partial September 11, 2026

Does time to emergence scale with the number of cells?

DESIGN §1.3 sweep 2 sets two readings against each other: if emergence is a hazard per cell-epoch, a world with more cells buys more tickets and events scale with cell count; if it is a hazard per world, cell count buys nothing. Square worlds from 32² to 256² separate them, and emergence gets more likely as the world gets bigger: 0, 0, 1 and 3 of 10 seeds transitioned, against the 0.06, 0.26, 1 and 3.7 a constant per-cell hazard predicts. Pooled, that hazard is 2.5 × 10⁻¹⁰ per cell-epoch (95% 0.7–6.4). The earliest transition of the whole programme, epoch 4 730, is a 256² run — but four events cannot measure an exponent, and two of them are still resuming from their snapshots.

Hypothesis

DESIGN §1.3 sweep 2 sets two readings of emergence against each other, and they predict opposite things about a bigger world. If a replicator is a lottery win — one cell somewhere stumbles into a tape that copies itself, and the rest of the world only has to not destroy it — then cells are tickets: the hazard of a first replicator is a rate per cell-epoch, so quadrupling the cell count quadruples the expected number of events in a fixed budget, and the smallest worlds should mostly never transition inside it. If instead emergence is a property of a world rather than of its cells — a global state the soup has to reach — then the hazard is a rate per run-epoch, every arm transitions about as often as every other, and cell count buys nothing.

The two readings differ in what a world is for. Under the lottery reading the world is a parallel search and its size is the search budget; under the per-world reading the size is a detail of the simulation and the clock runs at the same speed in all of them.

Method

Square tori of 64-byte tapes at 32×32, 64×64, 128×128 and 256×256 — width and height travel together, so the arms are 1,024, 4,096, 16,384 and 65,536 cells — with 10 seeds each and 20 000 epochs per run. Every other parameter is at its default, which means a mutation rate of 2^-12 and a neighbourhood radius of 1. 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 more samples; a run that never crosses is censored, not zero.

Result

Partial. Four of the 40 runs transitioned, and the counts rise with cell count. Two of the four — both in the largest arm — are still resuming from their snapshots after a deploy interrupted them, so they have not reported their own last epochs yet; their transitions are recorded and both fell well before the epoch each run has already reached, so neither the counts below nor the hazard depends on their finishing. The table is the recorded per-arm result; the phase diagram, the survival curves and the per-run rows below read the same runs live.

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.

Emergence gets more likely as the world gets bigger. Within the same 20 000-epoch budget the counts run 0, 0, 1 and 3 of 10 from the smallest arm to the largest. The two observables agree in the largest arm and only there: all three of its flagged runs also carry a replicator_count census of at least one replicating cell, the earliest of them sitting at several hundred replicating cells at its latest sample. Nothing below 256² has a census above zero at any sample.

The per-cell reading is the one that fits, and here is the test. If the hazard were a constant per cell-epoch, the expected number of events in an arm would scale with its cells multiplied by its epochs at risk — so, scaled from the single event at 128², the arms should show about 0.06, 0.26, 1 and 3.7 events. They show 0, 0, 1 and 3. If instead the hazard were constant per run-epoch, the four events would fall roughly evenly across the arms' observed lifetimes and about 2 of them would land in the two smallest arms, where none did — which happens by chance about one time in eight, so the per-world reading is disfavoured and not excluded. Pooled over all four arms, the hazard is 2.5 × 10^-10 per cell-epoch, exact Poisson 95% interval 0.7 to 6.4 × 10^-10, on four events and 1.6 × 10¹⁰ cell-epochs at risk; the two arms that have events sit at 3.1 × 10^-10 (128²) and 2.5 × 10^-10 (256²), each inside the other's interval. The sweep's own hazard table says the same thing per run-epoch, with every run censored where it was last watched: 0 events in 200 000 run-epochs in each of the two smallest arms, and intervals more than 200-fold wide wherever there is one event. Four events cannot measure an exponent. What they support is the sign — more cells, more emergence — not the claim that it is exactly linear in cell count.

The earliest transition of the whole programme is in the largest world. The 256² arm's fifth seed crossed at epoch 4 730, ahead of the mutation-rate sweep's 5 030 and the radius sweep's 5 910. That is one run, and it is not evidence that a bigger world runs faster per run beyond what the hazard already implies: a bigger world draws more events per epoch, so its first event tends to be earlier, and with one, one and three events the arms' medians — 15 560 against 15 800 — are not a time to emergence.

Below the largest arm, a collapse that is not a replicator. The one flagged run of the 128² arm collapses entropy_bits from the random-soup value to near 7.1, drives compress_ratio to 0.071 and lifts copy_rate off zero — with a census of zero at every sample. It is the disagreement class the mutation-rate and radius sweeps also report: a low-entropy attractor the substrate can fall into that is not a replicator by the test DESIGN §1.2 locks. Counted on the census rather than the detector the arms read 0, 0, 0 and 3, which is the same direction, steeper.

Size does not change the pre-emergence soup. Among the runs that never transitioned, the final compress_ratio sits between 0.93 and 0.96 in every arm, and the per-arm medians span 0.945 to 0.951 — less than the 0.937 to 0.956 the 32² arm's own ten runs span between them; entropy_bits ends within a few hundredths of 7.87 everywhere. A 1 024-cell soup and a 65 536-cell soup are the same soup. That is worth stating because the radius sweep found the opposite for its parameter — there the censored runs' floor moved monotonically from 0.951 to 0.859 as a cell's reach widened — so cell count acts on how often something happens, not on what the world looks like while nothing does.

A determinism cross-check, for free. Every parameter of the 128² arm is a default, so its ten runs repeat the mutation-rate sweep's 2^-12 arm and the radius sweep's radius = 1 arm. All three report the same single transition — the ninth seed at epoch 15 560 — and the same final compress_ratio, digit for digit, across three experiments run weeks apart. That is the guarantee DESIGN §1.1 makes, a run as a function of (params, seed), checked rather than asserted. It is also a caveat: that arm is not an independent sample, so this sweep holds 30 runs of new evidence and 10 of a repeat, and its single 128² event is the same event the mutation-rate finding already counted.

Two things keep this partial rather than published. The 20 000-epoch budget is short next to the epochs at which these transitions landed — three of the four fell after 15 000 — so every arm is heavily censored and the small arms may be slow rather than silent, which is exactly the reading a per-cell hazard predicts and cannot be told apart from it on four events. And a sweep of ours is only readable as a negative once the BFF positive control resolves its own census; until then an arm sitting at the censoring line says as much about the instrument as about the world.

What next

The direct extension is already running. The BFF positive control is a well-mixed soup of 2¹⁷ tapes — twice the largest arm here — and its three mutated seeds have all flagged, at 5 400, 9 750 and 14 250 epochs, which is the direction the per-cell reading predicts. It is not a fifth arm of this sweep: it drops the neighbourhood as well as raising the cell count, and its runs are unfinished, so it extends the programme rather than the curve. The lens for the scaling question is the survival and hazard section on the sweep's page, which spends a censored run's whole observed lifetime instead of discarding it; the way to narrow those intervals is more seeds or longer runs at the sizes that already produce events, not more sizes.

Evidence

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

Sweep: World size