这篇论文提出了Omega-S，一个只需权重矩阵的正则项，能让微调少忘旧知识，Llama-3-8B上保留率从62.9%提到84.1%，开销只增加4%。
Omega-S是一个即插即用的惩罚项，仅从权重矩阵计算，无需旧任务数据或Fisher矩阵。在Llama-3-8B上使用LoRA从代码微调到散文的实验中，Omega-S在10个种子中的9个上比无正则化保留更多原始能力。保留率从0.173提升到0.238（pass@1），即从62.9%提升到84.1%，且训练开销增加不到4%。实验表明，其实际生效机制是对节点度方差的正则化，而非名称暗示的拓扑目标。论文还量化了相同配置下保留率的标准差为0.104，并提供了全部代码和逐种子结果。
Omega-S: A Functional Resilience Index for LLM Fine-Tuning
Fine-tuning a large language model on new data degrades what it previously learned. We present Omega-S, a drop-in penalty computed from the weight matrix alone: it needs no previous-task data, no Fisher matrix and no stored copy of the old weights. It is three lines in an existing training loop and adds under 4% to the cost of a step. Retention. On Llama-3-8B with LoRA, fine-tuned from code to prose and measured by HumanEval over ten seeds, Omega-S retains more of the original capability than no regularisation on 9 of 10 seeds (0.173 -> 0.238 absolute pass@1; sign test one-sided p=0.011, Wilcoxon p=0.006), as a retention ratio, 62.9% -> 84.1%. It also beats tuned weight decay on 10 of 10 seeds (p=0.002) and tuned EWC on 8 of 10 (p=0.014), every arm re-measured in the same session. Mechanism, measured rather than asserted. Omega-S is topological by construction, its objective built from Tr(A^3), but we measured which of its four factors actually moves and three do not: their elasticity with respect to the weights is at or below 1e-4, against 9e-3 for the degree-variance term. As implemented, the composite reduces to a penalty on the variance of node degrees, which means row magnitude in square modules and directional alignment in non-square ones. We report this because a method whose name promises one thing and whose gradient does another should say so. We also enumerate the open design choices, including a contrast-preserving construction that does what it was designed to do and makes retention worse on all ten seeds. Repeating an identical configuration, same seed and same hardware, gives a standard deviation of 0.104 in retention ratio. We have not found this quantified for low-rank fine-tuning of language models, and it bounds every seed-paired comparison in this literature, ours included. Code, per-seed results and the full record of negative results are available.