Aurantium🍊

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NLP · AI for Math · Crypto

Mathematics · Contributions

Open source made new things possible for more people.

Mathematical research

24 research projects · 7 with generalizations · 8 with Lean formalization

10 positive resolutions · 3 counterexamples.

Howroyd’s 27-set conjecture · _{Posed in 2018 · 8 years ago · Extremal graph theory}

One known residue class → all three classes. Completed the two remaining branches of the 2018 conjecture, then went beyond the count to classify all 27 maximum independent sets. The remaining-branch theorem is formalized in Lean 4.

Original sources & proof method

• Original source: Andrew Howroyd, OEIS A297557 (2018).
• Prior work: John Machacek, Procedia Computer Science 195 (2021), proved the 3 ∣ d branch. Combining the two results completes the counting formula for d ≥ 9.
• Method: An exact weighted-cover certificate and a coordinate-reduction argument reduce the classification to three backgrounds with nine corner extensions each.
• Further results & materials: Explicit classification of all 27 sets, beyond the conjectured count.
• Verification: Lean 4 proof of the full d ≥ 10, 3 ∤ d statement; 89 rational certificate entries.

Lecture Hall polynomiality and algebraic extensions · _{Posed in 2018 · 8 years ago · Algebraic combinatorics / Commutative algebra}

Checks through length 12 → a proof for every length. Extended the result into a family of coefficient formulas, SAGBI-basis and toric-degeneration theorems. The polynomiality and signed-path core is formalized in Lean.

Original sources & proof method

• Original source: Lukas Katthän, The Lecture Hall Cone as a toric deformation (2018), prepared for the Osaka lattice-polytopes workshop.
• Prior work: The original paper proposed the polynomiality and algebraic-structure statements and checked lengths at most 12. Flajolet’s continued-fraction theory and Sokal’s exposition provide credited classical tools.
• Method: A signed Dyck-path identification, Toeplitz–Hankel identities and the defining recurrence give the all-index polynomiality proof.
• Further results & materials: Composition coefficient formulas and exact monomial counts; a minimal monic SAGBI basis, a Hilbert-basis correspondence and flat toric degeneration over every field, under the manuscript’s disclosed square-minor interpretation.
• Verification: Lean formalizes the genuine rational-function polynomiality and signed-path core. SAGBI and degeneration extensions have written mathematical proofs.

Two further results: a 2016 peak-polynomial conjecture · the sharp bound for K₆ chains

Peak polynomials: the last real root · _{Experimental Mathematics · 2016 journal version · 10 years ago · Real roots}

Partial interval results → the full last-real-root statement. An integral sign argument covers the missing noninteger interval, without bounds on peak count or spacing.

• Original source: Sara Billey, Matthew Fahrbach and Alan Talmage, Experimental Mathematics 25(2), 165–175 (2016); preprint first posted in 2014.
• Prior work: Billey–Fahrbach–Talmage supplied the recurrence and symmetry identities. Li’s coefficient theorem already implies right-side positivity and several local sign properties; the additional argument covers the remaining noninteger interval.
• Method: A self-contained integral representation and sign analysis across the full interval beyond the penultimate peak.
• Further results & materials: Signs on both sides of the largest peak, its simplicity, and the singleton case are included; overlapping consequences of Li’s theorem are credited in the manuscript.
• Verification: All-parameter mathematical proof, with supplementary exact finite checks and a separately recorded small-case reproduction.

K₆ chains: a false 22-colour bound replaced by the sharp bound 23 · _{Conjecture posed in 2025 · Graph colouring}

A conjectured bound of 22 → the correct sharp bound of 23. Refuted 22 with a 450-vertex graph, then established the uniform bound for the entire family, backed by a 16.6-million-record exact certificate.

• Original source: Furmańczyk, Gözüpek and Özkan, Packing coloring of graphs with long paths (2025), Conjecture 2.
• Prior work: The proposed common upper bound was 22 for the path-aligned K₆ clique chains.
• Method: A finite-state scheduling relaxation and decreasing-rank certificate exclude 22; an explicit periodic colouring supplies the upper bound 23.
• Further results & materials: Sharp uniform bound 23 for the full chain family and exact value 23 for every t ≥ 75. The minimum counterexample length and all shorter-chain exact values are not determined.
• Verification: A 16,646,562-record exact certificate with Python and C++ checkers. This is computational certificate verification, not a Lean formalization.

More mathematical work · 20 projects

NLP & information retrieval

Dense & Hybrid Retrieval · Knowledge Graphs · Retrieval-Augmented QA

Research interests in retrieval and question answering over text–graph knowledge bases, focusing on complementary retrieval channels, relation-aware entity disambiguation, and coordination between retrieval and reranking.

Current work uses STaRK-MAG to explore how entity text, attributes and graph relations can jointly guide candidate selection under both semantic and relational constraints.

Research tools

Research-to-Decision — an agent workflow that turns research into a choice. Built on BMad Deep Recon and OpenInquire’s cross-platform reading, it clarifies requirements, compares evidence and trade-offs, and explains when to choose an alternative.

Contributions

Selected merged contributions:

Sentence Transformers · _#4251
Sentence embeddings and semantic search. Fixed rescoring for reused floating-point USearch indexes to preserve retrieval ranking.

REA · _{#2422 · #2448 · #2429}
An agent-driven reverse-engineering platform. Contributed improvements to cross-platform compatibility and binary diagnostics.

awesome-python · _#3376
A curated collection of Python resources. Contributed improvements to website accessibility and navigation.

Spec Kit · _#4956
GitHub’s spec-driven development toolkit. Contributed improvements to Windows workflow reliability.

IPython · _#15435
The interactive Python shell used in Jupyter notebooks. Fixed FileLinks(Path(...)) to accept pathlib directory paths.

Oh My Zsh · _#14184
A framework for managing Zsh configuration. Clarified uv initialization aliases and parent-workspace membership.

KOReader · _#16220
A document reader for e-ink devices and Android. Contributed improvements to annotation workflows.

JSON5 · _#371
The JavaScript implementation of JSON5. Contributed improvements to serialization correctness.

Xarray · _#11711
Labeled arrays for scientific Python. Contributed improvements to documentation workflow reliability.

pydicom · _#2388
A Python library for DICOM medical imaging. Contributed improvements to data conversion and interoperability.

Developer Roadmap · _#10451
Developer roadmaps and technical guides. Contributed to the Python learning content.

Free Programming Books · _#13533
A multilingual collection of free programming resources. Improved the contributor documentation.

_{Star snapshots · 2026-10-17–23}

Selected activity

_{AI-assisted updates}

• _2026-10-22 · Sentence Transformers #4251 — Tom Aarsen clarified precision handling and expanded the regression tests before merging the fix.
• _2026-10-21 · IPython #15435 — merged by Carreau.
• _2026-10-20 · Oh My Zsh #14184 — merged by robbyrussell.
• _2026-10-18 · Xarray #11711 — follow-up thanks from VeckoTheGecko after merge.
• _2026-10-17 · awesome-python #3376 — approved and merged by vinta.
• _2026-10-17 · Xarray #11711 — approved and merged by headtr1ck.
• _2026-10-17 · Published Aurantium · Mathematics: 24 research projects, including 8 with Lean formalization.
• _2026-10-15 · JSON5 #371 — merged.
• _2026-10-14 · Spec Kit #4956 — merged.
• _2026-10-13 · REA #2448 — merged; CassieFox23 independently reproduced and verified the fix on Linux.
• _2026-10-13 · REA #2422 — merged; N0zoM1z0’s follow-up extension #2635 also merged upstream.
• _2026-10-12 · REA #2227 — maintainer morluto called the contribution “Brilliant”; an upstream implementation followed.

More activity

• _2026-10-15 · Developer Roadmap #10451 — merged.
• _2026-10-15 · Free Programming Books #13533 — merged.
• _2026-10-15 · Published the Lean supplement for Hardinian asymptotics at r = 2.
• _2026-10-14 · KOReader #16220 — merged.
• _2026-10-14 · REA #3224 — Kotlin source inventory support adopted upstream with co-author credit.

All projects · Merged contributions