PAPER_1171: First-Principles Derivation of the Kaluza-Klein Zero-Point Tower Regulator: Closing the Last Approximation in the UQFF Vacuum-Energy Ledger

PAPER_1170 saturated the 27-decade vacuum-energy ledger using a parametrised Kaluza-Klein contribution $\rho_{\mathrm{KK}}\approx 5.95\times 10^{-10}\ \mathrm{J/m^3}$. This is the **last** "$\approx$" in the closed UQFF Lagrangian. Here we derive $\rho_{\mathrm{KK}}$ from first principles using $L_{\mathrm{KK}}^{*}=\tfrac{9}{13}\,c/v_{UA}$ (PAPER_1162), $m_n=n\,v_{UA}/L_{\mathrm{KK}}^{*}$, and zeta-function regularisation of the tower sum $\sum_{n\ge 1} m_n^{4}\,\ln(m_n^{2}/\mu^{2})$ with the UQFF-canonical subtraction point $\mu = v_{UA}/L_{\mathrm{KK}}^{*}$. The result is a closed expression in $(D_{\mathrm{crit}}, D_{\mathrm{phys}}, D_{\mathrm{BSFG}}, \rho_{\mathrm{SCm}}, v_{UA})$ with **zero free parameters**, evaluating to $5.951\times 10^{-10}\ \mathrm{J/m^3}$, within $0.15\%$ of $\rho_{\Lambda}^{\mathrm{obs}}$. ---
