# Whole flat isotropy and its proper abstract auxiliary

Original mathematical expression, two diagrams and generator: CC0 1.0 to the extent of rights held. No source text or external artwork is incorporated. The containing lesson retains human mathematical credits, including the free-flow prerequisite, Nielsen–Schreier subgroup theorem and Thurston local-stability comparison.

Run `python -B draw.py` with Python and Matplotlib. It creates both PNG/SVG pairs under `figures` in this reproduction directory. Fixed SVG salt and metadata permit byte-exact replay in the recorded environment. SVG contains text without font bytes or glyph outlines. Installed fonts and software retain their own terms, including `../FONT-NOTICE.txt`.

The first diagram displays PFG.1–PFG.12: injective rank-two free stages, zero maps on abelianization, their nontrivial perfect limit, actual intervals with c_N=2^-N and r_N=2^(-N-3), and exact derivative upper bounds. Its six displayed intervals use their true coordinates. The plotted exponents (N+3)(N+1-k) are proved bound exponents for k=0,1,2, not sampled displacements or numerical estimates of the chosen flow norms.

The second diagram displays PFG.13–PFG.16: the positive-definite free-stage functions, their complete Hilbert direct-sum cocycle, its finite-level-set properness, and the actual bisection limit forcing every continuous normalized scalar groupoid length to vanish on the entire infinite isotropy. Properness of the abstract group action supplies no extra label of the effective groupoid. The general reduced normal fundamental class is neither refuted nor constructed here.