# Reproduce the characteristic-surface representation lesson

The original formal source proves the complete surface representation theorem with every repeated-factor jet, a polynomial loss depending only on dimension and degree, the exact algebraic area bound, an explicit anisotropic cover, safe-circle overlap division, global weighted repair, compact-carrier verification and the Hilbert density construction.

The learner source has four worked examples and ten complete exercise solutions. The independent calculations differentiate and integrate the actual normal kernel, full real graph metric, characteristic kernels, repeated differential operators and forward-transform jets.

Run Python with NumPy, mpmath and Matplotlib:

    python -B check_examples169.py
    python -B make_figures169.py

These calculations supplement the full proof. No TeX installation or compilation is needed. The figures are original scalar scientific plots and explicitly labeled real sections of complex ellipsoids. Their four-real-dimensional ambient metric is retained in the complete argument and exact geometry JSON; a three-dimensional embedding would obscure that metric. The SVG date is omitted and its hash salt is fixed. DejaVu fonts retain their own license.

Classical human sources: Lars Hörmander, The Analysis of Linear Partial Differential Operators II, §15.3, Theorem 15.3.3, printed pp. 291–296; The Analysis of Linear Partial Differential Operators I, §4.1, Theorem 4.1.12, printed p. 98. The direct original upper-area argument gives the needed bound without importing the full general zero-order/degree limit theorem as an unproved prerequisite.

Completing this packet does not complete the entire assigned course, Chapter 15 exercises/notes, Chapter 16 targets or the residual work in Chapters 10–13.