A Centered Geometric Framework for the Distinct-Prime Goldbach Problem

We develop a centered geometric and combinatorial framework for the distinct-prime Goldbach problem---the assertion that every even integer $2N \geq 8$ is the sum of two \emph{distinct} primes. The rigorous content is an exact geometric reformulation: for each $N \geq 4$, the problem is equivalent to finding an integer $M \in [1,N-3]$ such that $N-M$ and $N+M$ are both prime, or equivalently such that the L-shaped region between nested squares of side lengths $N$ and $M$ has area $(N-M)(N+M)$ with both factors prime. We define the centered sets $C_N=\{M: N-M \text{ is prime}\}$ and $D_N=\{M: N+M \text{ is prime}\}$ inside $\{1,\ldots,N-3\}$; then $C_N\cap D_N\neq\varnothing$ is exactly the distinct-prime Goldbach assertion for $2N$. We also study the diagnostic set $E_N$ of admissible half-differences arising from straddling prime pairs; its elements correspond to Goldbach partitions of nearby even numbers. A decomposition by upper primes expresses $|E_N|$ in terms of the half-differences generated by each prime $Q\in(N,2N)$ minus explicit collision sets. Under a fixed upper-prime collision hypothesis, explicit estimates of Rosser--Schoenfeld and Dusart imply positivity of the statistic $H(N)=\log^2(2N)-((N-3)-|E_N|)$ for all sufficiently large $N$, and finite computation verifies the remaining tested range. Positivity of $H(N)$ gives a pigeonhole result $C_N\cap E_N\neq\varnothing$. The distinct-prime Goldbach conjecture is not proved here; the remaining gap is a centering theorem, equivalently a uniform lower bound for $R(2N)=|C_N\cap D_N|$.