# Main figure captions

**Figure 1 | Accuracy–cost trade-offs for local learning.** Large points and error bars show mean ±
sample standard deviation across five initialization seeds; small translucent points show the
individual seeds. Wall time was measured on a single GTX 1080. **a,** CIFAR-10 test accuracy versus
wall time for width-64 residual MLPs trained for
five epochs. The black line is the mean-based nondominated frontier among local-learning methods.
BP is a nonlocal reference and is excluded from frontier construction. Labels give hidden depth.
**b,** SDIL versus canonical equilibrium propagation (EP) on exact 784–500–10 (d1) and
784–500–500–10 (d2) forward architectures and the same first 50,000 MNIST training examples.
Within each matched architecture, every method uses the same budget: 25 epochs for d1 and 60 epochs
for d2. EP retains its published raw-pixel, hard-sigmoid energy dynamics and relaxation
schedule; SDIL retains its z-scored, tanh feedforward dynamics. Thus architecture, examples, and
epochs are matched, but preprocessing and state dynamics are intentionally method-native. EP
training particles persist across presentations as published; test particles are freshly
zero-initialized and run for the full free-phase schedule at each evaluation.

**Figure 2 | Credit-assignment scaling in deep local-learning networks.** Mean ± sample standard
deviation across five seeds on the same CIFAR-10 width-64 residual MLP protocol. **a,** Test
accuracy versus hidden depth. **b,** Paired accuracy gap from the exact-BP model with the same
depth and initialization seed. **c,** Mean per-sample cosine between the teaching signal and exact
descent direction, averaged over the earliest third of hidden layers. Exact gradients are used only
for this diagnostic, never for local learning. “Scaling” here means preserving useful accuracy and
credit assignment as depth grows; flattened-CIFAR accuracy itself does not increase with depth.
Against the predeclared `-1`-point retention margin, the paired one-sided depth-5-to-60 test rejects
a drop of at least one point for SDIL (`p=0.0037`) but not DFA (`p=0.2310`); results for every
method are retained in the figure manifest.

**Figure 3 | Somato-dendritic innovation is necessary under predictable apical traffic.** Mean ±
sample standard deviation across five seeds on MNIST, using depth-3, width-256 networks for 15
epochs. **a,** The raw apical compartment mixes causal feedback with ordinary traffic predictable
from the same neuron's somatic state. SDIL subtracts the neutral-period per-cell prediction and
uses only the innovation in the local eligibility update. **b,** Test accuracy as predictable
traffic increases. The norm-matched raw control has the innovation's per-sample magnitude but
retains the raw direction. **c,** Final cosine between the signal actually used for learning and
the exact negative hidden-state gradient, averaged over all hidden layers. Exact gradients are
diagnostic only. At `rho=0.5`, raw and norm-matched raw feedback reach chance while innovation
retains `97.35%` accuracy and positive descent alignment.