docs(showpieces): Homotopy Basins

[ofloveandhate]

What

A third Showpieces gallery entry: Homotopy Basins

Rasterize a window of the coefficient plane of f(x;c) = x^d − d·x − c; at each pixel run the total-degree gamma-trick homotopy H = γt(x^d − 1) + (1−t)(x^d − dx − c) and track all d start roots with the DoublePrecisionTracker. Every channel of the image is tracked data:

• hue — phase of the start–end correlation s(c) = Σ ζᵏ·xₖ(c) (each path contributes start root × landing point; jumps exactly where the swept discriminant permutes destinations)
• brightness — the tracker's own step count: the blazing arcs are the swept discriminant, where the adaptive stepper piles up tiny steps
• stripes — level bands of log|s| (cosmetic domain coloring; whirlpools are the zeros of s)
• white speckle — path failures near singular systems

Four frames from two computations (~17M tracked paths, ~15 min on 12 cores): per degree (d = 5 teaching, d = 9 hero) a full-bleed render and an annotated map with labeled Re(c)/Im(c) axes and every arc center c = −(d−1)ζ marked on its ring. Each window contains the family's whole branch-point ring. The docs page derives the arc parameterization c(μ) = −8ζ(1+μ)^{−1/8} − μ, μ = γt/(1−t) — the gamma trick, photographed — and cross-links the Monodromy Loom (same mathematics from solution space).

Notes

• Everything fed to the function tree is exact (fractions.Fraction; pixel coordinates snapped to rationals); no randomness anywhere, so the tracked data is fully deterministic.
• bertini.recording(False): millions of throwaway raw-tracker solves, deliberately below the records seam.
• Raster showpiece, PNG only (the image is a raster of solves); registered in tools/refresh_doc_artifacts.py with a SLOW note — this is the heaviest doc artifact.
• Docs build verified (sphinx -b dummy, no new warnings). No ADR: docs-only gallery addition; the load-bearing choices are documented in the generator's docstring and the page itself.

🤖 Generated with Claude Code

[ofloveandhate]

I really enjoyed using Claude to help write this. But the wording is spongy and overdramatic. I will manually revise before accepting.

[ofloveandhate]

I will only accept this as part of a complete human-at-the-keyboard revision of the docs, which is badly needed. Using Claude got me off the ground. It's time to do the writing myself.