where
푠
푚
acts as a scaling parameter linking additive Goldbach structure
to quadratic prime-density growth. This formulation induces a fixed con-
straint interval
4
5
≤ 푠
푚
≤
50
17
.
Computational analysis across multiple scales shows that
푠
푚
remains
tightly bounded and numerically stable, with values clustering near an
empirical constant
푠
∗
≈ 1.139.
Equivalently, the framework yields the reconstruction identity
휋(2푚) =
√
2푚 푅(2푚) 푠
푚
,
demonstrating that the prime-counting function, Goldbach partition func-
tion, and normalization parameter are structurally coupled.
These results suggest that the Goldbach partition function behaves as a
renormalized quadratic prime-density model governed by a stable normaliza-
tion structure. The proposed framework validates the bounds and reveals an
intrinsic coupling between prime distribution and additive partition behav-
ior, providing a basis for refined asymptotic modeling and further theoretical
investigation.