We give an exact algorithm for continuous separable convex quadratic programming with an integer equality matrix and nonnegative variables. The number of rational arithmetic operations and comparisons is polynomial in the dimensions and the encoding length of the constraint matrix, independently of the right-hand side, linear costs, and all nonnegative quadratic weights. Intermediate rational encodings are polynomial in the complete input. This yields a strongly polynomial algorithm for every integer matrix class with polynomially bounded entry encodings, including arbitrary matrices with entries in $\{0,\pm1\}$. The local solver compresses a quadratic objective on an integer box and solves only the compressed continuous problem. Two-sided proximity on boxes with rational bounds transfers its solution to a nearby original optimum; integer optima are used only in the proof. A single linear program supplies an error bound and a dual potential, while tight-coordinate revelation and fixed feasible anchors ensure termination and control encoding lengths. An explicit lifting extends the result to convex piecewise quadratic functions of signed integer affine features, with operation counts independent of breakpoints and objective coefficients. Applications include continuous multicommodity flow with shared capacities and piecewise quadratic costs of aggregate congestion.

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