Part V Scope and formal evidence
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15 Scope boundaries and further questions
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The large-zero theorem settles eventual simplicity but does not classify the finite exceptional set. A remaining question is whether every zero of \({\mathsf D}_N\) is simple and every reduced zero of \(\Delta_N\) is positive. Both statements hold for \(N=2\) and \(N=4\). The argument here proves them only outside a compact set for general \(N\).
The leading correction in Theorem 11.8 may admit iteration to a full exponential expansion for \(\tau_m\). Such an expansion is an open problem, not a result of this paper. It could refine the entire correction in the zero-zeta comparison and lead to explicit special values.
The symmetry-orbit zero zeta has a meromorphic continuation with one pole. No functional equation is obtained. A branch dependent sum over the \(N\) roots of each reduced zero must specify logarithms on the rays.
The coefficient families in Section 7, Section 9, and Section 10 have exact formal parameter-addition laws. Their arithmetic behavior, including odd-prime congruences, dyadic weighting, and \(p\)-adic or Amice–Mahler interpolation, belongs to Paper IX. General bounded-operator semigroup, resolvent, and Feshbach theory belongs to Paper VI. Paper III owns general nonlinear operation trees, while abstract completion and occupancy questions belong to Paper X. None of those theories is inferred from the scalar complex-analytic results proved here.