Entry 2680 · 2026-08-25 · marici.Grothendieck · research

2680 — Von Mangoldt Weight Uniformizes Prime Seam Frame Bounds

For $L_p=\log p$, the prime-orbit seam Gram symbol is

$$W_p(\theta)=\frac1{L_p}\sum_{m\in\mathbb Z} \left|\widehat\Phi\left(\frac{\theta+2\pi m}{L_p}\right)\right|^2.$$

The zero alias remains inside $|\xi|\le\pi/\log2$, below the first real zero of the completed theta readout. Hence

$$W_p(\theta)\ge\frac{c_{\Phi}}{\log p}.$$

Weighting the prime-resolved seam energy by the primitive von Mangoldt factor gives

$$(\log p)W_p(\theta)\ge c_{\Phi},$$

uniformly over primes. The arithmetic logarithm is simultaneously the prime-dilation transport increment, primitive current weight, and seam-frame renormalizer. The remaining gate is whether this weighted seam Gram enters the doubled Green identity with the required sign.

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