Jihun Kim

Jihun Kim

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.

Jihun Kim

Let $O_3$ be the one-skeleton of the octahedron. We give an affirmative computer-assisted proof candidate for the question whether every geodesic CAT(0) space satisfies $O_3$-comparison. The comparison problem is first written as an anchored Gram semidefinite feasibility problem. A closed-image argument gives an exact strict alternative: failure is detected by a positive semidefinite signed stress. Compact normalization reduces a negative stress to an extreme ray, and a minimal-face tangent criterion bounds its rank by four. Ranks one and four admit solver-free proofs. Rank two is reduced analytically to a single chain normal form and is discharged by a coupled conditional-CN-defect identity; a finite planar atlas remains only as a regression check. Rank three is reduced to an exact finite classification: 28,672 labelled supports give 9,479 structural candidates and 255 symmetry orbits, all of which are proved empty, non-extreme, or CAT(0)-valid for their full continuous weight family. The final retained count is zero. The finite part uses integer, rational, and symbolic polynomial arithmetic only; no floating point optimization, tolerance, or bounded sampling enters the proof chain.

Jihun Kim

Let $O_3$ be the one-skeleton of the octahedron. We give an affirmative computer-assisted proof candidate for the question whether every geodesic CAT(0) space satisfies $O_3$-comparison. The comparison problem is first written as an anchored Gram semidefinite feasibility problem. A closed-image argument gives an exact strict alternative: failure is detected by a positive semidefinite signed stress. Compact normalization reduces a negative stress to an extreme ray, and a minimal-face tangent criterion bounds its rank by four. Ranks one and four admit solver-free proofs. Rank two is reduced analytically to a single chain normal form and is discharged by a coupled conditional-CN-defect identity; a finite planar atlas remains only as a regression check. Rank three is reduced to an exact finite classification: 28,672 labelled supports give 9,479 structural candidates and 255 symmetry orbits, all of which are proved empty, non-extreme, or CAT(0)-valid for their full continuous weight family. The final retained count is zero. The finite part uses integer, rational, and symbolic polynomial arithmetic only; no floating point optimization, tolerance, or bounded sampling enters the proof chain.

Jihun Kim

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.

Jihun Kim

Let $O_3$ be the one-skeleton of the octahedron. We give an affirmative computer-assisted proof candidate for the question whether every geodesic CAT(0) space satisfies $O_3$-comparison. The comparison problem is first written as an anchored Gram semidefinite feasibility problem. A closed-image argument gives an exact strict alternative: failure is detected by a positive semidefinite signed stress. Compact normalization reduces a negative stress to an extreme ray, and a minimal-face tangent criterion bounds its rank by four. Ranks one and four admit solver-free proofs. Rank two is reduced analytically to a single chain normal form and is discharged by a coupled conditional-CN-defect identity; a finite planar atlas remains only as a regression check. Rank three is reduced to an exact finite classification: 28,672 labelled supports give 9,479 structural candidates and 255 symmetry orbits, all of which are proved empty, non-extreme, or CAT(0)-valid for their full continuous weight family. The final retained count is zero. The finite part uses integer, rational, and symbolic polynomial arithmetic only; no floating point optimization, tolerance, or bounded sampling enters the proof chain.

Jihun Kim

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.