Where kappa is not: four places I looked and found nothing

I went looking for what carries the rule-determined part of boundary occupation (kappa) and mostly found places where it is not. Most of the entry is about what failed, including a prediction of mine.

The budget, and what it left

Over 2185 stationary rules, ln kappa has a standard deviation of 0.349. A variance budget (post hoc, no new runs) splits it roughly into 0.15 start-to-start noise, 0.47 to 0.54 predicted by the first eight letters of the rule, and 0.31 to 0.38 rule-determined but unexplained. Those shares are rough to about ±0.1: three different cohorts, no propagated uncertainty. They say nothing about why the median sits near 0.8 of the uniform-occupation reference.

Place 1: the end of the rule, and the letter counts

I registered four predictions before running the feature comparison and all four held. On 162 fresh stationary rules (training on 2185 census rules, no overlap), held-out R² is +0.62 for a lookup on the first eight letters, −0.11 for the last eight, and +0.08 for length, R count and run structure. Composition adds +0.008 to the prefix. An unregistered control (leave-one-out inside the training set, where lengths match) gave −0.07 for the suffix, so the suffix failure is not just a length-alignment artifact. What is left unsaid: the prefix lookup is a family-membership predictor and families overlap between train and test.

Place 2: the lead alone

This is the one I got wrong. I predicted that the lead (the initial run of R) alone would reach R² of at least 0.15, because the lead experiment found a slope of 0.15 to 0.24 in ln kappa per lead step. In the run-structure test it scored −0.004. Three of four registered predictions failed. Keys made of the first two, three and four runs reach 0.25, 0.34 and 0.48, against 0.62 for eight letters.

I should have seen it coming. The earlier result was conditional on area, and the c-law work later showed that conditioning on area reproduces the lead slope without any direct lead effect. A conditional association is not an unconditional predictor, and I registered the second as if it were the first. The prefix signal looks distributed over the opening, not one early number. A caveat I did not control for: run keys and letter-prefix keys differ in how finely they split the rules.

Place 3: dropout in the extend-lead arm

Only 53% of lengthened-lead mutants survived the filters. Reweighting by base lead and length changes nothing (mean change in ln kappa −0.058, naive −0.062), but worst-case bounds on the missing mutants run from −0.35 to +0.19. That interval includes both signs, so the registered failure of the extend-lead predictions is really “uninformative”, not “no effect” (notes). The shorten-lead arm (−0.198) is sturdier but its upper bound only just reaches zero (+0.015).

Place 4: lead-colour loops

The square-loop mechanism needed lead-colour 2×2 laps to be a real share of the steps. They are not: about 7.6% of loop steps at lead 6, with a bootstrap interval of 3.2% to 12.3% on 20 rules, and 2.3% over all leads (experiment). The interval is the new part; the earlier headline gave only the point value.

What would change my mind

That the unexplained third is a pile of noise I have mislabelled as rule-determined: the ICC behind it comes from a different cohort and a different start condition than the census spread. That the prefix signal is just ancestry, with training and test sharing families, so it would vanish on unseen families. I have not tested either, and both would shrink what I have said above.

Update, 2026-10-20 (later): the ablation

The per-position ablation was registered and run the same day. I predicted the information would be front-loaded, and that prediction failed: removing letter 5 costs the most held-out R² (0.40), letters 2 to 7 each cost 0.2 to 0.4, and only letter 8 is nearly free (0.07, interval includes 0). A permutation control that keeps the group count agrees. Because the lookup is a conjunctive key, this says the letters matter together, not that each contributes its own share; the losses add up to far more than 0.62. The original next-test paragraph below is left as written.

Update, 2026-10-20 (later still): it is the family, not the letters

Two further registered tests changed how I read the ablation. An additive model over letter positions scores 0.11 on the fresh rules and adding all pairwise interactions 0.35, against 0.62 for the lookup (additive). Then I held out whole 6-letter prefix families in cross-validation on the census (family hold-out). The lookup falls from 0.59 to −0.16, below simply predicting the mean, and the pairwise model from 0.30 to −0.69. I had predicted the pairwise model would keep some signal across families; it did worst of all.

So the “prefix signal” is family membership: rules that share an opening behave alike, and I have not found a letter-level rule that transfers to an opening the model has never seen. The paper already says its prefix test is transfer within families, and this makes that caveat the main point. Caveats: only 43 families of uneven size, held-out family means differ from the training mean (which drives much of the negative R²), and nothing here shows there is no structure within a family. The ablation numbers above should be read as “which letters distinguish families”, not “which letters cause the effect”.

Next test

A per-position ablation of the eight-letter lookup with a control that matches the number of groups, registered first. It would say which of letters five to eight carry the extra 0.15 over four runs, and whether run structure was ever the right unit.