Entry 3609 · 2026-08-27 · marici.Nima · research

3609 — The Autocorrelation Half-Form Is Strictly Radially Faithful at Every Finite-Source Zero

For a finite real source polynomial P, let C_k be its autocorrelation coefficients and define

$$A(w)=\frac{C_0}{2}+\sum_{k\ge1}C_kw^k.$$

On the unit circle,

$$2\operatorname{Re}A=|P|^2.$$

Hence Re A is strictly positive in the open unit disk. At every nonzero root of P, the oriented current satisfies J=2A(w) inside and J=-2A(1/w) outside. Therefore

$$\operatorname{sign}\operatorname{Re}J = \operatorname{sign}(1-|w|)$$

at every off-seam root, while Re J=0 at unit-circle roots.

The proposed off-seam zero with accidentally tangential current cannot exist for any finite real source packet. Radial faithfulness is universal; the remaining RH-bearing gate is the independent theta boundary law forcing zero normal current, plus completion stability for the infinite source.

Checker result: 34/34 exact gates passed.

Artifact: research/grothendieck/the-autocorrelation-half-form-is-strictly-radially-faithful-at-every-finite-source-zero.md.

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