# Slow decrease and entire Fourier division

The formal chapter proves the five equivalent slow-decrease and entire-division conditions, the upper-half-plane comparison, and finite-regularity perturbation stability. It includes real-ball propagation, full Phragmén–Lindelöf barriers, Lindelöf's exponential-type quotient argument, the complex polynomial-growth upgrade, local quotient control at zeros, Banach completeness and a uniform Baire exponent.

The learner chapter gives four examples and ten fully solved exercises. The two original illustrations show exact harmonic comparison functions and a finite atomic division identity with logarithmic windows around a transform zero. Their geometry description retains coordinates, weights, radii, constants and proof locators.

Run the figure program with Python, NumPy and Matplotlib to reproduce the PNG and SVG figures. Run the numerical program with Python, NumPy, SciPy and mpmath for finite supplementary checks. Its numerical report gives precision and errors. The general mathematical conclusions are established by the complete analytic proofs, not by numerical samples.

Original exposition, proofs, examples, solutions, figures, geometry and code: GPT-6.1 Sol (OpenAI), Ultra; CC0 1.0. The DejaVu font notice retains its own terms. References: Terence Tao, *246B, Notes 2* (2021) and *245B, Notes 9* (2009); Lars Hörmander, *The Analysis of Linear Partial Differential Operators I* and *II* (1983). No text or images from the referenced books are included.