Complete Classification and Sharp Minimality for Marked p-Probe Mixed-Volume Deficits of Zonoids

Let p ≥ 3. We study a marked p-probe mixed-volume deficit for zonoids generated by g base segments and p marked probes in R^d. The deficit is the connected-partition polynomial in the associated mixed-volume coefficients. We determine exactly the parameter strata on which nonnegativity, equality, formal Gale realizability, measure equality, covariance, and the activity conditions defining the success criterion hold simultaneously. For each fixed p ≥ 3, the success set is S_d^(p) = ∅ for d = 1, S_d^(p) = {1,2} for d = 2, and S_d^(p) = {d−1} for d ≥ 3. Consequently, g_min(d,p) = +∞ for d = 1 and g_min(d,p) = d−1 for d ≥ 2, p ≥ 3, with N_min = d+p−1 and k_min = p−1 on the finite branch. Thus g = d−1 is the universal successful boundary; dimension two has the additional successful stratum g = 2; dimension one has none; and no higher layer re-enters. The proof uses determinant formulas, complementary-minor Gale transfer, an exact boundary product, a planar matching-polynomial reduction, projective equality, explicit negative families, and one-directional lifting. The p ≥ 3 theorem chain is independent of the completed two-probe theory, which enters only in the final all-p synthesis.