The joint coefficient--scaling design of integer-forcing (IF) precoding couples a full-rank integer coefficient matrix with a positive diagonal scaling matrix, resulting in a mixed discrete--continuous optimization problem. We first distinguish the original full-rank formulation and its finite-field invertibility requirement from the unimodular restriction studied in the subsequent analysis. Under this restriction, we formulate coefficient selection as the unimodular minimum-sum basis problem (UMSBP). By parameterizing the scaling matrix with its normalized diagonal vector, we define a deterministic exact single-output UMSBP oracle whose preimages partition the scaling manifold into disjoint coefficient-labelled regions. The restricted objective is consequently represented as the lower envelope of fixed-coefficient quadratic functions. Based on this formulation, we derive an exact decomposition into fixed-coefficient scaling optimization, coefficient-consistency verification, cross-region exploration, and candidate comparison. We prove that exact UMSBP evaluation is NP-hard through a polynomial-time reduction from exact Euclidean CVP. For a fixed coefficient matrix, we derive an equivalent convex formulation, its normalized stationarity condition, and a matrix-balancing interpretation. Under the stated positivity and finite-projective-diameter condition

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