# Reproduce the two regularity exercise bridges

This original packet solves the two distinct Chapter15 exercises on printed page291. It proves the full positive-jet strength/distance comparison, the exact complex-zero/real-strength weight equivalence and the directional high-derivative estimate from the tube representation. Every reflected weight, all coefficient derivatives, exponential moments and differentiation justifications are retained.

The learner gives four worked examples and ten complete solutions. Its explicit complex-frequency heat solution attains spatial order1 and time order2, with exact derivative phases and factorial constants.

Run Python with NumPy, mpmath and Matplotlib:

    python -B check_examples175.py
    python -B make_figures175.py

The checks independently differentiate actual factored polynomials and their full jets, compute exact complex nearest points of affine symbols, extract complex coefficients on a unit torus, verify heat zeros and real physical operators, integrate the actual complex heat solution and all stated moments, and check the noninteger exponential supremum. These supplement the full proofs.

The two original PNG/SVG pairs retain exact distances, repeated-root collisions, complete four-real-dimensional characteristic coordinates, parameter measure and sharp derivative normalizations. Scientific plots display all relevant objects; a Blender scene would add no useful geometry. SVG dates are omitted and their hash salt is fixed. DejaVu fonts keep their own license. No TeX installation is required.

Human source: Lars Hörmander, The Analysis of Linear Partial Differential Operators II, two unnumbered exercises on printed p.291; cross-references Theorems11.1.7,11.4.1,11.4.8 on printed pp.65,85,90.

The two separate page300 arbitrary-distribution representation exercises, Chapter15 notes, all Chapter16 targets and every other assigned residual remain active.