# Can a Dynamic Internal Field Govern a Transformer's Cognition? Certifiability, not Superiority, in Homeostatic Compute Control

arXiv: https://arxiv.org/abs/2608.24319 · HTML: https://mlsanity.com/paper/2608.24319/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 33 of its 56 references that are in the map.
- First posted on arXiv 25 Aug 2026.

## Builds on (most cited first, 30 of 33)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Neural Turing Machines](https://mlsanity.com/paper/1410.5401/) (2014)
- [Adaptive Computation Time for Recurrent Neural Networks](https://mlsanity.com/paper/1603.08983/) (2016)
- [Transformers Learn Shortcuts to Automata](https://mlsanity.com/paper/2210.10749/) (2022)
- [GRAND: Graph Neural Diffusion](https://mlsanity.com/paper/2106.10934/) (2021)
- [Mixture-of-Depths: Dynamically allocating compute in transformer-based language models](https://mlsanity.com/paper/2404.02258/) (2024)
- [Looped Transformers as Programmable Computers](https://mlsanity.com/paper/2301.13196/) (2023)
- [Depth-Adaptive Transformer](https://mlsanity.com/paper/1910.10073/) (2019)
- [Exploring Length Generalization in Large Language Models](https://mlsanity.com/paper/2207.04901/) (2022)
- [The Parallelism Tradeoff: Limitations of Log-Precision Transformers](https://mlsanity.com/paper/2207.00729/) (2022)
- [The Illusion of State in State-Space Models](https://mlsanity.com/paper/2404.08819/) (2024)
- [PonderNet: Learning to Ponder](https://mlsanity.com/paper/2107.05407/) (2021)
- [The Right Tool for the Job: Matching Model and Instance Complexities](https://mlsanity.com/paper/2004.07453/) (2020)
- [Graph-Coupled Oscillator Networks](https://mlsanity.com/paper/2202.02296/) (2022)
- [AntisymmetricRNN: A Dynamical System View on Recurrent Neural Networks](https://mlsanity.com/paper/1902.09689/) (2019)
- [Learning Stable Deep Dynamics Models](https://mlsanity.com/paper/2001.06116/) (2020)
- [PDE-GCN: Novel Architectures for Graph Neural Networks Motivated by Partial Differential Equations](https://mlsanity.com/paper/2108.01938/) (2021)
- [Looped Transformers for Length Generalization](https://mlsanity.com/paper/2409.15647/) (2024)
- [Lipschitz Recurrent Neural Networks](https://mlsanity.com/paper/2006.12070/) (2020)
- [Differentiable plasticity: training plastic neural networks with backpropagation](https://mlsanity.com/paper/1804.02464/) (2018)
- [Coupled Oscillatory Recurrent Neural Network (coRNN): An accurate and (gradient) stable architecture for learning long time dependencies](https://mlsanity.com/paper/2010.00951/) (2020)
- [Length Generalization in Arithmetic Transformers](https://mlsanity.com/paper/2306.15400/) (2023)
- [Anti-Symmetric DGN: a stable architecture for Deep Graph Networks](https://mlsanity.com/paper/2210.09789/) (2022)
- [UnICORNN: A recurrent model for learning very long time dependencies](https://mlsanity.com/paper/2103.05487/) (2021)
- [Recurrent Equilibrium Networks: Flexible Dynamic Models with Guaranteed Stability and Robustness](https://mlsanity.com/paper/2104.05942/) (2021)
- [Backpropamine: training self-modifying neural networks with differentiable neuromodulated plasticity](https://mlsanity.com/paper/2002.10585/) (2020)
- [Oscillatory State-Space Models](https://mlsanity.com/paper/2410.03943/) (2024)
- [On the stability properties of Gated Recurrent Units neural networks](https://mlsanity.com/paper/2011.06806/) (2020)
- [Introducing Neuromodulation in Deep Neural Networks to Learn Adaptive Behaviours](https://mlsanity.com/paper/1812.09113/) (2018)
- [Learning to Dissipate Energy in Oscillatory State-Space Models](https://mlsanity.com/paper/2505.12171/) (2025)

## Abstract (arXiv)

An intelligent system does not merely reason: it governs its own reasoning - how much to compute, when to stop, which module to activate. Can that role be played by a dynamic internal field - a low-dimensional homeostatic state with explicit physics and certified stability - that modulates cognition without performing it? Ours is a field on the module graph governed by a family of PDEs on the graph Laplacian, advancing with an adaptive-depth reasoner. We certify the stability of the integrator of the whole family - an integrator certificate, not a closed-loop one. New, and proved here: a discrete Schur-Cohn criterion for Verlet with velocity coupling, necessary and sufficient per latent root, with no commutation hypothesis. The answer is threefold: substance no, structure only in part, certifiability yes. The type of the field's physics is irrelevant for accuracy: wave, diffusion, gated mixtures and a 2D Navier-Stokes substrate tie. A twenty-seed preregistered deconfounding campaign bounds the structural claim: at equalized caps the second-order effect is strong in one family (+0.087 [+0.042, +0.132], t=4.0) but is not detected in the other (+0.014 [-0.013, +0.040], n.s.), so part of the original contrast was capacity, not order; and a matched-interface GRU is indistinguishable in the first and nominally exceeds the field in the second (-0.035 [-0.067, -0.002]). What distinguishes the field is not capability but that its one-step operator admits an exact runtime stability check - a difference of kind, not of existence: learned recurrences carry certificates too, sufficient and conservative ones. A kill-gate with a positive control finds no evidence for the field as evidence accumulator (Delta AUC +0.0007 [-0.0065, +0.0079] vs a 0.03 threshold). A dynamic internal field is a viable, certifiable compute governor, but not an enhancer of cognition: it modulates, it does not think.