My work spans executive leadership in technology companies, independent research, and building with AI. I enjoy moving between big-picture thinking and getting things done.
Born in Germany, I’ve called Asia home for nearly two decades, including the past ten years in Singapore.
Current interests
Technology and human flourishing
The future of intelligence
Building with AI
Champagne
Research
· Computability · Computational Complexity
A Universal Turing Machine with 20 Instructions
We construct a universal Turing machine with 20 instructions, five states and five tape symbols, improving our earlier 21-instruction construction. A parity invariant simplifies printing and restoration, while a direct compiler periodically refreshes the program to keep its execution history short. For each fixed source machine, simulation takes cubic time and quadratic space when input length is bounded by running time, with astronomically large constants. Lean 4 proofs establish finite-input universality, halting equivalence, the resource bounds and recovery of the source machine’s final configuration.
Universal Computation With 144 Residue Classes: A One-State Linear Operator Algorithm
We construct an explicit universal one-state algorithm with 144 residue-selected rules acting on a single nonnegative integer. This is a significant reduction from the modulus of 396 reported by Kaščák in 1992: 252 fewer residue classes, a decrease of 63.6%. Every continuing rule gives an exact nonnegative successor on its entire residue progression. Lean 4 proofs establish universality under explicit computable input and output encodings and preserve nontermination.
How few instructions can a universal Turing machine use? We construct one with 21 defined instructions, five states and five tape symbols, modifying the 22-instruction machine of Neary and Woods. It simulates ordinary Turing machines in polynomial time and space, starting from finite input on a uniform blank tape. An explicit compiler and Lean 4 proofs establish universality, halting equivalence and the resource bounds. The supplement develops exact costs, output recovery and encoding-size optimisation.
Erdős Problem 278: Formulas and Complexity of Maximum Covered Density
How much of the integers can one cover by choosing a residue class for each of a given set of distinct moduli? We give exact formulas for structured families, prove hardness results for exact optimization and counting, and give an exact algorithm controlled by noncoprime-graph treewidth. A scoped Lean companion verifies structural results; the complexity classifications remain written proofs. The paper and companion include exact computational checks and reproduction instructions.
Erdős–Szekeres: Upper Bounds for Convex Heptagons and Octagons
We advance the Erdős–Szekeres convex polygon problem, a question dating back more than 90 years. We prove ES(7) ≤ 81 and ES(8) ≤ 319, improving the explicit 2016 bounds of Nassajian Mojarrad and Vlachos by 32 and 89 points, respectively. The proofs combine geometric arguments with exact computation and are formalized in Lean. The conjectured thresholds of 33 and 65 remain open.
Noncoverage for Distinct Odd Moduli with at Most Three Prime Divisors
We prove that distinct odd moduli cannot cover the integers when each modulus has at most three distinct prime divisors. Prime powers, residues, and the total number of primes across the family are unrestricted. The main theorem is formalized in Lean. A separate extension requires a modulus with at least four distinct prime divisors, all at most one billion.
Nine Prime Divisors in Odd Distinct Covering Systems
We show that any odd distinct covering system must involve at least nine distinct primes, strengthening our earlier bound of eight. The proof allows arbitrary prime powers and is formalized in Lean, together with two quantitative corollaries. The paper also examines the method’s limits at rank ten.
Eight Prime Divisors in Odd Distinct Covering Systems
We advance research on the Erdős–Selfridge odd covering problem and show that any odd distinct covering system must involve at least eight distinct primes, improving the previous bound of seven. The proof allows arbitrary prime powers and is formalized in Lean. A separate formalization shows that the paper’s specified scalar method cannot reach a nine-prime bound.
Seven Prime Divisors in Odd Distinct Covering Systems
The Erdős–Selfridge problem, dating to 1965, asks whether finitely many arithmetic progressions with distinct odd moduli greater than one can cover the integers. This paper advances the frontier on this longstanding problem by proving that any such system must have at least seven distinct prime divisors. A Lean formalization and an independent exact-rational checker accompany the proof.
