Cosmology Baryogenesis Staging Hypercharge Constraint Couples Higgs To Quarks

The hypercharge constraint

In the Standard Model of particle physics, hypercharge is the charge that distinguishes the weak force's partners. The Higgs field carries hypercharge, and its interactions with quarks and leptons are what give those particles mass. A machine-checked library of formal theorems, the framework's library, contains a declaration named hyperchargeConstraint_couples_higgs_to_quarks. In plain terms, this declaration establishes a constraint: the hypercharge values assigned to the Higgs boson and to quarks are not independent, but are locked together by the requirement that the theory's anomalies cancel.

This is a structural result about the Standard Model's charge assignments. The framework's library proves that the hypercharge of the Higgs doublet, the quark doublets, and the lepton doublets must satisfy a specific linear relation for the electroweak theory to be consistent. The declaration is a formal statement in the library's language, and it is proved in the machine-checked collection with no unproven assumptions. The declaration does not, however, claim that this constraint alone produces the observed matter-antimatter asymmetry of the universe.

In the framework's account, baryogenesis, the process that generated the excess of matter over antimatter, is a staged derivation. The library contains a staging module that holds small, honest targets, preventing the derivation from faking the missing mechanism. One invariant in that module is the sphaleron zero-protection obstruction: electroweak sphalerons conserve B-L, so if the sourced B-L charge is zero and sphalerons equilibrate, the surviving baryon number is zero. The hypercharge constraint is one piece of this larger puzzle, not the whole solution.

What the declaration does not claim is that the hypercharge constraint, by itself, explains the observed baryon asymmetry. It does not claim to derive the CP violation needed for baryogenesis, nor does it claim to specify the dynamics of the phase transition that would drive the universe out of equilibrium. The constraint is a necessary condition for the Standard Model's consistency, and it is a proved statement in the library, but the full mechanism of baryogenesis remains open.

/-- 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 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])

/-- **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)

/-- 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 declaration does not claim to derive the observed baryon asymmetry of the universe. It does not claim to specify the CP violation mechanism needed for baryogenesis. It does not claim to describe the dynamics of the electroweak phase transition.

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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