[Phase 2 write-up] Cumulant propagation under a compute budget: state-matched corrections, verified fast arithmetic, and a learned terminal corrector (submission #335413)

ARC White-Box Estimation Challenge 2026
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Submission ID: #335413. Graded 3.3425257575e-9 adjusted (raw final-layer MSE 2.4308e-8). This write-up corresponds to that one submission. It is the frozen #334316 cumulant estimator plus a learned terminal residual corrector; the propagated state is byte-identical to #334316.

Full write-up: writeup-phase2-final.pdf (PDF, 40 pages) (218.3 KB)

My Phase 2 nominations are #336611 and #335413. #336611 runs the same corrector on LordProcciuto’s public MIT-licensed e30_a estimator (item 8); the engine it runs is not mine, and its eligibility is up to the organizers.

What the document covers

  1. Method. Third-order cumulant propagation in the framework of Wu et al. (arXiv:2605.05179), carrying the covariance plus a factorized third-order state with a rank cap of 7,680. Two cumulant repairs were evaluated independently: a mixed covariance/third-cumulant pair term (ratio 0.958 on a frozen 32-network block) and a covariance-shaped fourth-order template (ratio 0.958).

  2. Correction fitting that matches the deployed state and the final loss. Small coefficient tables are fitted on the actual truncated float32 trajectory, using native central-difference responses of the deployed estimator. The objective is pooled final-layer MSE, not a moment’s own oracle. Official path 3.7978e-9 → 3.6473e-9, with training, development and official networks kept separate. The tested fits still show a generalization gap after independent optimizer checks: 200-network fits through a differentiable replica did not generalize even within the replica.

  3. Verified fast arithmetic inside the metered interface. A six-level alternative-basis schedule (Schwartz–Vaknin) and a selectively placed rank-48 4×4 recursion (Dumas–Pernet–Sedoglavic–Tichavský, arXiv:2609.12027). Each comes with an exact bill recurrence and an accuracy gate per component, and the changes that leave the bill unchanged are checked for byte-level parity. Official 3.6473e-9 → 3.5984e-9, with predictions unchanged up to float32 regrouping.

  4. At this operating point, the binding constraint was the 0.4 s residual-time gate, not the FLOP bill. A preregistered 44-candidate campaign found more than twenty candidates that improved the development score by a further 0.9–1.6%. All but the four I submitted pushed the forecast residual time above the 0.38 s planning margin. Two findings about the grader interface are documented for other participants: dtype metadata on the client/server path, and the local-to-official residual-time ratio.

  5. Negative results for the tested configurations, including matched-refit studies and preregistered gates:

    • the rank cap, tested in both directions;
    • relaxing and tightening which coordinates are retained;
    • low-rank fourth-order cores;
    • oracle ceilings of the pair fourth-order blocks;
    • hybrids with 504aldo’s MIT-licensed estimator.

    A later projected shared-feed regeneration implementation failed on eight independent synthetic development networks after matched refits (+23.12% adjusted score against the refitted control, 0/8 wins). A saved-result postmortem separates prediction displacement from error correction without attributing the failure to one component. The confirmation block stayed unopened and the candidate was not submitted.

  6. A covariance correction that reached the grader. I ran eight preregistered representation gates after 26 September. The eighth, a 28-coefficient correction to the off-diagonal covariance fitted to the final mean, lowered error on fresh networks from the official public split (−1.7%, 7/8) but cost wall time against the 120 s limit. I restricted it to layers 8–14 and combined it with a byte-identical reduction in copies that lowers residual time. That graded as #334316 (−1.22% against #333716, no failures). Paired timing shows that residual time follows the number of client calls, while wall time follows the arithmetic.

  7. A learned terminal residual corrector on the frozen estimator (Section 6). With the propagated state frozen, a 6,594-parameter per-layer network reads 33 per-neuron quantities that the estimator already computes. It transports two scalar messages through the weights, one of them gated by the ReLU pass probability, and corrects only the final-layer mean. It adds 0.13% to the bill and about 0.3% to wall time. It is trained on the organizer-provided public targets, with the scored quantity itself as the loss (per-network cost-weighted final MSE). Three preregistered steps were graded:

    step submission adjusted score
    pretraining on 256 networks #334717 3.392e-9
    terminal objective #334793 3.373e-9
    training pool of 760 networks #335413 3.343e-9

    Together that is −5.6% against #334316. Each step passed a development gate on 64 held-apart networks and an independent 64-network final block that was opened once, and the local final ratios predicted the official ratios to within 0.35 points. One intermediate candidate (384-network pool) passed its ratio and interval gates but not the 39/64 wins criterion on its sealed block (37/64), so I did not submit it. The variants I tested are documented:

    • capacity along the hidden axis and the layer axis (per-layer unshared weights overfit from epoch 3; even their best development checkpoint failed the ratio, interval and win gates against the shared corrector);
    • extra transported state;
    • the objective;
    • training horizon and learning-rate continuation;
    • ensembling;
    • more data: −0.9% from 256 to 384 networks, −0.6% from 384 to 760.

    The document also gives a quiet-host timing protocol. Its baseline-against-baseline bands passed in an idle block, after an earlier attempt on a loaded host failed.

  8. Transfer to a public engine (Section 6). I retrained the same two-message corrector architecture from scratch on the captured state of LordProcciuto’s public e30_a estimator (#335981, 3.116e-9; MIT, derived from 504aldo’s V29), while keeping the engine byte-identical. It graded as #336611 = 3.044e-9: score ×0.977 and raw ×0.975 against the engine, at +0.14–0.15% FLOPs on the two uncaptured baselines checked. Its final 64-network block had a raw-MSE ratio of ×0.978. A later holdout, fresh for this transfer project, scored ×0.976 under its recorded cost approximation. The write-up states plainly that deployment gate 3 failed and that I submitted anyway, outside the all-gates-pass protocol. The follow-ups stopped at different stages:

    • the seed ensemble failed its holdout gate;
    • the GRU stopped on development evidence;
    • scaling up with synthetic data failed its interim continuation rule.

    Monte Carlo validation of the targets covered two official networks at 2^25 samples. The further H2 holdout was not opened for this transfer project.

Fixed checks from 12–14 October are reported as negatives or as incomplete local validation, including the first-layer hard-gate control variate and an unsubmitted arithmetic candidate (v4). None of those checks produced an official improvement.

Every quantitative claim is tied to a receipt in the evidence index (Section 8 of the PDF).

AI use. LLM coding assistants (Anthropic Claude and OpenAI Codex) were used extensively for literature synthesis, mathematical and engineering hypotheses, derivation drafts, estimator and experiment code, log interpretation, preregistered campaign execution, and writing. The consolidated document and PDF tooling were also prepared with their assistance. LLM plausibility is not validation: quantitative claims are tied to recorded experiments or published sources. Explanatory interpretations and limitations are separated from measurements. I remain responsible for reviewing and submitting the final account.

Thanks to ARC and AIcrowd for running the challenge. Questions and counter-measurements are welcome.

hyojun_kwon