Deeper Calibration Scale Scans

Shot Total Scaling Three-Row Scan

probabilities before counts

Changing the shot total scales expected counts after the observed probabilities are fixed. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

Shot totals scale counts after probabilities are fixed

The true state and assignment matrix stay fixed. The observed probabilities are computed once, then shot totals scale expected counts.

pobs=(1320,720)p_{\text{obs}}=\left(\frac{13}{20},\frac{7}{20}\right)
Middle shot-total rowThe matrix-vector product is fixed before counting.true probabilitieszero 3/4one 1/4assignmentobserved probszero 13/20one 7/20over 40 shots26 zero reads14 one reads

Three shot totals keep the same observed fractions

The first two columns stay fixed across the scan. Only the count columns scale with the number of shots.

pzpoNzNo13207201371320720261413207205228\begin{array}{c|c|c|c}p_z&p_o&N_z&N_o\\\frac{13}{20}&\frac{7}{20}&13&7\\\frac{13}{20}&\frac{7}{20}&26&14\\\frac{13}{20}&\frac{7}{20}&52&28\\\end{array}
Shot-total scanThe middle row shows the count scale explicitly.true probabilitieszero 3/4one 1/4assignmentobserved probszero 13/20one 7/20over 40 shots26 zero reads14 one reads

The largest shot row keeps the same proportions

The large row has 52 expected zero reads and 28 expected one reads.

1320⋅80=52\frac{13}{20}\cdot80=52
Large shot-total rowThe probability row and count row agree.true probabilitieszero 3/4one 1/4assignmentobserved probszero 13/20one 7/20over 80 shots52 zero reads28 one reads

More shots do not change the calibration matrix

This scan claims expected counts only. It does not add sampling variance, Bayesian updating, or detector drift.

1320+720=1\frac{13}{20}+\frac{7}{20}=1
Shot-count scope auditThe helper checks integer expected counts.true probabilitieszero 3/4one 1/4assignmentobserved probszero 13/20one 7/20over 40 shots26 zero reads14 one reads