This is the summary of our algorithmic-contribution write-up; the full 16-page PDF is attached below, and every script and results file it references is public in the replication repository (MIT). Referenced graded submission: #314695 (raw 5.89e-6, multiplier 0.169, adjusted 9.97e-7).
Disclosure: this work was AI-assisted (Claude, Anthropic). The model proposed experiments, wrote the code, and drafted the documents; a human directed the research and reviewed the results. The write-up was independently reviewed twice against the underlying scripts and data; details in its Disclosure section.
The central finding. A deep moment-propagation chain behaves as an error-compensating dynamical system: its per-layer errors are anticorrelated and partially cancel, so the chained estimator attenuates zero-mean noise while amplifying small coherent biases (measured end-to-end at a ~16:1 exchange rate). This one property explains why step-exact Edgeworth corrections from true cumulants make the chained estimator worse, why corrections succeed only when fitted on the chain’s own rolled-forward trajectories (DAgger-style), why a variance-reducing shrinkage filter doubled the final error despite working as designed, and why layerwise bias re-anchoring failed the same way in radiant-allomancer’s independent experiments.
Contributions, briefly:
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Diagnosis. The depth-32 failure of k=2 propagation is error dynamics, not closure order: with true input moments a single closure step is already below a strong sampling estimator’s total raw error (1.16e-6 vs 1.61e-6); chained, the same closure is 52× worse, and re-anchoring the covariance to truth makes it worse still.
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Method. Per-layer linear corrections fitted on the chain’s own rolled-forward trajectories reach 7.4e-6 held-out at seed-resampled expectation, ~8.4× better than plain kprop at matched budget, graded as #314695.
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Ground-truth joint-cumulant tests. Using keenanpepper’s published full K3 tensors (evaluation-only, pre-registered locked-set policy): the joint structure missing from the closure is real and compressible, yet an analytic carrier of it, at reconstruction cosine 0.83, still loses 4.5× to a plain N=4096 MC probe, 10/10 IDs.
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An error budget for the graded artifact. Roughly half of #314695’s final error is probe-sampling noise, attributed by field (the pair field dominates, and its noise anticorrelates with κ3’s, so denoising κ3 alone makes things worse), with a measured denoising ceiling and three instrumented exploitation failures.
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Negative results with mechanisms, and an open problem: a lower-variance unbiased estimator of {κ3, E[zc²ᵢzcⱼ]} at fixed sample count is worth up to ~2× raw MSE to any probe-fed correction scheme; shrinkage-family estimators are excluded by the bias-amplification dynamics, not by their variance performance.
On the town-hall hypothesis. We confirm the premise quantitatively: the pre-activation covariance participation ratio contracts 128 → 5.2 (25×) over the 32 layers. We then tested the natural implementation (a subspace-split hybrid) and found it negative with a structural mechanism: sampling noise concentrates in the same low-rank subspace as the signal (λᵢ/N, direction by direction), so the complement carries only ~1.7% of sampling error and even a perfect analytic side is capped at ~2%. A method exploiting the low rank must either beat λᵢ/N in the leading directions or convert rank into FLOP reduction. Details and the depth-graded rank budget are in §6 of the PDF.
Relation to prior write-ups. We read pscamillo’s and evaaaz’s write-ups and radiant-allomancer’s before finalizing ours; §6 of the PDF details where our results corroborate theirs (three of radiant-allomancer’s negatives match ours mechanism-for-mechanism, from the opposite architecture) and where we tested their components in our setting. Particular thanks to keenanpepper, whose two public datasets (arc-whestbench-higher-moments-2026, whest-k3-tensors-2026) made the fitting targets and the ground-truth tests possible at all.
Sampling wins Phase 1’s raw metric, and the write-up says so plainly; the analytic track targets the regimes where sampling itself is unreliable. We would welcome cross-checks — in particular of trajectory-fitting against pscamillo’s falsification suite, and of the probe-noise decomposition (which applies to any probe-fed estimator, including hybrids) against radiant-allomancer’s two-budget method.
Henry_ARC-WhiteBox_Phase1_Writeup_Submission-314695.pdf (119.9 KB)