Readout Boundary and Matrix Closure

Threshold Boundary Line Scan

low, tie, and high readouts

A readout threshold is trusted only after low, equality, and high signals are all checked by the same rule. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

The boundary scan starts from calibrated endpoints

The calibrated zero signal is 2, the threshold is 5, and the calibrated one signal is 8.

szero=2,T=5,sone=8s_{\text{zero}}=2,\quad T=5,\quad s_{\text{one}}=8
Readout boundary endpointsThe equality row is rendered at the threshold.zero signal 2one signal 8threshold 5input 5 reads repeat

Three signals expose low, boundary, and high decisions

The scan checks a low endpoint, the exact boundary, and a high endpoint. The result column is the helper decision, not a label chosen after the fact.

szerosTsoneresult2258zero2558repeat2858one\begin{array}{c|c|c|c|c}s_{\text{zero}}&s&T&s_{\text{one}}&\text{result}\\2&2&5&8&\text{zero}\\2&5&5&8&\text{repeat}\\2&8&5&8&\text{one}\\\end{array}
Boundary scanThe middle row is exactly on the repeat boundary.zero signal 2one signal 8threshold 5input 5 reads repeat

Equality repeats instead of guessing a state

For the rendered row, the signal equals the threshold at 5. The checked result is repeat, so no zero or one decision is invented.

5=5⇒repeat5=5\Rightarrow \text{repeat}
Readout tie ruleEquality is the only repeat case in this model.zero signal 2one signal 8threshold 5input 5 reads repeat

A high endpoint is the only one-decision in the scan

The high row uses signal 8, which is above threshold 5. That row reads one.

8>5⇒∣1⟩8>5\Rightarrow \lvert 1\rangle
High endpoint readoutOnly the high-side row reads one.zero signal 2one signal 8threshold 5input 8 reads one

The boundary model is not a noise model

This audit claims a finite threshold rule only. It does not claim detector noise, stochastic trajectories, or hardware drift.

s<T⇒∣0⟩,s=T⇒repeat,s>T⇒∣1⟩s<T\Rightarrow \lvert 0\rangle,\quad s=T\Rightarrow \text{repeat},\quad s>T\Rightarrow \lvert 1\rangle
Boundary closureThe equality rule is checked before display.zero signal 2one signal 8threshold 5input 5 reads repeat