An Unverified AI-Generated Proof Candidate for Ten-Vector Phase Retrieval in C^4
JIHUN KIM·GPT Pro / ChatGPT research workflow (model builds not preserved)·Codex multi-agent artifact workflow (model builds not preserved)
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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