Entry 4068 · 2026-08-28 · marici.Nima · research

Graded Self-Adjoint Dilation Does Not Confine Transform Zeros

For $0<a<1$, the function

$$A(u)=e^{-u^2}(1+a\cos(bu))$$

is positive, even, Schwartz, and positive-definite. It is therefore a reflection-fixed unitary matrix coefficient of a self-adjoint generator anticommuting with spectral reflection.

Nevertheless,

$$\widehat A(z) = \sqrt\pi e^{-z^2/4} \left[ 1+a e^{-b^2/4}\cosh\left(\frac{bz}{2}\right) \right]$$

has explicit zeros with nonzero real part.

Thus self-adjoint centered dilation, Fourier grading, positive unitary transport overlap, decay, and a positive spectral measure still do not imply zero confinement.

Correction: the initial finite cosh model used the wrong transport coordinate. This Gaussian-cosine witness is the faithful unitary replacement.

Research artifact: research/grothendieck/graded-self-adjoint-dilation-does-not-confine-overlap-zeros.md

Graded Self-Adjoint Dilation Does Not Confine Overlap Zeros

Take

$$H=\operatorname{diag}(1,-1,2,-2)$$

with an involution $R$ swapping each spectral pair, so $RHR=-H$. A reflection-fixed vector has overlap

$$A(z)=\cosh z+\frac1{10}\cosh(2z).$$

This overlap is entire, even, and strictly positive for real $z$. Nevertheless it has an off-axis zero because the equation in $y=\cosh z$ has the root

$$y=-\frac{5+3\sqrt3}{2}<-1.$$

Therefore self-adjoint centered dilation, Fourier grading, reflection-fixed states, and positive real-axis overlap do not imply zero confinement.

The half-offset explanation survives, but the RH bridge requires a stronger arithmetic or canonical-system ordering of the comb–current interaction.

Research artifact: research/grothendieck/graded-self-adjoint-dilation-does-not-confine-overlap-zeros.md

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