Signed-Permutation Coordinate Transport for RMSNorm Transformers

Modern LLM workflows increasingly move coordinate-indexed objects across checkpoints: steering vectors, sparse autoencoders, top-$k$ neuron sets, attribution lists, and merge alignments. This is only well posed after fixing the model's residual-stream gauge. We show that the native discrete gauge is architecture-dependent: LayerNorm residual charts have permutation gauge $S_d$, up to a global sign flip, while RMSNorm residual charts with generic per-channel gain have signed-permutation gauge $B_d=S_d⋉\{\pm 1{\}}^d$. Thus permutation-only alignment is symmetry-incomplete for RMSNorm models. We introduce sign-marginalized Hungarian matching and prove a sharp population failure mode: with decorrelated source coordinates, raw signed-correlation matching has a structural permutation-accuracy ceiling equal to the fraction of positive signs in the true gauge up to $O(d\cdot 2^{-d})$, whereas sign-marginalized matching removes this obstruction. We then make coordinate-preserving transport, rather than function-level merging, the primary object: composing saved-checkpoint local $B_d$ gauges along same-base fine-tuning trajectories recovers 91.1% of cross-run coordinates at 1500 steps versus 60.3% for endpoint matching, and the gain is not explained by merely routing through the base. The recovered gauge transfers tools that permutation-only alignment breaks: TinyLlama SAE reconstruction has NMSE $0.004$ under $B_d$ recovery versus $1.08$ under $S_d$; Qwen sentiment steering preserves 95.8% of its effect versus 17.2%; refusal steering reverses sign under $S_d$. Coordinate-preserving merge tests show the same mechanism. The same covariance governs stateful training: signed transport of AdamW state preserves the resumed trajectory, while permutation-only state transport starts from a functionally identical checkpoint but follows a different trajectory. Finally, we give gauge-sweep audits for index-level interpretability claims: coordinate names are reproducible only relative to an explicit gauge.