From coloured plane groups to forward-time spacetime groups

Let Γ be a wallpaper group and H the subgroup preserving one selected colour. A perfect N-colouring is a transitive coset action on Γ/H. It extends directly to one forward-time clock precisely when H is normal and Γ/H is cyclic of order N:

H ◁ Γ and Γ/H ≅ C_N.

Equivalently, a surjection χ: Γ ↠ C_N assigns each spatial operation a phase χ(γ)/N in ℝ/ℤ. The colour has become a time offset. Conversely, any forward-time group with fixed spatial projection and finite phase image supplies such a cyclic character. Fix A at phase 0, B at phase 1/N, and so on, and let C_N = (ABC…) be one forward tick. A spatial generator with offset +k/N maps to C_N^k, with k taken modulo N. The identity and powers that split into several disjoint cycles are still powers of this one clock; none reverses time. For N = 2 every index-two subgroup is normal with quotient C₂, which is why the regular-cyclic row reproduces the classical count 46 exactly.

Forgetting every operation's time component places the 275 spacetime representatives into the 17 wallpaper projection categories. That does not say that a generic frame has the full projected wallpaper symmetry. Its guaranteed frame-preserving group is the zero-phase kernel H, often a proper subgroup of the projection Γ. Special frames may have larger stabilizers, and a particular motif may add accidental symmetries.

A genuinely coloured film carries two pieces of chromatic data: an internal finite colour action and the phase clock. In the simplest abelian case this is a cocycle into C_M × ℝ/ℤ. Time reversal adds the black–white involution, which acts on the clock by phase inversion. Non-cyclic perfect colourings—such as S₃ colour actions—would require motifs with a non-abelian internal state rather than a single scalar clock.

The 68-row clockwork/coloring correspondence gives the preceding, uncoloured case: all 17 one-colour products and 51 nontrivial canonical forward clocks are paired with their regular cyclic plane colouring, zero-phase kernel, static wallpaper plate, and associated real-crystal example.

No count of those coloured spacetime groups is asserted here. Such a census must first specify whether equivalence is taken inside a fixed parent, under the full affine normalizer, or under the Galilean re-basing used by the 275-class spacetime catalog.

Download the generated JSON census, summary CSV, orbifold audit CSV, or the 68-record forward manifest.

python3 scripts/generate_color_forward_census.py
python3 scripts/generate_color_forward_census.py --check
python3 -m unittest discover -s tests -p 'test_color_forward*.py'

The regular-cyclic cells were independently reconstructed with the pinned Senechal–Wieting subgroup engine: normal index-N kernels are grouped into affine-normalizer orbits, with stability checked across matrix-entry bounds one through six.

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