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# Convention settled before computation
T469 stores M(K)=E||X-c_K||_2, where c_K=E[X] and X has the stated uniform measure. The single symbolic parameter is region. Length, area, volume, arc length and surface area are normalized to probability according to the region's intrinsic dimension. Distances are Euclidean in the ambient space.
Selection: unit interval; unit square; rectangle of sides 1 and 2; equilateral triangle of side 1; disk of radius 1; regular pentagon, hexagon, heptagon and octagon of circumradius 1; unit cube; solid three-dimensional ball of radius 1; circle of radius 1; sphere in R^3 of radius 1. This is a short set of classical sampling benchmarks, including every regular polygon from 3 through 8, with the square and triangle in the unit-side convention specified by the family and the other four in T390's unit-circumradius convention. All region keys agree with T390 and T467 wherever applicable. No dimension or aspect-ratio grid is implied.
Type R, 100 significant digits for irrational entries, exact rational returns for interval/disk/ball/circle/sphere. The newer skill and artifact policy take precedence over the family proposal's suggestion to decimalize rationals. Rigour proven: closed forms in Arb throughout; independent numerical quadratures are checks, not certified error bounds.
Sources: Li and Qiu (2016), DOI 10.1155/2016/8371750, equation (23) and Examples 1-3; Bailey, Borwein and Crandall, Box integrals, DOI 10.1016/j.cam.2006.06.010, author manuscript equations (4), (8), (10), and section 7. These fix the rectangle, regular-polygon and cube conventions. Radial integration gives d/(d+1) for the unit d-ball. Constant radius gives exactly 1 for the circle and sphere.
The proposed geometric probability tag is absent. Existing probability theory, elementary geometry and integral tags fit. A shared new tag would apply to T467, T468, T469, T19 and T283; no other table is edited here.
Search evidence: screen-current.log, lookup-*.json, tags.html and neighbor-addresses.log. The two irrational sample values have no search hits; 2/3 occurs in unrelated families. The unique-title draft claim succeeded. No proposal request provenance belongs in the mathematical document.