Knob Scans and Boundaries

Readout Threshold Boundary Scan

below, tied, above

A calibrated readout boundary is scanned with low, tied, and high signals before count statistics. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

A readout threshold needs two calibration levels

The zero signal is 1/4, the one signal is 3/4, and the threshold is 1/2.

szero=14,sone=34,T=12s_{\text{zero}}=\frac{1}{4},\quad s_{\text{one}}=\frac{3}{4},\quad T=\frac{1}{2}
Readout thresholdThe marker is below the threshold.zero signal 1/4one signal 3/4threshold 1/2input 3/8 reads zero

A low signal reads zero

Signal 3/8 is below the threshold, so this finite model returns zero.

s=38<T⇒zeros=\frac{3}{8}<T\Rightarrow \text{zero}
Below-threshold signalThe signal marker sits on the zero side.zero signal 1/4one signal 3/4threshold 1/2input 3/8 reads zero

Low, tied, and high signals have different decisions

The equality row is deliberately not rounded upward. At the threshold the rule says repeat, because the readout boundary is ambiguous in this finite model.

sdecisionreason38zeros<T12repeats=T58ones>T\begin{array}{c|c|c}s&\text{decision}&\text{reason}\\\frac{3}{8}&\text{zero}&s<T\\\frac{1}{2}&\text{repeat}&s=T\\\frac{5}{8}&\text{one}&s>T\\\end{array}
Threshold tie rowThe tie row is rendered as a repeat decision.zero signal 1/4one signal 3/4threshold 1/2input 1/2 reads repeat

A high signal reads one

Signal 5/8 is above the same threshold, so the decision changes to one. The scan makes the boundary visible before any statistics are applied.

s=58>T⇒ones=\frac{5}{8}>T\Rightarrow \text{one}
Above-threshold signalThe high marker sits on the one side.zero signal 1/4one signal 3/4threshold 1/2input 5/8 reads one