Power-graph lengths under expanding cutoffs: expansions and conjugate thresholds

For y = |x|p with p > 1, we study Euclidean length in dilates of a compact convex window whose positive vertical-axis intercept is normalized to one. A single expansion in t = R−(1−1/p) treats signed, asymmetric one-sided boundary charts, without central symmetry, horizontal support or a smooth join. Generalized analytic charts give a convergent coefficient formula, while finite boundary data give a quantitative error bound. We evaluate the square constant for every exponent using gamma functions and explicit Laurent finite parts. Two corollaries distinguish the Hölder-conjugate contact threshold from the quadratic threshold at p = 2, which requires first-order cancellation and a nonzero quadratic coefficient. Smooth tilted ellipses and a nonsmooth tilted vertex exhibit the different regimes. The square logarithms are classical hypergeometric specializations. MSC 2020: Primary 41A60; Secondary 44A15. The ZIP contains LaTeX sources, vector-figure inputs, and fixed-input numerical checks.