The Toeplitz Boundary Index Is the Euler Logarithmic Derivative
For one prime, the depth-k unilateral valuation shift has defect projection
Pairing this boundary index with the connected Euler coefficient gives
Therefore the Toeplitz boundary map implements the Euler logarithmic
derivative. With z=p^{-s}, Mellin differentiation supplies log p, so every
prime power receives exactly the von Mangoldt weight log p.
The nested difference P_k-P_{k-1} has rank one: each prime-power depth adds
one boundary cell. The result explains the arithmetic coefficient but not RH;
Fourier exchanges the two valuation cones and reverses the relative boundary
orientation, whose global archimedean sewing remains open.
Research packet: the-toeplitz-boundary-index-is-the-euler-logarithmic-derivative.md
Graph admission: ev-000000008422-5f2ff01d-1557-41ea-8897-2b8875794073.