An Unverified AI-Generated Proof Candidate for Ten-Vector Phase Retrieval in C^4
Abstract
We record an unverified candidate obstruction to phase retrieval by ten fixed vector intensity measurements in complex dimension four. In the proposed argument, after Parseval normalization, a ten-vector phase-retrieving family would define a smooth embedding of CP^3 into the affine hyperplane H ≅ R^9 of normalized measurement outcomes. The rank-three normal bundle ν of this embedding satisfies p₁(ν) = −4u². On a projective hyperplane on which one rank-one measurement vanishes, the key step asserts that the projected coordinate direction gives an actual nowhere-zero section of ν. If this and the subsequent characteristic-class steps are valid, the restricted normal bundle splits as a trivial line plus an oriented real two-plane bundle, forcing its first Pontryagin class to be the square of an integral Euler class. On CP^2 this would require an integer k with k² = −4, a contradiction. If the full candidate argument is correct, combining it with Vinzant's explicit eleven-vector construction would show that eleven measurements are necessary and sufficient.Certificates
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