Fractal Cross Product: Theory, Differentiable Implementation and Application to Medical Image Analysis

The magnitude of the generalized Euclidean cross product is a Gram volume whose degree under common scaling is fixed by the integer dimension of the spanning frame. We formulate a generalized Fractal Cross Product (FCP) as a nonlinear radial deformation with a prescribed positive degree $D$, which may be non-integer. The scalar construction applies in any ambient dimension $m\geq k$, while its canonically oriented vector form requires codimension one. It recovers the classical generalized cross product exactly at $D=k$ and retains orthogonality, alternation, rotation equivariance, and $D$-homogeneity, but is generally not multilinear. For exact self-similar frame systems, the construction also obeys a scale-balance law at the similarity dimension. We derive a differentiable, dimensionless image response and a non-circular empirical accumulation exponent obtained by regressing raw angular Gram responses across patch widths. Binary64 calculations recover the finite-frame identities to roundoff, while raster experiments recover the Sierpiński-triangle value 1.5849625 at three resolutions. In five-seed medical-imaging comparisons, FCP-centered fusion increased mean area under the receiver operating characteristic curve from 0.7264 to 0.8135 and from 0.5843 to 0.6765 on the random and hospital-separated Retinal Image Database for Optic Nerve Evaluation partitions, respectively, and from 0.7403 to 0.7475 on FracAtlas. On FracAtlas, balanced accuracy increased from 0.6186 to 0.6721 and the harmonic mean of precision and sensitivity from 0.3669 to 0.4457. These results support the utility of the complete fusion framework, but do not isolate the effect of FCP from that of its complementary descriptors and fusion head.