Exact Coordinate-Balancing Calculus and Sharp Stability for Symmetric Lattice Cross-Covariograms

Abstract

For finite A, B ⊂ ℤ^d, write g_A,B(u) = |A ∩ (B − u)|. We study how this cross-covariogram varies as the coordinates of u are balanced along a fixed ℓ₁ shell. A centered exchange-fiber property, satisfied by lattice sections of compact convex sets invariant under signed coordinate permutations, decomposes every unit balancing transfer into parity-fiber endpoint exposures in {0, 1}, yielding the Schur order on each shell and a fiberwise equality certificate for comparable shifts. For equal Lee balls of radius t in dimension d ≥ 3 and comparable shifts on an active shell 2 ≤ s ≤ 2t, the overlap deficit is at least a closed-form constant times the number of unit transfers; a central-box section theorem evaluates it in lower-dimensional Lee-ball counts. Incomparable shifts may share a value, so equality is order-theoretic. The extremizers are characterized for 4 ≤ s ≤ 2t: each interior shell carries exactly ⌊s/2⌋ − 1 balanced-gap minimizing orbits, independently of d and t, with equality only across single such transfers; the two outermost shells form a boundary family. A structural theorem evaluates the sharp layer in every dimension for the Minkowski interpolants C_d + bB^d_∞ between cross-polytope and cube: the constant is determined by lattice-point counts of the same body two dimensions down, plus an explicit facet surcharge for transfers into a zero coordinate, and one inequality decides the minimizing transfer. Cubes and Lee balls are its degenerate regimes. Convex hyperoctahedral symmetry thus fixes the transfer sign but neither the sharp scale nor the minimizing orbit; reversals occur once convexity or coordinate-permutation symmetry fails.