# Gray–Scott with the six 442 time symmetries

**Numerical update, 4 September 2026:** The [verified atlas](scott-gray/)
now contains nonuniform periodic standing and rotating chemical waves whose
coordinate variants satisfy all six listed time characters. Each delivered
field passes independent forward return, all generator phase tests, and
actual timestep refinement. The reference fields come from two underlying
branches; additional standing spatial modes supply distinct patterns at
some of the same parameters. Coordinate copies can have extra symmetries
and are not counted as new branches. This is not a continuum existence proof.
[Evidence and spatial refinement](scott-gray/VERIFIED-ORBITS.md).

This note concerns the **six forward-time groups g94–g99** in this project's 442/p4 correspondence, using their actual affine operations from [`data/catalog.json`](data/catalog.json). All six are compatible with an autonomous, isotropic Gray–Scott equation. None requires reversing time. That establishes the absence of a symmetry obstruction; it does **not** establish that a nonstationary periodic orbit exists for a particular feed, kill, diffusion ratio, domain size, or seed.

A solver can search these six classes using ordinary forward integration plus a symmetry-twisted periodic closing condition. The difficult part is finding and refining a nontrivial recurrent orbit, especially one resembling the user's gliders. Repeating an attractive movie or imposing symmetry on its rendered frames is not, by itself, a periodic PDE solution.

## Equation and convention

Let the two concentrations be \(z=(u,v)\), on a square torus of physical side length \(L\):

\[
u_t=D_u\Delta u-uv^2+f(1-u),\qquad
v_t=D_v\Delta v+uv^2-(f+k)v.
\]

Here \(D_u,D_v>0\) and the feed \(f\), kill \(k\), and diffusion coefficients are spatially uniform and constant in time. Normalize position to \(x\in\mathbb R^2/\mathbb Z^2\), and let \(T>0\) be the proposed period. The normalized-coordinate Laplacian carries the factor \(L^{-2}\).

The catalog operation \((M,a,s,\tau)\) acts on spacetime by

\[
(x,t)\longmapsto(Mx+a,st+\tau T).
\]

For these six groups \(s=+1\). Invariance means

\[
z(Mx+a,t+\tau T)=z(x,t).
\tag{1}
\]

This is the convention used by the existing correspondence animations: a copy at \(Mx+a\) displays the motif's time \(t-\tau T\). In particular, the sign of the quarter-period shift distinguishes g96 from g97. Do not infer it from a clockwise-looking canvas without accounting for the inverted screen y-axis.

## Exact six groups

Write \(R(x,y)=(-y,x)\), \(C(x,y)=(x+1/2,y+1/2)\), and \(H(x,y)=(-x,-y+1/2)\). Coordinates are taken modulo one. Unit translations have zero time shift, and a full time translation is always \(T\).

| Catalog | Chaim short colour form | Phase-as-height space group | Extra generators \((g,\tau)\) | Instantaneous spatial kernel \(K\) | Phase image |
| --- | --- | --- | --- | --- | --- |
| g94 | 442 | P4, No. 75 | \((R,0)\) | p4 / 442 | \(C_1\) |
| g95 | ²4²4¹2 | P4₂, No. 77 | \((R,1/2)\) | p2 / 2222 | \(C_2\) |
| g96 | ⁴4⁴4²2 | P4₁, No. 76 | \((R,1/4)\) | p1 / ◦ | \(C_4\) |
| g97 | ⁴4⁴4²2 | P4₃, No. 78 | \((R,3/4)\) | p1 / ◦ | \(C_4\) |
| g98 | ¹4²4²2 | I4, No. 79 | \((R,0), (C,1/2)\) | p4 / 442 | \(C_2\) |
| g99 | ⁴4⁴4¹2 | I4₁, No. 80 | \((R+(1/4,3/4),3/4), (C,1/2)\) | p2 / 2222, generated by \(H\) | \(C_4\) |

The data sources are [`data/catalog.json`](data/catalog.json), [`data/clockwork-coloring-correspondence.json`](data/clockwork-coloring-correspondence.json), and [`data/clockwork-lifts.json`](data/clockwork-lifts.json). The corresponding browser entry is the [442 correspondence](correspondence-p4.html#g94). Superscripts in Chaim's short form give the **orders of generator colour permutations**, not the directed time shifts. The C4 short forms follow the source page's derived-rule convention; they are not presented as literal book rows.

For g99 the shifted rotation centre is essential. Its zero-phase half-turn is \(H\), rather than a half-turn about the origin. It is incorrect to obtain this record merely by adding a half-cell translation to the origin-centred g96 generators. The six records' catalog `render.ops` are the definitive source for programmatic validation; some compact display labels suppress coordinate-setting information.

The g96/g97 records correspond to different chronological phase actions even though their Chaim short form is identical. They have equal feasibility in an isotropic Gray–Scott system: a **spatial reflection** of an orbit conjugates the quarter-turn and swaps these handed classes, while continuing to run the PDE forward.

## The named overlay generators

The browser reuses the exact α, β, γ glyph outlines and their affine centers from the [442 correspondence](correspondence-p4.html). Its named generators are a different generating set from the compact \(R,C\) table above. In particular, α and β in g94–g98 use \(R^{-1}\), so their directed phases for g96/g97 are the inverses of the phases listed for \(R\).

| Group | α center; phase | β center; phase | γ center; phase |
| --- | --- | --- | --- |
| g94 | (0,0); 0 | (−1/2,−1/2); 0 | (0,−1/2); 0 |
| g95 | (0,0); 1/2 | (−1/2,−1/2); 1/2 | (0,−1/2); 0 |
| g96 | (0,0); 3/4 | (−1/2,−1/2); 3/4 | (0,−1/2); 1/2 |
| g97 | (0,0); 1/4 | (−1/2,−1/2); 1/4 | (0,−1/2); 1/2 |
| g98 | (0,0); 0 | (0,1/2); 1/2 | (−1/4,1/4); 1/2 |
| g99 | (−1/4,0); 1/4 | (1/4,0); 3/4 | (0,−1/4); 0 |

α and β are quarter-turns; γ is a half-turn. The g99 quarter-turn linear part is \(R\). Centers are in catalog coordinates, repeated by integer spatial translations. The numerical canvas displays `q[y*N+x]` with y increasing down the screen, so it plots these y coordinates directly. Consequently the g94–g98 quarter-turns look counterclockwise on this canvas, and the g99 quarter-turns look clockwise. This display convention changes neither the affine operation nor its directed phase. Each marked center is checked as a fixed point of the corresponding complete affine map.

The overlay marks the **requested** symmetry. During ordinary forward evolution, only the instantaneous spatial kernel is imposed on the starting field. A marked generator with a nonzero time shift is therefore a constraint to measure, not evidence that the moving field satisfies it.

## Feasibility by class

| Group | Compatible nonstationary search target | What is actually established |
| --- | --- | --- |
| g94 | A p4-symmetric breather or periodically rearranging fourfold pattern | The p4-fixed subspace is invariant under Gray–Scott evolution. Periodicity still needs to be found. |
| g95 | A two-phase pattern: quarter-turn plus half-period; half-turn visible at every instant | The group relations and evolution are compatible. A two-phase orbit in the p2-fixed subspace is a valid target. |
| g96 | Four-phase circulating or rotating packets | A relative return after one quarter-period under \(R\) is consistent and yields a full period after four segments. |
| g97 | The opposite handed four-phase arrangement | A reflected g96 solution gives the paired class without time reversal. |
| g98 | Two interleaved, fourfold arrangements displaced by half a cell and half a period | A half-period translation return is compatible with instantaneous p4 symmetry. |
| g99 | Four-phase arrangements with a half-period centred translation and instantaneous affine half-turn | The affine kernel and quarter-period action close consistently. Both the centring and rotation constraints must be checked. |

These are algebraic feasibility statements and suggested search targets, not six numerical existence results. Uniform stationary solutions satisfy every requested symmetry and must be excluded from a successful animation search. A spatially uniform nonstationary periodic solution can illustrate g94 but cannot realize a faithful nontrivial phase action at its minimal period. Likewise, a solution may have additional spatial or temporal symmetries: verifying the requested group establishes **at least** those symmetries, not necessarily that this is its full symmetry group.

The literature supplies evidence that periodic behavior is reasonable in Gray–Scott-type systems, but does not settle these six two-dimensional wallpaper classes. Gomez, Mei, and Wei prove periodic spike solutions for a classical Gray–Scott system on a one-dimensional interval, including stable and unstable cases. Their geometry and parameter scaling differ from this square-torus problem. [Author-hosted paper](https://math.unm.edu/~gomezd/docs/2019_hopf_gs_sc.pdf).

Farr and Golubitsky study rotating chemical waves on a circle, but their 1992 paper uses a **three-component reversible Gray–Scott variant**. It is relevant to the symmetry mechanism and is not an existence proof for the two-component equations above. [Author-hosted paper](https://www.asc.ohio-state.edu/golubitsky.4/reprintweb-0.5/output/papers/gray_scott_92.pdf).

## Why time reversal is different

The user's proposed reflection with a time **shift** is compatible with the equation. If a reflection \(J\) is paired with a nonzero shift, \(J^2=1\) forces that shift to be half a minimal period. The p4/442 family itself has rotations and translations, with no mirror generators; mirror examples would belong to a different wallpaper family.

A true time-reversing condition is a separate matter. Write the autonomous equation as \(\dot z=F(z)\), and let \(P\) be any purely spatial isometry action. Isotropy gives \(F(Pz)=PF(z)\). If an orbit obeyed

\[
z(-t+t_0)=Pz(t),
\]

differentiation would give

\[
-F(z(-t+t_0))=PF(z(t))=F(z(-t+t_0)).
\]

Thus \(F=0\) along the orbit: it is stationary. This elementary obstruction assumes the stated autonomous Gray–Scott dynamics and a purely spatial action on the concentrations. It does not apply to a forward time shift. **None of g94–g99 has this obstruction.**

## Why gliders need a periodic-orbit search

Bulatov's SymSim examples demonstrate symmetric Gray–Scott motion, including U-shaped gliders. His JMM presentation describes U-skates moving along straight paths and disappearing in collisions, and distinguishes ordinary periodic boundary conditions from symmetry-aware evolution. Spatial periodicity and attractive long-lived motion do not establish temporal recurrence. [Bulatov, SymSim presentation, JMM 2024](https://bulatov.org/symsim/240103_JMM/index.html).

A complete 442 pattern cannot generically be a single rigidly translating glider lattice with nonzero drift velocity \(c\): its quarter-turn rotates that velocity to \(Rc\), whereas one common drift would require \(Rc=c\), hence \(c=0\). Degenerate fields with continuous extra translation symmetries are the exception. Appropriate targets are local gliders in balanced arrangements, periodically turning/rearranging packets, rotating structures, or breathing worms. A glider in a periodic spatial box need not return to its initial chemical field after one traversal: interactions and deformation can prevent recurrence.

The user-supplied images are useful shape and palette references. They are not enough to identify an exact seed, feed/kill pair, or period. Browser presets should be described as glider-inspired unless an actual source preset or measured trajectory establishes otherwise. The upstream implementation is [Bulatov's `symhub` Gray–Scott application](https://github.com/vbulatov2011/symhub/tree/main/apps/symsim/gray_scott).

## A solver that preserves the physical question

Let \(\Phi_h(t,z_0)\) be the forward flow of a spatially discretized Gray–Scott equation. For a spatial map \(S\), define its action on fields by \((P_S z)(x)=z(S^{-1}x)\). Choose a representative with phase \(1/m\), and solve

\[
\Phi_h(T/m,z_0)-P_S z_0=0,
\qquad z_0\in\operatorname{Fix}(K),\qquad T>0.
\tag{2}
\]

| Group | \(m\) | \(S\) in equation (2) | Kernel generators on the unit square |
| --- | --- | --- | --- |
| g94 | 1 | Identity | \(R\) |
| g95 | 2 | \(R\) | \(R^2\) |
| g96 | 4 | \(R\) | Identity only |
| g97 | 4 | \(R^{-1}\) | Identity only |
| g98 | 2 | \(C\) | \(R\) |
| g99 | 4 | \(S(x,y)=(-y+3/4,x+1/4)\) | \(H\) |

For g99, \(S\) is the catalog's phase-1/4 operation and \(S^2H=C\) modulo the lattice, so this formulation includes the centring relation. For g95, \(S^2\in K\). In all six cases \(S^m\in K\), and equivariance extends a closed fundamental segment to a full orbit.

A practical sequence is:

1. Construct glider/worm/spiral seeds, and project the **initial state only onto the zero-phase kernel**. On a square grid whose width is divisible by four, every catalog rotation and fractional translation is an exact grid permutation.
2. Integrate the ordinary unforced Gray–Scott equation. Search the trajectory for small relative return errors at physically meaningful lag times. Parameters, seed geometry, and box size are independent search variables.
3. Refine a promising relative return using shooting, multiple shooting, or spacetime collocation, with the period as an unknown. Add a phase condition such as \(\langle z_0-z_{\rm ref},F_h(z_{\rm ref})\rangle=0\), a lower period bound, and a nonstationarity test. Matrix-free Newton–Krylov methods avoid forming a dense Jacobian.
4. Independently validate the complete candidate using forward integration, all catalog operations, and a finer timestep. Spatial refinement is needed before claiming continuum accuracy.

Forward integration preserves \(\operatorname{Fix}(K)\) when the discretization is equivariant. This avoids enforcing inconsistent instantaneous fourfold symmetry on g96/g97, where fourfold symmetry exists only after a time shift. General orbit continuation and treatment of continuous symmetries are documented in the [pde2path periodic-orbit guide](https://arxiv.org/abs/1908.00905).

An alternative is full spacetime collocation on \(\theta=t/T\): solve

\[
T^{-1}\partial_\theta Z-F_h(Z)=0,\qquad
Z(x,\theta+1)=Z(x,\theta),
\]

with equation (1) as exact linear constraints. Averaging an initial movie over the finite spacetime action is a useful initialization/projection onto those constraints. It does **not** solve the nonlinear Gray–Scott residual: in general, averaging and the cubic reaction \(uv^2\) do not commute.

## What the browser should report

Keep three statuses distinct: an unforced simulation, a candidate periodic orbit, and a numerically validated periodic orbit. A constructed symmetry preview can help explain an unsolved group, but it must say that it is a preview rather than a PDE solution.

For a computed orbit report at least:

- Feed, kill, diffusivities, domain size, grid size, integration timestep, period, and seed or reproducible preset.
- Full-period return error in **both** concentrations, before any seam blending or periodic replay.
- Maximum or RMS symmetry mismatch for **every** affine catalog operation and its specified time shift; include centring for g98/g99.
- Temporal variation/activity and spatial contrast, so a stationary field or a nearly constant movie cannot masquerade as a useful periodic pattern.
- Numerical status and the tolerances used. A small residual is an approximate numerical result, not a rigorous existence proof.

Use an error scale based on the orbit's nontrivial variation as well as absolute concentration error; normalizing only by \(u\approx1\) can hide an almost-dead pattern. Exclude arbitrarily short periods, check plausible period divisors, and repeat the return check with a smaller timestep. A seamless visual loop, symmetry averaging, residual measured only on a projected movie, or measuring only the displayed \(v\) channel is insufficient evidence.

This formulation makes all six searches concrete and testable while keeping the distinction between mathematical compatibility, attractive symmetry animation, and an actual Gray–Scott periodic orbit explicit.