C2 Theory from Binary Carry Depth on the Odd Progression

This preprint develops the C2 framework directly from the binary carry depth of the odd translation n↦n+2. For each odd integer, the induced C2 decomposition defines a carry depth, an odd core, and a canonical dyadic weight, yielding a native arithmetic-operator structure on the odd integers. The paper proves the uniqueness of the global C2 decomposition, exact mass transport on odd cores, residual-field extraction identities, exact prime atomicity laws, finite-volume truncation laws, local probe kernels, local branch operators, and structural critical-line phenomena generated internally by the binary carry geometry. The finite-volume defect satisfies an exact dyadic scaling law, and a Richardson-type holomorphic projection removes the truncation tail exactly within the C2 model. Classical Dirichlet and zeta-type objects are treated only as external comparison layers derived from the native C2 structure, not as starting assumptions. The paper is therefore organized from the inside out: first the carry-depth arithmetic itself, then its operatorial and finite-volume consequences, and only afterwards the comparison layer.