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# Digital coupled-learning scaling protocol
## Paper role
The paper has three experimental parts.
1. **Digital plug-in transfer.** Add the same local SDIL residual to the
existing digital local-learning backbones and measure recovery from a
matched teaching-channel imperfection.
2. **Digital coupled-learning scaling.** Isolate system size in a sparse
coupled-learning lattice. Compare ideal learning, fixed component
imperfection, same-RMS zero-mean noise, and SDIL under paired tasks and
component draws.
3. **Hardware-realistic simulation.** Use the reconstructed nonlinear CLLN,
the Appendix-C component model, and the nonideal CDS/autozero primitive.
Part 2 tests the scaling mechanism cheaply. Part 3 tests whether the local
operation survives the constraints of the proposed hardware substrate.
## Frozen ladder
- Topology: periodic square resistor grid.
- Side lengths: `4, 8, 12, 16, 24, 32`.
- Learnable edges: `2 * side^2`, from 32 to 2,048.
- Boundary geometry: the released 4-by-4 source and target layout scaled with
the grid side. The side-4 circuit is exactly the released layout.
- Task family: the 40 released Figure-5 ring-classification tasks: five input
diameters and eight label rotations.
- Initial state: tile each released 4-by-4 gate pattern over the larger grid.
- Pairing: every method receives the same task, initial gates, and component
draw. Component draws are nested within task and size.
- Size-dependent optimization: select the learning exposure using ideal
coupled learning only, then freeze it for every imperfect condition at that
size.
## Conditions
| Condition | Local teaching measurement |
|:--|:--|
| clean | ideal voltage-square difference |
| matched noise | ideal update plus a fresh random-sign version of the measured imperfection |
| raw bias | fixed per-edge gain, input-offset, and multiplier-offset errors |
| SDIL | raw task-period measurement minus the same edge's instruction-off measurement |
| constant calibration | raw measurement minus a frozen per-edge mean, used as a bias-specific baseline |
| overclamping | the published stronger-clamp correction, used after the core ladder is stable |
The SDIL subtraction reads local voltage drops and the local learning channel.
It does not read a device constant, a task loss, a downstream weight, or a BP
gradient. Every neutral observation and local scalar is counted.
## Primary endpoints
Accuracy remains the task endpoint. Predictor error and update residuals are
diagnostics.
- final classification error at a matched epoch budget;
- fraction of task/device pairs reaching zero classification error;
- restricted epochs and local updates to zero error, assigning the experiment
horizon to runs that do not reach the target;
- learning-curve area for classification error;
- wall time and neutral observations as implementation costs.
For each endpoint, plot task-clustered means and 95% paired bootstrap
intervals. Device draws are averaged within each task before resampling tasks.
## Scaling statement
Fit each predeclared endpoint against learnable edge count on log axes. Report
the two method slopes, their paired bootstrap interval, and the relative slope
change. The headline wording is selected from the observed sign:
- lower resource exponent: “SDIL reduces the updates-to-target scaling
exponent by X%”;
- lower error-growth exponent: “SDIL reduces imperfection-induced error growth
with network size by X%”;
- overlapping interval: report no resolved scaling-rate difference.
The plots may use the visual organization of Figure 4 in arXiv:2507.02092,
but every axis is an independent experimental intervention or a separately
named endpoint. Nodes, edges, and FLOPs from the same ladder are not presented
as independent confirmations.
## Gates
1. The side-4 linear circuit reproduces clean task learning.
2. Clean learning remains viable across the ladder after clean-only exposure
selection.
3. Fixed imperfection is worse than the same-RMS zero-mean control on at least
one task endpoint.
4. SDIL improves raw biased learning at every reported size and retains its
advantage at the two largest sizes.
5. The scaling conclusion is computed from all frozen sizes and task clusters,
including failed runs.
6. Hardware-realistic claims use Part 3 only.
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