A General Proof of the Fong-Tsui Conjecture

Abstract: We present a general proof of the Fong-Tsui conjecture for bounded operators on arbitrary complex Hilbert spaces. Specifically, we show that $|T|\leq|\operatorname{Re}T|$ implies that $T$ is self-adjoint. The argument combines a positive inverse of a Sylvester map with a spectral cutoff determined by the norm of the positive defect $|\operatorname{Re}T|-|T|$. A local vanishing lemma reduces the analysis to the classical squared self-adjointness criterion, while positivity of the defect yields a global norm contradiction. We formulate the argument as an abstract four-operator vanishing principle, without compactness, trace, or separability assumptions. We also establish a quantitative stability estimate: if $|T|\leq|\operatorname{Re}T|+\varepsilon I$ and $0\leq\varepsilon\leq|T|$, then $$|\operatorname{Im}T|\leq6|T|<sup>{7/8}\varepsilon<sup>{1/8}$$. The constant is independent of the dimension, and the exponent is not claimed to be optimal. LLMs were used to assist with proof development, algebraic calculations, numerical checks, and auditing of the arguments.