Emergent Spacetime and Wave-Scale Readout from Anonymous Complex Zero Closure: Minimal Readout, an Infinite Constructive Solution, Logarithmic Spacetime Mapping, and Post-Mapping Wavelength-Frequency Invariance

Starting only from an anonymous set of complex numbers identified under permutations and the complex square-zero closure (sum of squares equal to zero), this paper asks what can be read out internally, in order of minimal cardinality, without importing space, time, wavelength, frequency, or velocity as background variables. Two nonzero elements fix a gauge-invariant amplitude ratio 1 and relative angle of 90 degrees; three elements couple relative amplitudes and relative phases through a cosine-rule constraint. As the central construction, a countably infinite constructive solution built from integer shells with amplitudes proportional to 1/M_k and M_k-fold phase cycles is given; every shell closes exactly and the infinite sum converges absolutely when the sum of 1/M_k is finite. The same integer M_k is readable both from amplitude multiplicity and from the phase-cycle order. Logarithmic mapping, the unique continuous homomorphism from multiplicative scale ratios to additive coordinates, defines relative spacetime scales, and the wavelength-frequency product remains invariant after the spacetime mapping. Recombining the two readouts as W = X + iT and squaring yields a 1+1 dimensional Lorentzian quadratic form; within this special solution every shell-pair interval is null, so that scale self-duality, wavelength-frequency invariance, and the null readout are three expressions of one identity. Written in Japanese and English.