A perfect sequence has ideal autocorrelation — zero sidelobes at every nonzero shift. The array orthogonality property (AOP) requires that the sequence, when folded into a two-dimensional array, produces rows and columns that are mutually orthogonal. A sequence with both properties would be simultaneously ideal in the time domain (perfect autocorrelation) and the array domain (orthogonal rows/columns).
The paper proves no such sequence exists. The two constraints are individually satisfiable but jointly contradictory. Perfect autocorrelation forces the sequence's discrete Fourier transform to have constant magnitude — the energy is uniformly distributed across frequencies. The AOP forces the array's Fourier transform to have a specific block structure. The uniform-magnitude constraint and the block-structure constraint are incompatible for any sequence length that admits the array folding.
The proof is algebraic: the periodicity relations imposed by the array folding, combined with the constant-magnitude constraint from perfect autocorrelation, produce a system of polynomial equations with no solution. The nonexistence is not approximate or asymptotic — it's exact, for every length and every alphabet size.
The result draws a boundary in sequence design: you can have perfect autocorrelation or array orthogonality, but not both. The two properties optimize different aspects of the same sequence — temporal correlation and spatial structure — and optimization in one domain forecloses optimization in the other. The tradeoff is not a matter of engineering approximation but of mathematical impossibility. The perfect sequence for simultaneous time-frequency-space use does not exist.