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.22965031); as sixty three-tint images from which 512 grids are built by a six-bit selection (paper [G], data doi:10.5281/zenodo.22862110); and as level sets of a square-plate eigenmode (note [P], same deposit as [A]). This paper shows, by exhaustive check on the published data, that the three descriptions present one finite structure (the sense is made precise in §1).

Theorem 1 (finite-corpus identification). 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 4.0. Code: AGPL v3; commercial licence on request.

Version 1.2.0. Revised after reading: (1) what \"one object\" means is defined in §1, and the independence of the corpora is stated there ([G] was generated and classified before the axes were formalised); (2) Theorem 1 is labelled a finite-corpus identification and Theorem 2 a finite computation; (3) new Proposition 1: the four T0 families are a basis of F_2^4, embedded injectively by the parity map (rank 4, rank 5 with the all-dark figure; new script espace_F2_4.py), and the half-shift acts through a linear form, so the eight unified families are the complement of a hyperplane, 2^4 − 2^3; (4) the plate section is shortened and separates \"exact in the guided model\" from \"predicted, not measured\"; (5) new §6 on the two referents (256 and 360) used by the Carter encoding, as a use of the structure and not its origin; (6) §7 retitled \"Why 12 is the smallest admissible period\", with its three criteria stated; (7) §8 is a table sorting every statement by status. (8) §5: the dimension 13 is derived from the levels (5 diagonal + 8 orthogonal); the C1 reading now states its rule for axes through cell centres (continuous closure on the side of increasing coordinate, as in regle_parite.py of [A]), on which every C1 number depends — the previous text wrongly said only two keys were affected. (9) §7: the proof of Theorem 3 is read further — for any period N = 2^a M, the descent is a set of cycles of lengths r(d) = min{r : 2^r ≡ ±1 mod d}, one family per divisor d > 1 of M, with trees of height a above them; it terminates exactly when every r(d) = 1, i.e. M ∈ {1, 3}; the table gives the cycles for N = 10 and 18, and harmonie_N.py checks the structure for every even N ≤ 200. No number changed.

Version 1.2.1. Licence of the text, figures and data set to CC BY 4.0 (it read CC BY-NC 4.0 in 1.2.0); the general study of doubling modulo n is cited (doi:10.5281/zenodo.23088601). No other change.