# Frequency-selective singularities and smooth convolutions

The formal chapter proves a compact continuous but not C1 construction with one prescribed singular point, exterior logarithmic Fourier collapse, and a proper zero profile on selected centers. Its peaked smooth kernels, complete function space and Baire argument are fully proved. It then proves all directions of the smooth-convolution characterization, including a continuous nonsmooth factor in any small support ball.

The learner chapter contains four worked examples and ten fully solved exercises. The figures show a concrete smooth probability seed, its transform powers, and exact comparison functions for the evaluation weight. They do not depict a computed Baire-selected function.

Run the figure program with Python, NumPy, SciPy and Matplotlib to reproduce the original PNG and SVG figures. Run the numerical program with Python, NumPy, SciPy and mpmath to perform finite supplementary checks. Its numerical report gives the precision and measured errors. The geometry description records formulas, support, normalization, powers and proof locators.

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, *245B, Notes 9* (2009) and *246B, Notes 2* (2021); Lars Hörmander, *The Analysis of Linear Partial Differential Operators I* and *II* (1983). No text or images from the referenced books are included.