Combinatorial Properties of the Configuration Structure of the 6-Dimensional Hyperrectangle

This paper defines a 6-dimensional structure by adding a 4-valued discrete label axis Q to the 5-dimensional hyperrectangle, and derives combinatorial properties from its configuration structure. The main body (§1–§9) establishes: element counts via the general formula fⱼ (k) = C (k, j) ·2^ (k-j), spin classification into 8 types (s = 0, 1/2, 1, 3/2, 2, 5/2, 3, 7/2) from non-zero axis counts, Q-label transition enumeration (9 states), spin-1 configuration counting (13 states), spin-2 (3 types) and spin-3 (1 type) configurations, sign product P (n) = (-1) ⁿ, and signed area M (σ) from R-axis-containing faces. §10 presents an interpretation example mapping 62 of 63 derived states to SM+gravity particles. v3: Complete rewrite — all results re-derived self-containedly, definitions separated from physical interpretation, spin extended to 8 types (n=0, 1, 2, 3), Table B with explicit state counts for SM-unmapped configurations.