observed value sits near φ⁻⁴⁴, where φ is the golden ratio, and the expression φ⁻⁴⁴·(1−φ⁻⁸)² lands within the Planck band.

The declaration saturation_exponent is the integer 45, defined as the Light Field saturation threshold exponent from consciousness theory, with Θ_crit = φ⁴⁵. Its role is purely arithmetic. The framework defines the baryon asymmetry rung as −44, and a machine-checked theorem proves that −44 + 45 = 1. This exact relation is the entire content of the theorem: it is integer arithmetic on two defined rung assignments, each of which is itself a hypothesis-grade empirical match, not a derived physical quantity. The theorem does not prove that matter content links to the Light Field, nor does it force the magnitude of the asymmetry.

What the framework does prove is the sign of the baryon asymmetry. A chain from the recognition cost function through the golden ratio, three generations, chirality, and the Jarlskog invariant shows that the structural η_B is positive, meaning matter exists. The magnitude, however, is not derived. The structural scaffold is explicitly not the prediction, and the φ⁻⁴⁴ rung match is a separate hypothesis with a named falsifier: a tightened CMB measurement of η_B outside the φ⁻⁴⁴·(1−φ⁻⁸)² band. The saturation exponent's arithmetic relation is a compact way to state that the rung assignments are consistent, not a mechanism.

In plain terms, the saturation exponent establishes one proved fact: the number 45 completes the rung arithmetic so that the baryon asymmetry rung and the saturation threshold sum to 1. It does not establish why that should be so. The integer 45 can be written several ways, such as 44 + 1 or half the fermionic degrees of freedom minus one, but these are re-expressions of the same number, not independent derivations. The honest status is that −44 is a hypothesis-grade rung assignment whose support is the numerical match itself, and the saturation exponent is its arithmetic partner in that assignment.

/-- The Light Field saturation threshold from consciousness theory.
    Θ_crit = φ⁴⁵ (from Consciousness.LightField). -/
def saturation_exponent : ℤ := 45

/-- Rung arithmetic: η_B rung (−44) plus the saturation exponent (45)
    equals 1, i.e. η_B ≈ φ/Θ_crit at the rung level.

    HONEST STATUS: this is exact integer arithmetic on two DEFINED rung
    assignments, each of which is a HYPOTHESIS-grade empirical match. The
    theorem proves the arithmetic relation between the two definitions,
    not a physical mechanism linking matter content to the Light Field.
    Treat the "matter-consciousness link" as an interpretive gloss on the
    arithmetic, falsifiable through either rung. -/
theorem eta_B_times_saturation :
    eta_B_rung + saturation_exponent = 1 := by
  simp [eta_B_rung, saturation_exponent]

/-- The φ-rung exponent for the baryon asymmetry scale.
    φ⁴⁴ ≈ 1.568 × 10⁹, so φ⁻⁴⁴ ≈ 6.376 × 10⁻¹⁰.
    The observed η_B ≈ 6.1 × 10⁻¹⁰ is within ~4.5% (before the
    (1−φ⁻⁸)² prefactor). HYPOTHESIS-grade rung assignment. -/
def eta_B_rung : ℤ := -44

/-- The structural η_B: proportional to J_CP / g_★.
    SCAFFOLD ONLY — numerically ≈ 3×10⁻⁷, about 500× the observed value,
    because the order-one-suppressed washout constant c is not derived.
    Used solely for the sign theorem (η_B > 0) and the smallness bound. -/
noncomputable def eta_B_structural : ℝ := jarlskog_structural / g_star

/-- η_B is positive: matter dominates over antimatter.
    This follows directly from J_CP > 0 and is the genuine derived content
    (the SIGN of the asymmetry, not its magnitude). -/
theorem eta_B_positive : eta_B_structural > 0 := by
  unfold eta_B_structural
  apply div_pos jarlskog_positive
  norm_num [g_star]

What this page does not claim

The saturation exponent does not prove a physical link between matter content and the Light Field. The theorem does not derive the magnitude of the baryon asymmetry from first principles. The integer 45 is not independently derived from a mechanism that forces the rung.

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/BaryonAsymmetryDerivation.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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