Signed Prime Möbius Correlations and a Conditional Route to Binary Goldbach

We formulate a conditional route to binary Goldbach through a signed family of prime–Möbius correlations. A squarefree divisor identity separates the von Mangoldt convolution into a short-divisor contribution and an exact large-divisor remainder. Divisor switching retains the moving endpoint, and character orthogonality identifies the complete nonprincipal family, including induced characters. We prove explicit bounds for nonunit residue corrections, endpoint contributions, noncoprime divisor terms, proper prime powers, and an elementary positive singular-series reserve. These estimates yield a conditional completion theorem with an explicit threshold formula, provided that two independently specified analytic obligations are established with effective constants. The principal unresolved estimate is an upper bound for the real part of the full signed character contribution for every sufficiently large even target. We also derive its exact Hilbert-space projection and weighted Pythagorean identities. They prove orthogonality of a projection and its residual, while retaining the unknown correlation as a separate energy component. A least-prime-factor decomposition gives an exact recursive representation but does not close the coefficient family on smaller Goldbach targets. The manuscript identifies the remaining arithmetic input, its quantitative budget, and the additional finite verification needed for a complete proof.