Internal Cancellation Geometry and Legal Gap Spectra in Shogi

Fixing a legal candidate move in shogi produces a finite system of all legal replies. When a board reply vacates its origin, sliding action edges of the candidate side may be temporarily released and then deleted again when the reply is completed. We call this hidden overlap internal cancellation. This paper determines its legal fiber structure and one-dimensional geometry. A complete hand delta separates drops, quiet board moves, and captures; within a fixed visible observation row, the only remaining mechanism freedom is the internal-cancellation set. Distinct nonzero cancellation values have disjoint reply-origin carriers, so every legal trace becomes meet-closed after adjoining the empty set. Each nonzero value decomposes canonically into at most two terminal suffixes of origin-release ray bundles, and this bound is sharp. Exact cancellation and release/cancellation pairs admit decoder-relative finite certificates; retaining an exact reply carrier makes the gap-aware certificate complete, whereas tail-only data forgets the actor-to-vacancy gap. We then fix an exact candidate--reply carrier and tail-only shape and vary the forgotten gap. The coordinate-admissible values form an integer interval, but the locally legal spectrum need not: a candidate-source vacancy gives a legal puncture with spectrum \2,4\ inside \2,3,4\. More generally, candidate paths, destinations, material roles, and king safety impose typed footprint conflicts. For regular affine deformation shells, king-safety failures are decomposed into atomic step and ray attack channels. Every raw atomic channel has interval support; removing the support intervals of b potential blockers leaves at most $b+1$ components. Summing these channel budgets over the post-candidate and post-reply checkpoints bounds the number of legal-spectrum holes. A discrete total-variation identity exactly counts observed obstruction components. Legal local examples realize one and two separated king-safety holes, while a non-affine example shows that mobility rank alone gives no general bound. A public finite audit checks 3,952,800 empty-board attack traces on the 9×9 board and finds at most two support components for every standard shogi piece type, with sharp witnesses. Standard-initial-position reachability is kept separate from local legality throughout.