Certified Quotient Calibration with Weighted-Projective Orbits and Finite-Shot Guarantees
Periodic controls can make a quantum-calibration scan redundant, but compilation, noise, and measurement can invalidate the symmetry behind that reduction. We introduce Certified Quotient Calibration (CQC), which audits whether circuit amplitudes have a common positive grading and define weighted-projective-space (WPS) control orbits. From pilot counts it bounds the largest proposed within-orbit and smallest audited between-orbit squared Hellinger distances by U_in and L_out. CQC accepts only when these bounds meet a task tolerance, preserve audited alternatives, and imply a net saving after certification cost; otherwise it rejects or defers the reduction. The theorem bounds accepted orbit substitutions and [0,1]-valued calibration objectives on one confidence event. Its cost corollary authorizes a representative-only scan only when its certification-inclusive cost is below that of the full grid. Weighted-projective descent, Hellinger testing rates, and multinomial concentration are standard; the contribution is their combination into a finite-shot approve--reject--defer guarantee for task error and net calibration cost. Gate-level experiments on IQM Garnet and IBM Marrakesh, Kingston, and Fez exercise all three decisions in compiled one-, two-, and three-qubit families. In a preregistered two-qubit Bell-phase experiment on IBM Kingston, the pilot audit found U_in=0.01922<L_out=0.09030, and quotient and full scans selected the same correction orbit for both injected phase offsets. Fresh held-out counts met the prespecified 0.03 noninferiority margin. Including 18,432 pilot shots, the quotient procedure used 36,864 rather than 55,296 total shots, a 33.3% reduction. These results support the CQC decision path for the tested families, but not a native cubic pulse law, a device-independent WPS orbit, or a universal saving rate.