Cosmology Baryogenesis Staging Sphaleron Equilibrium B Bplus L Shift Invariant

The sphaleron wall

In the Standard Model of particle physics, sphalerons are field configurations that can convert quarks into leptons. They are central to baryogenesis, the effort to explain why the universe has more matter than antimatter. A key fact about these processes is that they conserve the difference between baryon number (B) and lepton number (L). The combination B minus L, often written B-L, is untouched by sphaleron interactions. This conservation law is the classical starting point for the Recognition Science staging module.

The framework's machine-checked library of formal theorems contains a declaration named sphaleronEquilibriumB_BplusL_shift_invariant. In plain language, it establishes a barrier: if the universe begins with zero net B-L charge, and sphaleron interactions reach equilibrium, then the final baryon number must be zero. This is the sphaleron zero-protection obstruction. The theorem makes precise that a non-zero relic B-L charge is a necessary condition for a non-zero final baryon asymmetry. Without a pre-existing B-L excess, sphaleron reprocessing washes out any baryon number that might have been generated.

The library also records the Standard Model's reprocessing coefficient for three generations: after electroweak sphaleron equilibration, the final baryon number B equals (28/79) times the initial (B-L). This factor is a derived consequence of the particle content. The staging module uses these results to structure the baryogenesis derivation, holding honest theorem targets that prevent the lane from faking the missing mechanism. The theorems force conditions like a non-zero B-L and a departure from sphaleron equilibrium when a non-zero final baryon number is asserted.

In Recognition Science, this sphaleron wall is not a claim about the framework's own cost function or forcing chain. It is a staging result that connects the framework's ledger-based picture to a concrete Standard Model obstruction. The declaration does not prove that the required B-L asymmetry was actually created in the early universe. It does not derive the observed baryon asymmetry from first principles. It only establishes the logical gate: zero sourced B-L plus sphaleron equilibrium forces zero surviving baryon number. The mechanism that creates the initial B-L asymmetry remains an open target within the framework.

/-- 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 etaB_chain_decomposition
    (etaBFromYield : ℝ → ℝ → ℝ)
    (BfinalFromRelicBL : ℝ → ℝ)
    (sphaleronReprocessingFactor : ℝ)
    (R relic : ℝ)
    (h_Bfinal : BfinalFromRelicBL relic = sphaleronReprocessingFactor * relic)
    (h_etaB : ∀ (R' YB : ℝ), etaBFromYield R' YB = R' * YB) :
    etaBFromYield R (BfinalFromRelicBL relic)
      = (R * sphaleronReprocessingFactor) * relic := by
  rw [h_Bfinal, h_etaB]
  ring

/-- **Content-independent zero-protection falsifier (B0).**  For *any* SM-like
    content `(N, nH)` with nonvanishing anomaly numerator/denominator, an
    observed nonzero baryon relic at sphaleron equilibrium forces a nonzero
    `B − L` source.  This upgrades the banked SM-specific `nonzero_relic_forces_BminusL`
    (factor `28/79`) to the whole family `reprocessingFactorOf N nH`, so the
    wall is not an artifact of the number `28/79`: no choice of generation or
    Higgs count escapes it.  Any baryogenesis claim must therefore source
    `B − L ≠ 0` upstream regardless of the SM content count. -/
theorem nonzero_relic_forces_BminusL_general
    (N nH : ℤ) (BmL : ℚ)
    (hnum : ((8 * N + 4 * nH : ℤ) : ℚ) ≠ 0)
    (hden : ((22 * N + 13 * nH : ℤ) : ℚ) ≠ 0)
    (h : reprocessingFactorOf N nH * BmL ≠ 0) : BmL ≠ 0 := by
  intro hz
  exact h ((obstruction_via_derivedFactor_iff N nH BmL hnum hden).mpr hz)

What this page does not claim

This answer does not claim the framework proves the observed baryon asymmetry was actually produced. This answer does not claim the framework derives the mechanism that creates the initial B-L asymmetry. This answer does not claim the sphaleron equilibrium theorem is a statement about the framework's own cost function.

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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