The worst case of the Karmarkar-Karp differencing method is 7/6 for three parts and 11/9 for four

Multiway number partitioning splits numbers into m parts, minimising the largest part sum. The Karmarkar-Karp differencing method (largest differencing method, LDM) is a classical heuristic for it, used for example by the RL framework veRL to balance micro-batches. For m ≥ 3 parts its worst-case ratio has been open since at least 2002, between 4/3-1/(3(m-1)) and 4/3-1/(3m), and to our knowledge the upper bound had a complete proof only for one tie-breaking rule. We prove that, under every tie-breaking rule, the ratio is 7/6 for three parts and 11/9 for four, and at most 4/3-1/(3m) for every m; the largest part is also below the average plus one largest number. The key idea is that small numbers cannot hurt: either they keep the final imbalance small, or the result equals that of a run on the large numbers alone. This reduces the exact ratio to instances with at most 3m numbers, which we settle for m = 3 and m = 4 by a computer search whose certificates are archived and re-checked without a linear-programming solver. Files: the paper (PDF); the arXiv source; a supplement (MIT license) with the search code, the archived certificates (m = 3, and m = 4 with up to ten items), the solver-free checker, separately written re-checking programs, logs, and a README mapping every computational claim to a command; and, as a separate file, ldm-ratio-v1.1-m4n11-certificates.zip, the certificate archive of the m = 4 search with eleven items (20 files), which unzips into the supplement's results/m4/dump/u/ and is checked with the command given in the README. With it the whole m = 4 proof can be checked from stored data. SHA256SUMS lists all four files.