Finite generation and the empty query

This is the finite-generation check of XXII-A–C CheungXXIIA,CheungXXIIB,CheungXXIIC. The auxiliary inner product is a proof device, not spacetime geometry.

The constant family $W_e=W_0$ has zero strengths:

independent of $S$. Its component order is an infinite antichain, so every comparable difference generating set is empty and $K=\{0\}$. Thus it fails (K2), despite base-class nonemptiness CheungXXIIC (prop:empty-query).