Invasion — grow the crossing from the left side by always opening the easiest link; it finds each draw's crossing point without a dial, its forced links and bursts are a second exponent, and the burst sizes on a carpet carry a ripple with the period of the hierarchy
no run lists this probe in builds_on:
Why this exists. Filed 2026-10-03 from the owner's question whether a self-organised mechanism could find the critical dial for us. The codifier named three (invasion, a thermostat, drive-and-burn), claimed five things for invasion from memory, and the owner asked for the claims to be audited and tested before filing. The audit is below; the computation was done by the codifier in the cloud workspace, not on the lab machine, and is a look, not a run: 400 draws per carpet at 729, no second reader, seeds and md5s in P-054-invasion-look/.
The construction. Give every link a random number. Start with the whole left side joined. Repeatedly open the lowest-numbered link on the edge of what is joined, whether or not that number is below any dial. The joined region takes easy links until none is left on its edge, is forced to accept one hard link, and then runs through another burst of easy ones. Stop when it reaches the right side.
- The forced links are the running maxima of the accepted numbers. The last of them, the hardest link the growth was ever forced to accept, is the draw's crossing point: no path from left to right can avoid a link at least that hard, and the invasion's own path shows one that needs nothing harder. This is the same number the lab's sampler finds by opening all links in order, and on the same random numbers it is the same number exactly (it is the bottleneck of the minimum spanning tree, which both rules compute). Wilkinson and Willemsen (1983) introduced the rule; Chayes, Chayes and Newman (1985) proved that the accepted numbers converge to the threshold on an infinite lattice.
- A burst is the run of accepted links between two forced links. On an infinite lattice at threshold the bursts have a power-law size distribution with its own exponent (the burst dynamics of Roux and Guyon, 1989). That exponent is not the sharpening rate; it is the second exponent the audit of 2026-10-03 (PG-01) found the record lacks.
The audit of the codifier's claims (what was said on 2026-10-03, then what the literature and the computation say).
| claim | verdict |
|---|---|
| It finds the same crossing point exactly. | Holds. A theorem, and 0 mismatches in 7,200 draws (square, T, X at 81 and 729) with both rules on the same numbers. Mean crossing points match the lab's record: T at 729 0.7866 (run 083's pooled mean 0.786), X 0.7221 (census p_med 0.7217), square 0.4999; T's 20–80 width 0.0247 (run 085: 0.0250). |
| It gives the route, and the route reads P-053's table. | Fails as stated. The invasion's own path from right to left is a bottleneck-optimal route, but it is one of many and it wanders: on T at 729 it visits 108 block-faces for 64 blocks, and 81% of those visits enter and leave a block by the same side (X: 81 visits, 75%; square: 38, 46%). It is the tree's path, not the short one. The table is read from each block's own events (P-053's recorder), not from this path. |
| It gives a second exponent. | Holds. Burst sizes between forced links, up to crossing, follow a power law with exponent 1.68 on the square at 729 (halves of the file: 1.678, 1.678; within ±5% of the fitted line over sizes 4 to 10,000). On the carpets the fitted exponent is 1.71 (T) and 1.71 (X), but see the ripple below, which makes a one-exponent fit window-dependent. |
| Bursts bunch at powers of 9, the size of the holes. | Holds in substance, wrong in the number. The ratio of the burst-size density to its fitted power law on T peaks at sizes near 25 and near 1,400 (1.70, 1.72) with troughs near 130–360 (0.75) and beyond 10,000 (0.5); on X the same peaks at 1.14 and 1.32, troughs 0.77 and 0.74; on the square no feature above ±5%. The period is a factor of about 55 in size, which is the hierarchy's factor in cells (64 kept cells per block, 84 links), not its factor of 9 in side. T's ripple is twice X's. |
| It may be cheaper at 6,561. | Overstated. 2.3 times faster than a sort-and-join rule here at 729 (0.036 s against 0.083 s a draw, single thread), but the lab's perc.c shuffles incrementally and does not sort, and the invasion joins 58% of T's cells before it crosses (square: 34%). About the same cost is the honest estimate; not sized at 6,561. |
| One draw can locate the threshold from the acceptance profile. | Not supported. On the square the profile (accepted over observed, per 0.01 of the number) is a step whose half-point moves 0.470 → 0.481 from 81 to 729, toward 1/2. On T it is a ramp from 0.90 at 0.41 to 0 at 0.81, half-point 0.678 → 0.666, against a crossing median of 0.786: it does not sharpen and does not sit at the threshold. Continuing the growth past the crossing to a quarter of the cells adds a new forced link in 1 draw of 400 on T, 4 of 400 on X, 11 of 200 on the square. The invasion's threshold is the crossing point and nothing beyond it. Withdrawn. |
| Drive-and-burn settles above the threshold: planar lattices do not recover from forest fires. | Citation holds, scope overstated. Kiss, Manolescu and Sidoravicius, Ann. Probab. 43(6), 2015 (arXiv:1312.7004): for any p above threshold on a planar lattice invariant under translation, rotation and reflection, removing the infinite cluster and reopening a fraction δ of the closed sites does not restore percolation, with one δ for all p. A carpet is planar and is not translation-invariant, so the theorem does not cover it, and "carpets will not recover because they are planar" was not a consequence of it. The question for carpets is open. |
| The thermostat buys nothing with one dial. | Reasoning only; untested. |
Citations checked 2026-10-03: Chayes–Chayes–Newman and Wilkinson–Willemsen as stated by Bock and Damron, "The acceptance profile of invasion percolation at p_c in two dimensions" (arXiv:1904.08893); Roux and Guyon as stated by Tizdast et al., "Dynamical crossover in invasion percolation" (arXiv:2003.02193); Maslov (1995) was named from memory and is not confirmed, so it is not cited.
The look, in numbers (seeds 101, 103, 104 at 81 and 729; 2,000 draws at 81, 400 at 729 on each carpet, 200 on the square; errors where given are from halves of the data).
| square 729 | T 729 | X 729 | |
|---|---|---|---|
| forced links per draw (mean ± sd over draws) | 2250 ± 483 | 2093 ± 551 | 1689 ± 406 |
| cells joined when it crosses (median share) | 0.34 | 0.58 | 0.49 |
| burst exponent, sizes 4–6,000 | 1.678 | 1.707 | 1.710 |
| ripple of burst density about the power law | none above ±5% | +70% at ~25 and ~1,400; −25% at 130–360 | +14%, +32%; −23%, −26% |
| forced links on row seams, relative to the seam's share of links | 0.92 | 0.80 | 0.90 |
The last row: a forced link on T sits on the seam between two blocks 20% less often than the seam's share of links would give, against 8% less on the square (the square's column seams read 0.36 and are not comparable: forced links pile up near the source side). About 2,100 seam records per carpet, so ±0.02. A look.
The lead, held with conviction. 1. The ripple is the hierarchy, and its period is the cell factor. Bursts are bounded by the holes: a burst fills a region between forced links, and the regions available on a carpet come in the sizes of its blocks. The ripple's amplitude measures how strongly the holes box the growth, and T's is twice X's. This is P-048's "ripple invisible at L = 9ᵍ" made visible, because bursts sample every size at once where the crossing points sample one size per run. It should repeat at ~75,000 (not reachable at 729) and its amplitude should not fall with L. 2. The burst exponent is a second exponent of the pattern, and twins share it. Patterns with one block polynomial (run 086) should have the same burst exponent and different ripples, since the ripple is where the holes sit and the exponent is how the blocks connect. 3. The forced links avoid the seams more on the pattern whose lag is smaller. T's forced links sit inside blocks; the ladder's premise is that what passes between blocks is small, and this reads that directly. Weakly held: one number per pattern at one size.
Where it breaks. - The route claim is gone. Nothing in invasion reads the table; P-053's recorder does. What invasion adds that the recorder does not is the bursts, and the ripple in them. - The ripple may be the source, not the holes. Growth starts from the whole left side, so early bursts are wide and shallow and late bursts are deep; a ripple of period 55 could be the distance structure from the source. The square has the same source and no ripple, which is the control, but a point source and a periodic strip would settle it. - The burst definition. Bursts here are the runs between successive running maxima, on a finite lattice, stopped at the crossing. Roux and Guyon's hierarchy and Maslov's time-directed avalanches are other definitions with other exponents; 1.68 on the square is this definition's number, and it is the carpet's control, nothing more. - Three sizes of nothing. The ripple has two periods inside 729. One more period needs 6,561 and the burst file for 8,000 draws at that size is large; the sizes alone need storing, not the links.
A first reductio pass (OPS.md). Limits: with no holes the ripple is absent (holds, the square); on the one-block pattern at L = 9 there is one burst. Invariance: the crossing point is invariant between the two rules (holds exactly). Twins: lead 2. Neighbours: H (#17, the Sierpinski carpet's second generation) has a period of 8 cells per level in base 3 and should show a ripple of period 8, not 55; that separates "the hierarchy's cell factor" from any other 55. Negation: the ripple is the source's; a point source on T shows it as strongly or it is gone.
What a first run would be. T, X and H at 729 with a point source at the middle of the left side and with the full side, 2,000 draws each, burst sizes stored, seeds sealed; the clause is the ripple's period on H (8 in cells per level against 55 on T and X) and its survival under the point source. Then the exponent and the ripple amplitude over the census at 729 as a new column beside 1/ν, since one draw is a tenth of a second. Not sealed. The program here is the codifier's and is not the lab's instrument; the lab's version takes perc.c's bond order so that control (i) is exact against the stored samples.
Kept out on purpose. The thermostat (one dial moved after each draw) and drive-and-burn (remove the spanning cluster whenever one forms) are named in PG-01's entry of this date and not filed: the first adds nothing to a sampler that has each draw's exact crossing point, the second organises to a state above the threshold on symmetric planar lattices, and whether a carpet recovers from the fire is a different thread.
118 source lines, no addenda.