A fixed assignment matrix maps true two-state probabilities into observed readout probabilities and integer expected counts. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

Prediction reuses the calibrated assignment matrix

The prediction scan uses the checked matrix with first row 9/10, 1/10 and second row 1/5, 4/5.

A=[9101101545]A=\begin{bmatrix}\frac{9}{10}&\frac{1}{10}\\\frac{1}{5}&\frac{4}{5}\end{bmatrix}
Prediction matrixThe same calibration rows feed all prediction rows.assignment matrixread zeroread onetrue zero9/109 shots1/101 shotstrue one1/52 shots4/58 shots

Three true states become three observed readout budgets

The scan pushes a true-zero state, an even mixture, and a true-one state through the same assignment matrix with the same shot total.

pzpopread zeropread oneNzNo1091011018212121120920119011545416\begin{array}{c|c|c|c|c|c}p_z&p_o&p_{\text{read zero}}&p_{\text{read one}}&N_z&N_o\\1&0&\frac{9}{10}&\frac{1}{10}&18&2\\\frac{1}{2}&\frac{1}{2}&\frac{11}{20}&\frac{9}{20}&11&9\\0&1&\frac{1}{5}&\frac{4}{5}&4&16\\\end{array}
Prediction scanThe middle row is the even mixture.true probabilitieszero 1/2one 1/2assignmentobserved probszero 11/20one 9/20over 20 shots11 zero reads9 one reads

Prepared zero still has visible assignment errors

The true-zero row begins with probability 1 on zero, but the observed one-read probability is 1/10.

910+110=1\frac{9}{10}+\frac{1}{10}=1
True zero through assignmentThe observed row follows the calibration matrix.true probabilitieszero 1one 0assignmentobserved probszero 9/10one 1/10over 20 shots18 zero reads2 one reads

Prepared one uses the second calibration row

The true-one row begins with probability 1 on one, and produces 16 expected one-read counts out of 20 shots.

15+45=1\frac{1}{5}+\frac{4}{5}=1
True one through assignmentThe expected counts are integer-checked.true probabilitieszero 0one 1assignmentobserved probszero 1/5one 4/5over 20 shots4 zero reads16 one reads

Prediction closure is finite-count arithmetic

This closure does not claim detector drift, deconvolution, or hardware fidelity. It only shows how a calibrated assignment matrix changes finite expected counts.

1120+920=1\frac{11}{20}+\frac{9}{20}=1
Prediction closureThe helper checks probabilities and integer counts.true probabilitieszero 1/2one 1/2assignmentobserved probszero 11/20one 9/20over 20 shots11 zero reads9 one reads