Prime Deficit and Euler Compression with a Compensated Eisenstein Probe

We develop two routes toward the Riemann hypothesis and isolate their analytic requirements. For the weighted von Mangoldt deficit, a subexponential bound on the accumulated positive part at critical normalization suffices for the Riemann hypothesis. A deterministic crossing estimate controls completed excursions with at most square-root-order event counts. In a separate Eisenstein-integer approach, we study a specified squarefree block of a compensated probe. A positive second-moment diagonal of size at least makes the proposed second-moment threshold demanding. Linear Poisson summation preserves the phases that this norm discards. Sextic reciprocity and the complete Gauss identities give the exact coefficient . We extract the contribution of sixth powers with a power-saving lattice error and identify the resulting prime-marked Möbius average. Estimates for that average, for the remaining frequencies, and for continuation below the original comparison line remain open.