Cosmology Baryogenesis Staging Sphaleron Equilibrium Zero Of Zero Bminus L

The zero-protection obstruction

Baryogenesis is the problem of explaining why the observable universe contains far more matter than antimatter. The Standard Model of particle physics offers a candidate engine: electroweak sphalerons, processes that can convert quark and lepton numbers into each other. But the engine has a built-in limitation. Sphaleron interactions conserve the quantity B-L, the difference between baryon number B and lepton number L. If the early universe began with zero B-L, and if sphalerons reached equilibrium, then the surviving baryon number must be zero. The framework's machine-checked library of formal theorems records this as the zero-protection obstruction: a sourced B-L charge of zero, combined with sphaleron equilibrium, forces the final baryon number to vanish.

The library states this result as a theorem named nonzero_relic_at_zero_BmL_forces_offEquilibrium. In plain language: if the final baryon number is not zero, then sphalerons cannot have been in equilibrium. The theorem is a formal statement in the framework's library, proved from the definitions of sphaleron equilibrium and the final baryon number, with no additional axioms. It is a conditional statement, not an assertion about the actual universe. It does not say that sphalerons were out of equilibrium, nor that baryogenesis happened. It establishes a logical link: a nonzero relic baryon asymmetry rules out sphaleron equilibrium given zero initial B-L.

The same library contains a companion theorem, nonzero_relic_forces_BminusL, which states that if the reprocessing factor times B-L is nonzero, then B-L itself must be nonzero. This is a simple algebraic consequence: a product of two numbers is zero only if at least one factor is zero. Together, these theorems form a staging scaffold for a baryogenesis derivation. They do not by themselves produce a baryon asymmetry; they only constrain what is possible. The framework's own documentation calls this file a staging module that prevents the baryogenesis lane from faking the missing mechanism.

What the declaration does not claim is as important as what it proves. It does not claim that the universe's B-L is zero, nor that sphalerons were in equilibrium. It does not claim that baryogenesis is impossible. It does not claim that any particular out-of-equilibrium mechanism, such as CP violation or a first-order phase transition, actually operated. The theorem is a logical guardrail: it tells a would-be derivation that if it wants a nonzero baryon number, it must either start with nonzero B-L or break sphaleron equilibrium. The framework's contribution is to make that guardrail explicit and machine-checked, so that any future derivation must respect it.

/-- 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 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])

What this page does not claim

The universe's initial B-L charge is zero. Sphalerons were in equilibrium in the early universe. Baryogenesis is impossible; the theorem only constrains one pathway.

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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Derived articles

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