Why does the c-law slope b fall with loop length? Circulations per birth versus reads per circulation

Registered 2026-10-20 00:55 (+13:00), base commit 6049d6a plus the uncommitted instrument (src/lattice_trace.rs, tests/lattice_trace.rs, examples/lattice_c_law_colours.rs), committed together with this file before any run on the 960-ant list. Nothing in this folder has been run on that list. Disclosure of what was seen before: the lattice c-law results (experiments/2026-10-20-lattice-c-law, slopes b and intercepts per stratum, per-lead medians), and on the square lattice only, the per-colour and lag-4 findings of experiments/2026-10-19-c-law-mechanism (r_k decays ~0.88 per read; 48-78% of circulating reads return after exactly 4 steps). No per-colour, lag or run statistic has been looked at on the triangular or hexagonal lattice. The tracer's own gates (below) ran on short random rules only.

Question

1/c = a + b * lead holds on three lattices (R^2 0.94-0.95 square and triangular), but b falls with the loop that the lead letter closes: 0.86 (hex R, loop 3), 0.70 (square, 4), 0.34 (triangular, 6), 0.28 (hex r, 6; per-lead medians, since the OLS is destroyed by one rule). The lattice study's first-principles reading, b = l_b / N_b (boundary cells per lap over births per circulation episode), was never measured on either side, and its three hypotheses (b grows with l, constant, ~4/z) are all disproven. A fitted b(l) would be a fourth guess. Instead: measure the two factors of b.

Definitions (frozen)

• Everything as in experiments/2026-10-20-lattice-c-law (windows, boundary step, birth, lead, strata, alpha_w blob filter, kept set). The tracer src/lattice_trace.rs uses the last octave only (steps (2^26, 2^27]), identical to the probe's window k = 26.
• Return: a step that starts on a cell last read exactly ell steps earlier; ell = loop length of the rule's first letter: square 4, triangular 6, hex r/l 6, hex R/L 3 (checked by stepping in tests/lattice_probe.rs, gate G3 of the parent).
• Circulation: a maximal run of consecutive returns (any colour, boundary or not) of length >= ell (one full lap re-read). eps = circulations per birth. R = mean run length of circulations (lengths capped at 63). C = total steps inside circulation runs per birth = eps * R. Cb = the same counting only boundary steps in those runs.
• Slopes are Theil-Sen over kept rules (primary, robust to the 1/c = 390 outlier) and the slope of per-lead medians (secondary), leads 1..5 for 1/c, and m = lead for C, Cb, eps, R. OLS is reported but is not primary.
• b_med = slope of per-lead medians of 1/c, recomputed from this run (not copied).

Gates (instrument; a fail voids the dependent claims)

• T1 (done before this registration, cargo test --release --test lattice_trace): square LTrace equals edge_trace::trace_blank in every field on 30 random rules; hex, triangular and square equal a HashMap re-implementation on 30 random rules each; two one-line mutations (lag off by one, return lag ell + 1) were each caught by the tests.
• T2 on the 960-ant list: for every rule not escaped on the 8192 board, births and boundary_steps equal ants.csv window k = 26 exactly, and sum_k edge_k = boundary_steps.
• T3 1/c from the tracer equals the parent's to 1e-12 (follows from T2; reported anyway).

Predictions (kept rules, per stratum; my odds in brackets)

• P1 (circulation accounts for the lead slope). Cb per-lead-median slope is within [0.8, 1.2] x b_med in all four strata. [45%] (If this fails the reads that grow with the lead are not in lag-ell runs, and the loop picture of the slope is incomplete off the square lattice.)
• P2 (reads per circulation). R per-lead-median slope divided by ell lies in [0.5, 1.1] in all four strata (this is l_b / l: the fraction of a loop that is boundary and survives a lap). [50%]
• P3 (the discriminator). b = eps * lambda with lambda = dR/dm. The drop of b with loop length is carried by eps, not by lambda: eps at lead 3 (stratum median over kept rules) is lower on triangular than on square by a factor >= 2.0, and lower on hex r than hex R by a factor >= 3.0. [55%] Rival (lambda carries it, N_b/l roughly constant): ratios of eps within [0.7, 1.4] of 1, with lambda/ell falling instead. Reported with the measured vector either way.
• P4 (primary robust slope). Theil-Sen b is within the parent's OLS bootstrap CI on square, triangular and hex R, and in [0.15, 0.45] on hex r. [60%]
• P5 (hex ring centre). Births into a cell whose neighbours are all already visited (hole_births, e.g. a 6-ring centre) per birth are >= 2x higher on hex r-lead than hex R-lead (median over kept rules), because only the r ring encloses a cell. [40%]
• P6 (letter-typed hex lead). Splitting hex R-lead kept rules by the killing letter (the first letter different from the lead: r, l, L) and fitting 1/c ~ lead per type raises the mean R^2 weighted by n from 0.69 to >= 0.80. [40%] Not registered: any claim about U-turn letters (turn 2 on square, 3 on hex); a second stage with a new loop length (period 2) would need its own registration.

Decision rule

P3 is the headline. Supported if both ratios pass; "rival" if both eps ratios are in [0.7, 1.4]; otherwise undecided with the measured vector. A gate failure (T2) on a lattice means no claim on that lattice.

Commands

bash experiments/2026-10-20-lattice-c-law-colours/scripts/run.sh   # fresh target dir, sha256 logged, writes results/traces.jsonl
uv run experiments/2026-10-20-lattice-c-law-colours/scripts/analyse.py

Results

Run 2026-10-20 (scripts/run.sh, 509 s wall on 6 threads, shared with another job; binary sha256 24ea23a2...a8701 in results/run.log; traces.jsonl 1.8 MB), uv run scripts/analyse.py -> results/analysis.{txt,json}, post hoc scripts/explore.py -> results/explore.txt. run.log says "dirty: 0 files" but the one-line driver fix (see NOTEBOOK 01:20) was uncommitted at run time (the git status under the GIT_DIR workaround is unreliable); it is committed with these results.

Gate T2 passed: 855 rules compared (105 escaped in the tracer or the parent), 0 mismatches in births, boundary steps, or per-colour sum.

Registered predictions

	verdict	numbers
P1 Cb slope / b_med in [0.8, 1.2]	FAILED	0.42 / 0.19 / 0.29 / 0.17 (square / tri / hex R / hex r)
P2 R slope / ell in [0.5, 1.1]	FAILED	0.08 / 0.07 / 0.20 / 0.10
P3 eps ratios >= 2.0 and >= 3.0	formally met (7.2, 75) but not interpretable	see below
P4 Theil-Sen b in parent CI	HELD	0.691, 0.347, 0.850; hex r 0.306 in [0.15, 0.45]
P5 hole births r/R >= 2	HELD	0.094 vs 0.0097 per birth (9.7x); hex r at lead 1: 0.155
P6 per-kill-letter hex R fits, weighted R^2 >= 0.80	FAILED (0.781)	kill l: R^2 0.941 (b 0.88); kill L 0.717; kill r 0.696 (pooled 0.689)

Why P3 is not evidence. My circulation definition (runs of lag-ell returns of any cell, boundary or not) counts interior loops too. C (steps in such runs) is 30-120 per birth on the square lattice while 1/c is 2-5, so eps (1-19 circulations per birth) and R are dominated by interior behaviour and say nothing about the boundary episodes in b = l_b / N_b. The boundary-only version, Cb, is what P1 tests, and it carries only 17-42% of the slope. The decomposition I intended (circulations per birth times boundary cells per circulation) is therefore not measured; P3 passing is an artefact of a bad estimator and I do not count it. The headline question is unanswered by the registered tests.

Post hoc (unregistered, results/explore.txt; all non-escaped rules, per-lead medians, no blob filter). Boundary reads per birth by the colour the cell had when read (1/c is their sum). The lead slope lives in reads at the circulating colours 1..m-1 and is partly cancelled by falling reads at the killing colour and later:

stratum	circulating reads added per extra lead step	slope of circ	slope of kill	slope of later	total slope
hex R (loop 3)	0.95 each (0.95, 0.96, 0.96, 0.94)	0.95	-0.02	-0.07	0.88
square (4)	0.85, 0.78, 0.79, 0.77	0.80	-0.05	-0.06	0.69
triangular (6)	0.65, 0.50, 0.37, 0.35	0.46	-0.08	-0.02	0.36
hex r (6)	0.69, 0.62, 0.53, 0.47	0.58	-0.11	-0.20	0.28

Since a cell is read once per colour, "reads per birth at colour k" is the fraction of born cells that are still boundary cells at their k-th read. So the c law reads: that fraction is a plateau of about 0.95 (hex R), 0.8 (square), 0.4-0.65 (triangular), 0.5-0.7 (hex r) across the lead colours, with a slow decay for the larger loops, and b is that plateau minus the loss at the killing and later colours. This restates the identity 1/c = E[N_b] and is not a mechanism, but it says what must be explained: why a ring cell stays a boundary cell through the circulating reads with probability ~1 for loop 3 and ~0.5 for loop 6. Two further observations. (a) The ring-circulation picture does not carry the circulating reads off the square lattice: the share of circulating reads with lag exactly ell is 0.67-0.87 square but 0.05-0.44 triangular, 0.28-0.40 hex R, 0.06-0.24 hex r (rising with lead). (b) Hex r-lead rules fill holes: 15% of births at lead 1 land in a cell whose neighbours are all visited (the ring centre), falling to 6% at lead 5; hex R and square sit at 1-2%.

Reading. Not explained. What this run adds: the slope is carried by circulating-colour reads at a plateau rate that falls as the loop lengthens; loop-return (lag-ell) circulation is the dominant carrier only on the square lattice, so the "2x2 loop" picture should not be transplanted to other lattices; hex r rules behave differently through hole filling; the kill letter's type matters on hex (l-killed R-lead rules are linear in lead with R^2 0.94 and the other two types are not). Next registered step, if pursued: a boundary-only episode segmentation (maximal runs of consecutive boundary reads of colours 1..m-1), and the plateau fraction itself as a function of ring geometry (outside neighbours per ring cell: square 2, triangular 1, hex R 4, hex r 3), which is a measurable prediction not yet made.