negative September 11, 2026

Does spatial locality buy emergence?

All 40 runs finished and 6 transitioned, but locality is not what decided it: the well-mixed arm transitioned as often as the best local arm, 2 of 10, and the tightest arm did worst. The speed half of DESIGN §1.3 sweep 3 is not supported — though 1 or 2 events per arm can only rule out an enormous effect. What the sweep does show is about the soup rather than emergence: among runs that never transitioned the final compress_ratio falls monotonically as a cell's reach widens, 0.951 to 0.859. The diversity half of the hypothesis is untested and stays open.

Hypothesis

DESIGN §1.3 sweep 3 asks what spatial locality buys, and it asks it twice. A neighbourhood radius decides who a cell can interact with: at 1, 2 and 4 a tape only ever meets the tapes inside that Moore neighbourhood, while radius = 0 is the well-mixed world — a partner drawn uniformly from the whole torus, the infinity arm of the design record. The first half of the hypothesis is about speed: a tight neighbourhood shelters a new lineage from the soup long enough to take hold, so a local world should reach the first replicator sooner and more often than a well-mixed one. The second half is about what happens afterwards: locality is supposed to keep a world diverse once a lineage exists, by stopping one winner from reaching every cell. This finding answers the first half only.

Method

One world shape and one parameter moving. A 128×128 torus of 64-byte tapes at the engine's default mutation rate, 2^-12, 10 seeds per arm and 20 000 epochs per run, with the radius taking 1, 2, 4 and 0 and every other parameter at its default. The dependent variable is the transition epoch of DESIGN §1.2 — the first sampled epoch at which compress_ratio falls below 0.6 and stays below it for three further samples. A run that never crosses is censored at 20 000 epochs, not zero.

Result

All 40 runs finished and 6 of them transitioned. The table is the recorded per-arm result; the phase diagram and the per-run rows below read the same runs live, and the sweep's own page adds the hazard table.

The two medians are taken over the runs of each arm that did not transition — the soup the arm sits in when nothing emerges — at the last sampled epoch.

Locality did not speed emergence. The well-mixed arm, which has no locality at all, transitioned as often as the best local arm and sooner than two of them. Ordered by how much of the world a cell can reach, the counts run 1, 1, 2, 2 with the well-mixed arm at the top of that list rather than the bottom, and the earliest transition of the sweep belongs to the widest local arm, not the tightest. The effect the hypothesis predicts — a local arm reaching the first replicator sooner or more often than the well-mixed one — is not in the data.

And ten seeds could not have seen it unless it were enormous. The widest spread in the sweep is 2 of 10 against 1 of 10, which is a one-sided Fisher exact p = 0.5 — the two arms are not distinguishable at all. With ten seeds an arm cannot reach p < 0.05 against another until it shows 4 transitions where the other shows none. The hazard section on the sweep's page says the same thing on the runs' own time at risk: every arm's 95% interval contains every other arm's estimate, from roughly 0.05 to 0.11 events per 10⁴ run-epochs, and each interval is 30-fold wide in the arms with two events and more than 200-fold wide in the arms with one. So this is a negative result about an effect of the size the hypothesis imagined, not a demonstration that radius does nothing: a factor of two hiding in here would be invisible.

One transition did not hold. Five of the six transitioned runs end collapsed, with compress_ratio between 0.07 and 0.31 at the last epoch. The sixth — the radius = 2 arm's flagged run — falls to a floor near 0.11, stays down for about two thousand epochs, then climbs back through the 0.6 line and ends at 0.91, next to the 0.894 to 0.904 of its arm's seeds that never moved. It is a genuine crossing by the locked rule and a reminder that a transition in the DESIGN §1.2 sense is a state a world can leave again.

What the sweep does show is about the soup, not about emergence. Among the runs that never transitioned, the final compress_ratio falls monotonically as a cell's reach widens — 0.951, 0.901, 0.860 across the three local arms — and the well-mixed arm sits at 0.859, at the end of that trend rather than apart from it. The arms' spreads do not overlap between radius = 1, radius = 2 and the pair above them, and entropy_bits orders the arms identically. Wider mixing leaves the tapes more alike than a tight neighbourhood does. That is a real, clean effect on the pre-emergence soup and nothing more: it is measured on the runs where nothing emerged, and the mutation-rate sweep already found a floor of the same kind that moved with its parameter while emergence did not.

A determinism cross-check, for free. The radius = 1 arm is the engine's default radius, so its ten runs repeat the parameters of the mutation-rate sweep's 2^-12 arm and the world-size sweep's 128² arm. All ten seeds reproduce: the same transition epoch in the one seed that transitions, and the same final compress_ratio, distinct_tapes, top_share and entropy_bits in every one of them, digit for digit, across three experiments run weeks apart. copy_rate agrees wherever all three recorded it, which for the mutation-rate arm was only the seed that transitioned. That is the guarantee DESIGN §1.1 makes — a run is a function of (params, seed) — checked against itself rather than asserted. It is also a caveat on this sweep: that arm is not an independent sample, so the sweep holds 30 runs of new evidence and 10 of a repeat.

What next

The second half of the hypothesis is untouched and stays open: whether locality stabilises diversity once a replicator exists is a question about the six transitioned runs' distinct_tapes and top_share series after their transition epochs, not about how many arms transitioned, and nothing here tests it. Six runs spread over four arms is too thin to read that way — the comparison wants several transitions per arm — so it is left for the sweep that has them. The hazard section on the sweep's page is the right lens for the speed question once more seeds exist: it spends a censored run's whole observed lifetime rather than throwing it away, and the way to move those intervals is more seeds or longer runs, which is what mutation rate, long runs and the BFF positive control are for. Until the control resolves, an arm that shows nothing here is still evidence about the instrument as well as about the world.

Since answered on more seeds. Emergence peaks at an intermediate reach reads the speed half again, as how often a replicator arises, at the emergent rate rather than the default, on 90 fresh seeds at each of seven reaches under a pre-registered rule. It finds an effect ten seeds could not have seen: radius 4 emerges more often than radius 1 and than the well-mixed world, and the curve over the finite reaches peaks near 4.

Evidence

Transition, flagged, census and arm are used here in the senses set out in the glossary.

Sweep: Neighbourhood radius