$D_\infty$ and the Goldbach Convergence Hierarchy

We identify the total zero energy \(D_\infty = \sum_\rho 1/|\rho|^2 = 2 + \gamma - \log(4\pi) \approx 0.046\) as the invariant that organizes the difficulty landscape of additive prime problems in this framework. For a \(k\)-fold additive problem (representing \(n\) as a sum of \(k\) primes), each independent summand contributes a factor of \(1/\rho\) to the zero sum in the explicit-formula picture, so the natural tail is heuristically \(\sum 1/|\rho|^k\). This sum diverges for \(k = 1\) (prime counting) and converges for \(k \geq 2\) (Goldbach and beyond). Binary Goldbach sits at the first convergent scale for primes—the borderline where the tail is summable but the constants are tight.

We present three layered claims. First, the Convolution Convergence Theorem (conditional sketch): under RH, the triangle inequality and absolute convergence of \(\sum_\rho 1/|\rho|^2\) yield the explicit bound \(|E(n)| \leq 2D_\infty\sqrt{n} + O(1)\) for the standard zero-sum form of the Goldbach error; comparing this to the Hardy--Littlewood main term shows \(|E(n)| < S(n)\) for all sufficiently large even \(n\), while any uniform statement down to \(n = 4\) still requires a separate treatment of the main-term approximation for small \(n\) (or appeal to numerical verification). Second, the \(k\)-fold Damping Hierarchy: classical ternary Goldbach (Vinogradov; completed by Helfgott) is consistent with \(D_\infty^{(3)} \approx 0.0012\) lying deep in the convergent regime, where even the Vinogradov–Korobov zero-free region suffices for large-\(n\) analytic approaches. Third, the Additive–Correlative Duality: twin primes, prime \(k\)-tuples, and other fixed-shift correlations are correlative problems whose standard zero-sum models retain the \(\sum 1/|\rho|\) scaling of prime counting—without the Goldbach convolution damping.

The number \(D_\infty \approx 0.046\) is small enough that, once \(n\) is large enough for the Hardy--Littlewood main term to dominate the error bound \(|E(n)| \ll \sqrt{n}\), the zero sum has limited headroom: in that asymptotic regime the zeta zeros cannot overwhelm the main term. Small \(n\) are outside the domain of that asymptotic comparison and are handled separately in any complete argument (computation or sharper sieve input).

Keywords: Goldbach conjecture, zero energy, convolution damping, additive–correlative duality, Waring problem

MSC 2020: 11P32, 11N05, 11M06, 11L20