Proton safety, Pati–Salam and triality

Unified representations carry wanted and unwanted channels together. The proton-safe projector exists only after reducing the colour–lepton symmetry that made the complete multiplet attractive in the first place.

A finite Pati–Salam reconstruction supplies a concrete source split, while an all-orders rank-one normal form protects part of it. The simplest U(1) and universal Z₂ protections fail. The first surviving family-universal cyclic witness is a Z₉ lift whose faithful low-energy quotient is baryon triality.

This is a live conditional route, not closure: global anomalies, heavy thresholds, the full finite triple and the third edge remain decisive gates.

Buying a complete toolkit gives both the tool you wanted and the tools packed beside it. Unified gauge multiplets behave similarly: symmetry relates desired leptoquark interactions to diquark partners capable of destabilizing the proton. The problem is not merely finding a useful mediator but explaining why its dangerous companions are absent or ineffective.

The simplest family-universal extra U(1) charges collapse to familiar combinations and do not isolate the safe source. A universal binary sign also fails for three families because of a weak anomaly. The first surviving cyclic construction is a \(\mathbb Z_9\) lift whose observable gauge quotient is baryon triality.

A quotient here is like identifying clock readings that differ by a full turn: the larger bookkeeping group retains information needed for anomaly cancellation even when the low-energy observer sees only three classes.

The typed candidate action now separates BF memory, defect, reader, bosonic bulk, fermion capacity and response terms. A frozen point is healthy in all defined local bosonic sectors. Its classical stress tensor is also known term by term, with the pure BF term carrying no metric stress.

The remaining gaps are not a vague request for more elegance. They are a microscopic wall origin, a chiral fermion domain, a phase reference, an anomaly-free family operator, curved stress matching, an interacting spectrum and a device-specific signal.