# K-Way Energy Probes for Metacognition Reduce to Softmax in Discriminative Predictive Coding Networks

arXiv: https://arxiv.org/abs/2604.11011 · HTML: https://mlsanity.com/paper/2604.11011/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 2 of its 5 references that are in the map.
- First posted on arXiv 13 Apr 2026.

## Builds on (most cited first, 2 of 2)

- [Sample as You Infer: Predictive Coding With Langevin Dynamics](https://mlsanity.com/paper/2311.13664/) (2023)
- [Introduction to Predictive Coding Networks for Machine Learning](https://mlsanity.com/paper/2506.06332/) (2025)

## Built on by (most cited first, 1 of 1)

- [Cross-Entropy Is Load-Bearing: A Pre-Registered Scope Test of the K-Way Energy Probe on Bidirectional Predictive Coding](https://mlsanity.com/paper/2604.21286/) (2026)

## Abstract (arXiv)

We present this as a negative result with an explanatory mechanism, not as a formal upper bound. Predictive coding networks (PCNs) admit a K-way energy probe in which each candidate class is fixed as a target, inference is run to settling, and the per-hypothesis settled energies are compared. The probe appears to read a richer signal source than softmax, since the per-hypothesis energy depends on the entire generative chain. We argue this appearance is misleading under the standard Pinchetti-style discriminative PC formulation. We present an approximate reduction showing that with target-clamped CE-energy training and effectively-feedforward latent dynamics, the K-way energy margin decomposes into a monotone function of the log-softmax margin plus a residual that is not trained to correlate with correctness. The decomposition predicts that the structural probe should track softmax from below. We test this across six conditions on CIFAR-10: extended deterministic training, direct measurement of latent movement during inference, a post-hoc decoder fairness control on a backpropagation network, a matched-budget PC vs BP comparison, a five-point Langevin temperature sweep, and trajectory-integrated MCPC training. In every condition the probe sat below softmax. The gap was stable across training procedures within the discriminative PC family. Final-state and trajectory-integrated training produced probes whose AUROC_2 values differed by less than 10^-3 at deterministic evaluation. The empirical regime is small: single seed, 2.1M-parameter network, 1280 test images. We frame the result as a preprint inviting replication. We discuss conditions under which the decomposition does not apply (bidirectional PC, prospective configuration, generative PC, non-CE energy formulations) and directions for productive structural probing the analysis does not foreclose.