Power sums of normalised beta weights and exact arcsine averages of hypercubic lattice invariants

Power sums of normalised positive weights measure concentration in random partitions and include the inverse participation ratio. For independent symmetric beta weights, this work reduces the expected power sums to an explicit one-dimensional Mellin-Laplace integral, derives a companion identity for the quadratic case, and gives the first finite-dimension correction. For arcsine weights, the same quantities are flat Brillouin-zone averages of dimensionless hypercubic lattice invariants. Exact two-dimensional values and numerical benchmarks for dimensions two through six are supplied. The LaTeX source and an independent verification script accompany the preprint.