Yang Ding

Zhengyang Tang, Xin Lai, Pengyuan Lyu et al.

Phones are becoming an important execution surface for general-purpose agents, but training open models for reliable phone use remains difficult because the environment that matters at deployment, real devices running real apps, is slow, stateful, side-effectful, and hard to reset or verify, while scalable mock environments only approximate real behavior. We present PhoneBuddy, a training recipe and open-model line for agentic phone use that combines a real-app environment with a mock-app environment, PhoneWorld, which reconstructs runnable mock apps from real GUI usage structure. PhoneBuddy first builds a shared supervised fine-tuning stage from trajectories collected in both environments, then compares real-app RL against mixed RL across both environments. Across a 150-task human evaluation on real phones spanning apps, mini-apps, and cross-app workflows, task success rate improves from 36.67\% after supervised fine-tuning to 40.67\% after real-app RL and 45.33\% after mixed RL. On AndroidWorld, the same progression rises from 60.3\% to 77.2\% to 83.2\%. These results show that mock-app training is not a replacement for real-app RL, but a complementary source of scalable, resettable, and automatically checked interaction. The gains are strongest on app and mini-app tasks, while long-horizontal cross-app workflows remain an important open challenge.

Yongchao Wang, Yang Ding, Jiansheng Yang et al.

We present a complete framework for the explicit factorization of $X^n-1$ over integer residue rings $\mathbb{Z}_{p^e}$ for arbitrary $n$ with $\gcd(n, p)=1$. Classical approaches face fundamental bottlenecks: polynomial Hensel lifting requires updating global cofactors (scaling with $n$), while direct multivariate Newton--Hensel iteration on the factor coefficients requires Jacobian inversion (scaling exponentially as $O(p^{(m-1)^2})$ per layer due to zero-divisors, where $m$ is the coset dimension). Our framework eliminates both bottlenecks through three contributions: (1)~the \emph{Ideal Derivation Modulo Principle}, which characterizes all factor coefficients as roots of a multivariate Dickson polynomial ideal derived via modular remainder extraction; (2)~a \emph{cofactor-free Hensel lift} that elevates a single seed factor from $\mathbb{F}_p$ to $\mathbb{Z}_{p^e}$ using a cached polynomial inverse computed once over $\mathbb{F}_p$; and (3)~a \emph{dual-track coefficient reconstruction} mechanism that recovers all remaining factors from the lifted seed's trace array via MED-based coset dispatch, with Newton--Girard inversion as the primary path and quotient-ring Gaussian elimination as an unconditional fallback when $p \leq m$. Empirical evaluation confirms the theoretical grand total algebraic complexity of $O(n + m^3 \log p + e \cdot m^2)$ for explicitly factoring $X^n-1$ over $\mathbb{Z}_{p^e}$, validating the near-constant per-layer lifting cost $O(m^2)$ to depths exceeding $e = 1000$. The framework yields speedups of $445\times$ (including runtime auto-seeding overhead) over SageMath's C-backed FLINT/Pari engine and $33.5\times$ over the V1 scalar lift.