# Verification report for derivative-zero half-plane counterexamples

The accepted result is a complete counterexample to the precise alternatives
in AMR-036-0017, not a classification of all admissible entire functions.
The example is f(z)=exp(iz)+z+2i. Its zero-free upper-half-plane property
follows from |z+2i|>=2>|exp(iz)|; its first derivative zeros are real;
all higher derivatives are zero-free exponentials. The logarithmic-derivative
condition for polynomial limits excludes f from that closure. The nonzero
first derivative has zeros, which also excludes both exceptional families.

For every d>=1, the family
exp(iz)+2(z+2i)^d+sum_(k=0)^(d-1) a_k(z+2i)^k, |a_k|<1,
has the same property with strict zero avoidance in the closed upper
half-plane. The finite geometric-sum estimate proves this uniformly in every
parameter and all derivative orders. Vandermonde evaluation then excludes
countable holomorphic covers of uniformly bounded parameter dimension.
Unbounded-dimensional unions and nonregular parametrizations are outside
that last statement.

The self-audit checked signs, zero derivatives, both boundary conventions,
the n=0 and n=d endpoints, all n>d, degenerate exponential parameters,
Hurwitz and differentiation closure, and the measure-zero dimension
argument. No mathematical gap remains known in this stated scope.

The exact regression script tests the domination bound for 1<=d<=64,
0<=n<=d and positive rational logarithmic-derivative values. It uses integer
and rational arithmetic without floating point. The general theorem rests
on the printed proof, not these finite tests. Normal and optimized Python
outputs are compared in the release package.

The source comparison uses the September 10, 2026 Eremenko note, its
companion example attributed to Lev Buhovski, and the Eremenko-Sodin survey.
The precise current family allows an arbitrary constant where the frozen
record allowed only modulus one; the main example escapes both versions.
Buhovski's earlier exp(iz)-1 example is credited and not counted as new.
No identical polynomial-perturbation construction was found in a bounded
search ending October 11, 2026; no absolute priority is certified.

AI assistance was used in this self-audited work. The report does not assert
independent expert review, peer review or proof-assistant verification.