# A Held-Out Transition-Pair Falsifier for Long-Horizon Non-Abelian State Tracking

arXiv: https://arxiv.org/abs/2606.07254 · HTML: https://mlsanity.com/paper/2606.07254/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 4 of its 8 references that are in the map.
- First posted on arXiv 5 Jun 2026.

## Builds on (most cited first, 4 of 4)

- [The Illusion of State in State-Space Models](https://mlsanity.com/paper/2404.08819/) (2024)
- [Structured Sparse Transition Matrices to Enable State Tracking in State-Space Models](https://mlsanity.com/paper/2509.22284/) (2025)
- [The Expressive Limits of Diagonal SSMs for State-Tracking](https://mlsanity.com/paper/2603.01959/) (2026)
- [On the "Induction Bias" in Sequence Models](https://mlsanity.com/paper/2602.18333/) (2026)

## Built on by (most cited first, 3 of 3)

- [A Calibrated Test of Internal Action Maps: State Signals Without Global Affine Closure](https://mlsanity.com/paper/2608.13626/) (2026)
- [The Automaton Underneath: The Additive Input Pathway Is a Parasitic Attractor for State Tracking in Householder Linear RNN](https://mlsanity.com/paper/2609.18966/) (2026)
- [Tracking States or Tracking Cosets? An Algebraic Account of Learned State Tracking](https://mlsanity.com/paper/2609.29951/) (2026)

## Abstract (arXiv)

State tracking exposes a sharp limitation of sequence models: the relevant signal is often not a summary of observed tokens, but an ordered latent state that evolves through non-commutative transformations. We introduce a held-out transition-pair falsifier for finite non-Abelian group tracking. The protocol forbids selected ordered generator pairs during training and requires the same local patterns during evaluation, blocking one direct local-transition memorization pathway. In a controlled S3 × S3 benchmark, a projected recurrent state model trained only on length-8 sequences produces error-free final-state predictions (perfect 250/250 per horizon) through evaluation horizons up to 1,048,576 tokens across five seeds. Matched native-readout baselines, including bag, GRU, and a single-configuration structured state-space model, remain near floor under the same protocol. Projection-matched GRU, structured SSM, and bag baselines equipped with analogous finite-group prototype readouts also remain near chance under the same split. Mechanism diagnostics show that hard projection coincides with low homomorphism error, low state-consistency drift, and non-trivial commutator separation, while softened projection collapses final-state accuracy. Clean-split audits verify zero verbatim reduced-word overlap and zero structural-template overlap between training and evaluation partitions. The evidence is scoped to this controlled finite-group falsifier rather than to a general architecture ranking. Within that regime, explicit projected non-commutative state composition acts as a useful inductive bias for long-horizon hidden-state tracking.