This is my Phase 2 algorithmic-contribution write-up. The PDF is attached (33 pages).
arc-whestbench-phase2-writeup-333534-Kuznetsov.pdf (823.5 KB)
Submission ID: #333534 (graded 3.7357e-9: final-layer MSE 2.03e-8 at 18.3% of the FLOP budget, no failed networks; 55th of 295 on the board today). It is the one submission this write-up corresponds to.
The method, briefly: ARC’s SIMPLE k = 3 cumulant chain (Wu et al., arXiv:2605.05179), re-derived and re-implemented from scratch, plus a scale-mixture κ₄ closure in O(n²) that only pays together with the C₀·D diagram the reference truncates, consistency cuts that remove computation and gain accuracy, old κ₃ sources in a rank-448 basis that rides the transport, one learned κ₄-diagonal corrector (mliston’s idea), and Strassen–Winograd as flopscope operations. It is 2.1 times more accurate than the reference algorithm at an eighth of its compute; §5.2 has the ablation of every component on 32 fresh networks.
Most of the document is about why this family of estimators behaves as it does. These are the parts I would most like read, each a measurement against Monte-Carlo or 1e9-sample truth with the script that reproduces it:
- A neuron’s error at the next layer is a random projection of this layer’s joint error through an independent weight column (Proposition 1). Odd orders have no coherent level (Proposition 2), and along the network’s own columns the errors are distributed as along random directions. That is why per-neuron correctors fail on k ≥ 3 chains.
- What a per-neuron correction can learn is set by the truncation order, not by depth: about two thirds of a Gaussian closure’s error at every shape from 256 × 8 to 1024 × 16, at most 4% of a k = 3 chain’s.
- That learnable part is a skew law, κ₃ᵢ ≈ c(l)·μᵢ. At every layer its slope is the projection of the previous layer’s post-activation partial trace on the mean, c(l+1) = (3q/n)⟨v, m⟩/‖m‖², exactly in expectation over the next weights and to about 1% on the truth at width 1024. The first value is derived: c(1)·n = 13.91, of which 9.82 comes from the projected per-neuron skews and 4.09 from the input’s radius, which a positively homogeneous network carries as an exact scale mode (inputs on the sphere give 9.2–9.7). v follows m because an amplitude mode along the mean takes over deep ReLU layers and a ReLU neuron’s response to a rescaling is its mean (Euler); with SiLU the alignment fades and the law with it. A Gaussian closure that carries the law with two constants per layer removes exactly its learnable part (63–65% at depth/width ≤ 1/32).
- κ₃ is not closed on the two slices the ReLU step reads. The all-distinct part is 28.8% of the next diagonal per layer at 1024 × 16; its share is derived (Proposition 3) and confirmed on the true κ₃ tensor by Monte Carlo (measured/predicted 0.94–1.01 at widths 64–256). In log terms about 72% of what carrying κ₃ buys at 1024 × 16 lives in that part.
- Five compressions of the κ₃ state looked affordable to a cost model and died on the meter for one reason: the construction of the compressed object costs what it saves.
- One order up, the missing accuracy is the young κ₄, above all the tree that hands κ₃ on through one covariance edge. Inside ARC’s own k = 4, three layers of κ₄ blocks plus one scale-mixture block keep 89% of its gain over k = 3 at 256 × 16. A reader of those trees cuts our raw error to ×0.70 on all ten public networks, but in our representation it costs about what it buys.
Concurrent work: il.miscusi (18248) and kai_leal_gonzalez (18246), both posted yesterday, reached related results independently — the same scale mode, removed inside an estimator, and a conditional form of the identity behind item 3. My measurements are dated 10-14 in the repository history; §6.4 of the PDF credits both and says what each part adds.
After the close I will publish the code under MIT (every estimator version, the float64 research chain with all its switches, every experiment folder with its results and scripts, and the full research journal) and the Monte-Carlo oracles used here under CC BY 4.0 (per-neuron moments of 30 public networks, full covariances of two, the synthetic shapes), and post the links in this thread. The repository includes scripts/error_anatomy.py, which measures the coherent and learnable shares of any estimator from its outputs alone.
Credits: Wu et al. and ARC’s MIT reference implementation (the skeleton); 504aldo (18218: the κ₄ → κ₃ channel, and an independent replication of the walls in items 4–5); jamesrahenry (18097, 18157: the coherence mechanism, first); mliston (18170: the offline corrector); hyojun_kwon (18154). No code of any participant was used.
Use of LLMs (as in §9 of the PDF): the research was led by me — the problem, the goals, the direction at every fork, which ideas to pursue and which results to accept, every submission — and the key ideas of the method came out of that direction. Claude Code (Anthropic), under my direction, wrote all of the code, ran the experiments and the pre-submission checks, kept the research journal, proposed derivations and measurements as the work went (each checked numerically before use), and drafted the write-up and this post; separate Claude instances audited the newest sections. No number here comes from a model’s statement; each is from a logged run.
Questions and counter-measurements are welcome.