Parity Figures as Moiré: What Is Exact and What Is Not

A short note placing the parity construction of [A] (doi:10.5281/zenodo.22965031) inside the established theory of moiré (Amidror), and separating three statements usually run together. (1) The parity figure of a union of band systems is exactly the multiplicative moiré of square gratings of transmittance ±1: exclusive-or on {0,1} is multiplication on {±1}. (2) Two printed transparencies laid on one another do not produce it: they produce the conjunction of two {0,1} gratings, which differs from the parity on 108 cells of 144 for every pair of first-generation families. (3) Two gratings of linear-polariser strips at 0°/90° do produce it, complemented, exactly (Malus's law: XNOR = 1 − XOR). (4) The doubling homothety of [A] is the second harmonic of the cosine carrier whose level sets are the axes, not of the square-wave figures, which have odd harmonics only. Every claim is checked by a script on the published axis catalogue. Nothing new is proved; no grating was printed and no polariser cut. The measurement suggested — two polariser gratings cut from the first-generation families, viewed against a light box — would show the fifteen images of the loom of [O] (doi:10.5281/zenodo.22986529), minus their tints, one pair at a time.

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Version 1.0.1. References [A] and [O] corrected (deposit title and concept DOIs); licence line aligned with the record. No claim changed.