Semiboundedness Criteria for H* (the B-Note): Stability, Instability, and the Reduction to a Unit-Window Excursion Question for ζ on the 1-Line

Abstract

Full proofs of semiboundedness criteria for 1D Schrodinger operators applied to H* = -d2/dy2 + V* + 1/4 with V*(y) = log|zeta(1+iy)| - R(y): a stability criterion (uniform unit-window L1 bound on the negative part, explicit constant), two instability criteria (deep-wide dips; concentrated mass against local positive background, all trial constants verified by quadrature), and the corrected reduction to a unit-window excursion question about zeta on the 1-line. Includes the recorded withdrawal of a mass-only dichotomy (killed by a bounded-depth long-well counterexample during careful write-up). Companion to the Q-HSTAR theory memo and the IMM Proofs Volume. Nothing in this note bears on the truth of the Riemann Hypothesis.