# The Equilibrium Is the Initialization: Lazy Identity Collapse in Physics-Structured Deep Equilibrium Reasoning

arXiv: https://arxiv.org/abs/2607.11116 · HTML: https://mlsanity.com/paper/2607.11116/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 23 of its 28 references that are in the map.
- First posted on arXiv 13 Jul 2026.

## Builds on (most cited first, 23 of 23)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Shortcut Learning in Deep Neural Networks](https://mlsanity.com/paper/2004.07780/) (2020)
- [Understanding intermediate layers using linear classifier probes](https://mlsanity.com/paper/1610.01644/) (2016)
- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [Designing and Interpreting Probes with Control Tasks](https://mlsanity.com/paper/1909.03368/) (2019)
- [Deep Equilibrium Models](https://mlsanity.com/paper/1909.01377/) (2019)
- [Adaptive Computation Time for Recurrent Neural Networks](https://mlsanity.com/paper/1603.08983/) (2016)
- [Scaling up Test-Time Compute with Latent Reasoning: A Recurrent Depth Approach](https://mlsanity.com/paper/2502.05171/) (2025)
- [Lagrangian Neural Networks](https://mlsanity.com/paper/2003.04630/) (2020)
- [ProofWriter: Generating Implications, Proofs, and Abductive Statements over Natural Language](https://mlsanity.com/paper/2012.13048/) (2020)
- [Gradient Starvation: A Learning Proclivity in Neural Networks](https://mlsanity.com/paper/2011.09468/) (2020)
- [Hierarchical Reasoning Model](https://mlsanity.com/paper/2506.21734/) (2025)
- [PonderNet: Learning to Ponder](https://mlsanity.com/paper/2107.05407/) (2021)
- [Multiscale Deep Equilibrium Models](https://mlsanity.com/paper/2006.08656/) (2020)
- [Looped Transformers are Better at Learning Learning Algorithms](https://mlsanity.com/paper/2311.12424/) (2023)
- [Implicit Deep Learning](https://mlsanity.com/paper/1908.06315/) (2019)
- [Can You Learn an Algorithm? Generalizing from Easy to Hard Problems with Recurrent Networks](https://mlsanity.com/paper/2106.04537/) (2021)
- [The CLRS Algorithmic Reasoning Benchmark](https://mlsanity.com/paper/2205.15659/) (2022)
- [Monotone operator equilibrium networks](https://mlsanity.com/paper/2006.08591/) (2020)
- [JFB: Jacobian-Free Backpropagation for Implicit Networks](https://mlsanity.com/paper/2103.12803/) (2021)
- [Stabilizing Equilibrium Models by Jacobian Regularization](https://mlsanity.com/paper/2106.14342/) (2021)
- [Port-Hamiltonian Neural Networks for Learning Explicit Time-Dependent Dynamical Systems](https://mlsanity.com/paper/2107.08024/) (2021)
- [Path Independent Equilibrium Models Can Better Exploit Test-Time Computation](https://mlsanity.com/paper/2211.09961/) (2022)

## Abstract (arXiv)

Deep equilibrium models promise input-adaptive implicit computation: harder problems should demand more solver iterations, and the solved equilibrium should encode the result of genuine iterative inference. We report a cautionary study of a port-Hamiltonian DEQ with a learned initialization on two reasoning tasks -- ProofWriter entailment over frozen DeBERTa embeddings and a BFS-verified graph-reachability benchmark -- in which the implicit computation is a silent no-op. Across tasks, seeds, and controlled ablation arms, the solved equilibrium equals the solver's start point to numerical precision, and bypassing the solver entirely changes test accuracy by +0.00 percentage points in 18 of 19 training runs. Controlled interventions falsify the tempting explanation: removing the anchoring term reproduces every result, and retraining with noise-decoupled starts yields a solver that converges to the noisy start while the decoder learns to ignore it. The single escaping run diverges instead (|h*-z0|=171), producing a co-adapted noise channel whose removal improves accuracy. Iteration counts are uncorrelated with ground-truth difficulty (r=0.009), and the full apparatus never outperforms a two-layer MLP on either task. We trace the mechanism to gradient starvation along two distinct routes, show that the standard zeroing ablation is confounded and gives wildly seed-dependent answers where the correct substitution test gives a stable zero, and distill a four-test diagnostic protocol for auditing claimed implicit computation. All experiments run on a single free Colab GPU; code, raw logs, and analysis scripts are released.