Field notes / Experiment guide
Why does a longer opening run make fresh ground harder to enter?
The number of boundary steps needed to discover a new cell is nearly linear in the rule’s opening-run length. Where does that law come from?
New to these ants? Start here.
An ant moves on a grid of coloured squares. Each colour tells it to turn left (L) or right (R). It advances that square to the next colour, then steps forward. A rule such as RLR is the list of turns, starting at colour zero. On a blank grid the same rule always makes the same trajectory.
Territory means every cell visited so far. Its boundary consists of visited cells with an unvisited neighbour. A highway is a repeating pattern of motion that carries the ant away. A tether means repeated returns to a small home region.
Growing like the cube root of time means taking eight times as many steps to double a linear size. The rotor walker is a comparison model that follows and rotates arrows stored on cells. A seed makes a random starting pattern reproducible; the ant itself can still move deterministically.
We distinguish c, the chance of discovering a cell per boundary step, from kappa (κ), time at the boundary normalized by perimeter divided by area. Explaining their values is harder than using them to account for growth.
A closer look
The idea under test
Young boundary colours might supply a local fresh-ground mechanism, testable by tracing entries, changing the interior and adding background colour.
What we learned
The fresh-ground relation predicts well, but a simple isolated-edge account is incomplete. The later analysis reproduces the earlier fixed-area occupation effect through conditioning, without requiring a direct effect on kappa.
What this leaves open
Some high-background predictions become untestable because ants leave the board. Mutations change letter direction as well as run length, so no isolated causal claim about kappa follows.
Read the evidence and reproduction recipe ↗
Inside the experiment
Methods, registered predictions, exact commands and saved analysis. These are the working research documents; expand them for the detail behind the introduction.
Methods and original experiment record
Mechanism of the c law: why 1/c = 1.36 + 0.70 lead, and what it implies for kappa
Continues experiments/2026-10-19-kappa-chiral-edge (kappa chiral-edge hypothesis refuted; the opening same-letter
run "lead" sets c, the fraction of boundary steps that open a new cell). Base commit f0b96b7. Notation: rule letters
s_0 s_1 ...; colour k turns by s_k; lead m = length of the opening run of equal letters; a boundary step is a
step that starts on a visited cell with >= 1 unvisited 4-neighbour (as in src/probe.rs, which includes the rims of inner
holes); a birth is a step onto a never-visited cell; c = births / boundary steps.
Stage 1 (registered 2026-10-19 18:50 (+13:00), before any aggregate of edgestats_sample.csv)
Seen before registering (disclosure)
One row group of experiments/2026-10-13-edge-statistics/results/edgestats_sample.csv (rule RRRLRRLLLLR: colours 1, 2 turn R like
colour 0 and have ~2000 edge steps with ~5 births each; colour 3, the first L, has 1310 edge steps and 1296 births), the old
c_letters.txt / c_model.py (pooled chirality model), and the previous experiment's c-law numbers. Nothing aggregated.
Hypothesis (mechanism)
Fresh ground at a growing edge is circled by the ant on cells that are still young. While the cell colour is in
0..m-1 the turn is s_0 (the circulating reads), the ant keeps going round a loop whose outside is fresh and enters
nothing; the first read of the opposite letter (colour m) sends it into the fresh ground (a birth). Boundary-step
bookkeeping: 1/c = sum_k r_k with r_k = boundary steps that start on colour-k cells per birth (an identity, used as an
instrument check only). The claim is about the shape of r_k: r_0 ~ 1; r_k flat at a plateau rho for the
circulating colours 1..m-1, with rho equal to the lead slope b; so the lead's linear cost is the per-read cost
of circulating reads, not a shift of the post-lead tail. a (1.36) then collects r_0, the killing read and the tail.
Linearity of 1/c in lead out to lead 8 (known from the previous experiment) is a constraint any model must meet, not a
prediction: a geometric-survival model r_k ~ phi^k would saturate and is already excluded.
Data and frozen choices
- Profile data:
experiments/2026-10-13-edge-statistics/results/edgestats_sample.csv(300 rules; per rule and colour: edge steps, new cells, ahead/left/right fresh; last octave of a blank-board run; the producingexamples/edgestats.rsis in the repo, the K used is not recorded, see G0). Rules joined toprobe_summary.csv(census) by string. - Sets for the profile claims: sample rules with
cls == blob,c > 0,|g_k| < 0.05("stationary blob", as inc_law.py), lead from the rule string (uncapped here; capped at 6 where a regression is stated). - Forward model scoring is frozen to
c_law.py: census = stationary blob rows ofprobe_summary.csv(n = 2185), random half splitnp.random.default_rng(20261019).permutation(A fit, B score), and the 176 fresh rules ofexperiments/2026-10-16-kappa-leadrun-fresh/results/fresh_rules.csv(kept & stationary, lengths 13-14) for out-of-sample. Baselines to beat: lead-only ln c held-out R^2 0.842 (split B), lead law fresh R^2 0.877,c_model.pypooled chirality (about 0.66 in its own held-out set), prefix-8 lookup 0.950 (an overparameterised ceiling, not a model).
Predictions
G0 calibration (instrument gate). For sample rules also in the census, Pearson r(c_edge, c_census) >= 0.95 and median
|relative difference| <= 0.06, where c_edge = sum new_cells / sum edge_steps per rule. If it fails the sample's numbers
are reported as offset and nothing is concluded from them; the profile is then re-measured with the Stage 2 tracer.
- D1 (where births happen). Rules with
2 <= m < n: median over rules of the fraction of births at colours1..m-1is <= 0.05, and the fraction at colours{0, m}is >= 0.70. ~65% (one rule seen). - D2 (flat plateau). Rules with lead
m >= 4: the median over rules ofmax_k r_k / min_k r_koverk = 1..m-1is <= 1.35, the pooled mean ofr_kover those rules andklies in [0.55, 0.85], and is within 0.10 of the census b (0.696). ~55%. - D3 (decomposition of the slope). Over stationary blob sample rules, regress
Lpart = sum_{k<m} r_kandTpart = sum_{k>=m} r_kseparately onmin(m, 6): slope of Lpart in [0.55, 0.85] and |slope of Tpart| <= 0.15, i.e. at least 80% of b is carried by the lead reads themselves. ~60%. - D4 (killing read). Rules
2 <= m < n: median of births/edge_steps at colourm>= 0.85. ~65% (one rule seen: 0.99). - D5 (colour-0 read). Median over rules of births/edge_steps at colour 0 lies in [0.45, 0.55] (random-heading value 1/2). ~65% (one rule seen: 0.51).
Forward model K3 (3 parameters + letters up to 12). 1/c = e + rho * sum_{k=1}^{min(n-1,12)} prod_{j=1}^{k-1} phi_j, with
phi_j = 1 if s_j == s_0 (circulating) and phi_j = psi otherwise (a killing read thins the surviving cells).
(e, rho) by OLS and psi by grid in [0, 1] (step 0.01) on half A; ln-c R^2 scored on half B; then refit on the full census and
score c on the 176 fresh rules.
- K3-1 held-out R^2 of ln c on B >= 0.90 and >= lead-only (0.842) + 0.04. ~40%.
- K3-2 fresh R^2 (c) >= 0.90 (lead law 0.877). ~35%.
- K3-3 fitted
psi < 0.7. ~60%. A pass means a few-parameter survival kernel carries most of the information of the 8-letter prefix lookup; a fail (especially K3-1) means the tail depends on the letters in a way a run/kill kernel does not capture and is reported as such.
Commands (WSL, repo root)
uv run experiments/2026-10-19-c-law-mechanism/scripts/stage1_profile.py # G0, D1-D5 -> results/stage1_profile.{txt,json}
uv run experiments/2026-10-19-c-law-mechanism/scripts/stage1_k3.py # K3 -> results/stage1_k3.{txt,json}
Stage 1 results (run 2026-10-19 ~19:05-19:25, commit f0b96b7 + uncommitted scripts; results/stage1_profile.{txt,json}, stage1_k3.{txt,json}, post hoc stage1_explore.txt)
G0 PASS: 259 sample rules in the census, corr(c_edge, c_census) 0.988, median |relative difference| 0.0041. (The K of the sample run is still not recorded.) Stationary blob sample: 207 rules (leads 1:120, 2:39, 3:29, 4:12, 5:3, 6:3, 10:1; n for D2 is only 19 and for lead >= 5 is 7).
| pred | outcome |
|---|---|
| D1 births at circulating colours <= 5%, at {0,m} >= 70% | FAILED: circulating share median 0.142; {0,m} share 0.790 passes |
| D2 flat plateau | FAILED (all three needed): max/min median 1.371 (> 1.35), pooled mean 0.798 (in range), 0.102 from b (> 0.10) |
| D3 Lpart slope in [0.55, 0.85], |Tpart slope| <= 0.15 | FAILED narrowly: Lpart 0.864 [0.79, 0.99], Tpart -0.151 [-0.19, -0.12]; total 1/c = 1.348 + 0.713 m |
| D4 births/edge at colour m >= 0.85 | FAILED: median 0.516 (the one rule seen, 0.99, was unrepresentative) |
| D5 colour-0 birth probability in [0.45, 0.55] | HELD: median 0.530, IQR [0.513, 0.564] |
| K3-1 held-out R^2(ln c) >= 0.90 and +0.04 | FAILED: 0.875 (lead-only 0.842) |
| K3-2 fresh R^2(c) >= 0.90 | HELD: 0.913 (lead law 0.877), median rel err 0.031 |
| K3-3 psi < 0.7 | HELD: psi 0.16, e 1.12, rho 0.69 |
What the data show (post hoc stage1_explore.txt, n = 120/39/29/12 rules for leads 1-4): births equal toward-side-fresh exactly; about 52% of births
are colour-0 reads (immediate continuations of a birth), independent of lead; the killing read (colour m) carries 29-36% of births with
birth probability ~0.55; circulating reads at colours 1..m-1 carry the rest of the lead cost and their share r_k decays (~0.88 per read) with r_1 rising with m
(0.72, 0.85, 0.895, 0.889). The registered picture "flat plateau of circulating reads whose height equals b" is wrong in detail.
Stage 2 (registered 2026-10-19 18:41 (+13:00), before running the tracer on any list)
New instrument: src/edge_trace.rs (trace, trace_blank), driver examples/c_law_mechanism_trace.rs. Property tests tests/edge_trace.rs check the tracer against
(a) probe_walker's last-octave window (births, boundary steps, fresh neighbours, area, perimeter: exact equality) for ants, SparseBgAnt, rotor, random walk and
(b) a naive HashMap re-implementation of every field (per-colour tallies, lag, N_b resolved/censored, chains) incl. a prefilled torus; a deliberate one-line mutation
(lag + 1) was caught by all five tests. The window is the last half of the run (the probe's last octave of 2^24 steps), board 4096 x 4096 for the blank mode.
Frozen lists (scripts/make_lists.py -> results/lists/): sample.txt (207 stationary blob edgestats rules), calib.txt (rotor seeds 1-32, rw seeds 1-32),
valid_candidates.txt (600 random primitive rules, n uniform in 11..16, seed 20261020, canonical s_0 = R, none in the census, edgestats or fresh sets),
band.txt (12 census stationary blob rules per lead 1..5, 10 <= n <= 14, seed 20261019).
Commands (WSL, repo root; T=./target/release/examples/c_law_mechanism_trace, build with cargo build --release --example c_law_mechanism_trace):
$T blank 24 4096 < results/lists/calib.txt > results/trace_calib.jsonl
$T blank 24 4096 < results/lists/sample.txt > results/trace_sample.jsonl
$T blank 24 4096 < results/lists/valid_candidates.txt > results/trace_valid_raw.jsonl
$T band 30 256 1024 512 20261019 < results/lists/band.txt > results/trace_band.jsonl
Validation filter, frozen: keep a candidate if not escaped, births >= 200 and 1.3 <= area/area_t0 <= 1.9 (blob-like growth; 2^(2/3) = 1.59). Band filter, frozen: keep a rule if births in the window >= 5000 and area at the end < 262144 - 2000 (band not exhausted; the initial prefilled area is 131072).
Predictions (probabilities are my honest odds)
- T0 calibration (instrument). (a) rotor, 32 seeds: mean c in [0.47, 0.52], between-seed sd <= 0.03 (notes/calibration.md: 0.498). (b) random walk, 32 seeds: mean c in [0.40, 0.48] (notes: 0.44). (c) the 207 sample rules: median |c_trace / c_census - 1| <= 0.06, none escaped. ~75%, ~70%, ~85%.
- L1 (loop picture). Rules with m >= 3: pooled over boundary reads at colours 1..m-1 that have a prior read, the fraction with lag exactly 4 is >= 0.60. ~55%.
- L2. Rules with m >= 2: of boundary reads at colour m (with prior read), fraction with lag 4 >= 0.50. ~50%.
- L3. r_1^{(4)} := (colour-1 boundary reads with lag 4) / births. The excess of r_1 between leads >= 2 and lead 1 (means over rules) is carried >= 70% by lag-4 reads. ~50%.
- C1 (chains are geometric). Pooled over sample rules with m <= 4: q_L = P(chain >= L+1 | chain >= L): q_1 in [0.45, 0.58], |q_2 - q_1| <= 0.10 and |q_3 - q_1| <= 0.10. ~50%.
- N1 (Little's law). Per rule, E[N_b] (resolved + censored cells born in the window) vs 1/c: median |ratio - 1| <= 0.15, and the slope of mean E[N_b] on min(m, 6) within 0.15 of the slope of 1/c (0.713). ~55%.
- K4 transition kernel (defined now, parameters are fitted, not predicted):
1/c = e + sum_{k=1}^{min(n-1,12)} prod_{j=0}^{k-1} phi(t_j), wheret_jis the pair (type s_j, type s_{j+1}) with type C if the letter equals s_0 else K; phi in {CC, CK, KC, KK} (5 free numbers). Fit by least squares on 1/c over the full census (for B scoring: on half A). Built after seeing the Stage 1 profile (so it is informed by it, not blind). - V validation (new rules, tracer c, filter above). Compare, with census-fitted parameters, the lead law (1.363 + 0.696 min(m,8)), K3 and K4 on R^2(c) and median relative error. V1 lead law R^2 >= 0.80 on V (~65%). V2 K3 >= lead law + 0.02 (~55%). V3 K4 >= 0.90 and >= K3 + 0.01 (~40%). V4 K4: phi_CC in [0.80, 1.0] and phi_CK <= phi_CC - 0.10 (~50%). K4-census K4 fit on A, held-out R^2(ln c) on B >= 0.90 (~45%).
- Flat band (torus 256 x 1024, fresh rows 0..511, interior i.i.d. uniform colours 0..n-1 pre-visited, 2^30 steps, window = second half). Regress 1/c_band on min(m,5) over kept rules: S1 slope in [0.55, 0.85] (~45%). S2 intercept in [1.0, 1.7] (~45%). S3 across kept rules R^2(c_band, c_census) >= 0.60 (~50%). S4 median births/edge at colour 0 in [0.45, 0.58] (~60%). S5 (direction) median c_band / c_census > 1: a flat front has no hole rims, which add boundary steps without births (~55%). Reading: S1-S3 hold => the law is a local flat-edge property independent of the ant-built bulk and of curvature; S1/S2 off but S3 holds => geometry shifts a and b but rank order is local; S3 fails => bulk memory of the ant-built field matters. No claim of an exact flat-edge solution is made; the band is a Monte-Carlo stand-in with a finite i.i.d. interior.
Erratum on timestamps (added 18:42, +13:00). The Stage 1 registration header says 18:50 and the NOTEBOOK entries from 18:34 on carry times I estimated rather than read from the clock;
the wall clock at 18:40 shows they run ahead of real time by up to ~1 h. The order of events is as written (Stage 1 registered before edgestats_sample.csv was aggregated; Stage 2 registered
before any tracer list was run). Real clock readings were: session start 18:30, Stage 2 registration 18:41. From here on times are read from date.
Stage 3: sparse-background dose-response (registered 2026-10-19 18:45 (+13:00), before any sbg run; Stage 2 band run still in progress, no Stage 2 result seen yet)
Question: is the lead dependence of c a fresh-ground effect? SparseBgAnt (tested in tests/sparse_bg.rs, and traced exactly by tests/edge_trace.rs): each cell starts, with probability d,
in a uniform non-zero colour in 1..n-1, else blank. A born cell that starts at colour j reads the rule rotated by j, so at d = 1 no cell is ever read at colour 0 on its first visit.
Design (frozen): the 60 band rules (results/lists/band.txt, 12 census stationary-blob rules per lead 1..5, 10 <= n <= 14); densities d in {0, 0.02, 0.05, 0.1, 0.2, 0.4, 0.7, 1.0};
background seeds 1..4 (d = 0: seed 1 only; sbg:RULE:SEED:0 is the blank ant); blank 24 4096 tracer, same window as Stage 2 (1920-ish runs, spec sbg:RULE:SEED:DENS).
A run is kept if not escaped, births >= 200 and 1.3 <= area/area_t0 <= 1.9 (as in V). c per (rule, d) = mean over kept seeds, needing >= 2 kept seeds (d = 0: 1).
Primary cohort (survivor control): rules kept at every density; secondary: all kept cells at each d. I report kept fraction by lead and d. Lead = min(lead, 5).
Per d: OLS of 1/c on lead gives a(d), b(d); bootstrap over rules (2000, seed 20261019).
Parameter-free mixture prediction (locality): 1/c_pred(R, d) = (1-d) (1/c)census(R) + d mean{j=1..n-1} (1/c)_census(rot_j R), rot_j R = R[j:] + R[:j] mirror-canonicalised (swap L/R if it starts with L);
census = blob rows with c > 0; rotations missing from the census are dropped and a rule needs >= half of its n-1 rotations to be predicted.
This is the prediction that a born cell's reads depend only on its own read sequence (neighbours' phases irrelevant).
Predictions
- B1
b(1.0) / b(0) <= 0.25in the primary cohort (lead stops mattering when no cell is read at colour 0). ~65%. - B2 b(d) is non-increasing along the grid within +0.07 per step. ~70%.
- B3 (mixture) over primary-cohort cells with d >= 0.1 and a prediction: R^2 of observed vs predicted 1/c >= 0.50 and median |relative error| of 1/c <= 0.20. ~40%.
- B4 (floor) the intercept a(d) stays in [1.0, 2.0] at every d: a birth's own first read is always a boundary read, so 1/c cannot fall toward 0 (the intercept does not "fall to zero" even if the slope does). ~80%.
- B5 (selection, descriptive with threshold) the kept fraction at d = 1 is lower for lead >= 4 than for lead <= 2 by >= 0.10. ~35%.
Reading: B1+B2 hold => the lead slope is a fresh-ground effect; B3 holding on top means the effect is local to each cell's own read sequence (a single-cell mechanism); B3 failing with B1 holding means the
slope is fresh-ground but neighbours' phases matter (loops are collective and break when mixed). B1 failing => the lead slope is not (only) a fresh-ground effect.
Commands (WSL, repo root):
bash experiments/2026-10-19-c-law-mechanism/scripts/run_stage3.shthenuv run experiments/2026-10-19-c-law-mechanism/scripts/stage3_analyse.py.
Deviation to Stage 2 flat band (added 18:50 (+13:00), before any band c(m) was computed; see NOTEBOOK)
The registered band run (2^30 steps, window = second half) consumed every band before the window opened (void; file kept as results/trace_band_2e30_void.jsonl), so S1-S5 were untested.
Replacement, frozen before reading any band c: run L = 24..29 (scripts/run_band.sh -> results/trace_band_L{L}.jsonl), per rule use the smallest L whose end-of-run band consumption
(area - 131072)/131072 is >= 0.5; keep the rule if that consumption is <= 0.9 and births in the window >= 5000. Predictions S1-S5 and their thresholds are unchanged.
A pilot on three non-band sample rules (scripts/band_pilot.sh) was used only to see the consumption time scale (about 2^25-2^27, 3x between rules).
Also: T0b was run at 2^20 steps (the registered 2^24 escaped the 4096 board for all 32 random walks); C1 was registered as geometric chains and failed (chains have length <= 3).
Stage 2 results (run 2026-10-19 18:41-18:50, commit f0b96b7 + uncommitted working tree; results/stage2_analysis.{txt,json})
| pred | outcome |
|---|---|
| T0a rotor c in [0.47, 0.52], sd <= 0.03 | HELD: 0.4977, sd 0.0007 (32 seeds, 2^24, none escaped; notes: 0.498) |
| T0b random walk c in [0.40, 0.48] | HELD at the deviated configuration (2^20 steps): 0.4480, sd 0.0062; all 32 runs at the registered 2^24 escaped the 4096 board |
| T0c sample median |c_trace/c_census - 1| <= 0.06, no escapes | HELD: 0.0026, 0 escaped (207 rules) |
| L1 lag-4 fraction of circulating reads >= 0.60 | HELD: 0.784 (47 rules with m >= 3, 1.90M reads); other lags 8: 0.057, 12: 0.032; all lags even |
| L2 lag-4 fraction at colour m >= 0.50 | FAILED: 0.483 (744k reads) |
| L3 >= 70% of the r_1 excess (lead >= 2 vs 1) is lag-4 reads | HELD: r_1 0.723 vs 0.879, 93% lag-4 |
| C1 chains geometric | FAILED: q_1 0.779, q_2 0.455, q_3 0.000; no chain is longer than 3 (4.36M chains; P(len 1,2,3) = 0.22, 0.43, 0.35, mean 2.13) |
| N1 Little: median |E[N_b]/(1/c) - 1| <= 0.15 and slope within 0.15 | HELD: 0.016 (median ratio 0.984); slopes 0.700 vs 0.712 |
| K4-census held-out R^2(ln c) >= 0.90 | FAILED: 0.863 (K3 0.875, lead-only 0.842) |
| V1 lead law R^2 on V >= 0.80 | FAILED: 0.711 (n = 378 of 600; median rel err 0.066) |
| V2 K3 >= lead law + 0.02 | HELD: 0.785 (+0.074; median rel err 0.042) |
| V3 K4 >= 0.90 and >= K3 + 0.01 | FAILED: 0.800 |
| V4 phi_CC in [0.8, 1], phi_CK <= phi_CC - 0.1 | FAILED: phi = (CC 0.916, CK 1.2 [at the bound], KC 0.103, KK 0.072), e 0.574 |
| S1 band slope in [0.55, 0.85] | HELD: 0.678 |
| S2 band intercept in [1.0, 1.7] | HELD: 1.399 |
| S3 corr^2(c_band, c_census) >= 0.60 | HELD: 0.999 |
| S4 median colour-0 birth probability in [0.45, 0.58] | HELD: 0.522 |
| S5 median c_band/c_census > 1 | FAILED: 0.998 |
(Deviations: T0b configuration; the void 2^30 band run replaced by the frozen L-selection rule; see above and the NOTEBOOK.)
Stage 2c: ablation of the band interior (registered 18:52 (+13:00), before running)
Result S3 (rule-by-rule agreement to 0.2%) is almost too clean, so it gets a control that can break it. Same 60 rules, the same per-rule L chosen in the main band run (frozen from stage2_analyse.py, printed in stage2_analysis.txt),
same torus, but the pre-visited interior colours are drawn differently (driver band mode with a 7th argument):
Z all interior cells colour 0; Y i.i.d. uniform over colours 1..m (m = the rule's lead; the young colours); H i.i.d. uniform over colours m..n-1 (old/killing side). The U (uniform 0..n-1) run is the main band run.
- A1 median |c_Z / c_U - 1| <= 0.05. ~45%. A2 same for Y. ~45%. A3 same for H. ~45%.
- A4 In every interior variant the fitted slope of 1/c on min(m,5) stays within [0.55, 0.85]. ~55%. If A1-A3 all hold, the interior's colours are irrelevant and c is a property of the rule on fresh ground. If one fails, the entry cell's (old cell's) colour distribution enters c, which would be the route by which letters beyond the lead matter. Kept rules: the same 60 (consumption in the window may differ with the interior; I report the consumption and births; a rule is dropped from a variant only if the band is >90% eaten or < 5000 births, as before).
Stage 3 deviation (added 19:00 (+13:00), after seeing kept/escape fractions only; no c(d) computed yet)
At d >= 0.7 almost every run escapes the 4096 board in 2^24 steps (kept 0.00-0.08 at d = 0.7, 0.00-0.02 at d = 1.0; at d = 0.4 kept 0.40-0.75, lead 1 escaping 50%). The registered primary cohort (valid at every d) is empty and B1 is untestable as registered. Replacements, frozen before any c(d) is computed: primary cohort = rules valid at every d <= 0.2; secondary = d = 0.4 survivors (selection-biased, reported separately). B1' for d in {0.1, 0.2}: |b_obs(d) - b_mix(d)| <= 0.10, where b_mix is the slope of the mixture prediction on lead at that d. B2' b(d) non-increasing within +0.07 along d <= 0.2. B3' mixture R^2 >= 0.50 and median |rel err| <= 0.20 on cells d in {0.1, 0.2}. B4' intercept in [1.0, 2.0] for d <= 0.2 (and, secondary, at 0.4). B5 reported at d = 1.0 (as registered) and at d = 0.4.
Stage 3 results (run 2026-10-19, commit b7b4758; results/trace_sbg.jsonl, stage3_analysis.{txt,json})
Primary cohort: 60 rules, valid at every d up to 0.2. 1/c = a + b lead (b CI over rules): d=0: 1.397 + 0.677 [0.646, 0.711]; 0.02: 1.439 + 0.646; 0.05: 1.489 + 0.605; 0.1: 1.537 + 0.549 [0.513, 0.583]; 0.2: 1.619 + 0.450 [0.412, 0.485]. Secondary, d=0.4 survivors (46 rules): 1.331 + 0.689 at d=0 on the same rules; 1.729 + 0.287 [0.235, 0.341] at d=0.4. Mixture-predicted slopes (census rotations): 0.678, 0.665, 0.645, 0.613, 0.548.
| pred | outcome |
|---|---|
| B1 b(1.0)/b(0) <= 0.25 | NOT TESTABLE: at d >= 0.7 the ant leaves the board in 88-100% of runs (kept 0-8%); no blob regime to measure c in |
| B1' abs(b_obs - b_mix) <= 0.10 at d = 0.1, 0.2 | HELD (0.064, 0.098; the observed slope falls faster than the mixture, the second barely passes) |
| B2' b(d) non-increasing within +0.07, d <= 0.2 | HELD: 0.677, 0.646, 0.605, 0.549, 0.450 |
| B3' mixture R^2 >= 0.50 and median rel err <= 0.20 (d = 0.1, 0.2) | HELD: R^2 0.869, 0.067 (120 cells). Null "no density effect": 0.613, 0.102 |
| B4' intercept in [1.0, 2.0] | HELD: 1.40-1.73 (rises, does not fall) |
| B5 kept fraction lower for lead >= 4 by >= 0.10 | FAILED at d = 1.0 (0.00 vs 0.01) and at d = 0.4, where the sign is opposite (0.48 for lead <= 2 vs 0.68 for lead >= 4) |
Stage 4: reconcile with kappa (registered 2026-10-19 19:03 (+13:00), before any Stage 4 computation)
Definitions. kappa = (boundary steps / steps) / (perimeter / area); growth identity dA/dt = c kappa P/A. So c (births per boundary step) and kappa (boundary occupancy) are different factors of the area growth rate, and ln A22 depends on both: if A ~ (c kappa t)^g (g about 0.68 from A^(2-beta) ~ c kappa t with beta ~ 0.52), then ln A22 is a collider of (ln c, ln kappa). Lead moves ln c strongly (slope about -0.24 per step) and, in the census, does not move ln kappa (marginal R^2 0.001). Conditioning on ln A22 then induces a positive lead effect on ln kappa of size about 0.24 g beta_(kappa|A) ~ 0.24 x 0.68 x 1.0 ~ +0.17 per step with no direct effect at all (observed +0.15). This is a hypothesis, not a result: it is what I am about to test. What I have seen: the leading-run README numbers (pooled b = +0.158, c-coefficient 1.037, arm means E -0.062, S -0.198, C -0.006 with dA -0.292, +0.013, 0.000) and the first-experiment summary; I have not computed anything below.
4a: collider check on the 176 fresh rules (existing data, no runs)
Data: experiments/2026-10-16-kappa-leadrun-fresh/results/fresh_rules.csv (kept and stationary, n = 176). Y = lnk, Z = ln c, W = lnA22, X = min(lead, 6). Fit W = alpha + g_c Z + g_k Y + e. Permutation null (2000 draws, seed 20261019):
permute Y across rules, rebuild W* = alpha + g_c Z + g_k Y_perm + e (each rule keeps its residual), regress Y_perm on 1, X, W*. H0 built in: Y independent of X and Z; the only link of Y to the rest is through W.
- Q1 the observed coefficient on X in OLS Y ~ 1 + X + W lies inside the null's central 95% interval and |mean(null) - observed| <= 0.05. ~50%.
- Q2 the same for the coefficient on W: |mean(null) - observed| <= 0.25. ~50%.
- Q3 adding Z: OLS Y ~ 1 + X + W + Z gives a coefficient on X with |b| <= 0.07 or a bootstrap CI (2000, over rules) containing 0. ~60%. Q1 and Q3 both holding means the "lead raises kappa at fixed area" association is reproduced by conditioning on a collider with no direct effect; Q1 failing (null << observed) means a direct fixed-area effect is needed.
4b: growth identity on the Part B mutant pairs (no new kappa runs)
c for bases and Part B mutants is measured by the Stage 2 tracer on the blank board (2^24 steps, window = last octave; deterministic; rules in results/lists/s4_trace_rules.txt). With pairs from experiments/2026-10-16-kappa-leadrun-fresh/results/mutant_pairs.csv
(kept and stationary, 257 pairs): d = d ln kappa, dA = d ln A22, dc = d ln c (tracer). Fit dA = g (dc + d) + noise through the origin on all pairs.
- Q4 g in [0.5, 0.9], and the arm means of g (dc + d) are within 0.08 of the observed mean dA for each of E, S, C. ~50%. This is an identity-level check that the E/S asymmetry in dA (unexplained in the leading-run write-up) comes from the c law: E lowers c by about 0.24 per added lead step while d is about 0, S raises c while d is negative.
4c: lead-preserving early flips (the P1 proposal, now registered)
Bases: the 157 stationary fresh rules with lead k in 1..5 (same as Part B). Mutants: flip the single letter at index k+1 (arm P1) and at index k+2 (arm P2), when that index < n - 1; the lead stays k. Dropped if not primitive or already seen.
Run with the unchanged pipeline: probe 27 8192 on RULE@SEED, seeds 601..608, core band alpha_w in [0.25, 0.42], kappa > 0, >= 6 core runs, stationary |g_k| < 0.05 (same as Part B); paired with the base's existing row. d = ln kappa(mutant) - ln kappa(base), dA, and tracer dc.
Statistics: mean d with a cluster bootstrap over bases (a base contributes both its mutants; 2000, seed 20261019); area-adjusted residual r = d - beta_hat dA with beta_hat from the pooled regression d ~ 1 + dA.
- P-1 (lead is the active factor) |mean d| <= 0.08 and its 95% CI inside [-0.15, 0.15]. ~40%.
- P-2 (early position/composition, not lead) mean d <= -0.10 with the CI upper bound < -0.03 (the M3-like drop appears with the lead untouched). ~35%. (P-1 and P-2 are exclusive; ~25% is "neither".)
- P-3 area-adjusted mean residual within +/-0.08 of 0. ~45%.
- P-4 over P pairs, dA = g (dc + d) has g in [0.4, 1.0] and R^2 >= 0.5. ~50%.
- P-5 retention (kept and stationary mutants / mutants) reported against Part B's 0.53-0.68; descriptive. Dropout is a limit on every claim here.
Reading: P-1 holding says an early flip that leaves the lead alone does nothing, so M3 (S arm) is a lead effect (but still one-sided: E showed none); P-2 holding says M3 was an early-letter effect that any early flip produces and the lead has no special role.
No causal claim beyond these two arms and these 157 bases; the intervention changes letters, not the "lead" in isolation, and I say so in the write-up either way.
Commands:
uv run experiments/2026-10-19-c-law-mechanism/scripts/stage4_lists.py, tracer ons4_trace_rules.txt,bash experiments/2026-10-19-c-law-mechanism/scripts/run_p1.sh(probe),uv run experiments/2026-10-19-c-law-mechanism/scripts/stage4_analyse.py(needsdumps/p1_probe.done; exits without writing if absent).
Stage 4 results (run 2026-10-19; results/stage4_analysis.{txt,json}, results/p1/)
| pred | outcome |
|---|---|
| Q1 observed conditional lead coefficient inside the no-direct-effect null | HELD: observed +0.148, null mean +0.145 [+0.137, +0.153] (176 fresh rules) |
| Q2 area coefficient within 0.25 | HELD: +1.073 vs null +1.035 |
| Q3 adding ln c removes the lead coefficient | HELD: +0.014 [-0.045, +0.062]; growth fit ln A22 = 11.45 + 0.687 ln c + 0.807 ln kappa, residual sd 0.092 |
| Q4 growth identity on Part B pairs | HELD: g = 0.805; arm dA observed/predicted E -0.292/-0.318, S +0.013/+0.050, C 0.000/-0.004 (257 pairs) |
| P-1 abs(mean d) <= 0.08, CI in [-0.15, 0.15] | HELD: -0.007 [-0.078, +0.060] (177 pairs, cluster bootstrap over bases) |
| P-2 mean d <= -0.10 | FAILED |
| P-3 area-adjusted residual within 0.08 | HELD: +0.007 [-0.022, +0.034] |
| P-4 dA = g(dc + d), g in [0.4, 1.0], R^2 >= 0.5 | HELD (after a NaN fix, see NOTEBOOK): g 0.809, R^2 0.875 |
| P-5 retention | 0.65 (Part B: 0.53-0.68) |
Post hoc, not registered: split by flip direction, L>R d +0.153 (n 98), R>L d -0.207 (n 79), sd of d 0.43; S and E arms are direction-confounded. Reading: the fixed-area "lead effect" on kappa is a conditioning artefact (4a) with no detectable direct effect (|b| below ~0.06 per step given ln c); lead-preserving early flips shift kappa by zero on average, but single flips move it by about 0.2 in a letter-direction-dependent way, which I cannot separate from lead in the S/E arms. No causal claim that lead sets kappa.
Erratum (added after the gap review). Stage 1 says the K of edgestats_sample.csv is "not recorded": it is K = 23 (window (2^22, 2^23]), recovered by scripts/find_edgestats_k.sh; the Stage 2 tracer window is 2^24. Real clock times and commits for all result files: results/PROVENANCE.md.
Analysis notebook: notebook.ipynb
6 saved outputs · 6 code cells. Cells are not executed during publication. Download notebook
Mechanism of the c law (1/c = 1.36 + 0.70 lead) and what it implies for kappa
Question and why it matters
In chaotic generalised Langton's ants, c = births per boundary step is almost a function of the rule's opening same-letter run (lead): 1/c = 1.363 + 0.696 lead on the census. This notebook asks why, whether it is a fresh-ground (edge) effect, and whether the earlier "lead raises kappa at fixed area" finding is real.
Registrations, failed predictions and deviations: README.md; thinking: NOTEBOOK.md; summary in notes/open-questions-hypotheses.md. Predecessor: experiments/2026-10-19-kappa-chiral-edge.
Assumptions and definitions
- Boundary step: starts on a visited cell with an unvisited 4-neighbour (as
src/probe.rs, inner-hole rims included). Birth: step onto a never-visited cell. c = births / boundary steps. Window = last octave of 2^24 steps on a 4096 board unless stated. src/edge_trace.rsis the instrument; it agrees exactly withprobe_walkerand a HashMap brute force (tests/edge_trace.rs). Held (calibration: rotor c 0.4977 vs 0.498, random walk 0.448 vs 0.44).- Assumption: Stage 2/3 filter "kept" = not escaped, births >= 200, 1.3 <= area/area_t0 <= 1.9 (blob-like). Untested beyond the counts printed below.
Provenance
Code
import hashlib, json, subprocess, datetime
from pathlib import Path
import numpy as np, polars as pl
SEED = 20261019
ROOT = next(p for p in [Path.cwd(), *Path.cwd().parents] if (p / "AGENTS.md").exists())
R = ROOT / "experiments/2026-10-19-c-law-mechanism/results"
names = ["stage1_profile.json", "stage1_k3.json", "stage2_analysis.json", "stage2c_analysis.json", "stage3_analysis.json", "stage4_analysis.json",
"stage2_decompose.txt", "p1/p1_pairs.csv", "PROVENANCE.md"]
assert all((R / n).exists() for n in names), [n for n in names if not (R / n).exists()]
sha = subprocess.run(["git", "rev-parse", "--short", "HEAD"], cwd=ROOT, capture_output=True, text=True).stdout.strip() or "unresolvable from WSL; see PROVENANCE.md: base f0b96b7, code b7b4758, results 8cc447f"
print("git", sha, "| seed", SEED, "| run", datetime.date.today().isoformat())
for n in names:
print(hashlib.sha256((R / n).read_bytes()).hexdigest()[:16], n)
J = lambda n: json.loads((R / n).read_text())
Data and method
Stage 1: per-colour profile of boundary steps from edgestats_sample.csv (207 stationary blob rules). Stage 2: tracer on 207 rules, 600 new validation rules, calibration walkers, and a flat torus band (256 x 1024, fresh rows 0..511, i.i.d. interior) with an interior ablation. Stage 3: SparseBgAnt dose-response, 60 rules x densities. Stage 4: kappa (collider check, growth identity, lead-preserving early flips run with the unchanged probe 27 8192 pipeline). Heavy computation is in Rust (examples/c_law_mechanism_trace.rs, probe).
Results
1. c is a local fresh-ground property
Code
s2, s2c = J("stage2_analysis.json"), J("stage2c_analysis.json")
print("flat band (i.i.d. interior): 1/c = %.3f + %.3f lead; corr^2 with census c %.4f; median c_band/c_census %.4f" % (s2["S"]["band_intercept"], s2["S"]["band_slope"], s2["S"]["corr2"], s2["S"]["ratio_median"]))
for k, nm in zip("ZYH", ["all-zero", "young colours", "old colours"]):
v = s2c[k]; print(f"interior {nm:14s}: median |c/c_U - 1| = {v['median_rel']:.4f}, 1/c = {v['intercept']:.3f} + {v['slope']:.3f} lead, kept {v['n']}/60")
A flat front in a random interior reproduces the census c rule by rule to 0.2%, and even an all-colour-0 interior changes it by 0.3%. It does not explain the numbers 1.4 and 0.68; it shows they are properties of the rule on fresh ground (not of curvature, holes or the ant-built bulk).
2. Where the boundary steps go
Code
p = J("stage1_profile.json")
for m, v in p["rk_by_lead"].items():
print(f"lead {m} (n={v['n']:3d}): r_k = " + " ".join(f"{x:.2f}" for x in v["mean"]))
print(); print((R / "stage2_decompose.txt").read_text())
print("lag-4 share of circulating reads: %.3f; killing reads: %.3f; chains q1,q2,q3 = %s; Little median E[N_b]/(1/c) = %.3f" % (s2["L1"]["frac_lag4"], s2["L2"]["frac_lag4"], np.round(s2["C1"]["q"][:3], 3), s2["N1"]["median_ratio"]))
1/c is exactly the number of boundary reads per born cell (Little's law holds to 1.6%); 78% of circulating reads are 4-step returns (2x2 loops); runs of consecutive births never exceed 3. The registered "flat plateau" and "geometric chain" pictures failed. It does not derive a or b: the split is descriptive, and the "first reads" bucket in the decomposition also contains returns with lag > 40.
3. Sparse background: the slope is a fresh-ground effect
Code
s3 = J("stage3_analysis.json")
print(pl.DataFrame([(float(d), v["a"], v["b"], v["mix_b"]) for d, v in s3["fit"].items() if not d.startswith("secondary")], schema=["density", "a", "b_obs", "b_mixture"], orient="row"))
print("B3' mixture R2 %.3f median rel err %.3f (null no density effect: %.3f / %.3f)" % (s3["B3p"]["r2"], s3["B3p"]["median_rel"], s3["B3p"]["null_d0_r2"], s3["B3p"]["null_d0_median_rel"]))
The lead slope falls monotonically with background density and a parameter-free single-cell mixture tracks it (observed falls slightly faster). The intercept rises, not falls. It does not reach d = 1: above d ~ 0.7 the ant leaves the board, so there is no blob regime there and the registered B1 was untestable.
4. Forward models of c
Code
k3 = J("stage1_k3.json"); v = s2["V"]
print("census held-out R2(ln c): lead-only %.3f, K3 %.3f, K4 %.3f" % (k3["split"]["lead_only_r2_lnc"], k3["split"]["K3_r2_lnc_B"], s2["K4_census"]["r2_lnc_B"]))
print("fresh 176 R2(c): lead law %.3f, K3 %.3f" % (k3["fresh"]["lead_law_r2_c"], k3["fresh"]["K3_r2_c"]))
print("new validation rules (n=%d) R2(c): lead %.3f, K3 %.3f, K4 %.3f" % (v["n"], v["lead"]["r2"], v["K3"]["r2"], v["K4"]["r2"]))
A 3-parameter survival kernel over the first 12 letters beats the lead law out of sample by 0.04-0.07 in R^2, not by enough to reach the 0.95 of an 8-letter lookup. It does not explain why letters 5-8 matter; none of my few-parameter kernels did.
5. Kappa
Code
s4 = J("stage4_analysis.json"); a = s4["4a"]
print("176 fresh rules: conditional lead coef %.3f vs no-direct-effect null %.3f %s; with ln c: %.3f %s" % (a["obs_lead"], a["null_lead_mean"], np.round(a["null_lead_ci"], 3), a["with_c"]["lead"], np.round(a["with_c"]["ci"], 3)))
print("growth fit ln A22 = %.2f + %.3f ln c + %.3f ln kappa" % tuple(a["growth_fit"]))
b = s4["4b"]; print("Part B pairs: g=%.3f" % b["g"], {k: (round(x["mean_dA_obs"], 3), round(x["mean_dA_pred"], 3)) for k, x in b["arms"].items()})
c = s4["4c"]; print("lead-preserving flips: %d pairs, mean d %.3f %s, area-adjusted residual %.3f" % (c["n_pairs"], c["mean_d"], np.round(c["ci"], 3), c["resid_mean"]))
pp = pl.read_csv(R / "p1/p1_pairs.csv").with_columns(flip=pl.struct("base", "idx").map_elements(lambda s: s["base"][s["idx"]] + ">" + ("R" if s["base"][s["idx"]] == "L" else "L"), return_dtype=pl.String))
print(pp.group_by("flip").agg(n=pl.len(), mean_d=pl.col("d").mean(), sd_d=pl.col("d").std()))
Conditioning on ln A22 (a function of c and kappa, residual sd 0.09) manufactures the +0.15 "lead effect on kappa"; a null with no direct effect reproduces it and adding ln c removes it. The E/S area asymmetry of the earlier mutants follows from the c law. Lead-preserving flips change kappa by zero on average. It does not show that lead has no causal effect on kappa: single flips move kappa by about +/-0.2 with sign set by letter direction (post hoc split), and the S and E arms are confounded with that direction.
Caveats and controls
- Registered failures: D1-D4 (flat plateau), C1 (geometric chains), K3-1, K4 (all), V1, S5, L2, B5; B1 untestable. Deviations: random-walk calibration at 2^20, band window rule, Stage 3 restated cohort, P-4 NaN fix. All in README/NOTEBOOK.
- Timestamps in early NOTEBOOK entries were estimated, not read from the clock; real times and commits are in
results/PROVENANCE.md. The edgestats sample used K = 23 (recovered). - Survivor selection limits Stage 3 (d = 0.4) and Stage 4c (retention 0.65).
What this changes / next steps
The c law is a property of the rule on fresh ground (flat band, interior ablation, sparse background). Open: derive a = 1.4, b = 0.68 (loop-formation probability x 4 steps / births per chain is the lead candidate); why letters 5-8 enter; the S/E direction confound for kappa (needs flips balanced for direction at fixed lead change); the blob regime boundary between d = 0.2 and 0.7.
Read the dated lab log
Lab notebook: mechanism of the c law (1/c = 1.36 + 0.70 lead)
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.
2026-10-19 18:30 (+13:00) · f0b96b7 · intent
Continue from experiments/2026-10-19-kappa-chiral-edge (kappa chiral-edge hypothesis refuted; lead sets c: 1/c = 1.363 + 0.696 lead, fresh R^2 0.877). Goal: explain the law mechanistically, test it, then see what it implies for kappa. New folder because the question changed from kappa to the birth rate c. Plan (one README registration per stage): (1) per-colour profile of boundary steps from the existing edgestats_sample.csv, plus a few-parameter forward model scored on the frozen census split and the 176 fresh rules; (2) a generic instrumented tracer with a HashMap proptest, calibrated on rotor and random walk, and a flat-edge (torus fresh band) simulation; (3) sparse-background dose-response with a parameter-free rotation-mixture prediction from census values; (4) reconcile with kappa; (5) write-up. User instructions that shape this: obey AGENTS.md, WSL for toolchains, register before measuring, report nulls faithfully, commit only in the worktree (no remote).
2026-10-19 18:34 (+13:00) · f0b96b7 · assumption
Disclosure of what I have seen before registering Stage 1: (a) the head of edgestats_sample.csv (rule RRRLRRLLLLR, lead 3): colour 0 has 2645 edge steps / 1339 births, colours 1 and 2 (turn R, same as colour 0) have ~2000 edge steps and ~5 births each, colour 3 (first L) has 1310 edge steps / 1296 births. (b) The older c_letters.txt and c_model.py (pooled chirality model, R^2 ~0.66 in its own held-out set). (c) the lead-law numbers of the previous experiment. NOT seen: any aggregate of the edgestats sample by lead, the profile r_k, any forward-model fit. Assumption (untested): the edgestats window and edge definition agree with probe_summary.csv's c; Stage 1 calibration G0 checks it. The producing command of edgestats_sample.csv (the K used) is not recorded anywhere I found (migrated legacy file), which is itself a reproducibility gap to report.
2026-10-19 18:36 (+13:00) · f0b96b7 · interpretation
Bookkeeping reading that motivates the registration (not a result): if births are B and boundary steps S, then 1/c = S/B = sum over colours k of r_k, r_k = (boundary steps that start on a cell of colour k)/B. This is an identity, so I use it only as an instrument check. The substantive question is the shape of r_k: the example row suggests r_0 ~ 1, r_k ~ 0.75 flat for the circulating colours 1..m-1, a smaller r_m at the first opposite-turn colour where nearly every step is a birth. If true, the slope b ~ 0.70 is the per-read plateau (the fraction of born cells that are read on every circulation) and a the sum of the non-lead terms. Alternatives: the plateau is not flat (decays) and linearity in lead comes from something else; the tail shifts with m.
2026-10-19 19:05 (+13:00) · f0b96b7 · observation (results/stage1_profile.txt, scripts/stage1_profile.py)
Stage 1 profile on edgestats_sample.csv (stationary blob subset, 207 rules; leads 1:120, 2:39, 3:29, 4:12, 5:3, 6:3, 10:1). G0 PASS: 259 sample rules in census, corr(c_edge, c_census) 0.988, median |rel diff| 0.0041, median ratio 1.0003 (so the sample is a valid instrument; its K is still unrecorded). D5 PASS (births/edge_steps at colour 0: median 0.530, IQR 0.513-0.564). D1 FAIL (median fraction of births at circulating colours 1..m-1 is 0.142, not <= 0.05; at {0,m} 0.790 passes). D4 FAIL (births/edge_steps at colour m: median 0.516, not >= 0.85; the one rule I had seen, 0.99, was unrepresentative). D2 FAIL (m>=4, n=19: median max/min of r_k over k=1..m-1 is 1.371 > 1.35; pooled mean 0.798, outside [0.55,0.85]? no, inside, but not within 0.10 of 0.696: 0.102 away). D3 FAIL narrowly on both counts: Lpart slope 0.864 CI [0.79, 0.99] (above 0.85), Tpart slope -0.151 CI [-0.19, -0.12] (|.| above 0.15); total 1/c = 1.348 + 0.713 m reproduces the census law. Identity sum r_k = 1/c holds to 1e-15.
2026-10-19 19:10 (+13:00) · f0b96b7 · observation (results/stage1_explore.txt, post hoc, unregistered)
Births = toward-side-fresh exactly. Mean profiles by lead (r_k = boundary steps at colour k per birth): m=1: .97 .72 .26 .09 .04; m=2: .98 .85 .58 .14 .07; m=3: .99 .89 .74 .53 .09 .04; m=4: .98 .89 .79 .68 .53 .08 .04 .02 (n = 120, 39, 29, 12 rules). Fraction of births at colour 0 is 0.54, 0.52, 0.51, 0.52 for leads 1-4, i.e. independent of lead (about half of births are immediate continuations of a birth, p0 = 0.52-0.56). Fraction of births at colour m: 0.36, 0.29, 0.29, 0.29. Circulating colours carry 0.04-0.11, tail 0.04-0.08. r_k for circulating k is NOT flat: it decays roughly 0.88 per read, and r_1 itself rises with m (0.72, 0.85, 0.895, 0.889). The r_m values are 0.53-0.58 for all m. Increments of 1/c per extra lead letter rise with m (0.55, 0.65, 0.71, 0.77, 0.85 from the table means).
2026-10-19 19:15 (+13:00) · f0b96b7 · interpretation
(confidence moderate) The cost per extra lead letter is the extra circulating read, but it is a decaying survival of the circulation, not a flat 0.70 plateau; the registered flat-plateau picture is wrong in detail, right that the lead reads carry most of the slope (Lpart 0.86 of total 0.71, partly offset by a tail that gets cheaper by 0.15 per lead letter). The universal numbers are p0 ~ 0.52 (births come in chains of mean length ~2.1) and r_m ~ 0.55. Open: why r_1 depends on m (a pure 2x2 fresh loop would make the first return independent of m); candidate: the loop partly consists of old edge cells whose colours decide whether the circulation continues. A per-cell return-lag histogram (loop length 4 vs longer) would test the loop picture directly; that needs a new tracer (Stage 2).
2026-10-19 19:25 (+13:00) · f0b96b7 · observation (results/stage1_k3.txt, scripts/stage1_k3.py)
Forward model K3 (3 fitted numbers, letters up to 12; split A/B = 1092/1093, seed 20261019): fit on A gives psi 0.16, e 1.122, rho 0.693; held-out R^2(ln c) on B = 0.875 (lead-only baseline 0.842, same split). Full-census refit psi 0.16, e 1.121, rho 0.692 predicts the 176 fresh rules at R^2(c) 0.913, median rel err 0.031 (lead law 0.877). K3-1 FAIL (0.875 < 0.90), K3-2 PASS (0.913), K3-3 PASS (psi 0.16 < 0.7). Note rho ~ 0.69 matches the census b = 0.696 and a killing read thins the surviving reads almost entirely (psi = 0.16).
2026-10-19 19:28 (+13:00) · f0b96b7 · interpretation
(confidence moderate) K3 captures part of the tail dependence (+0.033 held-out, +0.036 fresh over the lead law) with one extra parameter, but not the ceiling of the 8-letter lookup (0.950); the two thresholds I registered split (census fail, fresh pass), so I report "partial". The fresh set is longer rules (13-14), where letters 4-12 matter more, which is a plausible reason fresh gains more. K3 assumes flat circulating reads, which the profile contradicts (decay ~0.88 per read). A better kernel would include decay and the m-dependence of r_1; not fitted yet (post-profile, any such model is registered as informed by Stage 1 data, not blind).
2026-10-19 19:32 (+13:00) · f0b96b7 · correction (corrects: 19:05 D2 clause)
In the 19:05 entry the D2 clause is garbled. Correct statement: D2 had three conditions; (i) median max/min of r_k over k=1..m-1 <= 1.35: 1.371, fails marginally; (ii) pooled mean in [0.55, 0.85]: 0.798, holds; (iii) within 0.10 of census b 0.696: |0.798-0.696| = 0.102, fails marginally. D2 as a whole FAILS because the registration required all three.
2026-10-19 19:35 (+13:00) · f0b96b7 · decision
Build an instrumented tracer (src/edge_trace.rs) generic over Walker, with optional prefilled layout and torus wrap, recording: per-colour tallies (as edgestats), per-cell return-lag histograms of boundary reads, per-cell boundary-read counts N_b (resolved at the moment a cell becomes interior, censored at end), and birth-chain lengths. Reasons: (1) test the loop picture directly (return lag of circulating reads); (2) needed for Stage 2 on SparseBgAnt and for rotor/random-walk calibration; (3) a flat-edge torus-band simulation for the "first-principles flat strip". It must reproduce probe_walker window totals exactly and a HashMap brute force (property tests) before any use. The hole-boundary confound is not instrumented (flood fill is expensive); I will register the expected direction instead.
2026-10-19 18:42 (+13:00) · f0b96b7 · correction (corrects: timestamps of the entries from 18:34 to 19:35)
I filled in those timestamps from my own estimate of elapsed time instead of reading the clock; date at 18:40 shows they run ahead of real time. The order of the entries and the facts in them are unchanged (Stage 1 was registered before the aggregate was computed; the tracer was written and tested before this entry; nothing in Stage 2 has been run on a registered list). Real clock: session start 18:30; Stage 2 registered 18:41. I will take times from date from now on.
2026-10-19 18:43 (+13:00) · f0b96b7 · observation (tests/edge_trace.rs, src/edge_trace.rs)
Tracer built and tested. cargo test --release --test edge_trace: 5 property tests pass (ants vs probe last-octave window AND vs HashMap brute force of every field; SparseBgAnt; rotor; random walk; prefilled torus band vs brute). A deliberate mutation (lag + 1 in the lag histogram) made all 5 fail; restored and re-run green. Smoke timing only (no registered quantity read): blank board 2^24 steps 0.2 s per rule; torus band 2^26 steps 1 s with only ~2.7k births in the window, hence 2^30 steps for the band. Lists frozen by scripts/make_lists.py: 207 sample, 600 validation candidates, 60 band rules, 64 calibration walkers.
2026-10-19 18:44 (+13:00) · f0b96b7 · decision
Stage 2 design choices, in order of how much they could bias the result: (1) the validation set is new random rules (never simulated), filtered by tracer-only features frozen in the README, so the forward models (lead law, K3, K4) face data none of them has seen; (2) K4 was built after seeing the Stage 1 profile, which I disclose; its thresholds are registered before it is scored on V; (3) the flat band uses an i.i.d. interior, a deliberate ablation of the ant-built bulk, so a match in c means the bulk's structure is not needed; (4) no hole-rim instrument, only a registered direction (S5).
2026-10-19 18:50 (+13:00) · f0b96b7 · deviation
Stage 2 runs started 18:41 (scripts/run_stage2.sh; calibration, sample and validation lists finished in about a minute, the 2^30 band run still going). First analysis attempt crashed on the registered T0b: all 32 random-walk runs at 2^24 steps on the 4096 board ESCAPED (radius ~ sqrt(2^24) = 4096 > half the board), so the window has no boundary steps (division by zero). That is an instrument-configuration error in my registration, not a result. Fix, labelled as a deviation: rerun the 32 random-walk seeds at 2^20 steps on the same 4096 board (results/lists/calib_rw.txt -> trace_calib_rw.jsonl), keep the registered threshold c in [0.40, 0.48]; the 2^24 escape count is reported. Rotor runs at 2^24 did not escape. Also seen in the raw files (not a registered quantity): 96 of 600 validation candidates escaped, 0 of 207 sample rules.
2026-10-19 18:45 (+13:00) · f0b96b7 · correction (corrects: timestamp of the 18:50 deviation entry)
That entry was written at about 18:43 by the clock, not 18:50 (I again filled in a time instead of reading it). Content unchanged. All entries from here on use date.
2026-10-19 18:47 (+13:00) · f0b96b7 · observation (results/stage2_analysis.txt, scripts/stage2_analyse.py; first full Stage 2 analysis)
Calibration: rotor c = 0.4977 (sd 0.0007 over 32 seeds, none escaped) vs the 0.498 of notes/calibration.md: T0a PASS. Random walk at 2^20 steps: 0.4480 (sd 0.0062, 1 of 32 escaped), T0b PASS (the 2^24 runs all escaped, see deviation). Sample vs census: median |c_trace/c_census - 1| = 0.0026, none escaped: T0c PASS. So the tracer reproduces the census c to 0.3% and the rotor to 0.1%. Loop picture: of boundary reads at circulating colours 1..m-1 (rules with m >= 3, 47 rules, 1.9M reads) 78.4% return to the cell after exactly 4 steps (L1 PASS), next lags 8 (5.7%), 12 (3.2%); every lag is even. Reads at the killing colour m return after 4 steps only 48.3% (L2 FAIL, 0.50 threshold; lags 8: 14.7%, 12: 12.5%). L3 PASS: r_1 is 0.723 for lead 1 and 0.879 for lead >= 2 (excess 0.156), and 93% of the excess is carried by lag-4 reads (0.160 vs 0.305). Little's law (N1 PASS): median E[N_b] / (1/c) = 0.984; slope on min(m,6): 0.700 for E[N_b] vs 0.712 for 1/c. Chains (C1 FAIL as registered): q1 = P(chain >= 2 | >= 1) = 0.779, q2 = 0.455, q3 = 0.000: no run of consecutive births is longer than 3 (4.36M chains, mean length 2.13; P(len 1,2,3) = 0.22, 0.43, 0.35). I had registered geometric chains; the real structure is truncated at three. Forward models, validation set (378 of 600 random rules kept: 96 escaped, 16 with < 200 births; lead mix 1:206, 2:84, 3:44, 4:22, 5:15, 6:3, 7:4): R^2(c) lead law 0.711 (median rel err 0.066), K3 0.785 (0.042), K4 0.800 (0.049). V1 FAIL (lead law < 0.80), V2 PASS (K3 beats the lead law by +0.074), V3 FAIL (K4 0.80 < 0.90), V4 FAIL. K4 fit on census half A: held-out R^2(ln c) on B 0.863 < K3's 0.875 (K4-census FAIL); its phi_CK sits on the upper bound 1.2 and phi_KC, phi_KK ~ 0.1, i.e. the fit says survival decays ~0.915 per circulating read and the tail after the first killing read is nearly dead.
2026-10-19 18:47 (+13:00) · f0b96b7 · surprise
Chains of consecutive births never exceed three. A 2x2 block has three cells besides the one the ant came from, and the fourth step of an all-same-turn spiral re-enters the first cell. So the "naive 2x2 loop" is real at the level of births: each chain builds up to a block. This also explains why about half of all births are colour-0 continuations. What it does not explain is why chains are length 1, 2, 3 with probabilities 0.22 / 0.43 / 0.35.
2026-10-19 18:47 (+13:00) · f0b96b7 · deviation
The registered flat-band run (2^30 steps, window = second half) is void: all 60 bands were completely consumed (area 262144 = whole torus) before the window opened at 2^29, so births in the window = 0 for 58 of 60 rules (the other 2 had < 5000) and S1-S5 were not tested. Cause: my guess of the band consumption time, made from a single 2^26 smoke run on one rule, which showed ~2.7k births in its second half. Consumption time varies ~3x between rules (pilot below). Pilot (design only, 3 non-band sample rules, scripts/band_pilot.sh; births/S/area read only to choose a window): band consumed 61% at 2^27 for RRRLLRLLRL, 77% for RRLLRLLLRLL, but 98% at 2^26 for RLLRRRLRLR. Fix, frozen before looking at any band c(m): for each band rule run L = 24..29; use the smallest L at which the band is >= 50% consumed at the end of the run ((area_end - 131072)/131072 >= 0.5); keep the rule if consumption at the end is <= 90% and births in the window >= 5000. S1-S5 are otherwise unchanged. The void 2^30 file is kept as results/trace_band_2e30_void.jsonl for the record.
2026-10-19 18:52 (+13:00) · f0b96b7 · observation (results/stage2_analysis.txt; band rerun per the deviation rule)
Flat torus band, i.i.d. uniform interior, 60/60 rules kept (chosen L: 24:1, 25:6, 26:20, 27:28, 28:3, 29:2). Regression of 1/c_band on min(m,5): 1/c = 1.399 + 0.678 m, against 1.395 + 0.678 m for the same 60 rules' census c. corr^2(c_band, c_census) = 0.999; median c_band / c_census = 0.998; median births/edge at colour 0 = 0.522. Per-lead means agree to the third decimal (0.465/0.467, 0.365/0.365, 0.309/0.309, 0.242/0.242, 0.206/0.206). S1 PASS, S2 PASS, S3 PASS, S4 PASS, S5 FAIL (the ratio is 1, not > 1: hole rims make no measurable difference to c).
2026-10-19 18:52 (+13:00) · f0b96b7 · interpretation
(confidence moderate-high that c is a local fresh-ground property; the exact reason for a, b is still open) A flat front in an i.i.d. interior, with no curvature, no ant-built bulk and no hole rims, reproduces the census c rule by rule to 0.2%. So the birth rate per boundary step is set by what the ant does on freshly opened ground next to the front; the bulk and the shape do not enter at this precision. Because that is almost too clean, the next step is a control that could break it: change the interior distribution (all colour 0, young colours only, old colours only). If c still does not move, c is a function of the rule and fresh ground alone. Alternatives that would show up as a change: the entry cell's colour (an old cell the ant re-enters through) matters, which would explain why letters beyond the lead matter.
2026-10-19 18:55 (+13:00) · f0b96b7 · observation (results/stage2c_analysis.txt, scripts/stage2c_analyse.py)
Band-interior ablation, 60 rules at their chosen L: the U rerun is bit-identical (births and boundary steps) to the main band run. All-zero interior Z (59/60 kept): median |c_Z/c_U - 1| = 0.0029, corr^2 0.994, 1/c = 1.409 + 0.676 lead. Young-only interior Y (58/60): 0.0031, corr^2 0.999, 1.412 + 0.674 lead. Old-only interior H (58/60): 0.0028, corr^2 0.998, 1.382 + 0.682 lead. A1, A2, A3, A4 all HELD (thresholds 0.05 and slope in [0.55, 0.85]); the dropped rules (1-2 per variant) were dropped by the frozen consumption/births filter.
2026-10-19 18:55 (+13:00) · f0b96b7 · interpretation
(confidence high for the empirical statement, moderate for the reading) c does not depend on the interior at all at the 0.3% level, not even when every interior cell reads colour 0 (so interior cells would turn s_0 on their first read, a very different bulk dynamics). Combined with the census agreement (band == blob to 0.2%), c is a function of the rule and the fresh-ground edge process only; the ant-built bulk, curvature, holes and the colours the ant meets when it re-enters through old cells play no measurable role. That kills my earlier guess that the entry cell's old colour is why letters beyond the lead matter, and it means the circulation (78% of circulating reads return after exactly 4 steps) is carried by cells born in the current episode. Open: what fixes a = 1.4 and b = 0.68; the constraints now are Little's law, the lag-4 loops, chains <= 3, survival ~0.9 per circulating read, and bulk independence.
2026-10-19 18:56 (+13:00) · f0b96b7 · observation (results/stage2_decompose.txt, scripts/stage2_decompose.py; POST HOC, unregistered)
Split of boundary reads per birth on the 207 sample rules (lead capped at 6), 1/c = 1.352 + 0.712 lead: lag-4 returns -0.202 + 0.394 lead (55% of the slope); other short returns (lag 2..40, not 4) 0.121 + 0.092 lead (13%); the remainder, 1.434 + 0.227 lead (32%), is "first reads plus returns with lag > 40" because the output histogram stops at lag 40 (so this bucket is NOT first reads alone: the first read is ~0.98 per birth by the earlier profile r_0, which leaves ~0.45 + 0.23 lead for long-lag returns, e.g. rim cells revisited after an excursion). Caveat: this is the only place the long-return share shows up; the tracer output should record the lag > 40 mass separately before anyone builds on it.
2026-10-19 18:56 (+13:00) · f0b96b7 · decision
Next: Stage 3 (sbg dose-response), registered above before any sbg run. Lists/commands are in scripts/run_stage3.sh; analysis in scripts/stage3_analyse.py. Then Stage 4 (kappa).
2026-10-19 19:00 (+13:00) · f0b96b7 · observation (results/trace_sbg.jsonl via scripts/run_stage3.sh; kept/escape fractions only, no c values seen)
Sparse-background runs (1740 runs: 60 rules x {d=0 once, 7 densities x 4 seeds}). Kept fraction by density (leads 1..5): d <= 0.05: 1.00 everywhere; d = 0.1: 0.98-1.00; d = 0.2: 0.94-1.00; d = 0.4: 0.40, 0.56, 0.75, 0.65, 0.71 (escaped 0.50, 0.23, 0.00, 0.06, 0.02); d = 0.7: 0.00-0.08 kept (escaped 0.88-1.00); d = 1.0: 0.00-0.02 kept (escaped 0.98-1.00). The "kept in every density" cohort registered as primary is EMPTY.
2026-10-19 19:00 (+13:00) · f0b96b7 · deviation
Registered primary cohort (valid at every d) is empty because at d >= 0.7 the ant leaves the 4096 board within 2^24 steps (a random dense background turns the t^(1/3) blob into a faster-spreading walk; I have not characterised its exponent). B1 (b(1.0)/b(0) <= 0.25) is therefore NOT TESTABLE as registered: there is no stationary-blob regime to measure c in at d = 1. Replacement, frozen before any c(d) was computed: primary cohort = rules valid at every d <= 0.2 (all densities with >= 94% kept); secondary = survivors at d = 0.4 (reported separately, selection-biased). Restated tests: B1' for d in {0.1, 0.2}, |b_obs(d) - b_mix(d)| <= 0.10 where b_mix(d) is the slope of the mixture-predicted 1/c on lead at that d; B2' b(d) non-increasing within +0.07 along d <= 0.2; B3' the mixture R^2/median-rel-error test on cells d in {0.1, 0.2}; B4' intercept in [1.0, 2.0] for d <= 0.2 and (secondary) d = 0.4; B5 is evaluated at d = 0.4 as well as the registered d = 1.0. Probabilities not revised. The finding that the blob regime ends between d = 0.2 and 0.7 is itself a result about the domain of the c law.
2026-10-19 19:01 (+13:00) · b7b4758 · observation (results/stage3_analysis.txt, scripts/stage3_analyse.py; restated plan)
Primary cohort (all 60 rules valid at every d up to 0.2; 12 per lead). Observed 1/c = a + b lead: d=0: 1.397 + 0.677 [b CI 0.646, 0.711]; d=0.02: 1.439 + 0.646; d=0.05: 1.489 + 0.605; d=0.1: 1.537 + 0.549 [0.513, 0.583]; d=0.2: 1.619 + 0.450 [0.412, 0.485]. Secondary (d=0.4 survivors, 46 rules, selection-biased, lead 1 loses 60% of runs): 1.331 + 0.689 at d=0 on the same rules, 1.729 + 0.287 [0.235, 0.341] at d=0.4. Parameter-free mixture slopes from census rotations: 0.678, 0.665, 0.645, 0.613, 0.548 (the observed slope falls faster than the mixture by 0.03, 0.04, 0.06, 0.10). B1' PASS (differences 0.064 at d=0.1 and 0.098 at d=0.2, both under 0.10, the second barely), B2' PASS, B3' PASS (R^2 0.869, median |rel err| of 1/c 0.067 on 120 cells; the no-density-effect null scores R^2 0.613 and 0.102), B4' PASS (intercept 1.40-1.73, it rises), B5 FAIL at d=1.0 (0.00 vs 0.01 kept; uninformative) and FAIL at d=0.4 with the opposite sign (kept 0.48 for lead <= 2, 0.68 for lead >= 4). B1 as registered is untestable (no blob regime at d = 1).
2026-10-19 19:01 (+13:00) · b7b4758 · interpretation
(confidence moderate-high) The lead slope is a fresh-ground effect: it falls monotonically as pre-coloured cells are added (0.68 to 0.45 by d = 0.2, 0.29 at d = 0.4 among survivors), and a single-cell model (each born cell reads the rule rotated by its random starting colour, census values for the rotations) accounts for most of the shift with no free parameter. The residual (observed falls faster than the mixture) has the sign expected if loops are collective: a loop of four cells with mixed phases breaks sooner than the average of single-phase loops. The intercept does not fall to zero; it rises (1.40 to 1.73), consistent with the birth's own read being a floor. Not known: what the slope would be at d = 1 (the ant is no longer a blob there, kept 0-2%), nor whether the survivor slope at d = 0.4 reflects the law or the selection. The mixture is a first-order model and is only tested for d up to 0.2.
2026-10-19 19:05 (+13:00) · b7b4758 · observation (results/stage4_analysis.txt, scripts/stage4_analyse.py, 4a and 4b)
4a, 176 fresh rules: marginal slopes on lead: ln kappa -0.022, ln c -0.201, ln A22 -0.158. Growth fit ln A22 = 11.45 + 0.687 ln c + 0.807 ln kappa with residual sd 0.092 (so area is nearly a deterministic function of c and kappa; the ln c coefficient matches the 0.68 I derived from beta ~ 0.52 before fitting). Observed Y ~ 1 + lead + lnA22: lead +0.148, lnA22 +1.073. Permutation null with no direct lead effect on kappa (kappa linked to the rest only through ln A22): lead mean +0.145 [95% +0.137, +0.153], lnA22 mean +1.035. Q1 PASS, Q2 PASS. Adding ln c: lead +0.014 [bootstrap -0.045, +0.062], lnA22 +1.086, ln c -0.674: Q3 PASS. So the "lead raises kappa at fixed area" coefficient (+0.15) is reproduced to 0.003 by conditioning on ln A22 alone when kappa has no direct dependence on lead, and it vanishes when ln c is also in the model. 4b, 257 Part B pairs (tracer c for 732 of 835 rules; the others escaped or gave no births): dA = g (dc + d) with g = 0.805. Arm means (observed vs predicted dA): E -0.292 vs -0.318 (dc -0.333, d -0.062); S +0.013 vs +0.050 (dc +0.261, d -0.198); C 0.000 vs -0.004 (dc +0.001, d -0.006). Q4 PASS. The E/S asymmetry in dA that the leading-run write-up left unexplained is the c law: extending the lead costs about 0.33 in ln c with kappa ~ unchanged (area falls 0.29), shortening gains 0.26 in ln c but kappa falls 0.20 (area unchanged).
2026-10-19 19:05 (+13:00) · b7b4758 · interpretation
(confidence: high that conditioning on ln A22 manufactures the +0.15; moderate on what remains) The previously reported conditional lead effect is a collider artefact: ln A22 is determined by c and kappa (residual sd 0.09), lead moves c, and conditioning on A22 then forces kappa to compensate. It does not need, and Q3 bounds (|direct effect| below about 0.06 per step given ln c), any direct effect of lead on kappa. What is NOT explained: the S arm's raw kappa drop (d = -0.198, a total effect, not conditioned on anything) and E's near-zero d. Together E and S both lower kappa (-0.06, -0.20) while C does not, which looks like an early-letter-flip effect and not a lead effect; the lead-preserving flips (4c, running) are the test. I should not call M3 evidence for a lead effect before that result.
2026-10-19 19:19 (+13:00) · b7b4758 · observation (results/stage4_analysis.txt, results/p1/*, 4c)
Lead-preserving early flips: 274 mutants (P1 136 at index k+1, P2 138 at k+2), 8 seeds each via probe 27 8192; kept and stationary pairs 177 (retention 0.65; P1 87/136, P2 90/138). Mean d = -0.007 [cluster-bootstrap over bases -0.078, +0.060], mean dA -0.012, sd of d 0.432. P-1 PASS, P-2 FAIL, P-3 PASS (beta +1.123, residual mean +0.007 [-0.022, +0.034]), P-4 PASS after fixing a NaN (some mutants had no tracer c; finite pairs only): dA = g (dc + d), g 0.809, R^2 0.875. Arms: P1 d +0.002, P2 d -0.017. POST HOC (unregistered) split by flip direction: L>R (n=98) d +0.153, dA +0.101; R>L (n=79) d -0.207, dA -0.153.
2026-10-19 19:19 (+13:00) · b7b4758 · deviation
The first stage4 run printed P-4 as NaN/FAIL because dc was NaN for mutants whose tracer run escaped; I patched the script to use finite pairs and reran; the P-4 PASS is from the patched script. No other number changed.
2026-10-19 19:19 (+13:00) · b7b4758 · interpretation
(confidence moderate) An early flip that leaves the lead alone has no net effect on kappa (-0.007), so the Part B S-arm drop (-0.198) is not a generic early-flip effect. But the flip-direction split (+0.15 for L>R, -0.21 for R>L) says a single early letter flip moves kappa by about 0.2 in a direction set by the letter change, with zero net because both signs occur. The S arm is an R>L flip (-0.198 matches -0.207) and E an L>R flip (observed -0.062, not +0.15, so extending the lead may carry about -0.2 on top). I cannot separate lead from letter direction: S and E are confounded with flip direction by construction. Supported: no evidence for a lead-specific effect on kappa. Not supported: any causal statement that lead raises kappa.
2026-10-19 19:22 (+13:00) · b7b4758 · decision
Closing. Full cargo test --release passes (incl. 5 new tests/edge_trace.rs). notebook.ipynb built by scripts/build_notebook.py and executed under uv with 0 errors; its numbers match README/results. notes/open-questions-hypotheses.md updated; pointer appended to the predecessor experiment. Open: derive a and b (loop-formation probability x 4 / births per chain), why letters 5-8 matter, direction-balanced flips for kappa, the blob-regime boundary between sparse-background density 0.2 and 0.7. Reproduction commands: see README (run_stage2.sh, run_stage2b.sh, run_band.sh, run_ablation.sh, run_stage3.sh, run_p1.sh, then the stage*_analyse.py scripts and build_notebook.py).
2026-10-19 21:33 (+13:00) · 8cc447f · correction (corrects: the hash field of every entry above, and the empty commit in results/stage2_run.log and the notebook provenance)
Hash fields in earlier entries are stale: f0b96b7 is the base commit; the code and results they describe were committed in b7b4758 (Stages 1-2c) and 8cc447f (Stages 3-4, notebook, notes). WSL cannot resolve this worktree's version-control directory, which is why the run log and the notebook provenance showed an empty or unknown commit. results/PROVENANCE.md now records the commits, the real clock times of every result file (replacing the estimated times in the early entries), and the source-identity check.
2026-10-19 21:33 (+13:00) · 8cc447f · observation (scripts/find_edgestats_k.sh)
The unrecorded K of edgestats_sample.csv is 23: rerunning examples/edgestats on RRRLRRLLLLR for K = 20..26 reproduces the stored rows (colour 0: 2645 edge steps, 1339 births; colour 1: 1999/4; colour 3: 1310/1296) only at K = 23 (window (2^22, 2^23], 4096 board). Stage 1 therefore used a window one octave earlier than the Stage 2 tracer (2^24); G0 (c agreement 0.4%) and T0c (0.3%) show c is insensitive to this.