Cosmology Baryogenesis Staging Nonzero Relic At Zero Bm L Forces Off Equilibrium

The equilibrium obstruction

Baryogenesis is the name for the process that would explain why the observable universe contains matter but almost no antimatter. In the Standard Model of particle physics, one of the necessary conditions for baryogenesis is that the universe must have been out of thermal equilibrium at some point. If all relevant processes were in equilibrium, the reactions would run forward and backward at equal rates, and any net surplus of matter would be washed out. This is the third of the three Sakharov conditions, first stated by Andrei Sakharov in 1967.

The recognition framework's machine-checked library of formal theorems contains a declaration, nonzero_relic_at_zero_BmL_forces_offEquilibrium, that encodes a specific piece of this logic. The theorem takes as assumptions a final baryon relic that is nonzero, and an initial value for the quantity B minus L (baryon number minus lepton number) that is exactly zero. It then concludes that the sphaleron processes, which are the Standard Model's main baryon-number-violating reactions, cannot have been in equilibrium. The proof is a direct contradiction: if the sphalerons were in equilibrium and the initial B minus L were zero, the final baryon number would be forced to zero, which contradicts the assumption that it is nonzero.

The theorem does not claim that the universe actually was out of equilibrium, nor does it claim that baryogenesis happened. It only establishes a logical implication: a nonzero relic with zero initial B minus L is incompatible with sphaleron equilibrium. The result is a formal statement about the structure of the theory, not an empirical measurement. The framework's library also contains a related theorem, nonzero_relic_forces_BminusL, which shows that a nonzero final relic forces the initial B minus L to be nonzero when the reprocessing factor is nonzero. Together these theorems form a staging area: they mark what must be true for a complete baryogenesis mechanism, without pretending the mechanism itself has been found.

The value of this theorem is that it prevents a certain kind of shortcut. In the framework's development, the baryogenesis lane is an open problem, and this declaration is a guardrail that stops a would-be derivation from assuming both a zero initial asymmetry and an equilibrium sphaleron bath while still producing a baryon surplus. The theorem makes that combination impossible, so any future derivation must either supply a nonzero initial B minus L or a departure from equilibrium. That is a precise, checkable constraint on the space of possible baryogenesis mechanisms.

/-- 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)

/-- 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)

/-- 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)

What this page does not claim

The theorem does not claim that the universe was actually out of equilibrium. The theorem does not claim that baryogenesis actually occurred. The theorem does not provide a mechanism for producing the nonzero relic; it only states a necessary condition.

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)

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