One Object, Three Descriptions: Axes, Images and Grids of a Jacquard Pattern System

A Jacquard pattern system has been described in three separate ways: as sixteen families of straight axes on a 12 × 12 square combined by a parity rule (note [A], doi:10.5281/zenodo.22965032); as sixty three-tint images from which 512 grids are built by a six-bit selection (paper [G], data doi:10.5281/zenodo.22862111); and as level sets of a square-plate eigenmode (note [P], same deposit as [A]). This paper shows that the three are one object, by exhaustive computation on the published data. Theorem 1. Every one of the sixty images is a parity figure of axes: forty-eight cells of lines, the same for all sixty and equal to the nodal set of the degenerate guided plate mode cos(4πx/a) − cos(4πy/a); ninety-six cells carrying the parity of a union of the four families of level T0; and one of four tint permutations, which are the four natures of the system. The fifteen families of the loom are the fifteen non-empty subsets of the four T0 families (60/60 verified). Corollary. The half-shift theorem of [G] — exactly eight families of fifteen admit unified pairs, those with exactly one base of yang type — follows in two lines. Theorem 2. The parity map is injective on the 70 band systems visible on the triangle subdivision, and the sixteen family figures span a [1152, 13] binary code of minimum distance 288 with a unique minimum-weight word, T0 YANG MUT ⊕ T1 YIN (parity counted from the origin corner); on the cell reading the code is [144, 12] of distance 36. Theorem 3. On levels, the doubling homothety terminates without cycles if and only if the period is 2^a or 3·2^a, has a non-trivial fixed level (120°) if and only if 3 divides the period, and 12 is the smallest period with both and depth at least two. Nothing is measured on a plate; the plate reading is exact for guided edges and approximate for free ones. Every number is printed by one of five scripts from catalogue-axes.json and referent_360_v3.json, both included, together with the six figures (SVG) and the script that draws them. Licences. Text, figures and data: CC BY-NC 4.0. Code: AGPL v3; commercial licence on request.