Alex Chengyu Li

Contextual Theory Systems: From Paper-Bound Knowledge to Callable Scientific Theory

Scientific judgment about physical systems that available theories and models describe only partly usually depends on human interpretation and reasoning. Communication delays, inaccessible environments or time-critical streams of observations can prevent that human inquiry from b...

Four-Part Multiset Resolutions of Cylindrical Graphs

A multiset resolving partition distinguishes each vertex of a graph by the unordered multiset of its minimum distances to the partition parts. We prove that every prism formed from two corresponding cycles of length at least ten has multiset partition dimension four, using two si...

A Classification-Free Proof of the Root-Polytope Projection Theorem

Let Φ be a finite reduced crystallographic root system, let P_Φ = conv(Φ) be its root polytope, and let U be a nonzero subspace spanned by roots. Hopkins and Postnikov proved that the orthogonal projection of P_Φ onto U lies in κP_(Φ∩U) for some κ < 2. The published proofs conclu...

Independent-Set Threshold Games and Geodetic Removal on Odd Cycles

In the geodetic removal game, players select vertices until the convex hull of the unselected vertices, taken along shortest paths in the original graph, ceases to be the whole graph. We prove that the second player wins on every odd cycle of order at least five, resolving a conj...

A Permutation Grid with 155 Rook Placements

A permutation determines a crossword grid by placing one black square in each row and column. A complete rook placement chooses white squares so that every maximal horizontal or vertical white interval contains exactly one rook. Lewis and Won conjectured that the positive integer...

Four-Part Multiset Resolutions of Cylindrical Graphs

A multiset resolving partition distinguishes each vertex of a graph by the unordered multiset of its minimum distances to the partition parts. We prove that every prism formed from two corresponding cycles of length at least ten has multiset partition dimension four, using two si...

Reverse-Colexicographic Universal Cycles for Fixed-Weight Necklaces

A fixed-sum word is determined by all but its final symbol. Campbell, Janik-Jones and Sawada constructed universal cycles for these shorthand words and conjectured a reverse-colexicographic necklace concatenation description. We prove that description and generate the cycle in co...

Reverse-Colexicographic Universal Cycles for Fixed-Weight Necklaces

A fixed-sum word is determined by all but its final symbol. Campbell, Janik-Jones and Sawada constructed universal cycles for these shorthand words and conjectured a reverse-colexicographic necklace concatenation description. We prove that description and generate the cycle in co...

The Proof Engine Programme: Reliable AI-Assisted Mathematics and Accountable Research

AI-assisted mathematics creates an opportunity to organize research around two interacting, potentially overlapping roles. Reliable mathematical production turns questions into warranted, readable results. Structural research extracts general mechanisms, connects results, develop...

Boolean Antichains in Finite Lattices: Product Formulas and Matroid Applications

We study the enumeration of Boolean antichains in finite lattices through product formulas and matroid correspondences. The binomial transform of the labelled Boolean-antichain counts is multiplicative under lattice products, yielding an explicit positive convolution for every si...

Sharp Stability Thresholds and Fixed-Energy Laws for Colored Interlacing Triangles

In a depth-two colored interlacing triangle, each color occurs twice on top and once below, in top-bottom-top order. Earlier solutions by Shivam Patel establish eventual coefficient polynomiality and linear cumulants for the model introduced by Blitvić and Petrov. We determine sh...

Sharp Stability Thresholds and Fixed-Energy Laws for Colored Interlacing Triangles

In a depth-two colored interlacing triangle, each color occurs twice on top and once below, in top-bottom-top order. Earlier solutions by Shivam Patel establish eventual coefficient polynomiality and linear cumulants for the model introduced by Blitvić and Petrov. We determine sh...

Matchings and Independence in Bounded-Occupancy Token Graphs: Concurrent Independent Development

A token configuration assigns nonnegative occupancies of total size k to the vertices of a graph; a move transfers one token along one edge. This article records concurrent independent development of a matching proof of the independence-number conjecture of Reyes, Dalfó and Fiol....

Infinite paths in the composite-restricted visible lattice

We consider the nearest-neighbour graph on pairs of integers greater than one that are coprime and have at least one composite coordinate. We prove that this graph contains an infinite simple path, answering Erdős Problem 1212 affirmatively. More precisely, for almost every slope...

Infinite paths in the composite-restricted visible lattice

We consider the nearest-neighbour graph on pairs of integers greater than one that are coprime and have at least one composite coordinate. We prove that this graph contains an infinite simple path, answering Erdős Problem 1212 affirmatively. More precisely, for almost every slope...

Infinite paths in the composite-restricted visible lattice

We consider the nearest-neighbour graph on pairs of integers greater than one that are coprime and have at least one composite coordinate. We prove that this graph contains an infinite simple path, answering Erdős Problem 1212 affirmatively. More precisely, for almost every slope...

Proof Engine 2.0: An Evidence-Governed Method for Autonomous Mathematical Research

Artificial intelligence can assist mathematical research by generating arguments, checking proofs expressed in a formal language, and exploring possible questions. Autonomous research also requires a reliable account of how these activities contribute to an investigation. My earl...

A Classification-Free Proof of the Root-Polytope Projection Theorem

Let Phi be a finite reduced crystallographic root system, let its root polytope be the convex hull of Phi, and let U be a nonzero subspace spanned by roots. Hopkins and Postnikov proved that the orthogonal projection of the root polytope onto U lies in kappa times the root polyto...

The Exact Extremal Bound in Erdős Problem 848

For N ≥ 1, let A be a subset of [1,N] such that ab+1 is nonsquarefree for every a,b in A, with a=b allowed. We prove the sharp bound |A| ≤ floor((N+18)/25), with equality attained by the elements of the progression 7 modulo 25. Earlier work established the same formula only above...

The Exact Extremal Bound in Erdős Problem 848

For N ≥ 1, let A be a subset of [1,N] such that ab+1 is nonsquarefree for every a,b in A, with a=b allowed. We prove the sharp bound |A| ≤ floor((N+18)/25), with equality attained by the elements of the progression 7 modulo 25. Earlier work established the same formula only above...