arXiv subjects

Dhyey Dharmendrakumar Mavani

Publications and source records attributed to Dhyey Dharmendrakumar Mavani.

3 recordsLinked to original sources

arXiv2026

Formalizing chip-firing and Riemann--Roch for graphs in Lean 4

The Riemann--Roch theorem for graphs, due to Baker and Norine, is a foundational result establishing a powerful analogy between finite graphs and algebraic curves. We describe a complete formal proof of this theorem implemented in the Lean 4 theorem prover. Our formalization includes the existence and uniqueness of q-reduced divisors, a modified form of Dhar's burning algorithm, the bijection between acyclic orientations with unique source and maximal superstable configurations, and Clifford's theorem. We also include several challenges for future formalization.

math.CO↗

arXiv2026

Support Tokens, Stability Margins, and a New Foundation for Robust LLMs

Self-attention is usually described as a flexible, content-adaptive way to mix a token with information from its past. We reinterpret causal self-attention transformers, the backbone of modern foundation models, within a probabilistic framework, much as classical PCA is extended to probabilistic PCA. This reformulation reveals a key structural consequence of the underlying change of variables: a barrier constraint emerges on the parameters of self-attention. The resulting geometry exposes a degeneracy boundary where the attention-induced mapping becomes locally ill-conditioned, yielding a stability-margin interpretation analogous to the margin in support vector machines. This, in turn, naturally gives rise to the concept of support tokens. We further show that causal transformers define a consistent stochastic process over infinite token sequences, providing a rigorous probabilistic foundation for sequence modeling. Building on this view, we derive a Bayesian MAP training objective that requires only a minimal modification to standard LLM training: adding a smooth log-barrier penalty to the usual cross-entropy loss. Empirically, the resulting training objective improves robustness to input perturbations and sharpens the margin geometry of the learned representations without sacrificing out-of-sample accuracy.

cs.LG↗

arXiv2025

chipfiring: A Python Package for Efficient Mathematical Analysis of Chip-Firing Games on Multigraphs

This paper presents `chipfiring`, a comprehensive Python package for the mathematical analysis of chip-firing games on finite graphs. The package provides a robust toolkit for defining graphs and chip configurations (divisors), performing chip-firing operations, and analyzing fundamental properties such as winnability, linear equivalence, and divisor rank. We detail the core components of the library, including its object-oriented graph and divisor implementations, integrated Laplacian matrix computations, and an efficient implementation of Dhar's algorithm for determining the solvability of the dollar game. The `chipfiring` package is designed for researchers and students in graph theory, combinatorics, and algebraic geometry, providing essential algorithms and data structures for exploring these rich mathematical models. We describe the library's architecture, illustrate its usage with comprehensive examples, and highlight its specialized contributions compared to general-purpose graph libraries.

math.CO↗