6. The scored chain, #336537
Chain B starts from 504aldo’s published V29 and applies the same reading from the deletion side: it finds K=3 state that later reads barely use and stops computing it (Figure M, right track). Eight separate submissions, each one change on top of the previous, read network by network on the 50 leaderboard networks (ratios against the previous step, paired 95% intervals):
| step | origin | adj (×10⁻⁹) | ÷ previous [95%] | lower /50 | receipt | raw (×10⁻⁸) | C/B |
|---|---|---|---|---|---|---|---|
| 1 | V29 (504aldo) | 5.418 | #332957 | 2.133 | 0.254 | ||
| 2 | mixed | 4.449 | 0.821 [0.813, 0.830] | 50 | #334990 | 2.099 | 0.212 |
| 3 | ours | 4.286 | 0.963 [0.956, 0.970] | 47 | #335127 | 2.128 | 0.201 |
| 4 | ours | 4.258 | 0.993 [0.993, 0.994] | 49 | #335162 | 2.129 | 0.200 |
| 5 | ported (Puffi’s SSC) | 4.030 | 0.946 [0.942, 0.951] | 49 | #335185 | 2.014 | 0.200 |
| 6 | ours | 3.525 | 0.875 [0.867, 0.882] | 50 | #335300 | 2.067 | 0.169 |
| 7 | ours | 3.494 | 0.991 [0.987, 0.995] | 37 | #335301 | 2.090 | 0.166 |
| 8 | fitted corrector | 3.145 | 0.900 [0.890, 0.911] | 49 | #336537 | 1.853 | 0.169 |
| ours only: 3, 4, 6, 7 | 0.830 [0.820, 0.839] | 50 | four receipts composed |
The steps: (1) published V29; (2) leaf-8 Strassen (also found by ngmitam), our Strassen joins and dead-row trimming, with koushik_rudra’s V34 dead-unit pruning and the V37 cost blocks; (3) hub level 6, SVD ranks 8/16, shared basis 352; (4) wall-clock blocks, arithmetic unchanged; (5) Puffi’s SSC closure ported and its nine coefficients refit; (6) young-atom pruning, active-set trimming and twelve cost cuts; (7) feedback thin legs at rank 4; (8) a learned per-layer corrector. Steps 1–7 take C/B from 0.254 to 0.166 while raw moves from 2.13e-8 to 2.09e-8, mean adj −35.5%, lower on all 50 networks. The “ours only” row composes four receipts rather than being a separate submission, and step 7, #335301, is chain B itself, the graded control for #336537.
Step 6 carries the two mechanistic pieces:
- Active-set trimming. koushik_rudra’s V34 restricts the third-cumulant leg products to the rows of neurons that are almost never active. A column of the (2,1) slice enters every consumer only through its neuron’s first Hermite coefficient w₁ = Φ(α), so we extend the cut to columns: columns with w₁ < 0.014 skip the hub contraction and the shared-basis lift.
- Young-atom pruning. A young κ3 source born at layer b has one atom per neuron q of b, entering three rank-one terms. Its score is the norm bound s_q = ‖X₁,q‖ ‖Y₁,q‖ + ‖M_q‖ + ‖X₃,q‖ ‖y‖; at birth we keep the fewest atoms holding 99.5% of Σ s_q, rounded up to a multiple of 32, and zero the rest in every leg. As a single-block change on an earlier chain (at 99.6%) it moved adj 0.963× on the grader (#335175 → #335187).
Puffi’s response closure (SSC) was written for 256 × 32. We put its Sym(C⊗C) slices next to V29’s κ4 regeneration law at 1024 × 16, decouple the three uses of λ and refit the nine coefficients end to end on 12 public networks; on the grader it lowers raw 5.4% at C/B 0.200 before and after (#335162 → #335185). The cost work sets Strassen depth and leaves, residual and feedback SVD ranks, a narrower shared basis (352), rank-4 thin legs and buffer layouts.
Our pieces switched off inside the final chain (no corrector, mini networks 0–49, each arm against chain B on the same machine per network, 95% paired bootstrap):
| switched off | mean adj ÷ chain B [95%] | worse /50 | C/B | peak GiB |
|---|---|---|---|---|
| nothing: chain B, #335301 | 1 | 0.165 | 5.6 | |
| active-set trimming, with V34’s and V37’s row cuts | 1.160 [1.053, 1.227] | 49 | 0.199 | 7.2 |
| our column cut alone, both row cuts kept | 1.035 [1.032, 1.037] | 50 | 0.171 | 5.9 |
| young-atom pruning | 1.044 [0.949, 1.101] | 49 | 0.182 | 7.0 |
| representation (basis 384, ranks 16, six Strassen levels) | 1.049 [0.951, 1.109] | 48 | 0.190 | 7.9 |
| all three, with V34’s and V37’s row cuts | 1.278 [1.156, 1.355] | 49 | 0.234 | 14.7 |
The column cut alone is worth 1.035×, worse on all 50 networks and 1.034× on networks 4–49, outside the four on which chain B’s cost settings were tuned; all of it comes from compute (C/B 0.165 → 0.171, raw 0.998×). The young-atom and representation intervals cross 1 because network 6 gains strongly from removal; without it the ratios are 1.095 and 1.103. All three off peaks at 14.7 GiB (above the 8 GB grader cap).
State-carried residual corrector. At each layer one MLP, shared across layers, reads 118 inputs per neuron plus the layer’s one-hot: 35 chain statistics of this layer, 8 of the previous layer’s and a 32-dimensional state, both carried through W and W∘W, and 3 terms of the incoming correction. Three hidden layers of 128 GELU units output the layer’s new mean error and the next state (54,561 parameters). The error is carried forward linearly through the ReLU slopes and W and added to the final prediction, with chain B unchanged; the target is the error each layer adds along chain B’s own trajectory. Training data: 3,048 networks, the 1,000 of the public full split and networks 0–2,047 of Keenan Pepper’s public corpus. The 100 public mini networks chose the configuration. It gives 0.873× raw on the 100 mini networks, lower on all 100, and 0.887× on the grader’s 50 leaderboard networks, lower on all 50 (#336537 against #335301). Retrained from seeds 1 and 2 (the 1,000-network version, everything else fixed) it gives 0.881× and 0.882× on the mini networks. Artyom_Mazur bounds what a per-neuron correction can learn at 4% of a K=3 chain’s error; this one reads through W and W∘W, carries its state across layers and takes 11.3% off chain B’s raw on the grader.
Grader receipts for #336537: per-network raw 1.47e-8 to 2.36e-8 on the 50 leaderboard networks, C/B 0.166–0.178 on all 100, median wall time 85.5 s, slowest 102.1 s (limit 120 s), largest residual 0.357 s (limit 0.4 s), no failures.
7. Falsified closure hypotheses and structural boundaries (width 1024, official networks unless noted)
Each row states a common closure or cost hypothesis, the reading that falsified it and the structural invariant the reading reveals. If you are extending a K=3 chain toward fourth-order content, rows 11–20 sit closest to your work.
| # | hypothesis tested | reading | structural invariant revealed |
|---|---|---|---|
| 1 | Capping K=4 source age at 2, 3, 4, 6 layers keeps the gain | 109×, 67×, 41×, 12× worse | uniform lifetime cuts fail; sources must live long |
| 2 | K=4 sources in the shared basis at birth (rank 112) keep the gain | 3.72× worse (1.13e-8) | young sources stay exact; after two transports rank 256 is lossless |
| 3 | A rank 1–16 star crossed pairing keeps the gain | 18× to 15× worse | the diagonal star crossing must stay exact (3 matmuls) |
| 4 | A rank 1–16 SVD of the C·κ3 crossed kernel keeps the gain | 2.21× to 1.72× worse | the kernel stays near full rank (rank 16 keeps 11% of its energy at the worst layer) |
| 5 | Dropping all two-leg crossed pairings keeps the gain | 18× worse (5.47e-8) | crossed index pairings carry state content |
| 6 | Truncating (2,1,1) → K3 feedback CP columns to 128 keeps the gain | 6.76e-7, 223× worse, 195,308 units | column truncation loses the feedback state |
| 7 | Newborn feedback in a three-axis core keeps the gain | 5.10–5.57e-9 for k₃ = 64–256, against 3.90e-9 | core rank alone does not repair the newborn loss; one or two exact layers first give 4.17e-9 and 4.02e-9 |
| 8 | Keeping tree feedback exact and projecting the rest pays | 6.48–7.23e-9; a refit basis keeps 0–29% | tree and non-tree terms cancel when split (at n = 128, two terms of 61.6 and 63.7 sum to 0.9) |
| 9 | Removing one of the four K=4 output channels keeps the gain | 1,148× to 714,978× worse | all four channels close the recursion together |
| 10 | Star births plus same-side n³ slices keep the K=4 gain | 3.65× worse than full at 512, 6.3× at 1024 | omitted births gain weight with width |
| 11 | A learned final-layer readout on 1,000 networks pays | best 0.979×; re-propagated, 1.09–1.60× | carrying state beats output fits (section 6: 0.887× graded) |
| 12 | More K=3 cost work (section 6, step 3) keeps paying | adj −3.7%, raw +1.4% | the K=3 cost search we ran saturates near 3.7% |
| 13 | A sixth Strassen level in V29 (leaf 16) is cheap enough | C/B −5.4%, 7.3 GB, 1.25–1.9× wall time | the 8 GB and 120 s limits bind first |
| 14 | Truncated SVD of young K=3 sources (k = 16) pays | 256: cost −11%, raw +16%; 1024: over 180 s | cheap rank cuts lose accuracy and wall time |
| 15 | Patching K=3 with the K=4 chain’s C and κ4 keeps the gain | 0.29×; κ4 on layers 9–15 along 64 directions, 0.81× | memoryless patches miss the carried chain (0.08×) |
| 16 | Starting K=4 at layer 8 from an empty κ4 keeps the gain | 1.76e-7, 15× worse than seeded | late starts must inherit chain B’s κ4 slices |
| 17 | Old slice groups in a k = 64 energy basis keep the gain | 2.76e-7, 51× worse; 84% of the norm lost | energy picks the wrong readout directions |
| 18 | C·κ3 births as CP columns on chain B are cheap enough | 55,296 columns for a birth at layer 6; 110 + 271 units a pair | read cost grows with κ3’s width |
| 19 | One CP pool of 2,048–5,120 atoms for κ3 keeps the gain | 72× to 135× worse; uncapped, 0.994× | κ3 does not compress into a global pool of atoms |
| 20 | A late start with tree births only, or a covariance Krylov basis of rank 8/32, keeps the gain | 1.07× of chain B; 7.80× against 7.79× | the late-start gain sits in non-tree births, whose directions a covariance basis misses |
Read together, the rows repeat section 5 from the other side: cutting by uniform lifetime (rows 1, 16), by an energy or covariance basis (rows 17, 20), by a local fit (row 15) or by a first-order read weight removes state that a later read needed, and the compressions that keep the gain follow the reads, as the kernel’s own direction does and as old sources do when they enter a shared basis after their first sensitive reads.
8. Next to the other public write-ups
All eight public Phase 2 write-ups, each through the one graded submission it is bound to (the rules: “Each writeup must correspond to exactly one submission”; adj, raw and C/B read from each official submission page on 19 October, same grader, same 50 leaderboard networks):
| write-up | bound submission | recipe | adj (×10⁻⁹) | raw (×10⁻⁸) | C/B | #336537 ÷ entry: adj, raw, C/B |
|---|---|---|---|---|---|---|
| ours (18254) | #336537 | chain B with a corrector carried through W | 3.145 | 1.853 | 0.170 | |
| ours, corrector off | #335301 | chain B, the mechanistic steps alone (section 6, step 7) | 3.494 | 2.090 | 0.167 | 0.900, 0.887, 1.01 |
| @hyojun_kwon (18255) | #335413 | #334316 with a learned corrector, state unchanged | 3.343 | 2.431 | 0.138 | 0.941, 0.762, 1.23 |
| hyojun_kwon, corrector off | #334316 | K=3, factorized κ3 at rank cap 7,680, fitted covariance correction | 3.541 | 2.578 | 0.137 | #335301 ÷: 0.987, 0.811 |
| @Artyom_Mazur (18249) | #333534 | ARC’s simple K=3 rewritten, scale-mixture κ4 closure, per-neuron κ4-diagonal corrector | 3.736 | 2.032 | 0.184 | 0.842, 0.912, 0.92 |
| @Nurlanggs (18252) | #336225 | V29 with an all-distinct κ4 edge term and a fitted (3,1) law | 3.813 | 2.099 | 0.182 | 0.825, 0.883, 0.93 |
| @il.miscusi (18248) | #336317 | e30_a with a carried κ4 core and scale-mode extraction | 5.176 | 1.721 | 0.301 | 0.608, 1.077, 0.56 |
| @kai_leal_gonzalez (18246) | #334821 | V29 with a two-term (3,1) closure | 5.264 | 2.072 | 0.254 | 0.597, 0.894, 0.67 |
| @504aldo (18218) | #330093 | V29 (section 6, step 1 reruns it) | 5.418 | 2.133 | 0.254 | 0.580, 0.869, 0.67 |
| @DD_team (18250) | #335813 | V29-route closure with exact cost cuts | 27.33 | 3.206 | 0.853 | 0.115, 0.578, 0.20 |
#336537 has the lowest adjusted error of the eight, 5.9% below the next, @hyojun_kwon’s #335413, at 0.941× its adj and 0.762× its raw. Chain B alone, #335301, ranks first among the chains of all eight write-ups that run without a learned terminal corrector or a layerwise mean-correction table, at adj 3.494e-9, 1.3% below #334316 in adj and 18.9% in raw. Every chain the eight write-ups report below it carries one of the two: #336537; hyojun_kwon’s graded corrector steps #334717, #334793 and #335413, and #336611 (3.044e-9), the same corrector on LordProcciuto’s third-party e30_a engine; and the e30_a builds #335981 and #336226 (il.miscusi, 3.058e-9), which correct their means layer by layer with a ridge table of 87 features. LordProcciuto’s e30_a is a public MIT repository (GitHub - siming767-creator/whestbench-lordprocciuto: Team LordProcciuto's best WhestBench 2026 submission (e30_a, score 3.1163e-9), MIT · GitHub), graded as #335981 at 3.116e-9. #336611 and #336226 are cited in write-ups without being bound to them. Chain B keeps V29’s mean-correction table off; what it fits on public networks are the scalar closure constants of the propagation law (V29’s λ and reference ratios, SSC’s 9 coefficients), as every V29-derived chain does. #336317 holds the lowest raw in the table; #336537 reaches 0.608× its adj on 0.56× its compute. Write-ups quote multipliers against different baselines (2.1× against ARC’s reference algorithm for #333534; our 0.830 against chain B’s previous steps), so the graded numbers are the ones that stand side by side. C/B is each page’s “budget used” over the 50 leaderboard networks (the 0.169 elsewhere in this write-up averages all 100 graded networks).
- ARC (Wu, Lecomte, Winer, Robinson, Hilton, Christiano). The paper measures K=3 and K=4 at width 4–256 and 2–12 hidden layers and conjectures the bound c_K (L/n)^K (its Section 5). We run the same unaugmented K=4 at 1024 × 16, made storable, and give the first empirical confirmation at the challenge scale of the bound’s fourth- and third-order width exponents, n^−3.95 (3.68–4.08) and n^−3.08 (2.95–3.17) over widths 256–1024 (section 2). By ARC’s account this recursion is a deduction–projection estimator whose projection steps must keep what later computation reads; we measure every projection step that way, through the whole 16-layer chain, and meter the first compressed K=4 chains end to end (section 5).
- hyojun_kwon (18154, Phase 1; 18255, Phase 2). ARC’s K=4 converges where it is computable and hits a memory wall at 61.5 GB at 256 × 16. At 1024 × 16 our storage runs it within a 60.4 GB GPU peak, with 120 GB of births in pinned host memory. His Phase 2 write-up lists low-rank fourth-order cores and oracle ceilings of pair fourth-order blocks among its negatives, and binds a learned corrector on an unchanged state, the structure of ours: the side-by-side above compares the two tier by tier.
- @504aldo (18218, V29 and findings log). F29’s two cells flip with unaugmented K=4 (section 2); V29 is chain B’s starting point; V22’s shared basis carries to K=4 (section 3).
- kai_leal_gonzalez (18246). Inside V29 the exact (3,1) slice lowers final MSE 18.1% and a two-term closure recovers about a sixth; section 5 puts the same slice into chain B and sets it against full K=4.
- il.miscusi (18248). Carrying κ4 with memory makes a K=3 engine 2.25× worse until a scale mode is removed, then 0.380× graded: carried fourth-order state pays in a third chain as well, once it holds what later reads use.
- Artyom_Mazur (18249). At 256 × 16, three layers of κ4 blocks plus a scale-mixture block keep 89% of K=4’s gain. At 1024 × 16 a three-layer lifetime is 67× worse than full K=4 on network 0: width changes how long sources must live.
- Nurlanggs (18252). On V29 with an all-distinct κ4 edge term, oracle injections and memory cuts find that the all-distinct third cumulant needs rank about 3n/4 and the full depth, so the per-source leg representation of factorized K=3 ends near 3.8e-9. Section 5 reaches the same object from the K=4 side and names the next cut there: κ3’s representation.
- jamesrahenry (Phase 1 replication). A third-cumulant carrier with cosine 0.83 to the truth makes the final estimate 4.5× worse than its probe at 256 × 32, the split that section 5’s first-order read weight shows again at K=4 and 1024 × 16.
9. How it was developed, and how to reproduce it
I designed the methodology, propositions, and experimental architecture, led the research, approved every submission, and reviewed all experimental results, the PDF, and this post. LLM coding assistants were used for code implementation, cloud job orchestration, verification scripting, and manuscript editing. Every K=4 code path matches ARC’s reference implementation to 1e-15 at n = 40, every MSE is recomputed from saved predictions by verify.py, and all 1,935 numbers in the PDF sit in data/numbers.csv. I take full responsibility for the accuracy and completeness of this write-up and code.
Repository: GitHub - logxio/fourth-cumulant · GitHub (MIT). estimator.py is #336537 byte for byte, with its SHA-256 and the grader’s package versions in manifest.json. data/numbers.csv rebuilds every table and Figures 1–5. verify.py checks the estimator hash, refits the nine SSC coefficients and rescores saved predictions against the public truth. REPRODUCE.md lists every command with the value it must print, and THIRD_PARTY.md names the origin of each file.
python smoke.py # #336537 on one generated network: prediction and FLOP count
python make_tables.py all && python make_figures.py # every table and figure from data/numbers.csv
python verify.py # hash, SSC refit, saved-prediction rescoring
bash report/data/frozen_s16/run.sh # the pre-registered checks up to width 192, laptop CPU, about 40 min
Width-1024 K=4 runs need one GPU with at least 80 GB and 240 GB of host memory; everything else above runs on a laptop.
10. Three structural laws, and implications for mechanistic estimation and trained networks
The measurements on both sides of the 2⁴¹-FLOP budget reduce to three empirical laws, each stated with the reading that would refute it.
Law 1, unaugmented fourth-order scaling. The fourth-order gain grows with n/L, at the exponents of the bound Wu et al. (2026, Section 5) conjecture. At depth 16 full K=4’s error falls as n^−3.95 (3.68–4.08) and augmented K=3’s as n^−3.08 (2.95–3.17); at 1024 × 16 the ratio is 11.8× on all five official networks, and at n/L = 8 unaugmented K=4 wins on all six frozen networks (2.0–3.7×) where augmented K=4 loses 1.6–4.0× (section 2). This law would be refuted if unaugmented K=4 failed to beat augmented K=3 on frozen networks at n/L = 8.
Law 2, downstream readout observability. The value of carried state is what later layers read from it into the final estimate, where to first order a truncation δ incurs readout error <g, δ> with downstream readout vector g, rather than how exact it is where it is born (Proposition 2). Augmented K=3 handed the K=4 chain’s exact covariance and κ4 reaches 0.29 of its error, the K=4 chain 0.08; the exact (3,1) slice inside chain B recovers at most 10.0% of the K=4 log gain; a two-layer lifetime costs 109×, and a k = 64 energy basis loses 84% of the norm and ends 51× worse (sections 3, 5, 7). This law would be refuted if a memoryless fourth-order closure reached the K=4 chain’s 0.08× on official networks 0–3.
Law 3, two-family bottleneck. Beyond the budget, 96% of the K=4 units from layer 4 and about 2× of the error each sit in C·κ3 and non-tree births (27,390 and 30,705 of 60,549 units; removing either costs 2.08× and 1.95×), so the next cut is their representation; rank-4 non-tree pairs already take the best point from 75× to 15× the units at which it ties chain B (section 5). This law would be refuted if a K=4 subset without both families beat chain B’s adjusted error inside 1,024 units on official network 0.
Implications for mechanistic estimation and trained networks. The unit of mechanistic progress is carried state that later layers read rather than local closure order. Approaches that discard downstream readout alignment lose the chain’s gain, including memoryless fourth-order closures (0.29× against the chain’s 0.08×), local-norm truncation (84% of the norm lost, 51× worse) and uniform lifetime cuts (109×, 67×, 41×, 12× worse), whereas gains come from cross-layer factored transport (exact K=4 at 1024 × 16 on one GPU, Proposition 1), a dense old part of κ3 plus young factors (the measured next cut for the 87–88% of metered units spent in n⁴ reads) and readout-guided pruning. Full unaugmented K=4 at 1024 × 16 is the first empirical confirmation at the challenge scale of the fourth- and third-order exponents of the c_K (L/n)^K bound conjectured by Wu et al. (2026, Section 5), and, measured through the whole chain as the deduction–projection account asks of each projection step, the first end-to-end metered compressed K=4 chains reach 6.95e-9 at 525,045 units with rank-32 matrix pairs, 30.6% fewer than dense. Inside the budget that pruning multiplies the grader’s mean adjusted error by 0.830, lower on all 50 leaderboard networks, and brings #336537 to 3.145e-9, first among the eight public Phase 2 write-ups, while #335301 ranks first among their chains without a learned corrector or mean-correction table. As a testable prediction derived from these random MLP results for trained networks, learned weight anisotropy can concentrate higher-order deviations in selected directions, hence local-norm truncation may lose directions that downstream layers read; exact factored transport Q_{b->l} and the first-order readout inner product <g, delta> offer a mathematical framework to track them.
Credits
The K=3 and K=4 algorithms, their term tables and the reference code are ARC’s (Wu, Lecomte, Winer, Robinson, Hilton and Christiano, arXiv:2605.05179; mlp_cumulant_propagation, MIT, commit 93d091a4); every K=4 prediction in sections 1–4 evaluates them deterministically. Chain B starts from 504aldo’s V29 (MIT, commit 1a4083f) with its κ4 regeneration law and adaptive rule; it carries koushik_rudra’s V34 dead-unit pruning and the V37 cost blocks and final-layer row band, published in acco-cyber’s repository (MIT, one team), and Puffi’s SSC closure form. ngmitam also found the leaf-8 Strassen setting of step 2 and published it in the 18218 thread. Keenan Pepper’s public corpus trains the corrector. Ours: the storage and n³ forms that run full K=4 at 1024, every ablation, patch, late start and preregistration, the budget ledger and its reduction to two births, the matrix-pair compression and the end-to-end meter, active-set trimming, young-atom pruning, the cost work of steps 3, 4, 6 and 7, and the corrector.
Questions and counter-measurements are welcome. If you test the C_jk κ3_ijk term inside your own chain, find a C·κ3 read that does not widen with κ3, or get a compressed K=4 chain under 602 units, a reply here with the number would be the best outcome of this post.
Yan Su (AIcrowd: Susupro) · hi@yansu.me