The Pole-Selection Theorem for the Quotient Trace

Research Note 44 in the "Geometry of the Critical Line" programme. This note formulates the remaining Wall 3 question as a single target theorem — the Pole-Selection Theorem — and decomposes it into three lemmas. Lemma B (identity isolation) and Lemma C (normalisation transfer) are proved. Lemma A (carrier-pole control) is proved at the structural level from Epstein universality, finite-index thinning, and the Connes-style distributional quotient-trace framework: the regularised quotient trace is governed by the geometric carrier pole at s = 1, not by the multiplicative Haar point s = 0 or the arithmetic pole s = 1/2. Taken together, the three lemmas account for all factors of the Weil prime kernel and cross…