A Secondary Note on Split-Zero Globalization: Semimodule Classification, Multiplicative Reconstruction, and All-Orders Zeta Deformation

We extend the baseline globalization theory of the split-zero semiring S = G(R) = R ⊔ {τ}, with operations r ⊕ s = r + s, r ⊕ τ = r, τ ⊕ τ = τ, rs = rs, rτ = τr = τ, τ² = τ, for a nonzero commutative ring R with identity. Throughout, we assume the results of the companion note: the ring reflection pᵣ, the support character χᵣ, the classification of ideals, prime ideals, congruences, and localizations, and the supported-zero projector νᵣ(x) = 0ᵣx.

The main results are the exact classification of G(R)-semimodules as join-semilattice-indexed diagrams of R-modules; the identification of the cancellative subcategory with R-Mod; the Boolean-cubical form of free semimodules; the product theorem G(A × B) ≅ G(A) ×_𝐁 G(B); reconstruction from the multiplicative monoid together with the missing additive relations; the infinite-dimensionality of the multiplicative-monoid spectrum in the UFD case; the contracted monoid-algebra decomposition; conservative ideal/class/factorization results; and the full all-orders deformation theory of the shifted quotient-size zeta sheet, including universal jets, cumulants, arithmetic kernels, non-Eulerianity, Faulhaber interpolation, factorial interpolation, and the exact endpoint defect controlled by a two-point Stone model.