Parallel D-brane no-force identity ($f_3^8-f_4^8-f_2^8=0$)

Identical, parallel, equally oriented type-II D-branes preserve the same sixteen spacetime supercharges. Their annulus oscillator sum has the displayed numerator, which vanishes by the Jacobi abstruse identity. In the closed-string channel this is cancellation of attractive NS-NS exchange and repulsive RR exchange. The force vanishes in the flat supersymmetric configuration; anti-oriented branes, relative angles or supersymmetry-breaking backgrounds need not have this cancellation.