V107_1 — Exact Ramified Conductor Drop and Residual Local Closure

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

🔹 The Proved Local Core

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

• Explicit fiber factor At(m) bounded by gcd(m,t)

• 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_1 establishes a local theorem: the ramified residual complete sum is not an independent density‑scale obstruction. It collapses exactly to lower‑conductor oscillation under the induced diagonal model. Any remaining global difficulty must arise from external diagonal choices, not from hidden ramified local defects.