# Recovering singularities from convolution profiles

The formal chapter proves exact realization of one convolution profile carrier, both directions of the inverse singular-support criterion, the sharper closed-union localization, the singleton-indicator characterization of universal hull addition, the invertible difference-set bound, and the polynomial-profile and finite-jet results.

The learner chapter contains four original examples and ten complete graded solutions. Its figures show exact carrier-translation geometry and the singular loci of a concrete invertible input whose selected output hull is smaller than the general hull sum. The geometry description retains every coordinate, coefficient, support radius and proof locator.

Run the figure program with Python and Matplotlib to reproduce the PNG and SVG outputs. Run the numerical program with Python, NumPy, SciPy and mpmath for finite supplementary checks. It compares geometric formulas with independent linear programs, exact Fourier and Taylor identities with high-precision evaluation, and logarithmic averages and compact-bump transforms with independent physical quadratures. General mathematical conclusions rely on the complete analytic proofs. The selected nonsmooth convolution factor is proved to exist; its physical shape is not numerically asserted.

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.