Off-Center Ball Arithmetic: A Repositioning Extension for Midpoint–Radius Interval Arithmetic

In Arb's midpoint–radius (ball) arithmetic, every operation is evaluated at the midpoint of a ball. This record contains a manuscript and software treating the center of a ball as a policy: the center may be any point, and the radius is updated by the triangle inequality, with upward rounding, so that the enclosure is preserved. The manuscript proves soundness for every center, proves that the radius used is the smallest possible, shows that the midpoint minimizes the first-order propagated bound, and proves that re-centered computations compose soundly. It states three rules for use in a library following Arb's conventions (branch cuts, evaluation cutoffs, decimal printing). The software (C, built on FLINT/Arb, plus an exact-rational Python reference implementation) provides arb_set_center, arb_secant_step, midpoint_policy and a coefficientwise polynomial routine, arb_poly_set_center_round. Benchmark programs and raw CSV outputs are included for: real and complex basic operations at 64 to 32768 bits (re-centering costs about a factor two for addition and about 10–15% for multiplication, fused multiply-add, division and square root from 4096 bits upward); a recursive factorial product; the falling-factorial polynomial; and block splitting in polynomial multiplication, where rounding coefficient midpoints to the working precision reduced the number of blocks from 57 to 8 and the running time by a factor of about 3.3 for the square of a degree-10^4 exponential series at 333 bits, without loss of accuracy. Keywords: ball arithmetic; interval arithmetic; midpoint–radius; Arb; FLINT; arbitrary-precision arithmetic; rigorous numerics; polynomial multiplication Acknowledgment: AI assisted with code development, testing and experiments.