Sharp Joint Bounds for Reduced Responses from Open Divided Differences

We determine the exact joint range of two filtered responses of a scalar Schur complement from nine calibration samples. Divided differences remove polynomial contact terms and recover two nonnegative weighted spectral moments. Explicit one-point spectra realize every attainable pair while preserving all nine samples. The paper gives the endpoint conditions and a scalar quartic for computing the lower endpoint of the held-out contrast. For a positive vacuum moment, omitting any one sample on the stated grid makes the contrast unbounded. A counterexample shows that motion of the hidden block can leave all calibrations unchanged and still make the contrast unbounded, even when the curvature contrast is positive. The supplement provides four reproducible calculations.