Collinear Fractals GPU

Research explorer and companion software for collinear fractals, finite capture, and restricted polynomial roots.

Version: 0.2.0-alpha

Release line: v0.2.0-alpha GitHub-only prerelease. Author: Bernat Espigule

This repository contains a browser-based research explorer and multi-language reference implementations for canonical-coordinate inverse search associated with the collinear connectedness loci (\mathcal M_n).

The current public release is an alpha intended for reproducible exploration, finite inverse-word export, figure preparation, and independent inspection. It is not a formal proof checker; theorem-level proofs remain in the papers and thesis.

Quick visual summary

The browser explorer renders the parameter plane, dynamical plane, canonical traps, canonical enclosures, finite inverse-search data, and selected certificate JSON. The default original-attractor view now uses a prefix-cylinder visual renderer for (E(c,n)), with a seeded histogram preview available as an explicit alternative. Finite inverse-search status remains a separate certificate/status renderer, not the default visual overlay.

The explorer also includes curated example presets, share URLs, embed code, save-image export, undo/redo, comparison modes, palette controls, panel focus, and About/Cite and Support dialogs.

Curated example metadata lives in examples/, gallery metadata lives in gallery/, and reproducible figure job metadata lives in paper_figures/. Large generated images are intentionally kept out of git until they are curated, compressed, and tied to reproducible metadata.

Browser quick start

For the best experience, serve the repository locally:

python3 -m http.server 8000

Then open http://localhost:8000/ in a modern browser. Opening index.html directly also works for core rendering, but browser security rules may block loading example metadata from local JSON files.

The left panel renders the parameter plane. The right panel renders the dynamical plane, the canonical trap, the canonical enclosure, and the inverse-search tree. The selected certificate can be copied or downloaded as JSON from the sidebar.

The Canvas renderer refines progressively to single-pixel sampling. Visual controls such as prefix depth, histogram seed, histogram sample count, and layer opacity affect the drawing only. The selected-parameter verdict and exported certificate JSON always use the full selected search depth and node cap.

What is included

Path	Contents
index.html, index.css, explorer.js	Single-page Canvas/CPU browser explorer.
src/	Build-free browser ES modules for visual renderers and finite-search kernels.
workers/	Worker entry points for future certificate and histogram jobs.
javascript/	Node.js reference package and tests.
python/	Pure-Python reference package and tests.
swift/	Swift Package Manager implementation and tests.
mathematica/	Wolfram Language package.
matlab/	MATLAB static-class implementation.
maple/	Maple module implementation.
examples/	Curated example configurations, metadata, certificates, and notes.
gallery/	Gallery index and placeholder documentation for curated figures.
paper_figures/	Metadata and scripts for reproducible figure generation.
schemas/	JSON Schemas for certificates, examples, and figure metadata.
docs/	Implementation, validation, release, and QA notes.
qa/	Browser-engine, renderer, and kernel smoke tests.
tools/	Dependency-free static bundle validator and metadata-only benchmark scripts.

The browser explorer is currently a Canvas/CPU reference implementation. The project keeps the historical GPU name because it is the companion repository for the broader GPU-assisted exploration programme; WebGL/WebGPU acceleration can be added later without changing the mathematical API.

Feature status for v0.2.0-alpha:

Feature	v0.2 status
Prefix renderer for (E(c,n))	Implemented
Seeded histogram renderer	Implemented / smoke-tested
Certificate/status renderer	Implemented
Worker scaffold	Present
Typed-array optimized kernel	Experimental scaffold/wrapper only; reference kernel remains default
Active certificate inspector UI	Planned / not implemented
Level 0/1/2 atlas	Metadata scaffold
Curated thumbnails	Planned

Mathematical scope

For the collinear alphabet

A_m = {-m+1, -m+3, ..., m-1},

the explorer studies the marked-point condition

2c in E(c, 2n - 1),

which encodes connectedness of the original (n)-ary collinear attractor (E(c,n)). The inverse search works in canonical coordinates, prunes by a canonical enclosure, and detects canonical trap entry.

The repository uses the following theorem spine:

• marked point 2c;
• difference attractor E(c, 2n - 1);
• canonical coordinates, trap, and enclosure;
• finite inverse search and finite inverse-word export;
• finite-capture filtration Theta_k(n);
• bounded +2 boundary repair;
• lens-containment threshold n >= 20;
• off-lens witnesses for 2 <= n <= 19.

The software supports exploration, figure generation, finite inverse-word export, and independent inspection. The theorem-level proofs remain in the papers and thesis, using explicit inequalities and finite certificates.

Verdicts: Interior, Interior-offLens, Exterior, Undetermined

The repository deliberately keeps these labels separate:

• Interior: trap hit inside the parameter lens (X_n\setminus\mathbb R), matching the canonical trap framework.
• Interior-offLens: off-lens trap hit using the enabled off-lens rule. This is intentionally not merged with the in-lens label.
• Exterior: enclosure escape or complete enclosure-admissible tree exhaustion.
• Undetermined: selected depth or node cap reached.

The default search depth is:

k_max = 37

That default is shared by the web explorer and the companion packages.

Examples and gallery

The examples/ directory contains curated starting points for reproducible exploration:

Example	Purpose	Status
e_c4_overlap	Thesis Figure 3.1 example (c=(3+i\sqrt{11})/2).	Illustrative thesis example
e_c5_plane_filling	Thesis Figure 3.2 example (c=1+2i).	Illustrative thesis example
theta0_base_capture	Base-capture geometry for Theta_0(n).	Illustrative
trap_enclosure_n3	Interior and Exterior trap/enclosure certificates.	Finite-search certified
threshold_n20	Threshold/lens example linked to the finite-capture theorem.	Exploratory
hole_zoom_n13	Finite-capture zoom near an (n=13) hole.	Exploratory
off_lens_witnesses_n2_to_n19	Off-lens witness scaffold preserving Interior-offLens.	Exploratory
finite_capture_layers_n3	Early finite-capture layer visualization for (n=3).	Illustrative
level2_boundary_atlas	Level-2 boundary-atlas metadata and share-state scaffold.	Exploratory

Each example records parameters, expected status, reproducibility notes, metadata, and whether the case is illustrative, finite-search certified, theorem-certified, or exploratory. Certificate JSON is included only for curated cases where a compact finite-search export is already known.

Share URLs and reproducible states

The browser explorer can create share URLs and iframe embed code from the current state. The hash records the selected parameter, search limits, viewports, palette, comparison mode, visible layers, original-attractor renderer mode, prefix depth, histogram seed, histogram sample count, first-level piece coloring, and layer opacity.

Curated presets are loaded from examples/examples.json when the explorer is served over HTTP. If metadata loading is blocked, the explorer falls back to a built-in copy of the same public preset list.

Package tests

Run these from the repository root after cloning or downloading the bundle.

Browser engine smoke test:

node qa/explorer-prefix-smoke-test.js
node qa/render_smoke_tests.js
node qa/kernel_equivalence_tests.js

JavaScript:

cd javascript
npm test

Python:

cd python
python3 -m unittest -v test_collinear.py

Swift:

cd swift
swift test

Static bundle validation:

python3 tools/validate_bundle.py

The Wolfram Language, MATLAB, and Maple packages are included as reference implementations with matching formulas and defaults. See docs/QA_REPORT.md for the exact validation scope and runtime limitations.

QA status and limitations

The release tree includes docs/QA_REPORT.md, docs/VALIDATION.md, and RELEASE_CHECKLIST.md. The automated checks cover JavaScript, Python, Swift, browser-engine smoke tests, renderer smoke tests, kernel equivalence tests, formatting hygiene, staged Pages deployment, and static bundle validation.

The Wolfram Language, MATLAB, and Maple ports are reference implementations that should be checked in their native runtimes before stronger release claims are made.

The browser renderer is a research explorer. Original-attractor visual metadata is recorded separately from finite-search certificate fields. Certificate JSON exports are reproducibility artifacts; theorem-level proof relies on the mathematical text, finite certificates, and explicit inequalities rather than visual inspection.

See docs/RESPONSIBLE_USE.md for certification boundaries and responsible-use guidance.

Related mathematical work

This software accompanies and supports the following mathematical work.

1. Bernat Espigule, David Juher, and Joan Saldaña, "Collinear Fractals and Bandt's Conjecture", Fractal and Fractional 8(12), 725, 2024. DOI: 10.3390/fractalfract8120725.

2. Bernat Espigule and David Juher, "Finite Capture and the Closure of Roots of Restricted Polynomials", arXiv:2603.07397, 2026. DOI: 10.48550/arXiv.2603.07397.

3. Bernat Espigule, "Finite capture and the closure of roots of restricted polynomials", IHP audiovisual resource, 2026. DOI: 10.57987/IHP.2026.T1.WS3.016.

The 2024 paper gives the rectangle-covering/lens-local route and a global large-(n) result. The 2026 finite-capture work uses canonical traps, canonical enclosures, finite inverse search, and bounded-lag repair to obtain the sharp n >= 20 lens-containment threshold.

Citing

Use the metadata in CITATION.cff. A compact text citation is:

Bernat Espigule, Collinear Fractals GPU: companion software for collinear fractals, finite capture, and restricted polynomial roots, version 0.2.0-alpha, 2026.

No Zenodo DOI has been assigned yet. Add an archival DOI only after a future release is archived through Zenodo or an equivalent service.

Funding and acknowledgements

Parts of the mathematical framework, validation materials, examples, and companion software in this repository were developed during Bernat Espigule's doctoral research at the Universitat de Girona, supervised by Dr. Joan Saldaña Meca and Dr. David Juher Barrot.

This work was supported by the Spanish Ministerio de Ciencia, Innovación y Universidades through project PID2023-146424NB-I00, by the Generalitat de Catalunya through grant 2021 SGR 00113, and by the Universitat de Girona and Banco Santander Grant Programme for Researchers in Training, IFUdG 2022-2024.

The repository is maintained by Bernat Espigule. The funders, supervisors, and affiliated institutions do not necessarily endorse the software, the public explorer, or any results obtained with it.

Support

This repository is open-access research software accompanying my work on collinear fractals, finite capture, and restricted polynomial roots.

I am maintaining it while preparing my PhD defence and developing the next generation of reproducible figures, examples, and validation tools. Small sponsorships help support continued maintenance, documentation, public visualization, and research-software development.

Support is optional and does not affect access to the code, examples, documentation, issues, or citation materials.

License

Source code is distributed under the Apache License 2.0; see LICENSE.

Documentation, README files, examples, non-code figures generated by this repository, and non-code repository materials are distributed under Creative Commons Attribution 4.0 International unless otherwise stated; see LICENSE-docs.md and LICENSES/CC-BY-4.0.txt.