Boundary Depth Scans

Averaging Gate Limit Three-Row Scan

A residual budget is scanned below, at, and above the acceptance limit. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

Zero residual budget passes the gate

All five readings match the target, so the squared budget is zero.

0≤9⇒10\le9\Rightarrow1
Zero-budget averaging rowBudget, limit, and accepted bit are checked together.readings=0,0,0,0,0 Vtarget=0 Vaverage=0 Vresiduals=0,0,0,0,0squaredBudget=0gateLimit=9acceptedBit=1 bit

Equality at the limit still passes

One reading at three volts gives budget nine, exactly at the limit.

9≤9⇒19\le9\Rightarrow1
Gate equality rowThe helper enforces less-than-or-equal acceptance.readings=3,0,0,0,0 Vtarget=0 Vaverage=3/5 Vresiduals=3,0,0,0,0squaredBudget=9gateLimit=9acceptedBit=1 bit

One extra residual square rejects the row

Adding a one-volt residual raises the budget to ten, so the row fails.

budgetlimitaccepted0919911090\begin{array}{c|c|c}\text{budget}&\text{limit}&\text{accepted}\\0&9&1\\9&9&1\\10&9&0\\\end{array}
Rejected averaging rowThe accepted bit follows the exact budget.readings=3,1,0,0,0 Vtarget=0 Vaverage=4/5 Vresiduals=3,1,0,0,0squaredBudget=10gateLimit=9acceptedBit=0 bit