Keyed modulation is often judged secure by its key-space size and a key-less receiver's error rate, evidence that fails when the key is reused and the waveform is harmonically coupled. For a phase-keyed Fourier-curve constellation, whose $k$ tones share one data parameter, integer relations among harmonic indices yield data-cancelling mixed moments that expose key characters. A modular relation lattice characterizes them; for consecutive harmonics, third-order moments recover the relative phases and a fourth-order moment completes the key up to cyclic relabeling whenever its coefficient is nonzero, as in all evaluated settings. A non-data-aided relation-moment estimator with code-aided resolution of the relabeling turns this leakage into an attack without key-space enumeration. With one key per 168-symbol codeword, the attack meets a predeclared $0.1$ compromise criterion at eleven of twelve operating points of a $(3,6)$ LDPC-coded link; at the middle noise level, the eavesdropper's block error rate is below $0.04$, and $0.03$ for consecutive harmonics without artificial noise when a key spans only $28$ symbols. Tangent artificial noise and a harmonic set without relations below order four raise her error rate at intermediate reuse lengths but do not remove the one-codeword vulnerability. On a $2^{128}$-key grid at the same noise level, the cheapest tested refresh schedule whose $95\%$ lower confidence bound on her block error rate reaches $0.9$ costs $3.54$ fresh key bits per information bit, and no tested schedule on either grid meets even a $0.5$ target with less key than a one-time pad.

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