Unconditional Explicit Constants in the Goldbach Problem for Arithmetic Progressions

This paper, which is entirely unconditional, proves a sharpened almost-all theorem with fully explicit effective constants for the restricted weighted Goldbach sum R_{a,q}(N) := sum over p1+p2=N, p1 = a (mod q), of (log p1)(log p2), with q >= 1 and gcd(a,q) = 1, whose expected main term is M_{a,q}(N) = C_2 * S(N) * N / phi(q), where C_2 = 0.6601618... is the twin-prime constant and S(N) is the binary singular series.The results are organised around four pillars.(I) A complete character-pair decomposition of the second moment of the error E(N) := R_{a,q}(N) - M_{a,q}(N), extracting the exact diagonal constant G/(2*phi(q)), where G = prod_{p>2}(1 + (p-1)^{-2}) in [1.41320886, 1.41320899] is the Gallagher-Goldston constant.(II) A uniform minor-arc L^4 bound: integral over minor arcs of |S(alpha)|^4 dalpha <= kappa_safe * 2^A * X^3 / (log X)^A, with kappa_safe = 4.40, obtained by combining the complete Vaughan identity with the Bombieri-Vinogradov theorem in integral form, with an explicit derivation of kappa_explicit = C_V^2 * c_{L^2} = 4.004 before applying a rigorous 10% safety margin.(III) The effective almost-all theorem: #{N <= X even : |R_{a,q}(N) - M_{a,q}(N)| > C(A,q) * N * (log N)^{-3}} << X * (log X)^{-A}, with the explicit constant K := 2*C(1,4) <= 3.3624, obtained from C(1,4) <= 1.6812 via a Stechkin-type optimisation.(IV) A Pintz-type exceptional-set bound on {N <= X : R_{a,q}(N) = 0}.Every statement in the main body carries the tag [PROVED]. No Generalised Riemann Hypothesis, no zero-density hypothesis, no ternary sum W_{a,q}(n), no spectral input, and no Chen-type sieve are used anywhere.