Local Factorization of p-adic Gamma Sums

We revisit the proposed equality between discrete Fourier transforms of $p$-adic ${\mathrm{\Gamma }}_p$–values and $p$-adic $L$–derivatives for odd characters modulo a prime $p$. The clean identity is false in general. Building on Coleman reciprocity and the Gross–Koblitz formula, we prove an exact two-term decomposition: for each odd, nontrivial Dirichlet character $\chi (\mathrm{m}\mathrm{o}\mathrm{d}p)$, $${\mathrm{\Phi }}_p(\chi ):=\sum _{a=1}^{p-1}\chi (a){\log }_p{\mathrm{\Gamma }}_p\text{\hspace{0.17em}⁣}\left(\frac{a}{p-1}\right)=U_{1,p}{L}_p^{\mathrm{\prime }}(0,\chi )+U_{2,p}L(0,\chi ),$$ with constants $U_{1,p}\in {\mathbb{Q}}_p({\mu }_{p-1}{)}^{\times }$ and $U_{2,p}\in {\mathbb{Q}}_p({\mu }_{p-1})$ depending only on $p$ and the fixed branch of ${\log }_p$, but independent of $\chi$. Subtracting the $L(0,\chi )$–block yields a \emph{renormalized} local input $${\mathrm{\Phi }}_p^{ren}(\chi ):={\mathrm{\Phi }}_p(\chi )-U_{2,p}L(0,\chi )=U_{1,p}{L}_p^{\mathrm{\prime }}(0,\chi ),$$ uniformly in odd, nontrivial $\chi$. Plumbing these renormalized locals at every finite place into the Weil explicit formula (with the standard Li kernel at $\mathrm{\infty }$) reproduces exactly the classical Li coefficients. We also record a short, reproducible verification protocol; a tiny table for $p=5,7$ illustrates the $\chi$–independence of $(U_{1,p},U_{2,p})$.