ringmin

Ringmin is a computational-geometry research project on the minimum central circle problem: circles of radii 1,2,...,n are externally tangent to a central circle, and the goal is to minimize the central radius R.

The project has produced two public papers covering complementary sides of the problem: certified finite optimization and global asymptotic theory.

Navigate: Publications · How to cite · Mathematical scope · Quick verification · Certified results · Full certification and regeneration · Repository layout · Non-technical explainer (IT)

Publications

Paper I — finite problem and certified optima

arXiv: 2607.28654v2

Maurizio Falconi, Arranging circles of radii 1,2,...,n around a central circle: a Supnick TSP and certified finite optima, arXiv:2607.28654 [cs.CG], 2026.

The finite paper is accompanied here by the solver code, independent verifier, certificate artifacts, and paper source/PDF. The repository certifies the global optimum for n=3..14; larger-n candidate optima remain heuristic.

Paper II — global asymptotic theory

arXiv: 2609.13630

Maurizio Falconi, Minimum central circles: an effective characterization of the global asymptotic constant, arXiv:2609.13630, 2026.

The standalone sequel studies the asymptotic regime. It proves R(n) = Cn² + o(n²) and characterizes C_* by finite balanced-word linear programs with rigorous bracket width $(1/k+1/r)/\pi$. This gives an effective arbitrary-precision characterization in principle; an elementary formula and an efficient evaluation algorithm remain open.

The current explicit bounds give C_term + η_width ≤ C_* ≤ U₄, with the lower endpoint greater than $0.14056946887766098063257 > 1/8$. The four-block upper bound has genuine-permutation recovery and full feasibility. These post-v1 theorems have passed internal adversarial review; independent external acceptance is pending.

The two-paper publication architecture is preserved in the repository: the standalone asymptotic sequel and the conservative corrective v2 of the finite paper. See the publication history and finalization dossier. The former replacement candidate is superseded provenance, not the recommended next upload.

The historical v1 artifact is unchanged. The fixed-order feasibility oracle is a high-precision Simple Temporal Network check over all pairwise angular constraints; chain-only values are used only as lower bounds.

🇮🇹 Per una spiegazione semplice e non tecnica del problema e dei risultati principali, consulta il documento SPIEGAMI.md.

How to cite

Ringmin has two public papers. Cite the one corresponding to the result you use; cite both when referring to the project across its finite and asymptotic scope.

Paper I — finite problem and certified optima

Maurizio Falconi, Arranging circles of radii 1,2,...,n around a central circle: a Supnick TSP and certified finite optima, arXiv:2607.28654 [cs.CG], 2026.

Paper II — global asymptotic theory

Maurizio Falconi, Minimum central circles: an effective characterization of the global asymptotic constant, arXiv:2609.13630, 2026.

CITATION.cff remains the machine-readable citation metadata for the software and finite-certificate companion.

Mathematical scope

This project studies the geometric optimization problem of arranging circles of radii 1,2,...,n around a central circle while minimizing the central radius. Paper I treats the finite problem: it proves that the chain-ordering lower-bound problem is a fixed Supnick/anti-Monge TSP and uses explicit certificate artifacts plus an independent verifier to certify global optima for 3 <= n <= 14. Larger-n global optima remain uncertified.

Paper II treats the global asymptotic regime, proving the existence of the quadratic-scale limit and giving an effective finite-word LP characterization of its constant C_*. Current exact results, publication boundaries, and remaining open claims are recorded in PROJECT_KNOWLEDGE.md.

Environment

The submission-gate environment is pinned in requirements.txt and was:

• Python 3.14.3
• numpy==2.4.3
• scipy==1.17.1
• mpmath==1.3.0
• matplotlib==3.10.9
• pytest==9.0.2

Install:

python -m pip install -r requirements.txt
python -m pip install -e ".[test]"

pyproject.toml intentionally leaves runtime dependencies unpinned for normal editable development; use requirements.txt for the exact submission reproduction environment.

Quick Verification

These checks are intended for a quick local or CI smoke test. They do not regenerate the long-run certificates.

python -m pip install -r requirements.txt
python -m pip install -e ".[test]"
python -m pytest
python verify.py --start 3 --stop 8 --skip-frontier

The smoke verifier skips the frontier/progress-log audit. To run the complete certified scope from a fresh clone, restore the exact historical logs from their tracked, hash-checked archives first:

python scripts/frontier_logs.py restore
python verify.py --start 3 --stop 14

This restores original evidence; it does not rerun exhaustive searches. The reproduction dossier records a successful clean-source Windows run and its failure controls. POSIX execution and hosted CI for the current SHA are not inferred from it.

With pdflatex available, rebuild the standalone sequel artifact from a clean source-only tree and record its input/PDF hashes and metadata with:

python paper_assets/build_publications.py sequel

The sequel and corrective-v2 build notes identify their source bundles and distinct publication states. Each bundle compiles directly with pdflatex. Historical v1 source/PDF and citation metadata remain the original publication record.

Certified Results

The 30-digit values are generated in results/highprec.csv and copied into paper_assets/appendix_tables.tex.

n	R*	optimal cycle	floaters
3	0.260869565217391304347826086957	[3, 1, 2]	{}
4	0.844453589560855604347528524674	[4, 1, 3, 2]	{}
5	1.69549408120271081351328017371	[5, 1, 4, 3, 2]	{}
6	2.79491951889692485617024406797	[6, 1, 5, 3, 4, 2]	{}
7	4.15318955374381246513863858202	[7, 1, 6, 3, 4, 5, 2]	{}
8	5.7677942845896143026361805725	[8, 1, 6, 4, 5, 3, 7, 2]	{1}
9	7.72672655261128921886246177604	[9, 2, 8, 1, 5, 6, 4, 7, 3]	{1}
10	9.97990738586347760966552641468	[10, 2, 9, 4, 7, 1, 6, 5, 8, 3]	{1}
11	12.4887204871876588517468786264	[11, 2, 10, 4, 8, 6, 7, 1, 5, 9, 3]	{1}
12	15.2588704304484933617250043503	[12, 2, 11, 4, 9, 6, 7, 8, 5, 1, 10, 3]	{1}
13	18.3175630472173206282821941532	[13, 3, 1, 12, 2, 10, 6, 8, 7, 9, 5, 11, 4]	{1}
14	21.6653951822145150956462891793	[14, 3, 13, 2, 9, 8, 7, 10, 6, 11, 5, 1, 12, 4]	{1,2}

Certified means global optimality up to absolute tolerance 1e-10 in R. The bisection tolerances are 1e-12/1e-13, and the displayed binding structures are rechecked to 50 decimal digits.

Submission-Gate Verification

Run the independent verifier:

python verify.py --start 3 --stop 14

verify.py imports only the Python standard library and mpmath. It does not import src/ringmin. It checks three layers for every certified n:

1. Incumbent feasibility: reloads results/nNN/optimum.json, rebuilds the theta matrix at 50 digits, checks the saved witness angles, all pairwise angular constraints, Cartesian non-overlap, essential pairs, and floating circles.
2. Local optimality for the incumbent order: verifies feasibility at R* + 1e-12 and infeasibility at R* - 1e-12.
3. Global pruning certificate: reloads results/frontiers/nNN_frontier.json, verifies enumeration counts, canonicalization metadata, progress-log proof, frontier hash, top-excluded guard, and recomputes every frontier lower bound at 50 digits.

Current verifier summary:

n=03 incumbent=PASS local=PASS frontier=PASS eta=1.0e-12 frontier_size=1 total=1
n=04 incumbent=PASS local=PASS frontier=PASS eta=1.0e-12 frontier_size=1 total=3
n=05 incumbent=PASS local=PASS frontier=PASS eta=1.0e-12 frontier_size=1 total=12
n=06 incumbent=PASS local=PASS frontier=PASS eta=1.0e-12 frontier_size=1 total=60
n=07 incumbent=PASS local=PASS frontier=PASS eta=1.0e-12 frontier_size=1 total=360
n=08 incumbent=PASS local=PASS frontier=PASS eta=1.0e-12 frontier_size=1 total=2520
n=09 incumbent=PASS local=PASS frontier=PASS eta=1.0e-12 frontier_size=1 total=20160
n=10 incumbent=PASS local=PASS frontier=PASS eta=1.0e-12 frontier_size=4 total=181440
n=11 incumbent=PASS local=PASS frontier=PASS eta=1.0e-12 frontier_size=6 total=1814400
n=12 incumbent=PASS local=PASS frontier=PASS eta=1.0e-12 frontier_size=9 total=19958400
n=13 incumbent=PASS local=PASS frontier=PASS eta=1.0e-12 frontier_size=10 total=239500800
n=14 incumbent=PASS local=PASS frontier=PASS eta=1.0e-12 frontier_size=11 total=3113510400

The n=14 frontier was extracted from the existing K=50000 checkpoint heaps; no Stage-A rerun was needed.

Full Certification And Regeneration

The following commands are for full audit or artifact regeneration. They are not CI checks: the certified sweeps, especially n=13 and n=14, are long-running jobs.

Run tests:

python -m pytest

Regenerate certified searches:

python scripts/sweep_certified.py --start 3 --stop 13 --k 20000 --workers 8 --resume
powershell -ExecutionPolicy Bypass -File scripts/start_detached_sweep.ps1 -Start 14 -Stop 14 -K 50000 -Workers 8 -Resume

Run the full independent certificate verifier:

python verify.py --start 3 --stop 14

Regenerate high-precision values and certificate metadata:

python scripts/highprec_verify.py --start 3 --stop 14 --digits 50

Regenerate frontier certificates:

python scripts/extract_frontiers.py --start 3 --stop 14 --margin 2e-10

Run the float64-vs-mpmath calibration:

python scripts/calibrate_float64.py --start 8 --stop 14 --samples 100000 --workers 8 --batch-size 500

The current global maximum absolute deviation is 1.75137506176142662e-14, below the 1e-11 submission threshold.

Regenerate paper tables, figures, and report artifacts:

python scripts/refresh_heuristic_artifacts.py
python scripts/free_float_criterion.py
python scripts/patterns_table.py
python scripts/supnick_validity.py
python scripts/asymptotic_fit.py
python scripts/generate_figures.py
python scripts/build_report.py
python scripts/export_paper_assets.py

Compile the paper:

pdflatex -interaction=nonstopmode -halt-on-error -output-directory=paper_assets paper_assets/ringmin_paper.tex
pdflatex -interaction=nonstopmode -halt-on-error -output-directory=paper_assets paper_assets/ringmin_paper.tex

Run Logs And Data-Generation Hash

The M6 artifacts record generation_commit_hash in every results/nNN/optimum.json. The current data-generating commit recorded there is:

fea000523a1ec4193d8ba9c4637563fd65e86d1a

Long-run evidence:

• n=13 progress log: results/checkpoints/progress_n13_lb3.log
• n=14 progress log: results/checkpoints/progress_n14_lb3.log
• n=14 detached stdout: results/logs/sweep_14_14_20260610_221502.out.log
• n=14 detached stderr: results/logs/sweep_14_14_20260610_221502.err.log

Checkpoint .pkl files under results/checkpoints/ are intentionally ignored because they are large and regenerated by --resume; the frontier JSON files under results/frontiers/ are the portable pruning certificates.

Workflow And AI Assistance

The project was built in an AI-assisted workflow under the author's direction and final review. AI assistance supported research design, mathematical argument development, software implementation, tests, scripts, and repository artifacts. All certified numerical claims are backed by the independent verify.py verifier and the saved result artifacts.

Layout

• src/ringmin/: solver library and CLI.
• tests/: pytest coverage for geometry, evaluator, search, patterns, and SLSQP cross-validation.
• scripts/: reproducibility and artifact-generation scripts.
• verify.py: standalone mpmath/stdlib-only certificate verifier.
• results/: certified optima, frontier certificates, calibration data, and logs.
• figures/: regenerated per-n figures.
• paper_assets/: paper source, PDF, figures, tables, and appendix snippets.

License

MIT - see LICENSE.