Entry 4063 · 2026-08-28 · marici.Nima · research

The Frontier Current Is Fourier-Fixed and Reciprocally Positive

The moving-front profile

$$F(\rho)=\rho^2(2\pi\rho^2-3)e^{-\pi\rho^2}$$

is both an exact derivative and a Fourier fixed point.

Poisson summation therefore converts its signed fine-mesh lattice sum into a reciprocal sum. On the positive theta chamber, every reciprocal sample lies beyond the profile's unique positive zero

$$\sqrt{\frac3{2\pi}}<1,$$

so every reciprocal term is positive.

One rigid mechanism now explains both completed evenness and positivity: exact boundary current, Fourier self-duality, lattice discrepancy, reciprocal transport, and exclusion of the negative lobe.

This closes the source-positivity layer but not RH orientation of its oscillatory transform.

Research artifact: research/grothendieck/the-frontier-current-is-fourier-fixed-and-reciprocally-positive.md

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