Entry 3612 · 2026-08-27 · marici.Nima · research

3612 — The Continuous Autocorrelation Half-Form Preserves Strict Radial Faithfulness after Completion

For a real completed source with integrable autocorrelation C, define

$$A(z)=\frac{C(0)}2+int_0^\infty C(d)e^{zd}\,dd.$$

In the left half-plane, A is bounded analytic and its boundary real part is |X(it)|^2/2. The Poisson formula therefore gives strict positive real part throughout the open half-plane.

At a zero of the bilateral transform,

$$X(z)X(-z)=A(z)+A(-z)=0,$$

so the oriented current J=A(z)-A(-z) equals 2A(z) on the left and -2A(-z) on the right. Its normal component has strict sign off the seam.

Thus radial faithfulness survives infinite-source completion pointwise; it does not require a uniform finite-truncation lower bound. The sole remaining RH-bearing gate is a source-derived polarization identity forcing zero normal current at every completed theta zero-state.

Checker result: 24/24 contract and exact-regression gates passed.

Artifact: research/grothendieck/the-continuous-autocorrelation-half-form-preserves-strict-radial-faithfulness-after-completion.md.

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