One Image, Twelve Mathematics: A Cross-Family Benchmark and the U-Shaped Editability of Image Representations

An image is always a matrix, but it can be re-expressed in many ways: as a sum over a Fourier, cosine, or wavelet basis, as a sparse combination of atoms from an overcomplete dictionary, as a handful of geometric primitives, as a small neural function of position, as an explicit closed-form formula, or as a code in the latent space of a generative model. This preprint carries one image across twelve such families, on a common set of benchmark images and at a common parameter budget, and asks a question the fidelity literature does not: when you edit the parameters, what happens to the image? Two measured results and one finding follow. First, at an equal parameter budget the twelve families reconstruct comparably (peak signal-to-noise ratio 27-35 dB); fidelity does not distinguish them. Second, an editability-locality metric does: a single coefficient of a global transform is spatially spread (Fourier 0.16, DCT 0.23 concentration) while a coefficient of a space-localized transform is essentially local (wavelet 0.999, KLT 1.0). The finding is that editability is U-shaped across the abstraction spectrum: it is high at two poles, the designed-structure pole (geometric primitives, sparse atoms, space-localized transforms, where humans built local meaningful coordinates) and the learned-manifold pole (disentangled generative directions), and it collapses toward noise in between, where perturbing a raw neural-field weight or a constant of a brittle fitted formula destroys the image. A parameter is both stable and meaningful only when it indexes a low-dimensional manifold of plausible images with locally disentangled coordinates. The manuscript gives the twelve families with their governing equations, the cross-family benchmark on six images (two purpose-driven figures generated deterministically from the committed artifact), the U-shaped editability finding, and the honest scope: arbitrary photographs do not reduce to compact, faithful, human-readable equations, and the results say where a representation's parameters are, and are not, a usable handle. Code and committed artifacts (MIT): https://github.com/fsantibanezleal/CAOS_IMGLAB .