V107_3 — Exact Ramified Conductor Drop and Verified Local Closure

Description: This paper proves an exact ramified conductor‑drop identity for the zero‑frequency rational complete sum, with explicit verification of technical details. It shows that ramified contributions collapse precisely to lower‑conductor oscillatory sums, with fiber multiplicities computed prime‑power by prime‑power, even in entangled cases. The diagonal side, defined compatibly by the same fiber‑projection principle, obeys the same conductor‑drop law. Consequently, the ramified local defect reduces exactly to a lower‑conductor coprime discrepancy, controlled by the standard prime‑power stationary‑phase bound. The two‑adic sector vanishes entirely, since the admissible set is empty for even moduli.

🔹 The Proved Local Core

• Reduction of inverses from modulus t to modulus q

• Constancy of the additive phase on conductor‑drop fibers

• Exact ramified conductor‑drop identity (no coprimality condition required)

• Induced diagonal compatibility theorem

• Coprime lower‑conductor square‑root bound via stationary phase

• Weighted ramified defect closure at scale N1/2Qεmε

• Vanishing of the two‑adic component

🔹 Residual Local Closure

• Ramified defect collapses to lower‑conductor oscillation.

• Two‑adic contribution is identically zero.

• Final residual defect satisfies Dres(m)≪N1/2Qεmε.

🔹 Conclusion V107_3 establishes a verified local theorem: the ramified residual complete sum is not an independent density‑scale obstruction. It collapses exactly to lower‑conductor oscillation under the induced oscillatory diagonal model. Any remaining global difficulty must arise from external diagonal choices, not from hidden ramified local defects.