Emergent Condensate, Center of Mass, and Mass-Like Measure from an Anonymous Complex Standing-Wave Vacuum: An Anonymous Fourier-Mellin Readout of Condensation Strength, Finite Extent, and Vacuum-State Gauge Distortion

Without assuming localized particles, mass, a center of mass, or background spacetime, this paper starts from an anonymous set of complex standing waves under the complex square-zero closure (sum of squares equal to zero) and constructs a way to read out, when a localized condensate forms inside the homogeneous vacuum state field, its relative center of mass, finite extent, condensation strength, and the distortion of the surrounding vacuum-state gauge, without external coordinates or particle labels. An anonymous Fourier-Mellin transform over the empirical measure of log-amplitudes and phases is invariant under permutations, and its intensity is invariant under the common amplitude scale and common phase rotation. The cumulant expansion of the normalized condensate transform yields the relative center of mass from the first-order spectral phase and the finite extent and shape from the second-order spectral decay, giving a moment hierarchy (0th: condensation weight, 1st: center, 2nd: extent) proposed as a mass-like measure. Compensation of a nonzero closure residual and closure-preserving correlational rearrangement are distinguished as two types of background distortion. An appendix proves that a constant wavelength-frequency product across all shell pairs is equivalent to the power relation P proportional to L^alpha, classifies spacelike (alpha<1), null (alpha=1, the self-dual point of the previous paper), and timelike (alpha>1) readouts, and constructs an exact rational-exponent shell family with strict per-shell square-zero closure and absolute convergence for alpha<2. The connection from localized condensation to timelike readout is separated as an explicit hypothesis. Written in Japanese and English.