Cross-Entropy Is Load-Bearing: A Pre-Registered Scope Test of the K-Way Energy Probe on Bidirectional Predictive Coding
For researchers using energy probes to interpret predictive coding networks, this work shows that CE is a major empirical component of the probe-softmax relationship, limiting the generality of prior theoretical reductions.
This pre-registered study tests whether the K-way energy probe's reduction to log-softmax margin depends on cross-entropy (CE) loss. Replacing CE with MSE halves the probe-softmax gap, and CE training produces logit norms ~15x larger; about 66% of the gap is attributable to logit-scale effects, with 34% reflecting a scale-invariant ranking advantage of CE-trained representations.
Cacioli (2026) showed that the K-way energy probe on standard discriminative predictive coding networks reduces approximately to a monotone function of the log-softmax margin. The reduction rests on five assumptions, including cross-entropy (CE) at the output and effectively feedforward inference dynamics. This pre-registered study tests the reduction's sensitivity to CE removal using two conditions: standard PC trained with MSE instead of CE, and bidirectional PC (bPC; Oliviers, Tang & Bogacz, 2025). Across 10 seeds on CIFAR-10 with a matched 2.1M-parameter backbone, we find three results. The negative result replicates on standard PC: the probe sits below softmax (Delta = -0.082, p < 10^-6). On bPC the probe exceeds softmax across all 10 seeds (Delta = +0.008, p = 0.000027), though a pre-registered manipulation check shows that bPC does not produce materially greater latent movement than standard PC at this scale (ratio 1.6, threshold 10). Removing CE alone without changing inference dynamics halves the probe-softmax gap (Delta_MSE = -0.037 vs Delta_stdPC = -0.082). CE is a major empirically load-bearing component of the decomposition at this scale. CE training produces output logit norms approximately 15x larger than MSE or bPC training. A post-hoc temperature scaling ablation decomposes the probe-softmax gap into two components: approximately 66% is attributable to logit-scale effects removable by temperature rescaling, and approximately 34% reflects a scale-invariant ranking advantage of CE-trained representations. We use "metacognitive" operationally to denote Type-2 discrimination of a readout over its own Type-1 correctness, not to imply human-like introspective access.