Relation Cross-Scans

Detuning Boundary Cross-Scan

Exact candidate rows show accepted, boundary accepted, and rejected detuning decisions. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

Candidate square 1/4 has distance 3/4

The target square is 1. Candidate square 1/4 has distance 3/4 from the target, with tolerance 1.

∣14−1∣=34,accepted=1\left|\frac{1}{4}\mathbin{-}1\right|=\frac{3}{4},\quad accepted=1
Detuning row 1Each candidate cites its own LC mode source.candidate=1/4, distance=3/4, accepted=1

Candidate square 2 has distance 1

The target square is 1. Candidate square 2 has distance 1 from the target, with tolerance 1.

∣2−1∣=1,accepted=1\left|2\mathbin{-}1\right|=1,\quad accepted=1
Detuning row 2Each candidate cites its own LC mode source.candidate=2, distance=1, accepted=1

Candidate square 5 has distance 4

The target square is 1. Candidate square 5 has distance 4 from the target, with tolerance 1.

∣5−1∣=4,accepted=0\left|5\mathbin{-}1\right|=4,\quad accepted=0
Detuning row 3Each candidate cites its own LC mode source.candidate=5, distance=4, accepted=0

The boundary accepts equality and rejects the far candidate

Distance three fourths accepts. Distance one is exactly on the boundary and also accepts. Distance four is outside the same tolerance and rejects.

ωc⋅ωcωt⋅ωtdistancetolerancea14134112111151410\begin{array}{c|c|c|c|c}\omega_c\cdot\omega_c&\omega_t\cdot\omega_t&distance&tolerance&a\\\frac{1}{4}&1&\frac{3}{4}&1&1\\2&1&1&1&1\\5&1&4&1&0\\\end{array}
Detuning boundary cross-scanThe displayed row is the equality boundary.distance=1 is still accepted