# From Expert Reduction to Behavioral Divergence: Tracing Numerical State through Sparse MoE Inference

arXiv: https://arxiv.org/abs/2607.28097 · HTML: https://mlsanity.com/paper/2607.28097/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Efficient MoE training and inference systems, one of 12 directions in Mixture-of-experts.
- Builds on 9 of its 16 references that are in the map.
- First posted on arXiv 30 Jul 2026.

## Placed in

- Mixture-of-experts / [Efficient MoE training and inference systems](https://mlsanity.com/moe/efficient-moe-training-and-inference-systems/)

## Builds on (most cited first, 9 of 9)

- [DeepSeek-V4: Towards Highly Efficient Million-Token Context Intelligence](https://mlsanity.com/paper/2606.19348/) (2026)
- [DeepSpeed-MoE: Advancing Mixture-of-Experts Inference and Training to Power Next-Generation AI Scale](https://mlsanity.com/paper/2201.05596/) (2022)
- [mHC: Manifold-Constrained Hyper-Connections](https://mlsanity.com/paper/2512.24880/) (2025)
- [DSpark: Confidence-Scheduled Speculative Decoding with Semi-Autoregressive Generation](https://mlsanity.com/paper/2607.05147/) (2026)
- [Mixture of Quantized Experts (MoQE): Complementary Effect of Low-bit Quantization and Robustness](https://mlsanity.com/paper/2310.02410/) (2023)
- [NNSmith: Generating Diverse and Valid Test Cases for Deep Learning Compilers](https://mlsanity.com/paper/2207.13066/) (2022)
- [Value-and-Structure Alignment for Routing-Consistent Quantization of Mixture-of-Experts Models](https://mlsanity.com/paper/2606.05688/) (2026)
- [The Illusion of Equivalence: Systematic FP16 Divergence in KV-Cached Autoregressive Inference](https://mlsanity.com/paper/2604.15409/) (2026)
- [Fast exact summation using small and large superaccumulators](https://mlsanity.com/paper/1505.05571/) (2015)

## Abstract (arXiv)

Mathematically equivalent expert-reduction orders can produce observably different sparse-MoE executions. We isolate this effect in native DeepSeek-V4-Flash by freezing local MoE state and varying only aggregation semantics. Four schemes separate operand representation from accumulator precision. At one layer-5 fork, 720 A-mode orders yield 10 continuation basins; 720 B-mode orders form 360 exact structural classes and 11 basins. Under one Chinese prompt, the B classes split into 202 layoffs, 113 hiring, and 45 other continuations. Maximum-L-infinity B-branch selection separates 12, 24, and 36 of 50 prompts by 8, 16, and 32 tokens. Across 192 persistent trajectories per scheme, P32, A, and B change every native-reference route trajectory, while C preserves routes, token sequences, and texts. A separate 192-trajectory C check matches native MoE, post-mHC, next-router, and LM states bitwise. For one controlled B branch, exact post-mHC endpoint reconstruction reproduces the measured downstream trajectory. At the next decode boundary, exact FP64 reconstruction of the branch's full persistent state yields agreement for 301 downstream post-mHC states, 301 persistent-state checkpoints, 301 routes, predictions, and text over seven steps, given the same naturally generated next input. These controls identify post-mHC as an intra-token boundary and full persistent state as a cross-token continuation boundary. Identical tokens need not imply identical autoregressive state: divergence can survive a token boundary and become visible later. These results make expert operand conversion, accumulator precision, and reduction order part of a numerical compatibility contract for sparse-MoE runtimes and hardware backends. They establish controlled causal possibility, not deployment incidence; C's order invariance is limited to evaluated six-term states and schedules.