crystal

Walks from a fixed report with the same endpoint and the same tally are exactly the vertices of the maximal abelian cover of $K_4$, with periods $H_1(K_4;\Z)\cong\Z^3$. Realized by the harmonic projection, the only report-covariant linear map from oriented letters to periods, the cover is trivalent, three-dimensional, of girth ten, strongly isotropic and chiral. Its periods carry the report group as $\Lambda^2V\cong V\otimes\mathrm{sgn}$, so all twenty-four report permutations act as proper rotations, and edges along complementary letters are perpendicular, not opposite as the older dictionary of letters as directions assumed.

On the report qubit’s Minkowski space each letter generates a boost between its two reports, and the periods are exactly the tallies that generate rotations: the crystal’s metric is the norm of the observer’s rotation algebra. Read modulo 2 that metric is the Fano plane with a marked point, a clock; modulo 3 it picks out the four report axes, turned by the rotation group of a cube; modulo 7 its null directions are the eight points of the sky. That the crystal is physical space is a reading, and it costs one added clause, that the circulation of the observer’s live path of report changes is retained, which changes a finite statistic already at fourth order in time.