Write-up addendum (2026-08-10): the stabilizer schedule is state, not depth.
Following the deadline extension for mechanistic contributions, I’ve added a second dated addendum to my Phase 1 write-up (submission #314695, PDF re-rendered; the graded-time text remains unchanged, same convention as the 2026-08-04 erratum). Everything in it post-dates both the submission deadline and my private re-evaluation selections, and none of it was used in any graded submission. Scripts and committed results are in a separate folder of the replication repo: arc-whitebox-replication/post_phase1_state_keyed at master · jamesrahenry/arc-whitebox-replication · GitHub
The short version. The graded estimator’s only depth-32 dependency is its stabilizer coefficient table: 512 doubles indexed by layer number. I refit the same 16 correction slots as a function of measured per-layer state — participation ratio of the propagated covariance (the rank-contraction profile from §6, now used as a control signal), the ReLU operating point (mean/std of μ/σ), and off-diagonal correlation mass — with no layer index anywhere and 128 parameters instead of 512. Three results:
-
Parity at depth 32. Held-out 4.74e-6 vs the deployed table’s 5.21e-6 (5/10 per-net wins; a validation-selection caveat on the exact figure is stated in the addendum). The claim is parity with 4× fewer parameters, not superiority.
-
Both schemes extrapolate to depths 48 and 64 — and the reason is the finding. I expected index-clamping past row 31 to fail out-of-range. It doesn’t, because the chain’s state trajectory saturates past roughly L30: rank contraction completes, drift slows, and the last row stays approximately valid forever. The index table survives because it was secretly a state lookup whose state stops changing. Consistent with this, depth-64 error is lower than depth-32 — the saturated regime is effectively low-dimensional. State-keying gets the same transfer with zero fitted data past layer 32, which is the cleaner statement: the schedule is a function of where a net is in its rank-collapse trajectory, not how deep it is.
-
On 200 unseen public-split nets, the improvement concentrates in the hard tail. Overall 6.92e-6 vs 7.60e-6 (125/200 wins), but split by difficulty the improvement is 1.06× in the body vs 1.20× in the p90+ tail, corr(log difficulty, log improvement) = +0.31. Mechanistic read: part of what makes the hard nets hard is not irreducible variance but correction-model mismatch on nets whose internal state at layer l is atypical for layer l. This was directly prompted by @keenanpepper’s public-50 distribution analysis — it suggests the upper tail carrying ~36% of full-split error is, in part, the atypical-state-trajectory population, and yields a falsifiable prediction I haven’t tested: tail membership should be predictable from weights alone, via distance of a net’s state-descriptor trajectory from the ensemble band. No MC needed. If anyone wants to check that against the 512-net split before I do, please be my guest.
One negative result for the methods record, consistent with §5 of the write-up: a pooled fit (all layers at once, iterated) diverged within two refit iterations; the sequential fit-on-the-chain’s-own-trajectory structure was stable throughout. The keying can change; the trajectory-fitting requirement cannot.
Added at posting time (2026-08-11), since several post-close write-ups landed after this addendum was rendered: @amalgonim’s write-up (submission 324409, topic 18151) independently finds that Edgeworth corrections degrade the chained closure (3.8×) — §1c of my write-up, replicated with different machinery — and measures the closure family’s floor at 8.8e-7, putting a number on §5’s “well short of the sampling frontier.” Meanwhile both @jonah_butterbaugh’s write-up (submission 326094, topic 18147: exact Gaussian structure is “effective as a control for subtraction but fails as a direct predictor”) and @Cipo’s post-mortem (topic 18152: control variates built on inter-layer correlations) arrive at the role §8 proposed for this machinery — the analytic chain’s frontier value is as a control/null model, not a predictor. The state-keyed line above is aimed at exactly that role: a null you can evaluate per-net, cheaply, from the weights alone.
Henry_ARC-WhiteBox_Phase1_Writeup_Submission-314695_addendum-20260810.pdf (148.5 KB)