published October 01, 2026

Emergence peaks at an intermediate reach

DESIGN §1.3 sweep 13 put an exploratory pattern from sweep 12 to fresh worlds under rules fixed first: does a replicator arise more often at an intermediate reach than at the tightest one or in a well-mixed world? All 630 runs finished, 90 fresh seeds at each of seven reaches. Radius 4 emerged in 16 of 90 worlds, against 4 of 90 at radius 1 and 3 of 90 well-mixed (one-sided Fisher, Holm-corrected p = 0.0039 and 0.0027), and a logistic fit over the six finite radii bends down (radius² coefficient −0.114, Wald p = 0.0041) with its peak at radius 4.1. Both pre-registered hypotheses are shown. Four of the seven arms emerged exactly as often as sweep 12's did: different seeds and different worlds, the same totals.

Question

Rung 0 of the programme asks under what conditions a replicator arises at all. The radius sweep asked whether a cell's reach decides it and could not tell: ten seeds an arm, one or two transitions in each. The lineage-diversity sweep, built to ask something else, then saw radius 4 emerge four times as often as radius 1 and five times as often as the well-mixed world. DESIGN §1.3 sweep 13 puts that pattern to fresh worlds: does emergence peak at an intermediate reach?

Method

Sweep 12's world to the byte — a 128×128 torus of tapes held at 64 bytes, at the emergent mutation rate 2^-13, with lineage_rule = oriented — at radius 1, 2, 3, 4, 6 and 8 and 0, the well-mixed world. 90 seeds per arm, 20 000 epochs per run, 630 runs. The seeds are 91–180, not sweep 12's 1–90: the lineage rule changes no byte of a world, so those seeds at radius 1, 2, 4 and 0 would have replayed sweep 12's worlds and its exploratory counts with them.

The reading is pre-registered, with every number fixed before a run was read: docs/design_record.md, 2026-09-27, "Does emergence peak at an intermediate reach?". A run is emerged where it has a confirmed emergence epoch and reads replicator_share of at least 50% at some sample from it on — sweep 12's rule, unchanged, on the orientation-aware share. H-peak, the confirmatory test: radius 4 emerges more often than radius 1 and than the well-mixed world, two one-sided Fisher exact tests Holm-corrected at family level 0.05. H-shape: over the finite radii, a logistic regression of emergence on radius and radius² bends down, its radius² coefficient negative at a two-sided Wald p below 0.05, with its fitted peak strictly between radius 2 and 6 so that two of the radii read lie on each side of it. A curve that only rises or only falls across the radii is not a peak.

Result

H-peak shown H-shape shown Emergence peaks at an intermediate reach: radius 4 emerged in 16 of 90 fresh worlds, against 4 of 90 at radius 1 and 3 of 90 well-mixed, and the curve fitted over the finite radii peaks at radius 4.10. On one world: a 128×128 torus, tapes held at 64 bytes, 2^-13, 20 000 epochs.

Every count on this page is read from the sweep's stored runs at render time, through the same reading the sweep page draws.

A confirmed crossing that never read a share of one half after it is not emerged: the census that confirms a crossing is blind to reverse copiers, and the share clause asks that the world was, at some point, mostly replicators. Well-mixed is not a distance, so it stands at the end of the table and outside the fit.

The pre-registered reading final

Every run of the sweep has finished. A run emerged where it has a confirmed emergence epoch and reads replicator_share of at least 50% at some sample from it on, as in the lineage-diversity sweep.

Arm	Runs	Finished	Emerged	Rate
radius 1	90	90	4	4%
radius 2	90	90	9	10%
radius 3	90	90	7	8%
radius 4	90	90	16	18%
radius 6	90	90	6	7%
radius 8	90	90	3	3%
well-mixed	90	90	3	3%

H-peak

Radius 4 emerges more often than radius 1 and than the well-mixed world: shown

Two one-sided Fisher exact tests, Holm-corrected at family level 0.05: radius 4 16/90 against radius 1 4/90: one-sided Fisher p = 0.00385, Holm 0.00385; radius 4 16/90 against well-mixed 3/90: one-sided Fisher p = 0.00135, Holm 0.00269.

H-shape

Across the finite radii, emergence peaks at an intermediate reach: shown

A logistic regression of emergence on radius and radius² over the finished runs of radius 1, 2, 3, 4, 6, and 8, read where the radius² coefficient is negative at two-sided Wald p < 0.05 and the fitted peak lies between radius 2 and 6, with 2 of those radii on each side of it. The radius² coefficient is -0.114, Wald p = 0.00407, and the fitted peak is at radius 4.1. Well-mixed is not a distance and is left out of the fit: it emerged 3 of 90.

The same totals, not the same worlds

At radius 1, radius 2, radius 4, and well-mixed this sweep emerged exactly as often as sweep 12 did: 4 of 90, 9 of 90, 16 of 90, and 3 of 90. That is a coincidence of totals, not reused data. This sweep's worlds are seeds 91–180, runs 3673–4302, and sweep 12's are seeds 1–90, runs 3217–3576: no seed and no run is shared. The confirmed crossings differ: 10 against 12 at radius 2, 20 against 25 at radius 4, and 8 against 7 at well-mixed. At radius 1 both count 5 crossings: here seed 96 at epoch 17 990, seed 113 at epoch 10 480, seed 119 at epoch 13 000, seed 123 at epoch 10 520, and seed 143 at epoch 5 470; in sweep 12 seed 1 at epoch 5 030, seed 5 at epoch 7 000, seed 20 at epoch 6 210, seed 51 at epoch 17 250, and seed 82 at epoch 9 330. H-peak and H-shape read this sweep's runs alone; sweep 12's only suggested the question.

How it reads against the earlier findings

Does spatial locality buy emergence? read locality as not speeding emergence, on ten seeds an arm at the engine's default rate and on the detector's crossing alone. It could rule out an enormous effect and nothing smaller, and it saw no peak because one or two events an arm cannot draw one. This sweep reads nine times the seeds at the emergent rate and finds the effect that one could not see — not a monotone benefit of locality, which the tightest reach does not have, but a best reach in the middle. After emergence, one lineage takes the world at every reach is the other half: reach changes how often a replicator arises, while what sweep 12 read after one had arisen did not move with it — a single lineage held the world whatever the reach.

What is not claimed

• One world. Every run is 128×128, at one mutation rate, with tapes held at 64 bytes and a budget of 20 000 epochs. Where the peak sits in another world size, at another rate or with tapes that can grow is untested, and a reach of 4 means something different on a larger torus.
• Emergence is the replicator-world definition. A world counts where the detector confirmed a crossing and replicators then held half of it. A crossing alone is not what is claimed: every arm crossed more often than it emerged.
• No mechanism. Nothing here says why a reach near 4 is best: whether a tight neighbourhood starves a copier of partners, a wide one dilutes it before it takes hold, or something else. The sweep measured the curve, not its cause.
• Well-mixed is not on the radius scale. It is radius 0 in the parameters and a partner drawn from the whole torus in the world. It is compared with radius 4 in H-peak and left out of the fit, and it is not the point past radius 8.

Since carried to growable tapes. On growable tapes, a reach of 4 emerges more than ten times as often as radius 1 runs radius 4 on tapes of 64 bytes free to grow to 128, against the host–parasite sweep's control at radius 1 on the same seeds, under a pre-registered rule read off the stored worlds. The effect holds there. Its ratio of rates is larger than this page's, though the two sweeps read emergence on different instruments, so that is a comparison and not a test.

Evidence

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

Sweep: Locality and emergence