# An Exact Instrument for State Usage in Selective State-Space Models, and the Input-Driven Migration It Reveals

arXiv: https://arxiv.org/abs/2607.11796 · HTML: https://mlsanity.com/paper/2607.11796/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Core SSM architectures for language, one of 10 directions in State-space sequence models (S4, Mamba).
- Builds on 12 of its 16 references that are in the map.
- First posted on arXiv 13 Jul 2026.

## Placed in

- State-space sequence models (S4, Mamba) / [Core SSM architectures for language](https://mlsanity.com/ssm/core-ssm-architectures-for-language/)

## Builds on (most cited first, 12 of 12)

- [Mamba: Linear-Time Sequence Modeling with Selective State Spaces](https://mlsanity.com/paper/2312.00752/) (2023)
- [Transformers are SSMs: Generalized Models and Efficient Algorithms Through Structured State Space Duality](https://mlsanity.com/paper/2405.21060/) (2024)
- [The Hidden Attention of Mamba Models](https://mlsanity.com/paper/2403.01590/) (2024)
- [Falcon Mamba: The First Competitive Attention-free 7B Language Model](https://mlsanity.com/paper/2410.05355/) (2024)
- [Layer-Adaptive State Pruning for Deep State Space Models](https://mlsanity.com/paper/2411.02824/) (2024)
- [Model Compression Method for S4 with Diagonal State Space Layers using Balanced Truncation](https://mlsanity.com/paper/2402.15993/) (2024)
- [Model order reduction of deep structured state-space models: A system-theoretic approach](https://mlsanity.com/paper/2403.14833/) (2024)
- [Efficient Unstructured Pruning of Mamba State-Space Models for Resource-Constrained Environments](https://mlsanity.com/paper/2505.08299/) (2025)
- [Characterizing Mamba's Selective Memory using Auto-Encoders](https://mlsanity.com/paper/2512.15653/) (2025)
- [Hankel Singular Value Regularization for Highly Compressible State Space Models](https://mlsanity.com/paper/2510.22951/) (2025)
- [PerfMamba: Performance Analysis and Pruning of Selective State Space Models](https://mlsanity.com/paper/2511.22849/) (2025)
- [GHOST: Unmasking Phantom States in Mamba2 via Grouped Hidden-state Output-aware Selection & Truncation](https://mlsanity.com/paper/2602.11408/) (2026)

## Abstract (arXiv)

Selective state-space models such as Mamba route information through a bank of first-order modes whose input coupling is set by a learned selection mechanism. We give an exact instrument for measuring how a trained model uses these modes. Because the state matrix is diagonal, each channel's output decomposes exactly into per-mode contributions, and a per-(layer, channel, window) Gram tensor yields the exact output error of dropping any subset of modes, offline, at any budget. Validated against the reference implementation to a relative error of 2.3×10-7 on the Mamba-1 family where it is exact, the instrument predicts a layer's deployed pruning error to a median relative deviation of 5×10-7 over 4,464 configurations, its floor set by the reconstruction. Applying the instrument across the Mamba-1 family (130M--2.8B), the deployed 7B Falcon-Mamba, and Mamba-2, we find that trained models re-allocate their state space with the input: which modes carry the signal migrates across contexts, and at the most affected layers a per-input oracle roughly halves the output error of a fixed mode set. Frozen-signal counterfactuals attribute the migration primarily to the input-dependent write map Bt; the timestep usually identified with selectivity carries almost none of it. Input-scheduled mode pruning on this measurement outperforms static, Hankel-based, and layer-adaptive rankings at every scale from 130M to the deployed 7B Falcon-Mamba, and at half the state budget it matches the unpruned model. Because the scheduler reads each window's mode usage from a first pass, this demonstrates realizable headroom; we claim no deployed compute or memory saving.