AXIOM-TENSOR: Diagonally Modulated Signed-XOR Operators for Representation, Initialization, Optimization Geometry, and Exact Execution

This preprint presents a unified mathematical framework for diagonally modulated signed-XOR operators through a single component-summed shared-profile operator map. The framework connects representation theory, variance-preserving initialization, optimization geometry, exact execution, and adaptive representation while carefully distinguishing these concepts and their respective mathematical roles. The paper introduces ARCANA (Algebraic Rank-Corrected Activation-Normalized Amplitude), a variance-preserving initialization rule derived under explicit Latin-square assumptions, and proves an exact XOR-diagonal compiler, a fixed-component measure-zero representation theorem, a coherent-update identity, and an exact execution graph for fixed AXIOM operators. A binary-carry theorem further shows that fixed local stencil operators occupy only O(log n) signed-XOR diagonals despite having full ordinary matrix rank. The accompanying numerical experiments are intentionally limited to theory verification, including Gaussian projection, stencil reconstruction, and a small MNIST compatibility study. They are presented as consequence checks of the mathematical theory rather than evidence for universal dense-matrix compression or hardware acceleration. This work establishes the mathematical foundations of AXIOM-TENSOR. Large-scale model compression, runtime evaluation, and applications of ARCANA to large language models are reserved for future work.