The lattice's teeth: primitive reciprocal axes govern which Fourier components neural cellular automata can grow

Neural cellular automata (NCA) are trained on square grids by default. This preprint treats the lattice as an experimental variable and reports two results, together with the outcome of a pre-registered out-of-sample test of the second one. 1. A normalization artifact in cross-lattice comparisons, documented and retracted. Snapping periodic wave targets independently per wavevector onto each lattice's reciprocal box, then min-max normalizing, sets a three-wave target's variance to one of two narrow classes (near 1.5/36 or 1.5/20.25) depending on whether rounding preserves the wave set's closure. In a 310-training study this produced a 44% "advantage" of hexagonal over square substrates that reproduced in 12/12 restarts and survived an operator-conditioning control — while, apart from twelve trainings in a single panel that had genuinely learned, every wave training on every lattice had in fact converged to the predict-the-mean solution or stopped short of it (test error 0.97 to 1.62 times its own target variance, never explaining more than 3% of it). The advantage was the ratio of the two variances. The paper documents the failure, the diagnostic (per-condition loss/variance) and the repair. 2. A primitive-axis constraint on learnability. With closure enforced and a per-component readout, a trained NCA grows those Fourier components of its target whose wavevectors lie on a primitive reciprocal-lattice axis of its substrate — amplitude recovery 0.55 to 0.88 in restart means, per-restart minimum 0.50 — and fails to grow all others, recovery at most 0.11 in restart means (per-restart maximum 0.21), including components exactly representable along non-primitive (diagonal) directions. In the tested family, a substrate hosts a full pattern class only if its point-group symmetry admits the class. 3. Pre-registered falsification of the frozen quantitative form. An out-of-sample battery of 96 trainings (honeycomb, grid-size and wavelength, learned-kernel, misalignment and data-budget panels), with predictions frozen before the first training, partially refuted the constraint as stated: the pre-registered decider — predicted-versus-measured R2 of at least 0.8 — failed, at R2 = -0.39. The tooth has an angular half-width near 6 degrees rather than the calibrated 0.5, its snap-strain clause is inert, aligned recovery is graded by data budget and by the target's mode order rather than fixed, and the honeycomb gates on its embedding box's reciprocal axes rather than on its six-fold stencil symmetry. What survives out-of-sample is the gate itself: growth stays direction-selective, no geometry-forbidden component grows, and far-off-axis components remain dead (recovery 0.00); the exactly-representable-diagonal cell is a calibration result the battery did not re-probe. The constraint is real and soft, not sharp. Scripts and per-training JSONL records accompany the paper. The frozen prediction table and the analysis verdict (printing the R2 and the per-record violation count) are read-only entry points that reproduce the section-5 result from the completed records.