Standard-Model Anomaly Gate for Any Furey–Jordan Synthesis
Date: 16 October 2026 Branch: B11 Standard-Model representation synthesis Claim status: one-generation baseline verified; candidate combined modules must be checked against it
1. Left-handed convention
Use only left-handed Weyl fields:
Their hypercharges are
This convention prevents sign errors caused by mixing left- and right-handed anomaly formulas.
2. Local anomaly cancellation
For one generation:
and
3. Global (SU(2)) anomaly
Counting color copies, one generation has
left-handed weak doublets. The number is even, so the Witten \(SU(2)\) anomaly cancels.
4. Why this is a hard gate
A Furey representation match is not enough. After every projector, conjugation rule, selector and boundary condition, the surviving chiral module must reproduce these cancellations.
The gate must be rerun if the combined theory adds:
- mirror fermions;
- selector-charged fermions;
- extra \(U(1)\) factors;
- boundary-localized chiral modes;
- incomplete top-quark representations;
- family-nonuniversal charges.
Anomaly inflow can cancel a boundary anomaly, but then the bulk topological term and its quantized coefficient must be explicit.
5. Interaction with the Furey encoding
The current eight-state \(Cl(6)\) table encodes color and electric-charge-like sectors, not the full pre-Higgs chiral module. It cannot yet be submitted to this anomaly gate.
The staged encoding must add weak chirality and hypercharge before any anomaly-cancellation claim is admissible.
6. Stop-everything and falsification tests
Positive: the complete Furey-selected chiral module cancels every local and global anomaly without manually adding compensating mirrors, and any boundary anomaly has forced inflow.
Negative: a required multiplet is missing, the top sector remains incomplete, or cancellation needs arbitrary spectator fermions.
7. Certificate
0 uses integer hypercharge units to verify all displayed anomaly numerators and the even weak-doublet count.