Reply-Induced Deletion Geometry in Shogi : Incidence Structures, Fragility Spectra, and Co-Deletion Graphs

Abstract

We introduce a finite combinatorial structure for measuring how a legal Shogi move's immediate board-influence relation is destroyed by the opponent's legal one-ply replies. For a position $P$ and a legal candidate move $m$, rows index all legal replies and columns index the mover's action edges immediately after $m$. A binary entry records whether a reply deletes an action edge. The resulting reply-by-action-edge deletion incidence matrix unifies several distinct summaries: the reply-robust core is its zero-column set; edge fragility is encoded by column sums; a Reply-Fragility Spectrum is the histogram of positive column sums; a minimum reply-cover number is an exact set-cover invariant (equivalently, a transversal number only in the dual hypergraph); and pairwise co-deletion is the off-diagonal part of the integer Gram matrix $M^{\mathsf T}M$. We prove the associated identities and bounds and give explicit Shogi separation witnesses. In particular, positions reachable from the standard initial position can have the same number of legal replies, the same number of fragile edges, and the same minimum reply-cover number while having different fragility spectra; likewise, equal fragility spectra need not determine pairwise co-deletion geometry. The framework is descriptive rather than evaluative: it neither ranks candidate moves nor claims playing strength.