Concept
Turning the potential bound into balanced groups and label multiplicities
Since both parts of F are nonnegative, F<=2 implies Phi<=2M(b+w) and Xi<=2Qn. Each squared deviation is at most its whole sum. Thus c_u<=M/D+sqrt(2M(b+w)) and n_{t,g}<=n/r+sqrt(2Qn). Under 2(b+w)D^2<=M and 2Qr^2<=n, the square-root terms are at most M/D and n/r. Hence every label has at most 2M/D incidences and every group has at most 2n/r variables. If h is the largest label multiplicity, this gives hD<=2M. These are factor-two bounds, not perfect balance. They supply the previously conditional group and copy bounds.
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Updated 2026-11-07
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Math