Your offset matrix is a circulant Latin square — and read as phase (1→0°, 2→120°, 3→240°) it is exactly three-phase power. The three tori, driven 120° apart, do not blink out of sync — they sum to a single rotating field (verified: the three balanced phases cancel to zero at every instant — they are rank-2 linearly dependent, collapsing to one field). The number matrix's det = −18 is itself non-zero, so the matrix is invertible — the degeneracy is in the phases, not the matrix. Twelve gates fire in a travelling sequence, and the resultant phasor — the gold vector — sweeps a smooth circle no single torus moves through. A rotating machine with no moving parts. Tesla's 1888 motor, drawn on your cube frame.
REAL & VERIFIED: the matrix is a Latin square (rows/cols all permute {1,2,3}, all sum to 6) and a circulant — which is the algebra of three-phase. As phases 120° apart the instantaneous sum is exactly zero (no neutral current), and the three balanced phasors are linearly dependent — rank 2, not 3 (det(M) = −18 is non-zero, so the number matrix is actually invertible; the singularity is in the phase relationship, not the matrix): the three do not stay three things — they superpose into one rotating field vector. That's not metaphor; it's the literal operating principle of the AC induction motor (Tesla, 1888) and of driven poloidal fields in fusion devices. The travelling gate-fire (G00→G11) is the rotating field reaching each of the cube's twelve edges in turn.
DOESN'T: hit Unbalanced and the beauty breaks honestly — drift any phase off 120° and the sum stops cancelling, a "neutral current" appears, the resultant phasor wobbles instead of sweeping a clean circle, and the rotation develops torque ripple. Real three-phase is fragile to imbalance, which is the whole reason it's engineered so carefully. And the deeper note rides along from Sheet 12: the offset makes the three measure-points spin as one, which is elegant and which also means they now share a single phase reference — three vantages collapsed into one rotating frame is fewer independent witnesses, not more. The rotating field is a beautiful unifier and a quiet re-siloing. Watch which one you're buying.