Crystallographic Phase Retrieval: Periodic Obstructions and Weighted Rigidity

This paper disproves the standard-coordinate generic uniqueness assertions in Conjectures 4.7 and 4.11 and the support-separation assertion in Conjecture 4.8 of Bendory and Edidin, as stated. A nine-point support in the cyclic group of order 24 has eleven distance classes, yet a fixed rational orthogonal convolution produces a nonintrinsic partner for a Zariski-open set of real amplitude vectors. The initial mechanism is periodicity of the support. All convolution operators preserving a coordinate-support space are classified through its translation stabilizer. A stabilizer of order h supplies a spectral-unit torus of dimension floor((h−1)/2). For unions of cosets with uniquely represented quotient differences, the entire generic fiber is exactly one spectral-unit orbit; exceptional fibers admit a frequency-by-frequency classification. Explicit families exhibit ambiguity at arbitrarily small support density and with arbitrarily large dimension. The obstruction also survives the destruction of support symmetry. Puncturing periodic supports produces aperiodic eight-point supports with exactly four generic solutions, including the global sign ambiguity, despite a full-rank autocorrelation Jacobian. Two noncongruent supports admit positive, full-support partners throughout a nonempty Euclidean-open set of source amplitudes, disproving Conjecture 4.8 as stated. A discriminant determines whether the competing-support fiber contains zero, two, or four real vectors. Further punctured families retain positive-dimensional local fibers on supports with trivial translation and reflection stabilizers. Complementary results establish exact recovery through connected product graphs, closed inverses for two anchored families, and a rank-one characterization of weighted ambiguity between separated reflected Sidon sumsets. A calibrated coset yields an explicit reconstruction formula with a sharp stability constant. Completed digit-polygon shadow data are equivalent to cyclic diffraction data, with the necessary identities derived directly. The proofs are analytical and algebraic. Classical spectral units, graph recovery, and factor reversal are credited. The results do not provide a general classification of recoverable aperiodic supports or a universal sparse reconstruction algorithm; generic-basis uniqueness theorems remain unaffected. Supplements include editable LaTeX, six vector figures, reproducible figure code, exact symbolic checks, and a separate account of corrections and remaining questions.