Masses Mass Genesis T10 Universal Coupling Weak Charge Eq Zero Iff Decoupled

The decoupling test

In particle physics, the weak charge of a fermion, a number that sets how strongly it feels the weak nuclear force, usually comes with a separate list of which particles couple to which bosons. The declaration weakCharge_eq_zero_iff_decoupled in the Recognition Science framework's machine-checked library of formal theorems collapses that dictionary into one rule: a fermion has weak charge zero if and only if it does not couple to the W boson. The proof is a finite case check over the twelve fermions, so the statement is not a physical assumption but a derived identity inside the framework's own classification.

The rule is built from the framework's ledger, a discrete record of which species share a generation and a colour status. Two fermions are weak-doublet partners exactly when they share both and are distinct. The weak charge is then read off the partner relation: 1 if the partner has a smaller tilde-charge, -1 if larger, 0 if there is no partner at all. Since every fermion in the framework's list has exactly one partner, the charge is never ambiguous. The theorem weakCharge_eq_zero_iff_decoupled states that this charge is zero precisely when the derived W-coupling flag is false, which in turn happens exactly when no partner exists. In plain terms, a zero charge means the particle stands alone, with no doublet partner and no W coupling.

What the declaration does not claim is just as important. It does not assign numerical values to any measured weak charges, and it does not derive the existence of the W boson from first principles. The W boson is a declared object in the framework; the theorem only matches the derived coupling flag to the declared one. It also does not say anything about the Z boson, which couples to every fermion in the framework, including those with zero weak charge. The statement is a consistency check inside the framework's own definitions, not an independent physical law.

The payoff is a clean test for decoupling. If a fermion ever shows a zero weak charge, the framework forces it to have no W coupling, and conversely. That equivalence is what lets the framework treat coupling as a derived property instead of a separate input. It is a small but load-bearing step in the mass-genesis chain, where the pattern of charges and partners is supposed to fall out of the ledger's classification rather than be pasted in from experiment.

/-- **THEOREM (the Z falsifier).** A species's weak charge vanishes
exactly when it decouples from the W: the two carriers stand and fall
together on the doublet structure. -/
theorem weakCharge_eq_zero_iff_decoupled (f : Fermion) :
    weakCharge f = 0 ↔ derivedWCouples f = false := by
  cases f <;> decide

/-- **The weak-doublet relation, from the ledger's own classification.**
Two species are weak-doublet partners iff they share a generation and a
colour status and are distinct. Both clauses are banked predicates
(`genOf`, `couplesToColor`); nothing is imported from outside the
channel/predicate/generation structure. -/
def weakDoubletPartner (f g : Fermion) : Bool :=
  decide (genOf f = genOf g) && decide (couplesToColor f = couplesToColor g) &&
    decide (f ≠ g)

/-- **THEOREM (the partner is unique).** Each species has exactly one
weak-doublet partner: the matching is a perfect matching, six disjoint
pairs. -/
theorem doublet_partner_unique (f : Fermion) :
    (allFermions.filter fun g => weakDoubletPartner f g).length = 1 := by
  cases f <;> decide

What this page does not claim

The declaration does not assign numerical values to measured weak charges. The declaration does not derive the W boson's existence from first principles. The declaration says nothing about Z boson coupling, which is universal in the framework.

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/Masses/MassGenesis/T10UniversalCoupling.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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