Relation Cross-Scans

Finite Bin Basis Cross-Scan

Read each bin as a point-by-point multiply-and-sum ledger. The same alternating samples can have zero dc sum but a finite alternating-bin value of 4, showing why the chosen basis matters; real bin estimates depend on sampling, windowing, calibration, noise, leakage between bins, and uncertainty.

Finite dot-product bins are scanned across reviewed basis roles. 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 bin is still product-and-sum arithmetic

The alternating source multiplies sample by basis point by point. The dot sum is the visible finite bin value.

1+1+1+1=41+1+1+1=4
Alternating bin sourceThe role is accepted only with its checked basis.samples=1,-1,1,-1basis=1,-1,1,-1dotSum=4binRole=alternating

Changing the basis changes what the same samples show

The same alternating samples have zero dc sum but four alternating sum. The first row shows the dc basis on a flat signal.

samplesbasisroledot(1,1,1,1)(1,1,1,1)dc4(1,−1,1,−1)(1,1,1,1)dc0(1,−1,1,−1)(1,−1,1,−1)alternating4\begin{array}{c|c|c|c}\text{samples}&\text{basis}&\text{role}&\text{dot}\\\left(1,1,1,1\right)&\left(1,1,1,1\right)&\text{dc}&4\\\left(1,-1,1,-1\right)&\left(1,1,1,1\right)&\text{dc}&0\\\left(1,-1,1,-1\right)&\left(1,-1,1,-1\right)&\text{alternating}&4\\\end{array}
Finite basis scanOnly reviewed finite bin roles are used.samples=1,-1,1,-1basis=1,-1,1,-1dotSum=4binRole=alternating