Cosmology Baryogenesis Staging Lepton Yukawa Constraint Depends On Lepton Single

The lepton constraint

In particle physics, the baryon asymmetry of the universe is the observed excess of matter over antimatter. The standard model's electroweak sphaleron processes, which can convert between baryons and leptons, conserve the difference B-L between baryon number and lepton number. If a primordial B-L charge is zero and sphalerons reach equilibrium, the surviving baryon number is forced to zero. This is the sphaleron zero-protection obstruction, a barrier any theory of baryogenesis must overcome.

Within the Recognition Science framework, the machine-checked library of formal theorems contains a declaration named leptonYukawaConstraint_depends_on_lepton_singlet. This declaration establishes that a constraint on the lepton Yukawa coupling depends on a single lepton species, the lepton singlet. In plain terms, the framework's staging module for baryogenesis records that the lepton sector's contribution to the asymmetry mechanism is gated by one specific lepton state, not by a collective or averaged property of all leptons. The declaration is part of a curated staging area that holds small, honest theorem targets to prevent the baryogenesis lane from faking the missing mechanism.

The framework's library also proves related structural facts. One theorem shows that a nonzero relic B-L charge forces the system out of sphaleron equilibrium, and another shows that a nonzero reprocessing factor times B-L forces B-L itself to be nonzero. A general version of this obstruction holds for arbitrary generation counts N and Higgs doublet counts nH, as long as the derived factor's numerator and denominator do not vanish. These are formal consequences about the logic of the obstruction, not measurements of the universe's baryon content.

What the declaration does not claim is the full mechanism of baryogenesis. It does not assert that the lepton singlet Yukawa coupling has a measured value, nor that the framework has derived the observed baryon asymmetry from first principles. The staging module explicitly says its purpose is to hold honest theorem targets, not to provide a complete derivation. The sphaleron reprocessing coefficient 28/79 for three generations appears as a definitional choice, and the number 44 appears in several forced identities, but these are framework-internal structures, not empirical predictions with error bars.

The practical consequence is a discipline: the framework can prove constraints about the logical structure of the B-L obstruction while leaving the physical bridge from recognition events to the Standard Model's lepton sector as open work. A reader can rely on the formal statement about the lepton singlet dependence, and should not mistake it for a complete theory of why the universe has more matter than antimatter.

/-- B-L charge of SM leptons: B = 0, L = 1 -/
def leptonBL : ℚ := -1

/-- 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)

/-- **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

The declaration does not provide a complete derivation of the observed baryon asymmetry. The lepton singlet Yukawa coupling has no measured value asserted by this theorem. The framework has not derived the Standard Model's lepton sector from recognition events.

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.

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