17 Conclusion
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Cyclic projection turns the exponential into a finite differential basis, and the signed cycle produces the distinguished denominator \({\mathsf D}_N\). Its reciprocal powers yield Bell and Newton recurrences, an additive parameter law, Appell systems, and theta polynomials. Reduction by \(w=z^N\) then connects the reciprocal coefficients to a genus zero product and reduced-zero power sums.
Classical two term localization, specialized to the regular exponent polygon and combined with reflection symmetry, places all sufficiently large zeros on symmetry rays. The displacement, derivative and counting estimates yield the family-specific Hurwitz comparison for the orbit zero zeta.