The Goldbach Conjecture: A Categorical Structural Resolution by Odd-Square Paving and Paired Prime Spectra

This preprint proposes a categorical structural resolution of the strong Goldbach Conjecture as a corollary of the preceding Euler–Fourier–Riemann framework (doi:10.5281/zenodo.16887774). The additive problem is recast through the classical odd-square paving N² = 1 + 3 + ⋯ + (2N − 1), which provides, for every even target 2N, a complete mirror structure x ↔ 2N − x. Each odd layer is then equipped with its primitive Euler prime mark, additive displacement, and pair-centered logarithmic signature. Möbius inversion extracts the prime layer from Euler’s prime-power skeleton, while an exact two-parameter Fourier–logarithmic transform preserves arithmetic identity and selects mirror prime pairs through its zero mode. The Goldbach predicate is thereby represented equivalently as prime-mirror overlap, a positive convolution coefficient, a Fourier zero mode, and a non-vanishing paired prime spectrum. Within the categorical framework inherited from the preceding Riemann resolution, total annihilation of this paired spectrum is incompatible with the completed primitive Euler state. Goldbach is thus obtained as a structural corollary, without case-by-case search over prime pairs.