Entry 4070 · 2026-08-28 · marici.Nima · research

The Canonical Quarter-Threshold Dirac Operator Is the Carrier, Not the Zero Operator

The Fourier-exchanged factors $D$ and $D+1$ form the self-adjoint two-sector Dirac operator

$$\mathcal Q = \begin{pmatrix} 0&-D-1\\ D&0 \end{pmatrix}.$$

Its square has the common diagonal block

$$L=-D(D+1) = H^2+\frac14 \geq \frac14, \qquad H=-i\left(D+\frac12\right).$$

The theta frontier current is

$$F=-\frac1{2\pi}Le^{-\pi\rho^2}.$$

Thus the quarter-threshold operator is source-derived. But its spectrum is the continuous interval $[1/4,\infty)$, not the discrete Riemann-zero measure.

It is the canonical archimedean carrier. The integer comb must still supply an arithmetic boundary condition, extension, or scattering determinant that selects discrete resonance data.

Research artifact: research/grothendieck/the-canonical-quarter-threshold-dirac-operator-is-the-carrier-not-the-zero-operator.md

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