The prompt the figures were built from

I’m working in a project called “The Determinant Ladder” — a visual, non-academic exploration that starts from the determinant as a primitive (signed volume) and climbs three “dimensional directions” up to the 2026 counterexample to the Jacobian conjecture (the Alpöge / Claude Fable 5 result, dissected on Terence Tao’s blog, July 2026). Please read these two files first to absorb the project:

1. One_Formula_Many_Worlds.docx — the paper. It fixes the narrative voice (curious, honest, first-person-plural expedition; playful but calibrated: evidence is never called proof, numerics are labelled as instruments) and the figure numbering / captions.
2. determinant_ladder.py — the single Python module containing every figure as a function, organised under a numeric menu() with three “directions”: Direction 1 = SIZE of the matrix (formulas, Sarrus paths, rainbow port-fanned path diagrams, growth/crossing curves); Direction 2 = RICHNESS of the entries (determinant as a field, folds on det = 0); Direction 3 = DIMENSION of the map (the Fable counterexample: symbolic verification, collision wells, numerical collision hunting, escape/properness plots).

MY GOAL for this session: help me ADD MORE OF THE STORY as new Python-created visuals, keeping everything consistent with what exists.

House conventions to follow strictly:

• Every new visual is a function fig_<name>() in determinant_ladder.py, saving via the module’s _finish('<name>.png') helper (saves into ./figures/ and shows on screen for Spyder) and registered in the _CATALOG dict under the correct direction with the next free number. Never a standalone script.
• Only numpy / sympy / scipy / matplotlib. Pure functions, no globals, seeded randomness (np.random.default_rng with an explicit seed) so figures reproduce.
• Visual style: matplotlib defaults with the project palette (#1f77b4 blue, #d62728 red, #2ca02c green, #9467bd purple; families use cm.Reds/Oranges/ Greens/Blues/Purples), generous titles that state the mathematical point in one sentence, and captions that a curious non-mathematician can follow.
• Honesty rules from the paper apply to code comments and titles too: numerical searches are “evidence”, “hunts”, “probes” — never proofs; the 2D Jacobian conjecture is OPEN; the counterexample lives in n >= 3; the 4.273 drawability inversion must never be conflated with the 2D/3D conjecture boundary; credit is “Levent Alpöge with Claude Fable 5”, provenance pending formal review.
• After adding a figure: run it once headless (matplotlib Agg) to confirm it executes, then show me the PNG before we consider it done.

Backlog of visuals I already know I want (pick up wherever I say, or propose better versions):

1. Negative-term overlays: the odd shuffles of n = 3…5 drawn dashed over the positive rainbow, to see the cancelling pairs.
2. The n = 6 storm (360 positive paths) as pure texture — abstract-art mode, no letters, just the weave.
3. A “lane-consistent” variant of the port-fanning where each line keeps one berth for its whole journey.
4. A properness-probe heatmap for a DISGUISED Fable map, side by side with the original, hunting a conjugation-invariant picture of the non-proper locus.
5. An animated (GIF) walk of the Whitney-cusp fold, and eventually an animated rotation of the three collision wells.
6. A figure for the Bass–Connell–Wright degree-3 reduction idea, at cartoon level.
7. “Falling into itself” figure: Newton’s identities / Faddeev–LeVerrier — det’s coefficients built recursively from traces of powers with alternating signs; the additive invariant feeding the multiplicative one, log det(I+X) = tr(X − X²/2 + X³/3 − …).
8. “+C” figure: the Jacobian MATRIX field determines the map only up to an additive constant (DF = D(F+c)) — differentiation’s lost global datum — while det DF forgets far more; a visual of the information ladder map → DF → det DF with what each step discards.

When I ask for a new visual with a vague description, do what the project has done throughout: propose the crisp mathematical object my description is pointing at, name any standard concept I’ve reinvented, build it, and tell me honestly if my framing doesn’t correspond to real structure — then find the nearest thing that does.

If you update determinant_ladder.py, keep menu() backwards compatible (existing numbers keep their meaning; new entries get new numbers). If a change would also affect the paper’s claims or captions, list the affected paragraphs of One_Formula_Many_Worlds.docx at the end of your reply rather than editing the docx unasked.

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