Paper 125D: Multiplicative Packets, Defect Orbitals, and Radial Euler Cancellation Exact Three-Owner Identities, Bright–Dark Coordinates, and the Primitive Rank-One Boundary

We derive exact multiplicative decompositions that retain internal Möbius cancellation beforeestimating the near-resonance energy inherited from Paper 125C. Three prime-factor coordinatesgive finite packets with explicit support-boundary terms. Their individual packet energies have afixed logarithmic bound, while completed odd-numerator, odd-denominator packets at a commonfrequency cancel exactly. A cubic truncated Möbius identity is reorganized into components withnested support. Each reduced channel also factors into a universal double Möbius convolution and anabsolutely summable arithmetic correction. The associated Dirichlet series has at least two zeros atits density boundary, with an additional zero in the odd–odd case; a separate finite identity reducesthat case to two radial boundary bands. Ordered Euler packets satisfy a positive energy identityin the dilation-invariant continuous norm. Descending prime order minimizes the specified scalarpacket cost, not an arbitrary arithmetic energy. On an active primitive shell, integer radial samplingleaves only one coordinate, so further dilation packets cannot create an independent sampled radialprofile. Archived diagnostics and finite recomputations illustrate this distinction between improvedcoordinates and an actual bound. The signed interactions between distinct frequencies, the transferfrom continuous packet norms, and the return to the exact complement remain unresolved. The papersupplies exact identities and scoped component estimates, not a proof of the Riemann hypothesis.