From continuous charges to triality

Exact assertion

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.

What this commits us to

The construction is admitted only after the named additional input is supplied. The page records the dependency instead of hiding it inside the conclusion.

Logical anatomy

Verification route

1. Recover the declared objects and assumptions from the controlling documents.
2. Check the calculation, construction, obstruction or stopping rule on those same objects.
3. Compare the source conclusion with the exact assertion above; do not strengthen its verbs or enlarge its domain.
4. Apply the status rule and retain every named external dependency.

What would move or defeat this claim

Show that the named input is inconsistent, insufficient, unavailable, or fails to imply the claimed construction.

Source map

Controlling documents

• no minimal family universal anomaly free U1 can protect the proton safe S₁ source maskOpen controlling document →
• for an odd number of families any universal \(\mathbb Z_2\) that splits leptoquark from diquark triplets has the same mod two weak anomalyOpen controlling document →
• connected CKM and PMNS Yukawa graphs force any exact diagonal family \(\mathbb Z_2\) back to the universal anomalous classOpen controlling document →
• \(\mathbb Z_9\) is the minimal family universal cyclic lift that protects triplet companion mixing and its three family quotient is baryon trialityOpen controlling document →
• the \(\mathbb Z_9\) extension not its naive mod3 reduction realizes baryon triality and matches the Dai Freed mod9 anomalyOpen controlling document →