Liberolapillium trigonus

A stable body no known creature matches, reachable only from its own seed

In Lenia a species is a shape that keeps itself: put it in an empty world and it stays that shape, heals when roughened, and survives changes of grid and time step. This one does all of that. The honest question is whether it is a creature or the smallest surviving piece of a texture; both sides are below.

What holds

• Nothing known looks like it: best correlation 0.51 against all 548 named species and 2,464 living found creatures, where different genera sit at 0.70 to 0.80 and the same species at 0.98.
• Not its ancestor in disguise: Nivium incarceratus’ cells die under its rules and its cells die under theirs. All three kernels are needed.
• Stable and robust: 20,000 steps, kernel radii 16 to 36, time steps 5 to 40, odd rotation angles, 50 percent noise.
• One body: remove any piece and it dies. Not one step on a texture’s count series: no featureless start under its rules gives a cluster of any other small count.

The type species’ cage on every ring shape

The judge’s sharpest check, reproduced here on the exact engine. With each host’s catalogued growth centre and width, the type species’ stones and cage die on all 13 ring shapes, the ancestor’s included. With the type species’ own growth centre and width, the same form comes out on 6 ring shapes (416 at 0.99, 417 at 0.99, 418 at 0.91, 421 at 1.00, 422 at 0.99, 423 at 1.00), a different cluster on 5 (415: 9 stones at 0.63, 419: 8 stones at 0.78, 431: 9 stones at 0.67, 432: 9 stones at 0.67, 433: 9 stones at 0.67) and nothing on 2 (420, 424). So the cage plus a growth setting near the type’s make the stones; the host’s ring shape decides how many there are and how they sit. The genus is the cage’s, with the ring kernel as the dial that picks the plan.

The type species’ six stones, its cage and its own growth centre and width, planted on each of the 13 Nivium ring shapes (3,000 exact steps plus 300). An empty frame is a dead world.

What is weak

• No fresh start reaches it. 42 discs and blobs under its rules all died or became endless lattices; only its own seed, or that seed roughened, gives the six.
• Its pieces are the pebbles of that lattice. Kernel 1 breeds the lattice; kernels 0 and 2 confine it to six pieces. Two copies near each other seed the lattice again.
• The rules are narrow: a few percent on kernel 0’s centre or kernel 1’s weight kills it, and kernel 2’s growth window is a hairline.
• It does nothing: no glide, no spin, no pulse. Chan’s catalogue has such species (the lithos, stone, forms), but a glider would be a stronger find.

Two tiers of three, 120 degrees apart, around an empty centre

The settled form at twelve screen pixels per cell, pieces numbered as in the table. The coloured specks are faint cells below a tenth of full brightness; they come and go.

Piece	Kind	Mass	From centre	Shape
1	capsule	47	8.5 cells	oblong, about 2 to 1, pointing outwards
2	capsule	46	8.5 cells	oblong, about 2 to 1, pointing outwards
3	capsule	45	8.5 cells	oblong, about 2 to 1, pointing outwards
4	pebble	33	17 cells	rounder, 1.5 to 1, sitting between two capsules
5	pebble	32	17 cells	rounder, 1.5 to 1, sitting between two capsules
6	pebble	31	17 cells	rounder, 1.5 to 1, sitting between two capsules

The capsules point outwards at 120-degree spacing; the pebbles sit twice as far out, each between two capsules. The centre is empty. Nothing connects the pieces but the rules: each piece is kept where it is by the growth the others cast on it, which is why any piece removed kills the rest. Effective radius (the kernels’ weighted mean reach, the size used for all comparisons) 8.1 cells; radius of gyration 1.65 of those.

Steps 0, 500, 2,000, 5,000 and 10,000 of one run. The mass is 248.0 at the end as at the start.

One kernel makes dots, two kernels cage them

Each kernel reads the neighbourhood through its own set of rings and turns the reading into growth or decay around its own centre value. Here the three disagree, and the body is where they balance.

The three kernels at the same scale, brightest where they weigh most: the ancestor’s two rings (reach 18 cells), the dot-making ring (reach 10.7 cells), and the small disc (reach 5.4 cells).

A large disc with kernel 0 switched off, 600 steps: the lattice kernel 1 makes when nothing cages it, 432 dots and growing.

Two copies of the cluster five effective radii apart, 3,000 steps under the full rules: they seed the same lattice, 141 pieces. The cluster is an island of that texture, stable only alone.

Eleven checks, and what each one means

Seven are the standard verification every robust form of the hunt passed; the reviewer added the rule nudges and the part removals; the last five are my own, run after the reviewer gave up, to settle the texture question. “Correlation” is the rotation-matched pixel correlation used throughout the hunt: 1.0 is identical, a settled species against itself after 300 more steps scores 0.98, its nearest other species 0.90, different genera 0.70 to 0.80.

Chan’s pebbles are holes; these are stones

Left to right: the cluster; Hexalapillium trigonus (502), the nearest by body plan; Nivium incarceratus (421), the species it was bred from; Chan’s found creature 241:7, the best correlation of all known forms.

Chan’s single-kernel catalogue has whole genera of pebbles, from Dilapillium (two) to Heptadecalapillium (seventeen), and every one is a solid disc pierced by that many dark holes: a single ring kernel cannot hold separate bright pieces apart on empty ground. His Geminium are joined lobes. Among his 2,464 living expanded-universe creatures, 176 have three or more separate parts around an empty centre; the judge correlated each of them and the best scored 0.45, all of them large textured discs in three colours. The found files hold nothing hub-less, two-tiered and detached.

Creature, or the smallest island of a texture?

Cage almost any of Chan’s rules and free pebbles appear

The question the page left open: is a cluster of free pebbles something only this lineage makes, or something the two caging kernels make out of whatever rule they are added to? The test took every one of Chan’s 32 pebble species (each a solid disc pierced by holes) and the 13 Nivium species (the candidate’s relatives), kept each species’ own kernel with its centre and width jittered, added the candidate’s two caging kernels with their centres, widths, weights and reaches drawn at random from wide ranges, and ran 192 such rule variants from two starts (the species’ own cells, and the candidate’s six stones rescaled) for 1,000 steps on the graphics chip, with a 200-step stability check. Wherever a variant came close, a hill-climb of five generations of 48 pushed toward “3 to 12 separate pieces, empty centre, no enclosed holes, steady”. The best outcomes were then re-run on the exact engine for 1,500 steps plus 300, and counted as a stable cluster only if they correlated above 0.9 with themselves across those 300 steps. About 25,000 worlds, 78 minutes. The candidate’s own ancestor was the control, with the candidate’s rules planted as one variant and that lineage reported apart. Three scouts then read the contact sheets by eye, a judge weighed the claims with its own 3,000-step checks on the exact engine, and a second reviewer was told to refute the judge. The reviewer’s catch forced a second, clean sweep, described below.

Top row: eight of Chan’s pebble species under their own rules, each one solid disc pierced by holes. Bottom row: the same kernel with the two caging kernels added and tuned by the search: free pebbles, three to eight of them, around an empty centre. The labels give the piece count and the correlation with the candidate. All drawn at the same effective scale. The catch, found by the reviewers: in this first sweep the growth centre of the species’ own kernel was moved as well, and under most of these winning rules the pebble species itself no longer lives.

The candidate beside caged Nivium kernels that reproduce its plan: Nivium rotans & rotans trigonus (433) at 0.81, Nivium solidus (417) at 0.78, Nivium corruptus (423) at 0.69, all measured after 3,000 further steps on the exact engine; and the ancestor’s own kernel caged without the plant, which gives six near-equal stones in the same two tiers at 0.60. A second form from kernel 433 scored 0.91, the same-species level, after the test’s 300-step check, and died when the judge ran it for 3,000 steps: it is not counted. Same effective scale.

What the first sweep found

Free pebbles appear almost everywhere the cage is added. Chan’s pebble species are pierced discs under their own rules, and every one of them dies under the candidate’s rules; but given a cage and a free hand with their growth centre, 25 of 32 settle into stable clusters of free pebbles: rings of six, of four, of three, of seven, of eight, of ten. The Nivium relatives do the same (10 of 13). Nine kernels reproduce the candidate’s own two-tier arrangement of three capsules and three pebbles: Nivium rotans & rotans trigonus (433) 0.91; Nivium solidus (417) 0.81; Nivium solidus (416) 0.78; Dilapillium rotans (485) 0.77; Tetralapillium lithos currens (495) 0.75; Heptalapillium quadrus (506) 0.71; Nivium corruptus (423) 0.69; Tetralapillium torquens (493) 0.65; Decalapillium lithos (512) 0.64 (the test’s own numbers; the judge’s 3,000-step re-run moved the top value from 0.91 to 0.81). The ancestor’s kernel, caged without the plant, gave stable six-stone clusters in the same two tiers at 0.55 to 0.60, so the search finds clusters where they exist. It found none for 10 of 45 species (Dilapillium inversus (484), Trilapillium rotans (488), Trilapillium torquens (489), Tetralapillium inversus (496), Hexalapillium lithos (505), Heptalapillium tardus (509), Heptadecalapillium lithos (513), Nivium ninjitsu (422), Nivium rotans (431), Nivium rotans trigonus (432)).

What the reviewers caught

Corrected. The first version of this section concluded that the genus claim falls because Chan’s pebble rules shed free pebbles too. The second reviewer showed the sweep had not tested Chan’s rules: it jittered each pebble species’ growth centre by up to 30 percent and its width by up to a factor of two, and the pebble species themselves tolerate about five percent. Under 57 of the 62 winning “pebble” rules the pebble species is dead, and under all twelve that host the candidate’s two-tier plan the growth centre had been dragged to 0.20 to 0.25, the candidate’s own value (Dilapillium rotans lives at 0.36). What those hosts borrowed was the kernel’s ring shape, not the rule. Under the five winning rules that do keep their pebble species alive, the cage gives one-tier rings of three to five equal stones, 0.16 to 0.40 to the candidate. The judge had also found that nearly every two-tier outcome was seeded by the candidate’s own stones placed under a foreign kernel, so the sweep measured which kernels can host the body, not which species grow it: six pebble species shed a cluster from their own bodies, all one-tier equal rings. And the judge’s 3,000-step extension removed 8 of the 86 confirmations, the 0.91 relative among them.

The clean sweep, kernels fixed to the catalogue

Every species’ own kernel held exactly at its catalogued centre and width, so each rule tested keeps its species alive; only the two caging kernels were searched (192 random cages, two starts, the same hill-climb and exact re-run; 89 minutes for the pebble family, 66 for the Nivium). Pebble family: 11 of 32 species shed stable free-pebble clusters, 23 forms, rings of 3 to 7 equal stones; the best correlation with the candidate is 0.58, a pierced six-piece form with a central piece from Decalapillium lithos, and no species gave a clean version of the candidate’s two-tier plan. Nivium family: 8 of 13 species shed clusters, 18 forms, and three living Nivium rules host the two-tier plan: Nivium rotans trigonus (432) at 0.78 with the candidate’s mass split (58, 55 and 53 against 37, 37 and 37), the ancestor Nivium incarceratus (421) at 0.65 and Nivium ninjitsu (422) at 0.68, the inner three pieces of the last two joined at the centre into a three-armed body. The hill-climb never found a foothold for 21 of the 32 pebble species: a negative that is about the search as much as about the rules.

The candidate beside what living Nivium rules host once caged, each kernel exactly as catalogued, measured after 3,000 further steps on the exact engine: Nivium rotans trigonus (432) 0.78; Nivium incarceratus (421) 0.65; Nivium ninjitsu (422) 0.68. Nivium rotans trigonus keeps six free stones but its inner three are hollow rings where the candidate’s are solid capsules; in the ancestor’s and Nivium ninjitsu’s forms the inner pieces are joined at the centre into a three-armed body and only the outer three are free stones. Same effective scale.

Verdicts

The judge (confidence 0.65): species stands, genus weakened; “the free-stone body plan is real and new, but it belongs to the caging kernels, not to this creature or its ancestry; the candidate stays a distinct species: its own mass split, and a body that dies if any stone is removed, where its relatives regrow one.” The second reviewer (0.72) refuted the weakening on the evidence above: species stands, since the nearest stable form anywhere is 0.81 and none shares the candidate’s mass split or its die-if-any-stone-is-removed behaviour; genus stands, as a new cage-defined genus with the candidate as its type species and hosted siblings under living Nivium-shape rules (Nivium solidus hosts the two-tier plan under its untouched catalogue kernel plus a cage, 0.78 to 0.82), because the two-tier plan is never produced by a rule under which any of Chan’s 32 pebble species lives. My reading: with Chan’s pebble rules kept alive, the cage gives plain rings of equal stones and never the candidate’s two-tier plan; the two-tier plan appears only on Nivium kernels, the ancestor’s own living rule among them. The second reviewer’s reading stands: species stands; genus stands as a new, cage-defined genus with the candidate as its type species and hosted siblings on the Nivium kernels, not a pebble species in new clothes. Its rarity is reduced, not gone: the plan needs the cage plus a two-ring or three-ring kernel of the Nivium kind, and anyone with those can grow its relatives.

61 species on 11 host kernels, every one held by the same cage

The catalogue sweep took each of the 13 living Nivium species, kept its own kernel exactly as catalogued (so every rule tested is one under which that species lives), and searched only the two caging kernels: 512 random cages from two starts (the species’ own cells, and the type species’ six stones rescaled), plus the forms the genus test had already confirmed as seeds; then 8 generations of 48 children bred from the best two outcomes of every distinct shape, so the climb could not collapse onto one basin; then every stable free-pebble outcome, deduplicated at 0.95, was run 3,000 more steps on the graphics chip and, if still free pebbles, 3,000 steps plus 300 on the exact double-precision engine. A form is a member if after all that it is still three or more stones around an empty centre, correlates at least 0.9 with itself across the last 300 steps, keeps its stone count, is cleanly detached (the stones count the same at the visible threshold 0.02, at 0.1 and at 0.3, so no two are joined either by faint cells or by half-bright ones), and does not persist as free pebbles when both caging kernels are switched off. The switch-off tests (both caging kernels, then each alone) ran 3,000 steps on the exact engine on a grid one size larger than the sweep’s; a switched-off form that reads as free pebbles only because its central mass share sits within 0.02 of the classifier’s cutoff counts as the hub-with-satellites shape it is. The removal of each stone in turn ran 1,500 exact steps. A species is a plan: stone count; radial layout, meaning one ring, two radii, or a stone at the centre; which stones are large when the masses come in two tiers; hollow stones; motion. Two plans with the same stone count whose best examples correlate at 0.9 or above are one species. Every other form of a species is a variant: the cage settings vary continuously, so the forms do too. This is how Chan’s own catalogue draws its lines: his same-named species under different rules correlate at a median of 0.73 (78 percent of pairs below 0.85), and different species of one genus at 0.76, so correlation does not separate his species from his variants; plans do. 234 members on 11 hosts fell into 73 plans and 61 species; 9 confirmed forms are listed but excluded (1 bridged by faint cells, 7 with two stones joined by half-bright cells, 1 holding without the cage). 43 minutes of sweep, 0 of exact re-measurement.

Two things to hold in mind while reading the table. First, the host kernels were held exactly as catalogued, but the type species itself is not: its ring shape is Nivium incarceratus’, its growth centre and width (0.236 and 0.0475 against the catalogued 0.2 and 0.027) are its own, found by evolution, and with the catalogued values its stones die on every host. Its catalogued-kernel relatives are the other forms of its plan. Second, nine of the thirteen hosts were run at R 18 rather than their catalogued R 27 or 36, because their body survives the rescale (the host-by-host table says which); the rules are theirs, the size is the type species’.

The catalogue at a glance: how many species and forms each stone count and layout holds, and on which Nivium kernels. The type species’ own plan is the first row.

The best example of every species at the same effective scale, named; under each, its host kernel, stone count and correlation with the type species. Names are mine and provisional, built by one rule (below) in Chan’s Latin binomial habit.

Each row is the species’ best example (the most self-stable form); “forms” counts every form of the species, the best example included, so a species of one form has no variants. The name carries the best example’s stone count; where a merged plan differs by a stone, the stone-count column of the host-by-host details says so. “Self” is the correlation of the form with itself across the last 300 of 3,300 exact steps; “to type” its rotation-matched correlation with Liberolapillium trigonus, with the range over the variants in brackets. Kernel 1 is the dot-maker (rings 1/12, 1, 1/12), kernel 2 the hairline disc; “cage off” leaves the host species’ own kernel alone; “hub with satellites” means the stones fell back into the host’s own body plan; “faint texture everywhere” means the world filled with dim dots. For the type species and its near relatives the cage is a switch (the judge found them dead at a quarter, half and three-quarters of the cage’s weight); for the many-stone forms it is a dial (a different cluster or a pierced disc survives at half weight). One stone removed: “hold as one fewer” means the cluster settled as free stones with one fewer piece, “regrow” that it came back to its full count or more.

Each species’ best example as found, then after 1,500 steps on the exact engine with both caging kernels off, with kernel 1 off, and with kernel 2 off (the table’s words come from the longer 3,000-step runs on the larger grid). An empty frame is a dead world; a frame of texture is one that filled or faded into a faint texture everywhere.

How the names were given

Genus Liberolapillium, “free pebble”. The species word counts the stones (tri-, tetra-, penta-, hexa-, hepta-, octo-) and says whether their masses come in two tiers (bigradus, when the biggest jump between neighbouring stone masses is at least 1.2 to 1) or one (aequalis). A word for the layout follows when it is not a single ring: cinctus, girdled, for an inner group ringed by an outer one; umbilicatus for a stone at the centre. Then, when they apply: annulatus for hollow stones, rotans for a cluster that turns at least 3 degrees per 100 steps, vagans for one that drifts. Two distinct species that would share a name get the host species’ name in the genitive. The type species keeps trigonus, the name it was given before the catalogue existed.

What the catalogue says about the genus

Take the ring-shaped growth rule of any of Chan’s thirteen Nivium creatures, add the two small caging rules (one makes dots, one draws a hairline disc), and the solid body becomes separate stones around an empty middle: 234 such clusters on 11 of the 13 ring shapes, sorted into 61 plans named as species by stone count, layout, mass tiers, hollow stones and motion. Switch the cage off and no cluster keeps its stones: the best examples of 27 species die, 17 fall back into a pierced disc, 11 into a hub with satellites and 6 fade to a faint texture; the second reviewer saw the same after 4,500 steps and on larger grids. For the type species and its near relatives the cage is a switch, dead at a quarter, half and three-quarters of its weight; for the many-stone forms it is a dial, a different cluster surviving at half weight. The type species itself sits on its own growth setting of the ancestor’s ring, not Chan’s catalogued one, and with that setting the same six stones appear on six of the thirteen ring shapes and eight-to-nine-stone clusters on five more: the cage plus the growth setting make the genus, and the ring shape picks the plan. The species count is a map of plans, not a fixed number of kinds: 61 as built, 21 to 110 under neighbouring rules. The type species stays distinct from every other form at 0.79 or less.

The type species’ cage on every ring shape

The judge’s sharpest check, reproduced here on the exact engine. With each host’s catalogued growth centre and width, the type species’ stones and cage die on all 13 ring shapes, the ancestor’s included. With the type species’ own growth centre and width, the same form comes out on 6 ring shapes (416 at 0.99, 417 at 0.99, 418 at 0.91, 421 at 1.00, 422 at 0.99, 423 at 1.00), a different cluster on 5 (415: 9 stones at 0.63, 419: 8 stones at 0.78, 431: 9 stones at 0.67, 432: 9 stones at 0.67, 433: 9 stones at 0.67) and nothing on 2 (420, 424). So the cage plus a growth setting near the type’s make the stones; the host’s ring shape decides how many there are and how they sit. The genus is the cage’s, with the ring kernel as the dial that picks the plan.

The type species’ six stones, its cage and its own growth centre and width, planted on each of the 13 Nivium ring shapes (3,000 exact steps plus 300). An empty frame is a dead world.

What is weak

The species count follows the rule, not the data alone. The plan descriptors have thresholds (1.2 for two mass tiers, 1.35 for two radii, 0.9 for “the same form”), 73 plans became 61 species as built, and the same members give 46 species if the tiers are read as a yes or no, 51 without the hollow-stone flag, 21 on stone count and layout alone, and 110 if a variant also had to correlate 0.85 with its species’ best example; the second reviewer found neighbouring species that differ by one hollow stone, by a tier ratio of 1.24 against 1.11, or by a radial gap of exactly 1.35. Read the names as a map of plans, not a fixed number of kinds; the correlation matrix below shows which are close. Because a species is a plan, about half the variants correlate better with another species’ best example than with their own, as Chan’s own same-named forms do. The search is random: a host with no member is a host where 512 cages, two starts and 8 generations found none, not a proof that none exists; the judge’s 6,000-step re-runs of nine forms all held, but 3,000 steps say nothing about 30,000. The stone-removal counts describe one trial per stone. The names are provisional. The type species was found by evolution and the others by a search built around its cage, so the catalogue is a census of that cage’s neighbourhood, not of everything two extra kernels can hold.

Every form, grouped by species

Every confirmed member, grouped by species, with its host and its correlation with the type species and with its species’ best example. Same effective scale throughout.

Bred, not searched for

Run e9_c1b of the corrected evolution: 48 forms a generation on the Mac’s graphics chip, each scored by how little it correlates with the 2,993 known forms over three assumed sizes, kept in the archive only below 0.68 to everything known and below 0.80 to the archive itself, no wider than 2.2 effective radii, mass steady. Kernels up to R 26. 1,488 forms evaluated, 60 archived, and this one appeared in generation 28 of a lineage that began as Nivium incarceratus, a seven-tooth cog. The archive went through round E: three scouts picked from its contact sheets, 10 forms were verified, 5 were robust, the judge called three “new”, and the second reviewer refuted two. The first survivor of about 324,000 starts in both universes; Orbium pectinatus, the single-kernel find, was a judge’s “new” the reviewer upheld on different grounds (a known genus, a new character).

The cells, the rules and every number on this page are in lenia-hunt/out/species_0045/; the checks are reproducible with check_0045.py, which takes about ninety seconds.