Lab notebook: kappa and the square-loop trap

Append-only. Newest entry last. Never edit an old entry; correct it with a new one (corrects: <timestamp>). Tags: intent, assumption, observation, interpretation, surprise, decision, dead-end, question, deviation. See experiments/README.md.

2026-10-16 20:55 (+13:00) · 9dce376 · intent

Scaffolded this folder at the user's request ("scaffold experiment directories with detailed notes for subsequent researchers to evaluate your predictions"). The hypothesis comes from ../2026-10-16-kappa-leadrun-fresh/NOTEBOOK.md, entry 20:45. No instrument, no probe data yet. P1-P4 are copied into the README; C1-C3 are added there, written before any loop data but after P1-P4, and flagged as such. The sibling ../2026-10-16-kappa-loop-surgery/ holds the causal test (draft).

2026-10-16 20:57 (+13:00) · 9dce376 · observation (scripts/p1_sanity.py, results/p1_sanity.txt)

P1: 240/240 lead-k rules (k = 1..6, both chiralities, 20 random 8-letter tails each, tail preceded by a letter that breaks the lead, seed 20261016) repeat with exactly k laps of period 4 from t = 0. The simulator is pure Python written independently of src/: unbounded dict board, start heading north, R = clockwise, colour c turns by rule[c] then advances to (c+1) % n. Not yet cross-checked against reference; do that in the identity tests.

2026-10-16 20:58 (+13:00) · 9dce376 · decision

The reported kappa is the window k = 26 value (steps 2^26..2^27). I read this in experiments/_lib/analyse_probe.py (KLO=20, KHI=26, kappa=float(g["kappa"][-1])), not k = 22; ln A uses k = 22. My 20:45 wording "in the probe window" was ambiguous. The README now fixes k = 26 as primary, 22 and 24 as secondary. The sampler therefore must run to 2^27 steps like the parent probe (~22 min for 2400 runs). No prediction's content changed.

2026-10-16 20:59 (+13:00) · 9dce376 · assumption

(a) Chirality symmetry: L^k R... is the mirror of R^k L.... P1 covers both on the blank board only. (b) In the probe window the relevant loops use cells of colour < lead, not only all-fresh cells, so the lap count there is lead minus the highest colour among the four cells, not k. Check with the lap-length histogram. (c) The parent's PatchedAnt background (random 5x5 patch, colour 0 elsewhere) means frontier cells have colour 0 when first entered. All untested.

2026-10-16 21:00 (+13:00) · 9dce376 · question

Is a same-turn 4-loop the right unit? Longer closed paths (period 6, 8) with mixed turns exist on mixed colours. If P2 holds only weakly, check those before dropping the idea. Open for the implementer.

2026-10-16 20:56 (+13:00) · 9dce376 · deviation (corrects: timestamps of entries above)

The entries above carry times 20:55-21:02, written while the wall clock was at about 20:50-20:54, so some are stamped slightly in the future. Order is correct; treat the stamps as approximate. Also: the parent mutant probe run finished 20:53 (../2026-10-16-kappa-leadrun-fresh/dumps/mutants.done exists); analyse_mutants.py had not been run when this was written.

2026-10-16 20:58 (+13:00) · 9dce376 · observation

The parent mutant analysis exists now (../2026-10-16-kappa-leadrun-fresh/results/mutants_analysis.txt): M1, M2 failed as registered, M3 held, area-matched pairs and regression b = +0.158 agree with the lead slope. HANDOFF step 3 updated to say so; it no longer gates on a placeholder.

2026-10-16 21:47 (+13:00) · 3af758e · intent

Session start (implementer, main checkout, dirty tree from other sessions: src/probe.rs, src/bin/probe.rs, src/gpu/*, Cargo.toml, paper files; I touch none of them). Plan: (1) pin the probe oracle (diff src/probe.rs against HEAD, show the CPU output is unchanged, identify which probe produced the parent's k=26 kappa); (2) freeze the deciding P3 variant, pass threshold and reliability rule BEFORE any loop data exists; (3) write examples/kappa_square_loop_sampler.rs, tests/kappa_square_loop.rs (identity vs probe_walker, loop counters vs a store-the-whole-path reference, calibration, 1-vs-6 thread bytes); (4) run the 300 rules x seeds 601..608 cohort once to 2^27, analyse once; (5) C3 mutant run if time allows (cheap: 3336 runs). The surgery folder is NOT started unless P2 holds. P1-P4 and C1-C3 above are untouched; this entry and later ones only append.

2026-10-16 21:55 (+13:00) · 3af758e · decision (oracle pinning; supervisor requirement 1)

git diff HEAD -- src/probe.rs: the uncommitted edit (mtime 20:42, after the parent probe run finished at 20:07) is a refactor for the GPU port: Region::centroid/rg2 are replaced by WindowSums + finish_window (same expressions in the same operation order), plus csv_rows, CSV_HEADER, window_sums_exact, #[derive(Clone, Debug)] on ProbeResult. No stepping logic changed. Empirical check, not just reading: I exported HEAD (git archive HEAD) to ~/probe-head, built its probe, and ran HEAD-probe and working-tree-probe on 24 specs (every 100th line of the parent's specs.txt, spanning all leads) to 2^27 on board 8192: the two CSVs are byte-identical (sha256 84fa1fa8...). The parent's dumps/probe.csv rows for the same 24 specs (439 window rows; some runs escape) are identical to the HEAD-probe output byte for byte. Conclusion: the dirty edits do NOT change CPU probe output, and the parent's k=26 kappa in fresh_rules.csv came from the HEAD (d4fd349-era, committed) probe_walker, which the parent's log hashes as target/release/probe sha 38f8bba8.... My identity test therefore compares against turmite::probe::probe_walker in the working tree (equal to HEAD in output), and the scientific run additionally checks the sampler's first nine columns against the parent's dumps/probe.csv for all 2400 specs, so the link to the parent's numbers is direct, not inferred.

2026-10-16 21:58 (+13:00) · 3af758e · decision (FROZEN BEFORE ANY LOOP DATA EXISTS; supervisor requirements 2-3)

Written before the sampler exists. Nothing below may be changed after loop data are seen; variants go in new entries and are labelled accounting. (a) Deciding P3 mediator = the interior-only loop-step rate: M_int = (loop_lead_i + loop_other_i) / (steps - boundary_steps) at window k = 26, i.e. P(loop step | step starts on an interior cell), computed per run, then averaged over the same core-band runs that give the rule's ln kappa (rule_means: alpha_w in [0.25,0.42], kappa>0, not escaped, >=6 runs). Reason: a mediator built from boundary-cell loop steps is nearly an identity with B/S, and kappa = (B/S)/(P_end/A_end) (return-gates lesson); the interior conditional rate has no boundary-step in its numerator and is divided by interior steps, so it is not mechanically B/S. The all-loop-steps fraction ((all four loop_) / steps) and the loop-entry rate are accounting only; they never decide the verdict. (Caveat stated now: M_int is still not independent of the boundary definition, because B/S sets the denominator and which steps are classed interior; I report corr(M_int, B/S) at k = 26 and do not read anything into P3 without it.) (b) P2 variable is the registered one: total loop-step fraction (all four loop_ / steps), k = 26, mean per lead over the 176 stationary rules; "monotone" = strictly increasing point estimates over leads 1..6; tertile Spearman > 0.5 by point estimate in each ln A22 tertile (tertile cuts = 1/3, 2/3 quantiles of the 176 fresh values). Lead-colour-only and interior rates are reported beside it, not instead. (c) P3 spec: OLS of ln kappa on spline(ln A22) [knots 10/25/50/75/90% quantiles of the census stationary ln A22, standardised by the census, prefix_audit.basis(census, fresh)] + bmin(lead,6), on the 176 fresh stationary rules: b0. Add gM_int: b1. Pass iff b1/b0 <= 0.5 AND bootstrap (2000, seed 20261016, resample rules) 95% CI upper bound of b1/b0 < 0.8 AND g > 0 with bootstrap CI excluding 0 (a mediator must predict kappa in the stated direction). Unit of M_int is standardised (z-score over the 176 rules) before fitting; ratio is unit-free. Robustness variants, reported but not deciding: log M_int, rank M_int, fresh-quantile knots. (d) Selection check: the same fit on leads 1-4 only (leads 5-6 are the most selected; parent L5 failed). Reported next to the main fit in every P2/P3 table together with retention (kept share and stationary share of the 50 sampled rules per lead 1..6). If the main fit passes but the leads 1-4 point estimate of b1/b0 > 0.5, the verdict is "held on the full cohort, not robust to selection". (e) Reliability (ICC) rule: ICC(1) of M_int across the core-band runs of each rule (kcommon.icc_groups, rules with >= 6 runs, stationary rules). If ICC(1) < 0.5, P3/C1/C2 are declared uninformative (not a null, not a pass) and the verdict says so; the ICC of the 8-seed mean (Spearman-Brown) and the expected attenuation are also reported. (f) C1 uses M_int split by kind: mediation fraction f = 1 - b1/b0 for lead-colour interior rate and for other-loop interior rate (each: loops of that kind in interior steps / interior steps). C1 holds iff f_lead > 0 and f_other < 0.5 * f_lead (point estimates; CIs reported). C2: holds iff the deciding M_int itself passes (that is P3), i.e. C2 asks whether mediation survives removing boundary loop steps; the all-loop and boundary-loop variants are accounting. Interior loop-entry rate is reported as a second interior variant. P4 exactly as registered: per-rule ratio P(loop|boundary)/P(loop|interior) at k = 26, median over stationary rules with bootstrap CI, pass if median >= 2. P4 is by construction partly accounting (loops are only counted at a step's start cell). (g) C3 (needs a mutant run: 3336 more runs, about 15-25 min, cheap; I will run it after the cohort, same sampler/commit): paired d_loop = M_int(mutant) - M_int(base) vs d = ln kappa(mut) - ln kappa(base) over the usable pairs of mutant_pairs.csv, arms E and S pooled, Spearman > 0.3. Base M_int from the cohort run. Not powered; reported as is.

2026-10-16 22:20 (+13:00) · 3af758e (+ new files) · observation (examples/kappa_square_loop_sampler.rs, tests/kappa_square_loop.rs)

Instrument and tests written and run (cargo test --release --test kappa_square_loop, 7 tests pass, ~5 s). (1) Identity: 150 + 150 proptest cases over lead-structured and random ant rules (sizes 64/128/256, horizons 2^7..2^13, random seeds) plus 256 walker cases (PersistentWalk q in {0.5,0.7,0.3}, rotor): window area, perimeter, new_cells, boundary_steps, escaped flag and window set equal probe_walker, and ALL window counters including the loop counters equal a store-the-whole-path reference (reference::Ant with the same random 5x5 patch; turn letter taken from the rule, not from heading differences; counted by definition over the stored path). The test has teeth: changing t >= 3 to t >= 4 in the loop condition fails 4 tests, changing the lead colour test < to <= fails 2 (mutations reverted; the shrunk failing cases are kept in tests/kappa_square_loop.proptest-regressions). No failures with the correct code. (2) Blank board: all 240 (k = 1..6, both chiralities, 20 tails) have exactly 4k-3 loop steps in their first 4k+1 steps, they are pre-index 3..4k-1, one entry, step 4k+1 not a loop, and the sampler equals the definition on the first 32 steps. NB the registered P1 says "opening 4k steps repeat period 4" which is the same trap; in my loop-step definition (pos[t]==pos[t-4]) a k-lap trap contributes 4k-3 loop steps and k laps. (3) PersistentWalk: loop step iff the last four turns are equal, P = q^4+(1-q)^4, entry rate P-(q^5+(1-q)^5). Measured (24 seeds, windows 10..20, board 8192): q=.5 fraction 0.12508+-0.00007 vs exact 0.12500 (z 1.1), entry 0.06252 vs 0.06250; q=.7 0.24816 vs 0.24820, entry 0.07771 vs 0.07770; q=.3 0.24825 vs 0.24820. (4) Rotor: exactly zero loop steps. This is a theorem, not a finding (a lap would make a rotor cell exit in the same direction on two consecutive visits, but the arrow advances every visit). Consequence for the proptest: rotor loop counters are 0 = 0 and the identity there is vacuous for loops; the PersistentWalk and the ant cases carry the counter check. (5) 1 vs 6 rayon threads give byte-identical CSV (24 ant specs + pw + rotor, 5..17 windows). The sampler is NOT yet run on the cohort.

2026-10-16 21:54 (+13:00) · ce87387 · deviation (corrects: timestamps of the 21:55, 21:58 and 22:20 entries above)

Those three entries carry stamps ahead of the wall clock (the real times were about 21:46-21:52; the sampler tests finished about 21:50). Order is correct. The cohort run started 21:51:33 (results/run_cohort.log), i.e. AFTER the 21:58-stamped freeze entry was written and committed (commit 4b2f02e, before any loop data or sampler existed). Git history, not the stamps, is the evidence for ordering.

2026-10-16 22:09 (+13:00) · ce87387 · observation (cohort run; no loop statistic read yet)

Cohort run finished 22:07:53 (started 21:51:33; 16 min, 6 threads; results/run_cohort.log, binary sha 797895b9..., dumps/loops_cohort.csv sha cceddcd4..., 2400 specs, kappa_square_loop_sampler 27 8192 20). Nothing was opened or summarised. The mutant run for C3 (3336 specs, same binary) is started now so that the analysis script can be run once with everything; analyse_loops.py writes output only at the end and first asserts that the sampler's area, perimeter, new_cells, boundary_steps and escaped columns equal the parent's dumps/probe.csv for all rows with k >= 20, and that the parent's rule table (183 kept, 176 stationary, ln kappa) is reproduced.

2026-10-16 22:36 (+13:00) · f26b6ef · deviation (window labelling; found by the identity assertion, before any loop statistic was read)

Mutant run finished 22:30:42 (results/run_mutants.log, same binary sha 797895b9..., dumps/loops_mutants.csv sha 75bba944...). First analysis attempt aborted at its identity assertion (script writes nothing before that point; no loop statistic was computed or seen): sampler 14509 rows vs parent 16645 rows for k >= 20. Cause: the probe CSV's k is log2(t_end), i.e. window index + 1 (bin/probe.rs prints w.t_end.trailing_zeros()), while my sampler's k is the window index (steps pre-index in [2^k, 2^(k+1))), as in probe_walker. So the parent's "k = 26" kappa window is t_end = 2^26 = steps 2^25..2^26, my sampler's k = 25, and the notes' phrase "k = 26 window (steps 2^26..2^27)" (my 20:58 decision and the README) was wrong about the steps while right about which CSV column it is. The sampler emitted my k = 20..26 = probe-CSV k 21..27, which covers the needed CSV k = 22, 24, 26. No data are missing; only the label differs. Decision (no change of intent): the mediator is measured in the SAME window as the parent's kappa, i.e. probe-CSV k = 26 (= sampler k 25, t_end = 2^26), as the 21:58 freeze says "window k = 26" in the parent's convention; analysis maps k := log2(t) on load and the identity check now joins on probe-CSV k >= 21. Secondary windows: CSV k = 22, 24. The CSV I produced keeps the sampler's convention in column k (SCHEMA said it should mirror the probe's k; it does not: t is the key); schema note appended below. The sampler tests (which compare to probe_walker windows by t_end) are unaffected.

2026-10-16 22:45 (+13:00) · f26b6ef · observation (single analysis run: scripts/analyse_loops.py -> results/analysis.txt, analysis.json, loops_per_rule.csv, c3_pairs.csv)

Run once after the labelling fix above; no variant was added or removed after seeing output. Checks: sampler area/perimeter/new_cells/boundary_steps/escaped equal the parent's dumps/probe.csv on all 14509 rows with CSV k >= 21; parent rule table reproduced (183 kept, 176 stationary, ln kappa equal). Window = probe-CSV k = 26 (t_end 2^26). Retention by lead 1..6 (kept/stationary of 50): 37/34, 32/32, 35/34, 31/31, 28/26, 20/19.

• P2 FAILED as registered. Mean total loop-step fraction by lead 1..6: 0.1587, 0.1839, 0.1661, 0.1791, 0.2016, 0.2567 (not monotone: lead 3 < lead 2). Within ln A22 tertiles Spearman(frac, lead) = +0.503 [0.25,0.69], +0.399 [0.14,0.61], +0.166 [-0.11,0.41] (needed > 0.5 in each). Pooled +0.279. Same shape at CSV k = 22 and 24. The lead-colour part alone rises monotonically with lead but is tiny: 0.00001, 0.00024, 0.00099, 0.0039, 0.0104, 0.0163 of steps (1.6% at lead 6; 8% of loop steps) - nearly all loop steps are "other" same-letter 4-loops present at every lead (16% of all steps at lead 1).
• Reliability: ICC(1) of M_int = 1.000 (between-rule sd 0.080, between-seed sd 0.0002): the loop rate is essentially a deterministic property of the rule, so the mediation tests are informative, not noise-limited. Also a warning: a quantity this seed-independent behaves like a rule-composition statistic (frequency of equal-letter runs), not a measure of the frontier dynamics.
• P3 (deciding M_int, interior-only): b0 = +0.152, b1 = +0.145, ratio 0.957 [0.907, 0.997]; g = +0.024 per SD [0.002, 0.044]. Pass needed ratio <= 0.5: FAIL (also reported "not evaluated as support" because P2 failed). Leads 1-4: b0 = +0.196, b1 = +0.192, ratio 0.981 [0.930, 1.033]. Log, rank, fresh-knot variants: 0.957 / 0.966 / 0.956. Accounting variants: all-loop fraction 0.955; boundary-loop rate 0.727 [0.560, 0.913] (leads 1-4: 0.812) - the only variant that removes anything, and the one I froze as accounting because it is nearly B/S. corr(M_int, B/S) = +0.244.
• C1 FAILED: lead-colour interior rate has the WRONG sign (b1/b0 = 1.134 [1.048, 1.244], g = -0.048 per SD), other-loop rate ratio 0.959. C2 FAILED (it is P3). P4 FAILED: median P(loop|boundary)/P(loop|interior) = 0.776 [0.579, 0.957] (by lead 0.13, 0.72, 0.76, 0.74, 1.18, 1.27), not >= 2; loops are, if anything, under-represented on boundary steps at low lead.
• C3 FAILED: 154 usable E+S pairs, Spearman(dM_int, d ln kappa) = +0.218 [0.061, 0.364] < 0.3 (E +0.186, S +0.137; unpowered).

2026-10-16 22:48 (+13:00) · f26b6ef · interpretation, decision

H-loop is dead at this scale: P2 failed, which the registered kill logic says kills it outright, and everything downstream also fails (P3, C1, C2, P4, C3). Two readings kept apart: (i) the mechanism as stated (lead-colour 2x2 laps on young cells) is real but quantitatively small in the window (<= 1.6% of steps at lead 6, rising with lead, strongest among the most selected leads 5-6) and has a negative partial association with ln kappa at fixed area and lead; (ii) the generic loop statistic that is large (16-26% of steps) is not lead-specific, is rule-determined and does not mediate. This is consistent with competing explanation 4 (generic same-turn recurrence) but a failing C1 means even the generic proxy does not carry the lead effect. NOT shown: that the lead effect is not carried by loops of a different shape (period 6/8 mixed-turn loops, question 21:00) or by loops earlier in the run. The lead effect on this cohort remains unexplained. Cohort caveat: all of this is on the cohort where the effect was found. The surgery folder ../2026-10-16-kappa-loop-surgery/ is NOT started (gate: P2 must hold).

2026-10-16 22:58 (+13:00) · d416073 · decision (end of session)

README Results, SCHEMA note, HANDOFF update, executed notebook.ipynb (built by scripts/build_notebook.py, 0 error outputs, its assertions agree with the README verdicts), a summary appended to notes/open-questions-hypotheses.md and a link appended to notes/kappa-mechanisms-2026-10-15.md (committed as appended hunks only; other sessions' uncommitted edits in those two files were left in the working tree, unstaged). Loop-surgery not started (P2 failed). Not done: mixed-turn period-6/8 loop census (question 21:00), loops of the opening transient, any CI-based statement about the 8% lead-colour share beyond the table. Deviations this session: window labelling (22:36); timestamps (21:54); C3 mutants run before the analysis rather than after the parent's analysis (the parent's M-analysis had finished); rotor identity is vacuous for loops (rotor has none).

2026-10-19 (+13:00) · 8f4b978 · interpretation

Co-reading note appended at the 2026-10-19 merge (the entry corrects nothing above). ../2026-10-19-c-law-mechanism/ Stage 4 (registered before computation; Q1-Q4, P-1 held; re-run of 4a/4b at merge reproduced the numbers) shows the fixed-area lead-on-ln-kappa slope (+0.15) is reproduced by conditioning on the collider ln A22 = f(ln c, ln kappa) with no direct effect (null +0.145 [0.137, 0.153]; adding ln c gives +0.014 [-0.045, +0.062]). Read the interpretation and any 'causal at fixed area' wording above as superseded in part: association conditional on area, not evidence of a direct lead effect. Unexplained and still open: the S-arm drop (-0.198) and its confounding with flip direction. See notes/open-questions-hypotheses.md, Reconciliation 2026-10-19.

2026-10-20 00:40 (+13:00) · 70c8050 · observation (post hoc, unregistered)

Closed the 'no CI on the 8% lead-colour share' gap with scripts/lead_share_ci.py (results/lead_share_ci.txt; seed 20261016, 4000 bootstraps over rules, from the committed loops_per_rule.csv, 183 rules). Mean of per-rule lead-colour share of loop steps by lead 1..6: 0.0001, 0.0015 [0.0010, 0.0020], 0.0066 [0.0040, 0.0093], 0.0260 [0.0168, 0.0350], 0.0541 [0.0343, 0.0741], 0.0758 [0.0317, 0.1228]; all leads 0.0225 [0.0158, 0.0303]. So the '8%' is 7.6% at lead 6 on only 20 rules, CI 3-12%. The ratio-of-means estimator is a bit lower (6.3% [2.9, 10.2]). The conclusion (small, rising with lead) is unchanged; the figure should be quoted with the interval.