AI-assisted research
AI for Math
I use modern AI systems as collaborators in mathematical research: to explore conjectures,
search for constructions, organize long case analyses, write verification software, and
stress-test proofs. My current projects are in combinatorics, especially permutation patterns
and graph labelling. Each item below links to the strongest public record available —
arXiv or UC eScholarship for preprints, and GitHub or Zenodo for manuscripts, code, data, and certificates.
Public preprints
Papers currently available as public preprints (arXiv, eScholarship, HAL). Links to reproducibility materials are included when available.
PreprintNumber theory · 2026
All four Liouville sign patterns occur in the binary decompositions of every integer N ≥ 11
Proves that every integer N ≥ 2 outside {2,3,4,5,6,9,10} is a sum a+b
with (λ(a), λ(b)) equal to any prescribed pair of signs, removing the
Generalised Riemann Hypothesis and friability conditions from Mangerel's theorem for this existence
statement. The key step is a one-exclusion rigidity argument at primes; the main results are
formalized in Lean 4 with only the standard axioms.
Also deposited in HAL (hal-05763509, awaiting moderation).
arXivPermutation patterns · 2026
Residual structure and growing inversion-monotonicity regions for 1324-avoiding permutations
Studies the inversion distribution of 1324-avoiding permutations. The revised manuscript
proves monotonicity through k ≤ 2n+6 (computer-assisted), obtains fixed-defect structure and
growing regions, and records the remaining unrestricted conjecture as open.
A UCLA eScholarship deposit has been submitted and will be linked here when its public item URL is issued.
arXivGraph labelling · 2026
Trees of odd order with at most two vertices of degree two are edge-graceful
Proves a new case of Lee's edge-graceful tree conjecture using zero-sum block partitions
and Langford sequences, with symbolic checks and an independently re-checked census of
2,245,070 trees.
Extended by the manuscript in preparation on edge-graceful trees with few vertices of degree two (below).
PreprintCombinatorial trees · 2026
Nonexistence of order-18 Leech trees and of Leech spiders of order at least five
Proves that no Leech tree of order 18 exists and that no spider of order at least five is a Leech tree,
supported by exact searches, clean-room replication, and independently checkable certificates.
Manuscripts and active projects
Work that does not yet have a public preprint is labelled by its present evidence level.
In preparationPermutation patterns
The two ends of the inversion spectrum of 1324-avoiding permutations
A companion manuscript on fixed non-inversion defect at the top end and an exact difference identity at the bottom end.
In preparationGraph labelling
Super edge-magic labellings of spiders with equal legs
Constructive and finite results on equal-leg spiders, with a self-contained verification suite. A revision adding new infinite families is in preparation.
In preparationGraph labelling
Edge-graceful trees with few vertices of degree two
Work in progress on extending the two-vertex case above to trees with more vertices of degree two. The public repository and archive hold an earlier draft whose general proof is being revised; the construction archive and independent checker remain usable.
Computer science and AI
Work outside mathematics that evaluates AI systems themselves.
PreprintAI evaluation · 2026
How far can a commercial decision model's probabilities be trusted? A calibration audit of Jev against open and general-purpose classifiers
Audits the probabilities returned by TypeSafe's Jev 1.13.0 on SST-2 and AG News: how their
calibration depends on the task and the question form, how the 0.01 quantisation limits
routing, what temperature scaling and conformal prediction can repair, and how the comparison
with open zero-shot classifiers depends on the benchmarker's choices of checkpoint and label
descriptions.
How the work is reported
ProofStatements supported by a complete mathematical argument.
Certified computationFinite claims backed by code, hashes, and independent checks.
CandidatePromising constructions or formulas that still need proof or audit.
Open questionUnresolved claims kept separate from proved results.
AI systems contributed to exploration, coding, checking, and drafting. Mathematical claims are
presented according to the evidence actually available, and project repositories preserve the
reproducibility and audit record.