# Isolated atoms and separated singularities

The complete proof establishes compact convolution singular-support calculus, a common carrier core from isolated ordinary-support points, the finite-support singleton criterion, a positive atomic measure with any prescribed compact convex carrier, and the finite joint maximum law for separate singularities.

The learner chapter contains four worked examples and ten complete graded solutions. The figures show actual contracted boundary nets, an embedded segment's isolated atoms and boundary limits, overlapping ordinary supports with separate singularities, and point singularities that cancel when separation fails. Marker sizes do not represent atomic masses. The vertical rows in the second figure distinguish distributions rather than physical coordinates.

Run the figure program with Python and Matplotlib to reproduce PNG and SVG outputs. Run the numerical program with Python, NumPy, SciPy and mpmath for finite supplementary checks. It uses independent linear programs, physical derivative pairings, atomic series tail bounds, and separate physical and differentiated compact-bump Fourier integrals. General conclusions rely on the complete analytic proofs. The selected nonsmooth convolution factor is not numerically constructed.

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 those books are included.