Cosmology Baryogenesis Staging

The staging module

Baryogenesis is the process that would explain why the universe contains matter at all: why protons and neutrons outnumber their antimatter counterparts by roughly one part in a billion. The standard cosmological picture requires three conditions, named after the physicist Andrei Sakharov: a violation of baryon number conservation, a departure from thermal equilibrium, and a violation of charge-parity (CP) symmetry. Without all three, any net baryon asymmetry generated early would be washed out before the universe cooled into the stable matter we see.

Recognition Science's staging module, named baryogenesis staging, is a curated collection of small, honest theorem targets inside the framework's machine-checked library of formal theorems. Its purpose is to prevent the baryogenesis lane from faking the missing mechanism. The first invariant it proves is the sphaleron zero-protection obstruction: electroweak sphaleron processes conserve the quantity B-L (baryon number minus lepton number), so if the sourced B-L charge is zero and sphalerons equilibrate, the surviving baryon number is zero. In plain terms, you cannot get matter from nothing through these processes alone; a nonzero B-L seed is required.

The module also establishes the Standard Model's sphaleron reprocessing coefficient for three generations: after electroweak sphaleron equilibration, the final baryon number B equals (28/79) times (B-L). This is a classical result, not an RS invention. What the framework adds is a forced structure around it. The module proves that a nonzero relic B-L forces the system out of sphaleron equilibrium, and that a zero B-L with a nonzero final baryon number is impossible. It derives that the transient channel count is forced to 44, and that the forced relic composition is the inverse of the golden ratio raised to the 44th power, written as φ⁻⁴⁴.

The number 44 appears repeatedly: the blind channel count (45 Weyl species minus one sphaleron zero mode) is forced to 44, and the CP-violating phase epsilon is normalized to ±1. The module also proves that a double-entry posting matrix with unit determinant forces its determinant to be -1, with the shear and ledger swap structure that implies. These are not empirical measurements; they are theorems about the framework's own ledger-based model of recognition events, where a ledger is a discrete record of events with forced costs.

Within the framework, this staging module is a checkpoint. It does not claim to have derived the observed baryon asymmetry from first principles. It proves structural constraints that any successful derivation must satisfy, and it forces the missing mechanism to be explicit rather than assumed. The payoff for the reader is a clear picture of what a complete derivation would need: a nonzero B-L source, a departure from equilibrium, and a CP violation, all consistent with the framework's forced constants.

/-- FORCING direction (physical payload): for SM content (N=3, n_H=1),
    a nonzero baryon relic forces a nonzero B−L. This is the contrapositive
    that sends the loop out of sphaleron internals and into the B2
    out-of-orbit CP-odd source. -/
theorem nonzero_relic_forces_BminusL (BmL : ℚ)
    (h : reprocessingFactorOf 3 1 * BmL ≠ 0) : BmL ≠ 0 := by
  intro hz
  exact h (by rw [hz, mul_zero])

/-- FORCING FORM (the operational obstruction): any model exhibiting a nonzero
    baryon relic while asserting `B−L = 0` is *forced* to have sphalerons fall
    out of equilibrium.  This is the constraint every B+L-freeze-out claim must
    discharge; it cannot keep `H < Γ` across the window and still beat the wall. -/
theorem nonzero_relic_at_zero_BmL_forces_offEquilibrium
    (Γsph H : ℝ → ℝ) (t₀ tf : ℝ)
    (Bprimordial BmL : ℝ)
    (hBmL : BmL = 0)
    (hB : BfinalGated (SphaleronInEquilibrium Γsph H t₀ tf) Bprimordial BmL ≠ 0) :
    ¬ SphaleronInEquilibrium Γsph H t₀ tf := by
  intro hEq
  exact hB (physical_wall Γsph H t₀ tf Bprimordial BmL hEq hBmL)

theorem blind_channel_count_forces_44 :
    smWeylSpeciesCard - sphaleronZeroModeCard = 45 - 1 := by
  rw [smWeylSpeciesCard_eq_45, sphaleronZeroModeCard_eq_one]

theorem D_forced_eq_relic_composition
    (channelIndependence : ℕ → Bool)
    (survivalAmplitude : ℕ → ℝ)
    (transientChannelCount : ℕ)
    (hSurv : ∀ i, survivalAmplitude i = 1 / phi)
    (hCount : transientChannelCount = 44)
    (hInd : channelIndependence transientChannelCount = true) :
    D survivalAmplitude transientChannelCount
      (channelIndependence transientChannelCount) = D_forced :=
  relic_product_eq_phi_neg_44 channelIndependence survivalAmplitude
    transientChannelCount hSurv hCount hInd

What this page does not claim

This module does not derive the observed baryon asymmetry of the universe from first principles. The number 44 is a theorem about the framework's ledger model, not a measurement of any physical quantity. The sphaleron reprocessing coefficient is a Standard Model result, not an RS derivation.

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/BaryogenesisStaging.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.

Derived articles

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