Cosmology Baryogenesis Trajectory From Phi Ladder Eta B

The asymmetry ladder

Baryogenesis is the physics problem of why the universe contains matter but almost no antimatter. The standard measure is the baryon-to-photon ratio, written η_B, which observations put near 6 × 10⁻¹⁰. In the Recognition Science framework, this number is not fitted: it emerges as the 44th step of a ladder whose rungs are set by the golden ratio φ ≈ 1.618.

The ladder works on a temperature axis. The framework defines a sequence of temperatures T_k = T_GUT · φ^(−k), where T_GUT is the grand-unification scale and k counts rungs downward. At each rung, the asymmetry is η_B(T_k) = φ^(k − 44). At the top of the ladder, k = 0, the asymmetry is negligible, just φ^(−44). As the universe cools by one rung, the asymmetry multiplies by exactly φ: η_B(T_{k+1}) / η_B(T_k) = φ. After 44 such steps, at k = 44, the ratio reaches 1, which the framework calls the recognition-complete threshold. The late-time value φ^(−44) ≈ 6 × 10⁻¹⁰ is the one that matches the observed baryon-to-photon ratio.

The framework's machine-checked library of formal theorems proves three structural facts about this ladder. First, the per-rung ratio is exactly φ. Second, the trajectory is strictly positive at every rung. Third, at rung 44 the value is exactly 1. These are proved with no unproved assumptions in the framework's own logic. The framework also names five channels that can violate baryon number: sphaleron, electroweak, QCD, leptogenesis, and neutrino mass, and proves there are exactly five such channels in its model.

In Recognition Science, the ladder is not an independent postulate. It derives from the same forcing chain that produces the golden ratio as the unique self-similar scaling and fixes the number of spatial dimensions. The temperature rungs are the framework's way of connecting that abstract chain to cosmology: the same φ that governs recognition costs also sets the pace at which matter accumulates.

What the declaration does not claim is equally precise. It does not prove that baryogenesis actually happened this way in the real universe; it provides a formal trajectory, not an observation. The match between φ^(−44) and the measured η_B is an empirical check, not a theorem. The framework does not derive the value of T_GUT or the detailed particle physics of the five channels. And the framework does not claim that the ladder is the only possible baryogenesis mechanism; it is one trajectory within the framework's cosmology.

/-- η_B grows by exactly φ per temperature rung. -/
theorem etaB_ratio (k : ℕ) :
    etaB (k + 1) / etaB k = phi := by
  unfold etaB
  have hpos : (0 : ℝ) < phi ^ k := pow_pos phi_pos k
  have hpos44 : (0 : ℝ) < phi ^ 44 := pow_pos phi_pos 44
  have hphi_ne : (phi : ℝ) ≠ 0 := phi_pos.ne'
  field_simp
  rw [pow_succ]
  ring

/-- η_B(T_44) = 1, the recognition-complete threshold reached at gap-45. -/
theorem etaB_at_gap45 : etaB 44 = 1 := by
  unfold etaB
  have h : (0 : ℝ) < phi ^ 44 := pow_pos phi_pos 44
  exact div_self h.ne'

/-- η_B is strictly positive on the whole trajectory. -/
theorem etaB_pos (k : ℕ) : 0 < etaB k := by
  unfold etaB
  exact div_pos (pow_pos phi_pos k) (pow_pos phi_pos 44)

theorem bViolationChannel_count : Fintype.card BViolationChannel = 5 := by decide

What this page does not claim

The declaration does not prove that baryogenesis actually occurred this way in the real universe. The framework does not derive the value of T_GUT or the detailed particle physics of the five channels. The framework does not claim this ladder is the only possible baryogenesis mechanism.

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

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