Directional Numbers: Local Neighborhoods, Transport, and Stateful Scores

Directional Numbers develops a finite-alphabet framework for local neighborhoods, relative focus, directional trajectories, graph transport, and stateful scores. Starting with a ternary alphabet inside an integer lattice, it establishes dimensional extension and score-path correspondence, then separates routing instructions from carried directional state. Results include signed-permutation symmetry, gauge covariance, global transport triviality, spanning-tree normalization, anchored-score composition and reversal, and a sharp minimum-probe count for distinguishing permitted frame transformations. A connection Laplacian supplies a consistency test, including an exact quarter-turn triangle spectrum. Recoverable packet transitions and a sharp residual-record bound characterize information retention for finite many-to-one maps. Projection multiplicity, tagged dimensional possibilities, distributed translation, and generative descriptions are stated with explicit hypotheses. The manuscript includes proofs and a standard-library Python suite providing 44 reproducible finite checks. These checks supplement the proofs; they do not replace peer review or establish research priority. This mathematical preprint distinguishes coordinate re-expression, graph holonomy, and intrinsic metric curvature.