We reformulate Goldbach's Conjecture in terms of the midpoint $C = N/2$ and a symmetric offset $x$, so that $N = (C-x) + (C+x)$. For prime pairs exceeding $3$, the offset $x$ is confined to explicit residue classes modulo $6$ determined by $C bmod 6$; this is an elementary mod-$6$ wheel restriction, and we state exactly how the restricted search relates to the conjecture. We show that shielding a center $C=P_n$ by the first $n$ primes guarantees prime pairs only for $n le 3$, and we explain why this is a limitation of trial division rather than a demonstration of the sieve parity barrier. Under Hardy--Littlewood heuristics we derive the expected number of valid offsets and a Poisson-type model for the smallest one, and compare both with an exact computation over every center $C in [10^5, 2cdot 10^5]$. The counting prediction is accurate to within $1%$. The leading-order prediction for the mean smallest offset is too large by $11$--$34%$; we trace this to the point offset $x=0$ (the case $C$ prime) and to the regularity of the sifted set of candidates, and a model incorporating both agrees with the data to within $2%$. We also confirm by computation that every center $5 le C le 10^6$ admits a valid offset, and we report the smallest offset at the centers $C = 6cdot 10^{j}$ for $0 le j le 23$, which reaches heights far beyond the exhaustive range. The note is expository; no progress toward a proof is claimed.