UMMIC: Mother Map--Mellin--de Branges Unified Theory of ``Information Conservation--Phase Density--Sampling Stability'' (With Complete Proofs)

Starting from mother map kernels satisfying moderate axioms, under the Mellin isometry on L^2(\mathbb R_{+},dx/x) and the spectral dictionary of de Branges--Kreĭn canonical systems, we establish three parallel and closed main lines: (I) a Noether-type flux continuity equation (information conservation) with \Lambda(s)'s logarithmic potential as potential function; (II) ``phase density = spectral shift derivative = relative spectral density'' consistency (CCS) under self-adjoint scattering settings; (III) engineering-oriented Nyquist--Poisson--Euler--Maclaurin (EM) three-fold decomposition with non-asymptotic error closure. Accompanying this, under the parallel framework of positive-weight spectral density scale d\mu(E)=\rho(E)\,dE and phase coordinate v_\phi(E)=\delta(E)/\pi, we introduce the relative spectral density \rho_{rel}(E)=\xi'(E)=-\tfrac1\pi\delta'(E) and its integral coordinate v_{rel}(E)=\xi(E), unifying engineering and spectral theory scales, and on this basis unify the proofs of Landau sampling/interpolation necessary density thresholds, Wexler--Raz tight/dual sufficiency conditions, and Balian--Low impossibility (Mellin/Weyl version), thereby welding ``mother map--scattering--frame/sampling--error theory'' into a non-asymptotic hard-core system.