# 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; - local teaching-signal reads to zero error, counting both the task-period edge signal and every SDIL neutral edge signal; SDIL therefore pays two local scalar reads per updated edge in this ladder; - 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 resource count without a resolved exponent change: “SDIL uses X% fewer local teaching-signal reads at N edges”; - 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. ## Frozen calibration-budget stress test After the six-size core confirmation, run one supplementary control at side length 32. Reuse all 40 tasks, component seeds `20260830,20260831,20260832`, 600 epochs, and learning exposure `0.01 s`. Train only the static-calibration method, increasing its instruction-off calibration observations from 16 to 256. No learning hyperparameter or component draw is reselected. The output is `results/coupled_ladder/p4_constant256_side32.json`. Compare its final error, stable-zero fraction, and error AUC with the frozen 16-observation static calibration in `p2_confirm_side32.json`. This control tests whether the static baseline was limited by sampling error. It is supplementary and does not replace any point in the frozen scaling fit. The 256 calibration reads per edge are included in its observation cost. The frozen control is complete. Final error changes from `0.271875` with 16 observations to `0.303125` with 256 observations. The paired task-bootstrap improvement is `-0.03125` with a 95% interval of `[-0.059375, -0.003125]`. Stable-zero fraction changes from `0.233333` to `0.200000`, and local scalar reads to the censored target increase from `9,347,072` to `9,573,717`. The frozen decision is `sampling_limited=false`. Sources are `results/coupled_ladder/p4_constant256_analysis.json` and `results/coupled_ladder/p4_constant256_source.csv`. ## Completed overclamp confirmation The overclamp ladder is complete for all six sizes, 40 tasks, and three component draws. At 2,048 edges, clean overclamp, imperfect overclamp, and overclamp plus SDIL have final errors `0.00625`, `0.46875`, and `0.008333`. Their stable-zero fractions are `0.95`, `0.025`, and `0.933333`. SDIL reduces the excess final-error growth slope by `98.991%` with a paired task-bootstrap 95% interval of `[97.356%, 100.000%]`; the corresponding stable-failure result is `98.157% [94.865%, 100.000%]`. The error-AUC excess slope changes from `0.124788` to `-0.000104`. Sources are `results/coupled_ladder/p3_overclamp_scaling_analysis.json`, `results/coupled_ladder/p3_overclamp_scaling_source.csv`, and `results/figs/figure_clln_overclamp_scaling_confirmation.pdf`.