Reading orientation
Begin with cyclic projection and the signed differential cycle. The reciprocal chapters then build parameter polynomials, Appell systems, and theta polynomials from a unit series. The second half follows the same denominator through large zero geometry, canonical products, and zero zeta functions.
The visual atlas at the end of this volume supplies twenty-six worked plates. The separate Lean guide follows the released declarations and their hypotheses. All sixty-eight proof bodies and ninety-five numbered records of the selected manuscript are retained.
Abstract
Fix a positive integer \(N\). Root of unity projection of the exponential produces residue-class sections that, for \(\lambda\ne0\), form a basis for \(y^{(N)}=\lambda^N y\). The signed sections for \(y^{(N)}=-y\) yield the denominator \({\mathsf D}_N\) and its reduced form \(\Delta_N\). We derive reciprocal coefficient polynomials, Bell and Newton recurrences, an additive parameter law, and residue section Appell and theta families. For \(N\geq2\), \(\Delta_N\) has order \(1/N\). We specialize classical two term zero theory for exponential sums to the regular exponent polygon. In this equal-weight family, conjugation symmetry combined with unique near-model localization places all sufficiently large zeros exactly on the symmetry rays, and direct expansion gives their leading displacement and derivative sign. The classical perimeter law yields zero-counting asymptotics with explicit leading constants and bounded error. These results give genus zero products, the standard diagonal Fredholm representation, zero-power-sum formulas for reciprocal and logarithmic coefficients, nearest-zero coefficient asymptotics, and meromorphic continuation of the symmetry-orbit zero zeta by comparison with the Hurwitz zeta function. Complex powers are interpreted either formally or through a fixed logarithm on a zero free domain. The paper treats only this cyclic analytic family. Arithmetic interpolation, general nonlinear trees, general resolvent theory, and abstract completion are reserved for Papers IX, III, VI, and X.
Formal scope and evidence
The preserved Paper IV v1.0.0 release records full scoped Lean coverage of the sixty-eight theorem like manuscript results under their original hypotheses. This volume changes presentation and adds worked explanations. It does not create a new formal release.
Imported standard fact 11.1 is used in the ordinary manuscript growth argument. The Lean growth-order proofs establish that step separately. Open questions, branch choices, multiplicities, and the hypotheses of coefficient asymptotics remain explicit. Current finite computations support the worked models. The compiler and kernel evidence remains the preserved September release evidence.