Time symmetries from local rules

Which of the 68 animation symmetries a homogeneous local rule can make by itself, from a random start · Wed 28 Oct 2026 (week 44) · for you, then for Vladimir Bulatov, Chaim Goodman-Strauss and Scott Vorthmann

17 of the 51 clockwork groups are now made by a local rule we ran and a second agent re-ran — one phase per site, the same coupling table in every cell, a uniformly random start, and no steering. 5 more were reached with the reach re-measured, Hopf theory predicts 3 that no rule of ours reached, 3 are named in the literature and nowhere else here, 15 were touched without being confirmed, and 8 are open. Those 17 + 5 + 3 + 3 + 15 + 8 add to 51: each group sits in exactly one tier.

17confirmed by rules we ran
3predicted by Hopf theory only
5reached, reach corrected
8open clockworks

Card 4 below is the 333 film from the Tue 27 Oct 2026 (week 44) call. On the bare triangular lattice it is a needle: 27 random starts in 200 found it — 13.5%, at a 12 × 12 torus with identical frequencies (results/osc-333-reach.json), and the companion page measures the same rule against a stricter test — one handedness on every triangle and the bond average at its floor — and gets 28 of 300, about one in ten. Designing the cell instead of the coupling turned it into the generic outcome — 38 of 40 starts, which is a different rule rather than a better measurement of the same one — and the same trick delivered 12 more groups the bare lattices could not reach. That fourth card has a page to itself: Repelling clocks make 333 is the slow explanation of it (tailnet copy).

Costs follow the Mon 26 Oct 2026 (week 44) convention: (g, τ) is a symmetry iff q(g·x, t + τT) = q(x, t). A product group is a wallpaper group × time: every operation costs nothing, so a still picture or one that blinks in unison lands there. Those 17 are drawn on the map but never counted as a find — 2 of them a rule of ours did land on, and they are still marked “no clock”.

All 68, by parent wallpaper group

Each cell shows the group in Symmetries of Things notation, linked to its catalogue row, with its kernel — the wallpaper group of one frame — and its clock order underneath.

confirmed by a rule we ran reached, reach corrected predicted by Hopf theory named in the literature, not run here a search landed here, not confirmed open product group — no clock a product group a rule of ours landed on — still no clock, never counted

p1 1 group

op1 · clock 1no clock

p2 3 groups

2222p2 · clock 1no clock · ours
2₁2₁2₁2₁p1 · clock 2touched
c222₁2₁p2 · clock 2reach corrected

pm 1 group

**pm · clock 1no clock

pg 1 group

××pg · clock 1no clock

cm 2 groups

*×cm · clock 1no clock · ours
~*×½p1 · clock 2literature

pmm 6 groups · 1 confirmed

*2222pmm · clock 1no clock
*~2~2~2~2p2 · clock 2touched
*2₁~2₁2₁~2₁pm · clock 2touched
c*222₁2₁ᵃpmm · clock 2automata, oscillators
c*222₁2₁ᵇpmg · clock 2touched
c*22₁22₁cmm · clock 2theory

pmg 6 groups · 1 confirmed

22*pmg · clock 1no clock
22~*p2 · clock 2touched
2₁2₁~*pg · clock 2touched
2₁2₁*pm · clock 2touched
c22₁*ᵃpmg · clock 2oscillators
c22₁*ᵇpgg · clock 2touched

pgg 4 groups

22×pgg · clock 1no clock
22×½p2 · clock 2touched
2₁2₁×pg · clock 2touched
c22₁×¼p2 · clock 4reach corrected

cmm 6 groups · 2 confirmed

2*22cmm · clock 1no clock
2*~2~2p2 · clock 2touched
2₁*2₁~2₁cm · clock 2touched
c2₁*22ᵃpmm · clock 2oscillators
c2₁*22ᵇpgg · clock 2touched
c2*2₁2₁pmg · clock 2oscillators

p4 6 groups · 1 confirmed

442p4 · clock 1no clock
4₂4₂2p2 · clock 2designed cells
4₁4₁2₁p1 · clock 4open
4₃4₃2₁p1 · clock 4open
c44₂2₁p4 · clock 2open
c4₁4₃2p2 · clock 4open

p4m 6 groups · 2 confirmed

*442p4m · clock 1no clock
*4₂~4₂2pmm · clock 2touched
*~4₂4₂~2cmm · clock 2open
*~4~4~2p4 · clock 2designed cells
c*44₂2₁ᵃp4m · clock 23 families
c*44₂2₁ᵇp4g · clock 2touched

p4g 6 groups · 3 confirmed

4*2p4g · clock 1no clock
4₂*~2pgg · clock 2designed cells
4₂*2cmm · clock 2reach corrected
4*~2p4 · clock 2designed cells
c4₃*2pmm · clock 4oscillators
c4₁*2pgg · clock 4literature

p3 4 groups · 2 confirmed

333p3 · clock 1no clock
3₂3₂3₂p1 · clock 3designed cells
3₁3₁3₁p1 · clock 3reach corrected
r33₁3₂p3 · clock 3designed cells

p3m1 2 groups · 1 confirmed

*333p3m1 · clock 1no clock
*~3~3~3p3 · clock 2designed cells

p31m 4 groups · 2 confirmed

3*3p31m · clock 1no clock
3*~3p3 · clock 2designed cells
r3₂*3p3m1 · clock 3oscillators
r3₂*~3p3 · clock 6open

p6 6 groups · 2 confirmed

632p6 · clock 1no clock
6₂3₂2p2 · clock 3theory
6₄3₁2p2 · clock 3theory
6₃32₁p3 · clock 2designed cells
6₅3₂2₁p1 · clock 6designed cells
6₁3₁2₁p1 · clock 6reach corrected

p6m 4 groups

*632p6m · clock 1no clock
*6₃~3~2₁p31m · clock 2open
*~6~3~2p6 · clock 2open
*~6₃32₁p3m1 · clock 2literature

The eight best clockworks, running here — what you can press

Each demo starts from a uniformly random set of phases and integrates the rule in your browser — nothing is pre-baked. Restart gives a new random start; Check it applies the one operation that costs time and prints how well the moved picture matches the live one after the stated wait, together with the wait it measures for itself. A hand is a site’s phase; the ring’s shade is the same phase, so colour and motion say the same thing. Each card says how long it takes to settle: a demo takes more steps a frame while it is settling and slows to a watchable rate once it has.

1

6₃32₁ kernel p3 · clock order 2

Every one of 40 random starts reaches it, and the clock is a sixth-turn of the pinwheel — the cleanest find on the page.

The rule
Kuramoto with a second harmonic (h₂ = 0.15) on a triangular lattice whose cell carries a six-arm pinwheel; one coupling per bond class (11 classes, 23 bonds a site), chosen by a linear program: ±1 in the pattern − − + + − − + − − − +.
The clock
a sixth-turn of the six-arm pinwheel about the cell’s centre costs half a period. That is the operation Check it applies, written as the integer matrix [[0,-1],[1,1]] on the cell index.
Random starts that reach it
40 of 40 at a 12 × 12 torus (100%, 95% interval 91%–100%)
Settles on screen in
about 8 s from a typical random start (measured at 19.8 fps with all eight live, results/demo-settling.json).
Checked by
a second agent re-ran the rule from its own code: confirmed, with the same Heun integrator this page steps with, so the reach above is comparable. The page’s JavaScript reproduces the family’s own vector films/gap-1/vectors/pin6_tri_m3_q0-0_p0p272727-0p0909091_rc1_a0_h0p15_L12_designed.npz.
2

3*~3 kernel p3 · clock order 2

Three mirrors of the cell cost half a period each and not one turn costs anything — the clock lives on a reflection. The only mirror decoration every random start reaches.

The rule
Kuramoto with a second harmonic (h₂ = 0.4) on a triangular lattice whose cell carries a six-site rosette; one coupling per bond class (8 classes, 13 bonds a site), chosen by a linear program: ±0.0769 in the pattern − + − − − + − +.
The clock
a mirror reflection of the six-site rosette costs half a period. That is the operation Check it applies, written as the integer matrix [[1,1],[0,-1]] on the cell index.
Random starts that reach it
40 of 40 at a 12 × 12 torus (100%, 95% interval 91%–100%)
Settles on screen in
about 11 s from a typical random start (measured at 19.8 fps with all eight live, results/demo-settling.json).
Checked by
a second agent re-ran the rule from its own code: confirmed, with the same Heun integrator this page steps with, so the reach above is comparable. The page’s JavaScript reproduces the family’s own vector films/gap-2/vectors/gap2-top1-g233-kuramoto.json.
3

4₂4₂2 kernel p2 · clock order 2

The square-lattice twin of 333: a quarter-turn pinwheel with a half-period clock.

The rule
Kuramoto with a second harmonic (h₂ = 0.15) on a square lattice whose cell carries a four-arm pinwheel; one coupling per bond class (10 classes, 18 bonds a site), chosen by a linear program: ±1 in the pattern − + − + + + − + − −.
The clock
a quarter-turn of the four-arm pinwheel about the cell’s centre costs half a period. That is the operation Check it applies, written as the integer matrix [[0,-1],[1,0]] on the cell index.
Random starts that reach it
39 of 40 at a 12 × 12 torus (98%, 95% interval 87%–100%)
Settles on screen in
about 7 s from a typical random start (measured at 19.8 fps with all eight live, results/demo-settling.json).
Checked by
a second agent re-ran the rule from its own code: confirmed, with the same Heun integrator this page steps with, so the reach above is comparable. The page’s JavaScript reproduces the family’s own vector films/gap-1/vectors/pin4_sq_m2_q0-0_p0p142857-0p285714_rc1p25_a0_h0p15_L12_designed.npz.
4

r33₁3₂ kernel p3 · clock order 3

The 333 film from the Tue 27 Oct 2026 (week 44) call, now reached from almost every start instead of 27 in 200 on the bare lattice.

The rule
Kuramoto with a second harmonic (h₂ = 0.15) on a triangular lattice whose cell carries a three-arm pinwheel; one coupling per bond class (7 classes, 22 bonds a site, a phase lag of 0.125π), chosen by a linear program: ±1 in the pattern + − − − − − +.
The clock
a third-turn of the three-arm pinwheel about the cell’s centre costs a third of a period. That is the operation Check it applies, written as the integer matrix [[-1,-1],[1,0]] on the cell index. The cell drawn here is the √3 × √3 superlattice of the film from Mon 26 Oct 2026 (week 44), so every slide of the drawn cell costs nothing; the leading r in the symbol is the slide by one step of the underlying triangular lattice, which costs a third.
Random starts that reach it
38 of 40 at a 12 × 12 torus (95%, 95% interval 84%–99%)
Settles on screen in
about 11 s from a typical random start (measured at 19.8 fps with all eight live, results/demo-settling.json).
Checked by
a second agent re-ran the rule from its own code: confirmed, with the same Heun integrator this page steps with, so the reach above is comparable. The page’s JavaScript reproduces the family’s own vector films/gap-1/vectors/pin3_tri_m0_q0p333333-0p666667_p0p142857-0p285714_rc1p5_a0p3927_h0p15_L12_designed.npz.
5

*~3~3~3 kernel p3 · clock order 2

Three mirrors, each costing half a period, on the triangular rosette; no turn costs anything.

The rule
Kuramoto with a second harmonic (h₂ = 0.4) on a triangular lattice whose cell carries a six-site rosette; one coupling per bond class (6 classes, 8 bonds a site), chosen by a linear program: ±0.125 in the pattern − − + − − +.
The clock
a mirror reflection of the six-site rosette costs half a period. That is the operation Check it applies, written as the integer matrix [[1,0],[-1,-1]] on the cell index.
Random starts that reach it
12 of 40 at a 12 × 12 torus (30%, 95% interval 18%–45%)
Settles on screen in
about 12 s from a typical random start (measured at 19.8 fps with all eight live, results/demo-settling.json).
Checked by
a second agent re-ran the rule from its own code: confirmed, with the same Heun integrator this page steps with, so the reach above is comparable. The page’s JavaScript reproduces the family’s own vector films/gap-2/vectors/gap2-top4-g231-kuramoto.json.
6

*~4~4~2 kernel p4 · clock order 2

All four mirror lines of the square cell cost half a period; not one turn costs anything.

The rule
Kuramoto with a second harmonic (h₂ = 0.4) on a square lattice whose cell carries an eight-site rosette; one coupling per bond class (8 classes, 10 bonds a site), chosen by a linear program: ±0.1 in the pattern − − − + + − + −.
The clock
a mirror reflection of the eight-site rosette costs half a period. That is the operation Check it applies, written as the integer matrix [[0,1],[1,0]] on the cell index.
Random starts that reach it
10 of 40 at a 12 × 12 torus (25%, 95% interval 14%–40%)
Settles on screen in
about 11 s from a typical random start (measured at 19.8 fps with all eight live, results/demo-settling.json).
Checked by
a second agent re-ran the rule from its own code: confirmed, with the same Heun integrator this page steps with, so the reach above is comparable. The page’s JavaScript reproduces the family’s own vector films/gap-2/vectors/gap2-top3-g131-kuramoto.json.
7

4*~2 kernel p4 · clock order 2

Its mirrors are all glides — four of them: reflect the rosette, slide half a cell along the diagonal, and the picture comes back half a period later.

The rule
Kuramoto on a square lattice whose cell carries an eight-site rosette; one coupling per bond class (6 classes, 11 bonds a site), chosen by a linear program: ±0.0909 in the pattern − + − + − +.
The clock
a glide — a mirror reflection of the eight-site rosette, slid half a cell along the cell diagonal — costs half a period. That is the operation Check it applies, written as the integer matrix [[0,1],[1,0]] with the half-cell offset [0.5,0.5] on the cell index.
Random starts that reach it
6 of 40 at a 12 × 12 torus (15%, 95% interval 7.1%–29%)
Settles on screen in
about 11 s from a typical random start (measured at 19.8 fps with all eight live, results/demo-settling.json).
Checked by
a second agent re-ran the rule from its own code: confirmed, with the same Heun integrator this page steps with, so the reach above is comparable. The page’s JavaScript reproduces the family’s own vector films/gap-2/vectors/gap2-top5-g135-kuramoto.json.
8

4₂*~2 kernel pgg · clock order 2

The hardest confirmed find: it turns up in 3 of 60 random starts, and it still comes back when kicked.

The rule
Kuramoto on a square lattice whose cell carries an eight-site rosette; one coupling per bond class (6 classes, 11 bonds a site), chosen by a linear program: ±0.0909 in the pattern − − − + + −.
The clock
a quarter-turn of the eight-site rosette about the cell’s centre costs half a period. That is the operation Check it applies, written as the integer matrix [[0,-1],[1,0]] on the cell index.
Random starts that reach it
3 of 60 at a 12 × 12 torus (5.0%, 95% interval 1.7%–14%) Measured over 60 starts at L=12 with 30,000 steps (five times the search's integration time); 40 starts at the search's own 6,000 steps gave 0 and none locked.
Settles on screen in
about 11 s from a typical random start (measured at 19.8 fps with all eight live, results/demo-settling.json).
Checked by
a second agent re-ran the rule from its own code: confirmed, with the same Heun integrator this page steps with, so the reach above is comparable. The page’s JavaScript reproduces the family’s own vector films/gap-2/vectors/gap2-top6-g133-kuramoto.json.

The other 9 confirmed finds

Same bar — a rule we ran, re-run by a second agent — but their demos would need a different engine (three coupling shells, or a cellular automaton), so they are listed rather than animated. Where more than one family confirmed a group the row shows every family’s reach, because the headline figure is the best of them. Two of them, 3₂3₂3₂ and 6₅3₂2₁, are there because the verifier found them: round two had set each aside for want of an identification on the bigger torus, and the verifier’s own runs of the same rule reached them at a 24 × 24 torus, every hit matching the catalogue signature (families/gaps/verify/step2_basins_L24.json). Their reach is measured at L = 24, not 12, so it is not comparable with the rows above it.

SymbolParentClockFound byRandom starts
c*222₁2₁ᵃpmm2cellular automata, phase oscillators32 of 40 (80%)
phase oscillators 32/40; cellular automata 165/308
c22₁*ᵃpmg2phase oscillators18 of 40 (45%)
c2₁*22ᵃcmm2phase oscillators40 of 40 (100%)
c2*2₁2₁cmm2phase oscillators23 of 40 (57%)
c*44₂2₁ᵃp4m2cellular automata, limit-cycle lattices, phase oscillators30 of 30 (100%)
phase oscillators 19/40; cellular automata 7/64; limit-cycle lattices 30/30; best on the class's best_rule (K=-3, alpha=1.1, beta=2), L=12x12
c4₃*2p4g4phase oscillators37 of 40 (93%)
3₂3₂3₂p33designed decorations3 of 12 (25%)
r3₂*3p31m3phase oscillators31 of 40 (78%)
6₅3₂2₁p66designed decorations5 of 12 (42%)

Reached, with the reach corrected

The verifier rebuilt the group and its time costs exactly, and then measured a reach that did not match the search’s claim: four of the five came out lower, and c222₁2₁ came out higher — 13% against a claimed 4.2%. In two of them the arithmetic closes exactly, because the missing starts went to the enantiomorph: 3₁3₁3₁ lost 14 of 40 to 3₂3₂3₂ and 6₁3₁2₁ lost 12 of 40 to 6₅3₂2₁ — which is also how those two became finds of their own, two rows up. The search counted a mirror image as a hit; an energy match cannot tell the two apart, and the detector can.

The two families do not measure reach the same way. The designed-decoration rows ask for a locked state (the spread of dθ/dt across sites under 1e‑7), which is stricter than the searches’ own test (the phase pattern drifting less than 1e‑3 over 400 further steps), so a row can be identified without being counted. 4₂*~2’s 3 of 60 needed 30,000 steps: at the search’s 6,000 none of 40 starts locked at all.

SymbolFamilyMeasured reachClaimedvs the claim
c222₁2₁phase oscillators5 of 40 (13%)4.2%higher
c22₁×¼phase oscillators34 of 40 (85%)100%lower
4₂*2designed decorations8 of 40 (20%)47%lower
3₁3₁3₁designed decorations11 of 40 (28%)74%lower
6₁3₁2₁designed decorations19 of 40 (48%)78%lower
6 claims withdrawn by the re-run, across 4 groups
SymbolFamilyWhy it was withdrawn
olimit-cycle latticesRe-run independently as a product-group control, with the clock order read off the state by hand (a clock above 1 needs some op to hold to better than 2e-3 turns, so it cannot come from an unlocked run). Refuted as a product rule. From 30 of its own random starts at L=12 × 12 this rule reaches states with clock orders…
oreaction–diffusionthe group identification itself. The rule, its attractor, the reach and the 'product group, not a clockwork find' verdict all reproduce; the label o (kernel p1, one point operation) does not.
*×limit-cycle latticesidentified-lc.json reports this row with product: false and clock_order: 2. The catalogue's *× has clock_order 1 and tau = 0 on every one of its 2 ops — it is one of the 17 product groups named in the brief. A film with a clock above 1 cannot be this group, so the row is internally impossible: either the id or the…
~*×½phase oscillatorsthe claimed group was not reached at the second torus size, so the period could not be checked on two tori. halving the RK4 step (dt 0.0125) and re-seeding: 0 of 12 starts reach the same group (0 locked; the halved step covers half the integrated time in the same step budget, so a lower locked count there is expected).
~*×½cellular automatathe verifier's stepper, written from meta_json.rule alone, reproduces the family's own exported cycle frame by frame: yes. group by hand: the film's symmetry operations were found by brute force over D₄ × the torus × the time shift and each candidate was confirmed on all 48 frames; the costs of the cosets of the…
3*3limit-cycle latticesidentified-lc.json reports this row with product: false and clock_order: 4. The catalogue's 3*3 has clock_order 1 and tau = 0 on every one of its 6 ops — it is one of the 17 product groups named in the brief. A film with a clock above 1 cannot be this group, so the row is internally impossible: either the id or…

How it works, plainly

The detector

Take the film’s sites, put them on their own primitive lattice, and find that lattice’s own symmetries exactly by solving R G Rᵀ = G over the integers — so a √3 supercell, an unreduced basis or a supercell torus all give one answer. Then try every one of those point operations combined with every translation that carries site 0 onto a site, and keep the ones that permute the site set. For each kept operation, read off the one time shift that makes the moved film equal the live one; that shift is its cost. The resulting table of (operation, cost) is matched against the catalogue’s 68 rows operation by operation, not by name. Its suite is 292 tests, reported passing by the agent that wrote it (symdetect/README.md); they are not re-run by this page’s build, which has no pytest.

The search

FamilyWhat it isRulesOutcome
oscphase oscillators (Kuramoto–Sakaguchi, 7 lattices, 3 shells)2,959 of 7,252stage A cut short by Modal contention; stage B used a-priori rules
cacellular automata (outer-totalistic, cyclic, Greenberg–Hastings, Generations)280,692283 rules whose attractor is a clockwork crystal; 150 films exported
lclimit-cycle lattices (Stuart–Landau, Brusselator)1,3033,628 runs; 628 crystals (17.3%); 101 classes, 64 with a time cost
rdreaction–diffusion fields (CGL, two-layer Brusselator, FitzHugh–Nagumo)—every attractor found was a product group: one confirmed, 2222
gap-1chiral pinwheel decorations, couplings from a linear program234 of 5945 targets reached; stage A stopped at 234 of 594 configurations
gap-2mirror decorations, couplings from a linear program—6 clockworks reached, 5 of them confirmed
gap-3cellular automata run backwards (inverse design)—7 groups hit, all with a measured basin of zero from random starts

Two rounds. Round one swept rule space: bare lattices, every coupling sign, 280,692 cellular automata, 1,303 limit-cycle rules. It landed on 31 of the 68 groups — 20 of them clockworks (results/identified-{osc,ca,lc,rd}.json) — but most of the clockworks stayed out of reach, because an oscillator sitting on a rotation centre forces a mirror and kills the clock. Round two swept rule shape: decorate the cell with a free orbit — a pinwheel with no site on a centre, or a mirror rosette — and then solve for the couplings with a linear program that makes the wanted state beat every competitor. 13 of the 22 confirmed-or-reach-corrected clockworks above come from round two, all eight live cards among them.

AgentModal appSpent
rdtslr-rd$1.08
osctslr-osc$0.98
gaps-t3tslr-gap-3$0.69
catslr-ca$0.66
lctslr-lc$0.64
gap-t2tslr-gap-2$0.59
verifier-rdtslr-rd-verify$0.18
gap-1tslr-gap-1$0.17
total—$4.99

$4.99 of Modal compute, against the $20 ceiling you set. Each figure is each agent’s own estimate from its containers, cores and wall time at Modal’s published rates — not a billed number; launch-time estimates summed to $5.49. From results/modal-spend.jsonl.

The one-sign lesson

On the triangular lattice the same rule with one sign flipped gives two different universes. With attraction the clocks pull into unison: the film is a still picture blinking in step, and its group is the product o — no clock. With repulsion unison becomes a repeller and the 120° arrangement becomes the attractor, which is r33₁3₂. Every clockwork on this page is an attractor of its rule: that is what makes it reachable from a random start, and it is the opposite of the published Brusselator film with the same group, where the clockwork is unstable.

Theory and the literature

A Hopf bifurcation on a square or hexagonal lattice has a short list of branches, and each branch is an exact Fourier field whose group can be computed rather than searched. That list is the independent prediction of what a smooth local rule should be able to reach. Of its 19 rows, 14 have a sound signature; 5 are drifts that match an ITA number but not the cost signature (*×, **, *632, *~6₃32₁), and are marked provisional. A provisional row is never counted as a prediction: that is why the KPI tile above says 3 and not 4 — *~6₃32₁ was being quoted as a prediction until this was fixed on Wed 28 Oct 2026 (week 44), and it now appears only as a literature item. No stability inequality was re-derived here: Silber & Knobloch’s 1991 tables could not be opened.

PlanformLatticeGroupKindStatus here
Travelling rolls (TR)square**product *provisional — cost signature did not match
Standing rolls (SR)squarec*222₁2₁ᵃclockworkconfirmed by a rule we ran
Travelling squares (TSq)square*×product *provisional — cost signature did not match
Standing squares (SS)squarec*44₂2₁ᵃclockworkconfirmed by a rule we ran
Alternating rolls (AR)squarec4₃*2clockworkconfirmed by a rule we ran
Alternating rolls, opposite handednesssquarec4₃*2clockworkconfirmed by a rule we ran
Standing cross-rolls (SCR)squarec2₁*22ᵃclockworkconfirmed by a rule we ran
Travelling rolls, co-moving framesquare*2222productpredicted by Hopf theory
Travelling squares, co-moving framesquare*442productpredicted by Hopf theory
Travelling rolls (TR)hexagonal*×product *provisional — cost signature did not match
Travelling / rotating hexagons (TH)hexagonalr3₂*3clockworkconfirmed by a rule we ran (see the note below the table)
Travelling hexagons, opposite sensehexagonalr3₂*3clockworkconfirmed by a rule we ran (see the note below the table)
Standing hexagons (SH)hexagonal*632productprovisional — cost signature did not match
Standing / regular triangles (ST)hexagonal*~6₃32₁clockworkprovisional — cost signature did not match
Oscillating triangles (OT)hexagonal6₂3₂2clockworkpredicted by Hopf theory
Oscillating triangles, opposite sensehexagonal6₄3₁2clockworkpredicted by Hopf theory
Oscillating triangles on a static hexagon backgroundhexagonal6₂3₂2clockworkpredicted by Hopf theory
Wavy rolls I (WR I)hexagonalc*22₁22₁clockworkpredicted by Hopf theory
Wavy rolls II (WR II)hexagonalc*22₁22₁clockworkpredicted by Hopf theory

Two pairs of rows carry the same group for what their names call the two senses of handedness — alternating rolls and its mirror both read c4₃*2, travelling hexagons and its reverse both read r3₂*3. That looked impossible, so it was checked twice, and it is right: the two “opposite” films are not mirror images. Mapped onto its partner, the alternating-rolls pair agrees to 6.3e-16 under a pure translation by a quarter of the cell, and the travelling-hexagon pair to 2.0e-15 under a 60° rotation with a translation and a time shift — proper motions both, which cannot change a group. c4₃*2 and r3₂*3 are in any case each their own mirror image, like every centred or rhombohedrally centred space group: of the 68 forward groups 56 are their own mirror image and only 4 genuine enantiomorphic pairs exist (3₂3₂3₂ / 3₁3₁3₁, 6₂3₂2 / 6₄3₁2, 6₅3₂2₁ / 6₁3₁2₁, 4₁4₁2₁ / 4₃4₃2₁). The oscillating-triangle pair (6₂3₂2 / 6₄3₁2) is a mirror pair — nothing proper relates those two films (best residual 0.866) — and the detector splits it, which is the control. Scripts: tests/independent/mirror_planform_films.py and tests/independent/enantiomorph_partners.py; the claim in results/literature.json that the opposite-handed alternating rolls are c4₁*2 is the thing that is wrong, and c4₁*2 stays a literature row on a different item — the four-phase tiling of the 4:1 resonance cell. A * in the Kind column marks a row whose own product/clockwork flag disagrees with the group it was identified with; the group is authoritative, and all three such rows are provisional anyway.

ItemWhat it isWhere it landsFrom random startsOur check
Equivariant Hopf bifurcation, square latticeE(2)-equivariant PDE restricted to the square latticec4₃*2 · c4₁*2 · c*44₂2₁ᵃ · c2₁*22ᵃ · oyesverified here
Equivariant Hopf bifurcation, hexagonal latticeΓ = D₆ ⋉ T² acting on C⁶r3₂*3 · r33₁3₂ · 6₂3₂2 · 3₂3₂3₂ · *~6₃32₁ · 6₃32₁ · *632yesverified here
The H/K theoremAny Γ-equivariant dynamical system with a periodic solution u(t)—n/aunverified
Oscillatory Turing patterns in reaction-diffusionTwo coupled reaction-diffusion layers (Lengyel-Epstein / Brusselator), one Hopf-unstable and one Turing-unstabler3₂*3 · r33₁3₂ · *632 · *~6₃32₁reportedpartly verified
Parametric (Faraday) waves and vibrated granular layersA fluid layer shaken vertically (Zhang-Vinals / Navier-Stokes with parametric forcing) and a granular layer shaken vertically at…c*44₂2₁ᵃ · c2₁*22ᵃ · c*22₁22₁ · *~6₃32₁yesverified here
Frustrated XY antiferromagnets as rotating phase-oscillator patternsH = J Σ_⟨ij⟩ cos(θᵢ - θⱼ) with J > 0 (antiferromagnetic) on the triangular, kagome, honeycomb and J₁–J₂–J₃ square latticesr33₁3₂ · r3₂*3 · 6₂3₂2 · 6₄3₁2 · *~6₃32₁ · opartlyverified here
Kuramoto and Sakaguchi lattices: twisted states, chimerasθ̇ᵢ = w + K Σ_j sin(θⱼ - θᵢ - α) on a ring, torus or lattice, local or nonlocal couplingosee noteour own argument
Cellular automata that crystallise from random soup; discrete time crystalsCyclic cellular automata (n colours, advance when a neighbour holds the next colouro · r33₁3₂ · r3₂*3 · c4₃*2see notetargets verified here
Resonantly forced complex Ginzburg-Landau (2:1, 3:1, 4:1)∂A/∂t = (μ + i ν) A - (1+i β)|A|² A + (1+i α) ∇² A + γ Āⁿ⁻¹, the n:1 resonancer33₁3₂ · 6₂3₂2 · c4₃*2 · c4₁*2yesverified here
The 9 literature items in full, with references

Equivariant Hopf bifurcation, square lattice

The pattern. Five/six branches with maximal isotropy: travelling rolls, standing rolls, travelling squares, standing squares, alternating rolls (plus submaximal standing cross-rolls) Standing squares: the two roll sets in phase. Alternating rolls: the two roll sets a quarter period apart. Travelling squares: both sets carry the same travelling phase.

Where it lands. alternating rolls = c4₃*2 (mirror image c4₁*2); standing squares = c*44₂2₁ᵃ; standing/cross rolls = c2₁*22ᵃ; travelling rolls = o (product, drift-removable)

From random starts. Yes for whichever branch the cubic coefficients make stable - a square-lattice Hopf bifurcation from a uniform state selects one branch and generic small random perturbations of the uniform state land on it (modulo domain walls on a large box).

Our check. Group identifications verified numerically here; the original 1991 tables were not opened (paywalled) - names and amplitude vectors are from citing papers and from its own re-derivation.

Reference. Silber & Knobloch, Nonlinearity 4 (1991) 1063; Rucklidge & Silber, arXiv patt-sol/9704002 (opened: confirms alternating rolls have the Z₄ quarter-period-plus-90° symmetry plus two orthogonal mirrors)

Equivariant Hopf bifurcation, hexagonal lattice

The pattern. travelling rolls, standing rolls, travelling (rotating) hexagons, standing hexagons, standing/regular triangles, oscillating triangles, wavy rolls I and II, patchwork quilt Travelling hexagons: the three carriers share one travelling phase (and a 1/3-lattice translation costs 1/3 of a period). Oscillating triangles: the three standing-roll sets sit at temporal phases 0, 1/3, 2/3. Standing triangles: up- and down-triangles exchange every half period.

Where it lands. travelling hexagons = r3₂*3 (contains r33₁3₂ - the same group as the repelling-clock 333 film); oscillating triangles = 6₂3₂2 (contains 3₂3₂3₂); standing triangles = *~6₃32₁ (contains 6₃32₁); standing hexagons = *632 (product - not a find)

From random starts. Yes for the stable branch. In rotating convection the chiral-symmetry-broken oscillating hexagons are only reached 'by carefully controlled initial conditions' (Echebarria & Riecke), with bistability against a chaotic state.

Our check. Group identifications verified numerically here.

Reference. Roberts, Swift & Wagner, Contemp. Math. 56 (1986) 283 (unverified); Echebarria & Riecke, arXiv nlin/0002038 (opened)

The H/K theorem

The pattern. Two groups describe a periodic pattern: K = the subgroup fixing u(t) at every instant (the symmetry of one frame - the catalogue's 'kernel'), and H = the subgroup of symmetries that map the trajectory to itself, possibly with a time shift (the full space-time group). H/K is always cyclic, and it is realised as a phase shift group - exactly the catalogue's clock_order. H/K cyclic of order m means the generating operation costs 1/m of a period: m=2,3,4,6 give the 1/2, 1/3, 1/4 and 1/6 costs seen in the 68 forward groups.

Where it lands. This is the bridge between the bifurcation literature and the catalogue: a pair (H,K) with H/K = Zₘ is a forward space-time group whose kernel is K and whose clock order is m. A product group is exactly H = K × (time).

From random starts. n/a (a theorem).

Our check. unverified (reference not opened). The cyclic-quotient statement is standard; the exact realisability conditions are not re-derived here.

Reference. Buono & Golubitsky, Models of central pattern generators for quadruped locomotion I: primary gaits, J. Math. Biol. 42 (2001) 291-326 (states the H/K theorem); Golubitsky & Stewart, The Symmetry Perspective, Birkhauser (2002), ch. 3 (unverified - neither opened; statement from standard usage)

Oscillatory Turing patterns in reaction-diffusion

The pattern. 'Twinkling eyes' - oscillating Turing spots on a hexagonal lattice; oscillating hexagons and squares; the 'three-phase oscillatory interlacing hexagonal lattice' of Yang, Dolnik, Zhabotinsky & Epstein (2002), where the hexagonal spot lattice splits into three √3 × √3 sublattices Twinkling eyes: neighbouring spots oscillate out of phase; the three-phase interlacing hexagonal lattice has the three sublattices at 0, 1/3, 2/3 of the period (this is the phase relation reported as 'three-phase'; the exact assignment per sublattice is unverified).

Where it lands. Three-phase interlacing hexagonal lattice: r3₂*3 / r33₁3₂ if the three sublattices of the √3 × √3 superlattice carry 0,1/3,2/3 (verified numerically for exactly that site pattern). All-in-phase twinkling spots: *632 (product - not a find). Antiphase neighbouring spots on a honeycomb pair: *~6₃32₁ (verified).

From random starts. Reported to arise in simulations from small random perturbations of the uniform state (unverified - the PRLs are paywalled and it could only open the Brandeis repository records).

Our check. Partly: identifications verified numerically for the stated phase patterns; the experimental phase relations are unverified.

Reference. Yang & Epstein, Oscillatory Turing patterns in reaction-diffusion systems with two coupled layers, Phys. Rev. Lett. 90 (2003) 178303, record: https://scholarworks.brandeis.edu/esploro/outputs/journalArticle/Oscillatory-Turing-patterns-in-reaction-diffusion-systems/9923958439601921 (opened, abstract only); Yang, Dolnik, Zhabotinsky & Epstein, Spatial resonances and superposition patterns in a reaction-diffusion model with interacting Turing modes, Phys. Rev. Lett. 88 (2002) 208303, record: https://scholarworks.brandeis.edu/esploro/outputs/journalArticle/Spatial-resonances-and-superposition-patterns-in/9923958301301921 (opened, abstract only); Vanag & Epstein, Pattern formation in a tunable medium: the BZ reaction in an AOT microemulsion, Phys. Rev. Lett. 87 (2001) 228301 (unverified)

Parametric (Faraday) waves and vibrated granular layers

The pattern. Subharmonic stripes, squares and hexagons at f/2; hexagon <-> triangle alternation; f/4 states (squares, hexagons) in granular layers; superlattice (SL-I, SL-II) patterns under two-frequency forcing A subharmonic pattern inverts every drive period: peaks become valleys, so a half-period time shift composed with a half-lattice translation (stripes, squares) or with the 2-fold rotation (hexagons -> triangles) is a symmetry. Granular f/4 states repeat only every four drive cycles and are reported as hexagons alternating with triangles.

Where it lands. Subharmonic squares: c*44₂2₁ᵃ (verified for the checkerboard-antiphase pattern). Subharmonic stripes: c2₁*22ᵃ / c*22₁22₁ (the latter verified for the 2 × 2 stripe-antiphase site pattern). Hexagon/triangle alternation: *~6₃32₁ (verified). Superlattice states need their own supercell detection.

From random starts. Yes. These are primary pattern-forming instabilities of a uniform flat state: random (thermal or mechanical) noise is the generic initial condition in both experiment and simulation, and the selected pattern fills the container.

Our check. Identifications verified numerically for the idealised phase patterns; the experimental pattern sequences are from arXiv abstracts/records I opened.

Reference. Melo, Umbanhowar & Swinney, Hexagons, kinks and disorder in oscillated granular layers, arXiv patt-sol/9507003, Phys. Rev. Lett. 75 (1995) 3838, doi:10.1103/PhysRevLett.75.3838 (arXiv record opened); Park & Moon, Square to stripe transition and superlattice patterns in vertically oscillated granular layers, arXiv nlin/0101005, doi:10.1103/PhysRevE.65.051310 (arXiv record opened); Moon, Swift & Swinney, Role of friction in pattern formation in oscillated granular layers, arXiv cond-mat/0308541 (arXiv record opened); Kudrolli & Gollub, Physica D 97 (1996) 133 (unverified); Mueller, Phys. Rev. Lett. 71 (1993) 3287 (unverified); Tse, Rucklidge, Hoyle & Silber, Spatial period-multiplying instabilities of hexagonal Faraday waves, arXiv nlin/0005066 (opened: the subharmonic instability of standing hexagons produces a state with a spatio-temporal symmetry and 1:3 scale ratio)

Frustrated XY antiferromagnets as rotating phase-oscillator patterns

The pattern. Triangular 120° three-sublattice state; kagome q=0 and √3 × √3 states; honeycomb with J2; incommensurate spirals for J₁–J₂–J₃ Triangular: 0, 1/3, 2/3 of a turn on the three sublattices. Kagome q=0: the three sites of the unit cell at 0, 1/3, 2/3. Spirals: a continuously varying phase q.x - no finite sublattice structure.

Where it lands. Triangular 120°: r33₁3₂ for the clock-hand (vector) rendering, lifting to r3₂*3 for a scalar single-harmonic rendering (both verified; the extra mirror appears only because a single cosine harmonic is time-reversible). Kagome q=0: 6₂3₂2 (chirality-reversed 6₄3₁2) - verified. Honeycomb antiphase: *~6₃32₁ - verified. Incommensurate spirals: o after a change of frame - not a find.

From random starts. Partly. The triangular 120° state is reached from random starts only sometimes - the project's own measurement is 13 in 100, the rest freezing with patches of opposite handedness, rising to 74 in 100 with one symmetrisation and 100 with the full projector. Kagome is worse: the classical kagome XY antiferromagnet has a huge degenerate ground-state manifold, so gradient descent from random starts lands on a disordered member, not on q=0 or √3 × √3, unless further couplings (J2) select one.

Our check. Identifications verified numerically; the reach-from-random numbers for the triangular case are the project's own (26 Oct ledger), the kagome statement is from the arXiv record I opened plus standard degeneracy arguments.

Reference. Kakizawa, Misawa & Shinaoka, Monte Carlo study on low-temperature phase diagrams of the J₁–J₂ classical XY kagome antiferromagnet, arXiv 2310.15811, Phys. Rev. B 109 (2024) 014439, doi:10.1103/PhysRevB.109.014439 (arXiv record opened: J2 selects q=0 vs √3 × √3); Huse & Rutenberg, Classical antiferromagnets on the kagome lattice, Phys. Rev. B 45 (1992) 7536 (unverified); Miyashita & Shiba, J. Phys. Soc. Jpn. 53 (1984) 1145 (unverified)

Kuramoto and Sakaguchi lattices: twisted states, chimeras

The pattern. Twisted states (θ = q.x, a rigid phase winding); chimeras (a coherent domain coexisting with an incoherent one); splay states A twisted state has a linear phase ramp; a chimera has no spatial period at all in its incoherent part.

Where it lands. Twisted states: o. A twisted state is frequency-locked with a linear phase profile, so it is a rigid drift and a change of frame makes it static - not a clockwork. Chimeras: no spatially periodic attractor, so no space-time group. the famous Kuramoto patterns are exactly the ones that do not land in the 51 non-product groups.

From random starts. Twisted states on a ring have sizeable basins (the sync-basin literature measures them); chimeras need prepared initial conditions in finite systems.

Our check. The twisted-state=product-group argument is its own (same argument as travelling rolls, verified numerically for the travelling-roll case).

Reference. Kuramoto & Battogtokh, Coexistence of coherence and incoherence in nonlocally coupled phase oscillators, arXiv cond-mat/0210694 (arXiv record opened); Wiley, Strogatz & Girvan, The size of the sync basin, Chaos 16 (2006) 015103 (unverified - the arXiv hits were the 2017 and 2025 'revisited' papers, Delabays et al., arXiv 1706.00344, doi:10.1063/1.4986156, which it did open); Abrams & Strogatz, Phys. Rev. Lett. 93 (2004) 174102 (unverified)

Cellular automata that crystallise from random soup; discrete time crystals

The pattern. Cyclic CA from random soup self-organise into spiral cores and rotating defects, not into one periodic crystal; Greenberg-Hastings likewise. Life-like rules from soup end in still lifes plus period-2 blinkers. Discrete time crystals break the discrete time-translation symmetry of the drive (period 2T response).

Where it lands. A spiral-core attractor has no global space-time group. A uniform period-2T time crystal is spatially uniform, so its group is o-like (the broken symmetry is the drive's time translation, not a space-time one). Only a CA whose attractor is a strictly periodic tiling with a phase gradient - a 3-colour cyclic rule on the triangular lattice settling to 0,1/3,2/3 sublattices, or a period-4 rotor on the square 2 × 2 cell - would land in the catalogue: those idealised targets are r33₁3₂/r3₂*3 and c4₃*2 respectively (verified).

From random starts. Cyclic CA and Greenberg-Hastings are defined to be run from random soup; the question is whether any rule crystallises instead of spiralling. Unknown.

Our check. Identifications of the idealised targets verified numerically; the claim that no known CA rule crystallises into a non-product space-time group is a negative claim from the literature it could reach, not a proof.

Reference. Shalizi & Shalizi, Quantifying self-organization in cyclic cellular automata, arXiv nlin/0507067 (arXiv record opened); Bello & Sivakoff, Cyclic cellular automata and Greenberg-Hastings models on regular trees, arXiv 2108.06404 (arXiv record opened); Fisch, Gravner & Griffeath, Threshold-range scaling of excitable cellular automata, Statistics and Computing 1 (1991) 23 (unverified); Greenberg & Hastings, SIAM J. Appl. Math. 34 (1978) 515 (unverified); Else, Bauer & Nayak, Floquet time crystals, arXiv 1603.08001, Phys. Rev. Lett. 117 (2016) 090402 (arXiv record opened); Khemani, Lazarides, Moessner & Sondhi, On the phase structure of driven quantum systems, arXiv 1508.03344, Phys. Rev. Lett. 116 (2016) 250401 (arXiv record opened); Mi et al., Observation of time-crystalline eigenstate order on a quantum processor, arXiv 2107.13571, Nature 601 (2022) 531 (arXiv record opened)

Resonantly forced complex Ginzburg-Landau (2:1, 3:1, 4:1)

The pattern. n stable phase states separated by 2π/n: 2:1 gives two-phase standing patterns and fronts; 3:1 gives three-phase domains and three-armed structures; 4:1 gives four-phase rotating spirals and, at stronger forcing, standing two-phase patterns The n phase states differ by exactly 2π/n in the oscillation phase: that IS a clock of order n. In a periodic tiling of the n phases the generating lattice/point operation costs 1/n of a period.

Where it lands. If the three phases of the 3:1 resonance tile a triangular/hexagonal lattice: r33₁3₂ or 6₂3₂2 depending on whether the phases sit on sublattices (translation costs 1/3) or on roll sets (rotation costs 1/3) - both verified. If the four phases of the 4:1 resonance tile a square 2 × 2 cell: c4₃*2 / c4₁*2 - verified. Observed four-phase rotating spirals are not periodic tilings, so they carry no space-time group.

From random starts. The experiments and simulations are run from a uniform oscillating state with noise, so yes for the domain/spiral states - but they coarsen into domains and spirals rather than a crystal. A periodic tiling of the n phases is what the Modal search should look for (e.g. by adding a short-range repulsion between like phases).

Our check. Identifications verified numerically; the 4:1 rotating-spiral and standing-two-phase findings are from the arXiv/record abstracts I opened.

Reference. Lin, Hagberg, Ardelea, Bertram, Swinney & Meron, Four-phase patterns in forced oscillatory systems, arXiv nlin/0003047, Phys. Rev. E 62 (2000) 3790, doi:10.1103/PhysRevE.62.3790 (arXiv listing and the UT Austin chaos-group record opened; the PDF itself did not extract); Lin et al., Resonant phase patterns in a reaction-diffusion system, Phys. Rev. Lett. 84 (2000) 4240 (unverified); Coullet & Emilsson, Strong resonances of spatially distributed oscillators, Physica D 61 (1992) 119 (unverified); Elphick, Hagberg & Meron, Phys. Rev. E 59 (1999) 5285 (unverified)

Three mechanisms in print can do this: a Hopf bifurcation on a lattice; a subharmonic parametric instability (Faraday waves, vibrated granular layers), where the pattern inverts every drive period; and frustration — a repulsive coupling whose ground state is a multi-sublattice phase pattern, which is the 333 family. Travelling rolls, twisted states, chimeras, standing hexagons and uniform time crystals all land in product groups.

What is open

8 clockworks were reached by nothing here: no rule of ours, no Hopf branch, and no literature item we read identifies a reported state with that group. Four of the eight — 4₁4₁2₁, 4₃4₃2₁, c44₂2₁, c4₁4₃2 — do appear in results/literature.json, but only inside our own notes on the square-lattice branches, as the chiral subgroup of a mirror-symmetric planform or in a detector hit list. That is a mathematical consequence, not a report of anyone seeing one, so they stay open. c4₁*2, which is named in print — as one of the two four-phase tilings of the 4:1 resonance cell — was wrongly in this list until Wed 28 Oct 2026 (week 44) and is now a literature row.

4₁4₁2₁4₃4₃2₁c44₂2₁c4₁4₃2*~4₂4₂~2r3₂*~3*6₃~3~2₁*~6~3~2

15 more were touched: a search returned them, but the re-run could not confirm the group or measured a basin of zero from random starts, so they stay unproven.

2₁2₁2₁2₁*~2~2~2~222~*22×½*2₁~2₁2₁~2₁2₁2₁~*2₁2₁*2₁2₁×c*222₁2₁ᵇc22₁*ᵇ2*~2~22₁*2₁~2₁c2₁*22ᵇ*4₂~4₂2c*44₂2₁ᵇ

Sources

Every number on this page, and the file it comes from
  • Confirmed, corrected and refuted verdicts, reach and time costs: results/verified-osc.json, verified-ca.json, verified-lc.json, verified-rd.json, verified-gaps.json — each written by an agent that rebuilt the rule in its own code.
  • What each search found: results/identified-*.json; the rule records and the films: films/<family>/.
  • The detector: symdetect/ with symdetect/README.md; tests/test_symdetect.py (292 tests, inherited from that agent’s run).
  • Hopf planforms: results/theory-hopf.json and .md. Literature: results/literature.json and .md, condensed for the table above by src/assemble.py.
  • The handedness check: results/mirror-planform-check.json and results/enantiomorph-check.json, from the two scripts in tests/independent/.
  • How long each demo takes to settle, and the steps a frame it uses: results/demo-settling.json, from src/measure-settling.mjs.
  • Modal spend: results/modal-spend.jsonl.
  • The eight demos’ rules and the JavaScript port: demos/demos.json, demos/ops.json, src/engine.mjs, checked against the families’ own vectors by tests/engine.test.mjs.
  • The 68 groups, their symbols, kernels and parents: animated-groups-fable/docs/data/clockwork-coloring-correspondence.json and catalog.json.
  • The 333 numbers quoted here are the 333 page’s own: 2026-10-28-repelling-clocks-333/notes/numbers.md.

Companion pages: Repelling clocks make 333 · Symmetric animation analogs · the catalogue of all 275 groups: animated-groups-fable.