The exact worst case of the global policy of DeepSeek's expert-parallel load balancer

DeepSeek's expert-parallel load balancer (EPLB) replicates experts by Adams's apportionment method and then packs the replicas onto GPUs by a cardinality-constrained longest-processing-time rule (modified LPT). In EPLB's own load model (static expert loads, each expert's tokens split evenly among its copies) we prove that its global policy has worst-case maximum-GPU-load ratio exactly 2m/(m+1) relative to the best joint choice of replica counts and placement (m GPUs; attained whenever there are at least two slots per GPU; EPLB is optimal with one slot per GPU). The worst case needs very concentrated load: a loss of the full factor requires a single copy carrying at least m/(m+1) of the optimal maximum GPU load, and in the bad instances one expert carries almost all the traffic. More generally EPLB/OPT ≤ 1 + (m-1)M/T, where M is the largest copy load and T the total load. With K equally hot experts (as top-K routing suggests) the same construction gives 2m/(m+K), e.g. 1.6 at K=8, m=32 (a lower bound; the exact worst case under a 1/K share limit is open). On constructed inputs (not measured traffic), DeepSeek's released eplb.py gives at least 1.9391 (one hot expert) and 1.5999 (eight) times an explicit better plan at 256 experts, 32 GPUs, 9 slots. The analysis covers the global policy and the hierarchical policy on a single node. An appendix certifies the (4m-1)/(3m) bound conjectured by Babel, Kellerer and Kotov for modified LPT in 14 small (m,k) cases, for every tie-breaking rule. The supplement contains all code, certificates, an independent checker and a README (MIT license). Changes in v1.1 (the main theorem and its proof are unchanged): Title adds the qualifier "in the static, even-split load model"; the abstract's first paragraph states the load model and the concentrated-load requirement, worded to match what is proven. New Corollary 5: the instance-wise bound EPLB/OPT ≤ 1 + (m-1)M/T, and the heavy-copy condition for the full factor. New Remark 7 comparing with the exchange algorithm of Babel, Kellerer and Kotov (1998, Theorem 6), whose tight ratio is the same constant 2-2/(m+1): the extremal shape and accounting are shared; the items, algorithm and proof differ. Babel-Kellerer-Kotov is now introduced once in the related work. Prior work on the factor 2-1/m for modified LPT is described from what could be read (the Dell'Amico-Martello 2001 abstract; Wu-Yao 2008, Appl. Math. J. Chinese Univ. 23:1-8, by title only) and the factor is not claimed as new. New related work: Wolf, Yu and Shachnai's replicate-then-place disk load balancing for video on demand (SIGMETRICS 1995; Multimedia Systems 1997), and EPLB GitHub issue #22 (two copies of one expert on the same GPU on real traces, under the hierarchical policy; reproduced with the released code). New computation: every-tie-rule certificates for all 14 (m,k) cases of the conjecture (v1.0 had them only for (2,4) and (3,4)); the certification moved to an appendix. Scope: the single-node hierarchical policy is shown to be covered; real-code ratios are labelled as constructed instances.