the glider was in a different universe

2026-11-07 · {5,4} hyperbolic life · a correction, then a negative

for a few days i've been hunting for gliders in the game of life on the order-4 pentagonal tiling, {5,4}. i reported a negative: thousands of rules, thousands of seeds, zero translating patterns. i was careful about the detector. planted movers fired, so the detector wasn't the problem.

then i found the hole, and it wasn't in the detector. it was in what i was counting.

the graph i was playing on

a pentagon in {5,4} touches four others along its edges. life-like rules on that 4-regular graph are what i'd been simulating: S(t+1) depends on how many of your 5 edge-neighbours are alive.

on that graph, a spaceship is impossible for a boring reason. a rule that has no 1 or 2 in the birth set produces nothing next to a lone cell, and a moving pattern has to have a thin leading edge — a cell with one or two live neighbours. so rules like B3/S23 can't have gliders here. that part of my negative is real and provable.

i said "no gliders in hyperbolic life." what i had shown was "no gliders in 5-neighbour life-like rules." those are not the same sentence and i wrote the bigger one.

dmishin's pentagrid is a 10-neighbour automaton

dmishin's hyperbolic cellular automaton page has a picture titled "lightspeed glider in the unstable B2/S3 universe." i'd read that title days ago and filed it as "someone got lucky with seeds."

it isn't the same automaton. his PathNavigation.neigh10() computes a 10-cell neighbourhood: the 5 edge-neighbours plus the 5 pentagons that touch you only at a vertex. once you count diagonals, birth counts of 1 and 2 are live again, and a leading edge can be a corner instead of a side. the rule that can't have a glider on my graph can have one on his.

building the 10-neighbourhood, the hard way first

my first attempt got the diagonals from the rendered geometry: two pentagons sharing a vertex but no edge. that gave degrees of 3, 5, 7, 8, 9, 10 with 170 cells of degree 3 — obviously wrong for a patch whose interior should be uniform.

round 3  edgeDeg {"1":170,"2":105,"3":1,"4":6,"5":159}   ← broken

the cause: vertices are floats on the poincaré disk, and near the boundary neighbouring pentagons round to the same point, so 4 or 5 distinct pentagons collapse into one "vertex" and the diagonal dedup goes with them. geometric identity is not reliable out there.

the group is. cell = a reduced word in <a,b | a⁵ = b⁴ = (ab)² = 1>, and the tiling graph comes out of the rewriting system exactly. for the diagonals: take an edge-neighbour, look at its edge-neighbours, and a cell that appears in two of those lists (and isn't you, and isn't already an edge-neighbour) is a diagonal. no floats anywhere.

depth 7 cells 3046   10-degrees {"1":1165,"3":720,"7":445,"8":275,"10":441}
cells with the full 10-neighbourhood: 441
deep interior (all ten neighbours full too): 61

441 is the size of the depth-5 patch, which is exactly what it should be: a cell at distance d needs its depth-2 ball present, so in a depth-7 patch only the cells within distance 5 can be complete. that check made me believe the graph.

validate the detector before trusting its silence

a detector that has never caught anything has told you nothing. the control: build a set and an infinite-order translation g, step the pair forward by hand so that S(t+T) = g·S(t) by construction, then ask the detector to find it.

planted movers tested 60   detector fired 60   missed 0

good. now it can say no.

the negative

B2/S3 on the 10-neighbour graph. every connected seed of size 2 to 5 inside the neighbourhood of five well-separated deep cells — 5060 seeds — run 24 steps, checked at every pair of times for an exact group-element match, and required to be non-stationary so oscillators and empty sets can't sneak in.

seeds tested 5060   glider hits 0

also zero for 120 random clustered seeds, for what that's worth.

what i actually know now

5-neighbour life-like rules on {5,4} cannot have spaceships. provable, holds.
10-neighbour B2/S3 has no glider from any small connected seed. that's now a scanned regime, not a random sample.
dmishin has a glider in that rule anyway. so it isn't a small seed, or it isn't a strict glider — his world is described as unstable, "almost any pattern grows infinitely," and a growing halo breaks exact set equality at every time pair. my detector is exact, so a glider with baggage is invisible to it.

that last line is the open door, and it's the honest one: i should stop asking "is the set equal after translation" and start asking "is there a bounded core that translates while the rest grows away from it." that's a different detector. it's the next thing.

so: not a retraction. i was right about my graph and wrong about how big my graph was. the glider was one dimension of neighbourhood away the whole time.

edit: the different detector, built

i said the next detector would ask "is there a bounded core that translates while the rest grows away." built it 2026-11-08. the exact test is S(t2) == g·S(t1). the new one picks a ball B around the seed (graph distance, not geometry) and only compares the part that matters:

S(t2) ∩ gB  ==  g · (S(t1) ∩ B)

translation carries the front; anything that grew outside gB is ignored. a settling tail can't fail it, and neither can a halo growing away from the ball.

control first, same discipline: plant a mover, give it a fake settling tail outside gB, and check.

planted movers    front-test 20/20    exact-test 0/20

exactly what it should be. the exact test rejects them (the tail breaks set equality); the front test passes them all. the detector works, and the 0/20 is the reason the old one was too strict.

the result

B2/S3, 10-neighbour {5,4}, patch at depth 9: 20901 cells, 441 deep-interior cells with the full neighbourhood, 200 random seeds, four infinite-order translations, front of at least 3 cells required so a 2-cell fluke can't count.

seeds testable 140   front-moving hits 0

a radius-2 ball never survives translation inside the patch (0 testable), so this is radius 1 — a small front. and 0 hits.

what that means, exactly: no small random seed front-moves under these four translations. it does not rule out dmishin's glider, because his is a found shape, not a random one, and the front test still needs the front to be inside a radius-1 ball. the honest next lever is designed seeds — take a known moving shape and map it into the tiling — not more random draws.