arXiv subjects

Evan Chen

Publications and source records attributed to Evan Chen.

At least 19 recordsLinked to original sources

arXiv2026

Fresh Memory, Stale Plans: Dependency-Scoped Validation for Distributed LLM-Agent Memory

Distributed LLM-agent teams can read the latest shared facts and still act on an obsolete plan. A planner may derive an action from requirement $r_3$, another agent may commit $r_4$, and an executor may receive $r_4$ without replacing the plan derived from $r_3$. We call this \emph{stale-plan execution}: state freshness does not establish that the plan authorizing an action remains valid. We introduce PlanFence, a dependency-scoped action-validation protocol. Plans cite the exact public records they used, and an executor validates only the records that can affect the pending external action, replanning once or blocking when validation is incomplete. In 30 controlled live workflows with a post-plan revision, a freshness-only executor acts on the obsolete plan in every task, whereas PlanFence completes all tasks without an invalid action. Controlled replay reveals two conditional boundaries: proactive synchronization yields lower coordination stall at low churn, while PlanFence avoids repeated update-path coordination as churn grows and avoids validating unrelated state as the shared keyspace grows. These are controlled safety and systems-cost results, not general task-accuracy gains.

cs.AI↗

arXiv2026

Record compositions of alternating permutations and noncommutative symmetric functions

Amdeberhan, Shareshian, and Stanley recently proved that a function $\varphi$ arising in the theory of partition Eisenstein series counts the alternating permutations of $\{1,\dots,2n\}$ with a given `record' partition, and they asked whether there is a similar theory for record compositions, suggesting a role for noncommutative symmetric functions. Here we solve their open problem by showing that the number of alternating permutations of $\{1,\dots,2n\}$ with record composition $(\alpha_1,\dots,\alpha_\ell)$ is \[ \prod_{j=1}^{\ell}\binom{2s_j-1}{2\alpha_j-1}E_{2\alpha_j-1}, \] where $s_j=\alpha_1+\dots+\alpha_j$, $E_k$ is an Euler number, and the record composition of $w=a_1a_2\dots a_{2n}$ (so $a_1>a_2 \dotsb$) lists the factor lengths obtained by cutting $a_1a_3\dots a_{2n-1}$ before each left-to-right maximum other than the first. These numbers are the coefficients of a natural lift of the degree-$n$ sprout symmetric function with seed $\sec(\sqrt{t}\,)$ to noncommutative symmetric functions, expanded in products of noncommutative power sums of the first kind. An analogous refinement holds for every sprout sequence whose seed is given by the exponential formula. AxiomProver autonomously produced and verified the results in this paper in Lean.

math.CO↗

arXiv2026

Four-digit Kaprekar dynamics in odd bases

Start with four digits, arrange them in both descending and ascending order, subtract, and repeat. This simple process is known as the Kaprekar routine, famous in base ten for sending every nonconstant four-digit string to $6174$. We show that in every odd base $B>3$, the four-digit Kaprekar map has an unexpectedly rigid structure. After at most three iterations, every nonconstant orbit enters an explicit triangular region $\mathcal{T}_B$, and on this region the map is conjugate to projective doubling: \[ \{[r],[s]\}\longmapsto \{[2r],[2s]\}. \] This gives a complete finite description of all nonconstant terminal cycles, including an explicit formula for their lengths and counts. In particular, the longest terminal cycle has length at most $(B-1)/2$, and equality can occur only when $B$ is prime. For primes $p>5$, equality occurs precisely when the least positive $m$ with $2^m\equiv\pm1\pmod p$ is $m=(p-1)/2$. The results proved here were first formulated by Schwartz and Thakur. As a test case for AI-assisted formal mathematics, AxiomProver produced Lean/mathlib formalizations of these results.

math.NT↗

arXiv2026

Thakur's hypotheses on power sums of $\mathbb{F}_q[t]$

In his 2009 paper, Thakur posed three conjectural hypotheses for the degrees of the power sums \[ S_d(k)=\sum_{\substack{a\in \mathbb F_q[t] \text{ monic}\\ \deg a=d}} a^{-k} \qquad\text{and}\qquad s_d(k)=-\deg_t S_d(k). \] For prime fields $q=p$, we prove Hypotheses H1 and H2, giving a unique greedy description of the extremal term in Carlitz's formula and establishing the recursion \[ s_d(k)=s_{d-1}(s_1(k))+s_1(k). \] As consequences, the prime-field recursion gives the strict Newton-polygon convexity used in the prime-field Carlitz-Goss Riemann-hypothesis theorem, and it recovers Thakur's nonvanishing theorem for positive multizeta values over $\mathbb F_p[t]$. We also prove Hypothesis H3 for all finite fields $q=p^f$, establishing the monotonicity \[ s_d(k)<s_d(k+1)\qquad (p\nmid k). \] We provide Lean formalizations of the arguments in this paper, generated by AxiomProver.

math.NT↗

arXiv2026

Routing Without Training: Controllable-Ratio LLM Offloading via Reliability Gating

Local-cloud collaboration is a practical way to deploy large language models under resource constraints, but existing methods often rely on trained routers or collaboration-aware finetuning that tie routing behavior to a particular operating regime. In this work, we show that such training may be unnecessary: the local model's own inference-time agreement across sampled responses already provides a strong signal for deciding when to trust local execution and when to offload to a stronger cloud model. We propose CARGO, a training-free routing framework that estimates this agreement through prompt-varied sampling, applies Bayesian early stopping for sample-efficient uncertainty control, and supports arbitrary target collaboration ratios through lightweight deployment-time calibration. Across diverse reasoning and question-answering tasks, multiple local LLM families and scales, and both pretrained and finetuned local models, CARGO consistently outperforms other training-free baselines and in several settings surpasses supervised learned routers. These results suggest that effective and adaptable local-cloud collaboration can emerge directly from the local model's intrinsic response behavior, without requiring an additional trained router.

cs.AI↗

arXiv2026

Reciprocals of Partition Polynomials

Ballantine--Beck--Feigon--Maurischat introduced the subsum polynomial \[ \operatorname{sp}(\lambda,x):=\prod_i (1+x^{\lambda_i}) \] attached to an integer partition $\lambda$, and studied rational functions obtained by summing reciprocals of these polynomials over natural classes of partitions. They posed ten conjectures which naturally divide into coprimality and divisibility questions, special-value and recurrence formulas, and coefficient-shape problems. We prove all of the conjectures in the first two families: the ordinary and binary coprimality/divisibility conjectures, and the odd and ternary special-value/recurrence conjectures. AxiomProver autonomously produced Lean/mathlib formalizations and machine-checkable proofs of these six conjectures, and also discovered a counterexample to the statement as printed; the corrected form remains open.

math.CO↗

arXiv2026

ABC implies that Ramanujan's tau function misses almost all primes

Lehmer conjectured that Ramanujan's tau-function never vanishes. In a related direction, a folklore conjecture asserts that infinitely many primes arise as absolute values of Ramanujan's tau-function. Recently, Xiong showed that these prime values form a subset of the primes with density at most $2/11$. Assuming the $abc$ Conjecture, we prove the stronger upper bound \[ S(X):=\#\{\ell\le X:\ \ell\ \text{prime and } |\tau(n)|=\ell \text{ for some } n\ge 1\} = O(X^{13/22}), \] which implies that Ramanujan's tau-function misses a density 1 subset of the primes. We give a heuristic suggesting that $S(X)$ should nevertheless be infinite, with predicted order of magnitude \[ S(X)\asymp \frac{C X^{\frac{1}{11}}}{(\log X)^2}. \] The main engine in this note was formalized and produced automatically in Lean/Mathlib by AxiomProver from a natural-language statement of the problem.

math.NT↗

arXiv2026

On the paucity of lattice triangles

A rational triangle $T$ (one whose angles are rational multiples of $\pi$) unfolds to a translation surface ${X_T}$. The lattice triangle problem asks to classify those $T$ for which ${X_T}$ is a Veech (lattice) surface, which means that the $\operatorname{SL}_2(\mathbb R)$-orbit of ${X_T}$ is closed in its stratum (so its projection to moduli space is a Teichm\"uller curve). The most mysterious regime is the "hard obtuse window" (largest angle in $(\pi/2,2\pi/3]$), where it is conjectured that no lattice triangles exist. Using an arithmetic reformulation of the Mirzakhani-Wright rank obstruction, we prove a quantitative theorem that rules out all but a proportion $n^{-1+o(1)}$ of the triangles in this window with denominator $n$. The main technical result in our proof was autoformalized by AxiomProver in Lean (using mathlib).

math.DS↗

arXiv2026

Almost all primes are partially regular

For odd primes $p$, we let $K_p:=\mathbb{Q}(\zeta_p)$ be the $p$th cyclotomic field and let $\omega$ denote its Teichmuller character. For $\alpha>1/2$, we say that an odd prime $p$ is partially regular if the eigenspaces of the $p$-Sylow subgroup of $\operatorname{Cl}(K_p)$ under the Galois action vanish for all characters $\omega^{p-2k}$ with \[ 2\le 2k \le \frac{\sqrt{p}}{(\log p)^{\alpha}}. \] Equivalently, $p\nmid \operatorname{num}(B_{2k})$ throughout this range. We prove that a density-one subset of primes is partially regular in this sense. By Leopoldt reflection, this yields a partial Vandiver Theorem: for a density-one set of primes $p$, the even eigenspaces $A_p(\omega^{2k})$ vanish for all even $2k$ satisfying the inequality above. This result has consequences for Kubota-Leopoldt $p$-adic $L$-functions, congruences between cusp forms and Eisenstein series, and $p$-torsion in algebraic $K$-groups. The theorem proving partial regularity for almost all $p$ is fully formalized in Lean/Mathlib and was produced automatically by AxiomProver from a natural-language statement of the conjecture.

math.NT↗

arXiv2026

Dead ends in square-free digit walks

We study "dead ends" in square-free digit walks: square-free integers $N$ such that, in base $b$, every one-digit extension $bN+d$ is non-square-free. In base $10$, the stochastic independence model of Miller et al. suggests that infinite square-free walks occur with probability near $1$, corresponding to an asymptotic dead-end density of $\approx 5.218\times 10^{-5}$. We prove that the true asymptotic dead-end density satisfies \[ c_{\mathrm{dead}} \approx 1.317\times 10^{-9}, \] roughly a factor of $\sim 4\times 10^4$ smaller than the prediction. For every base $b\geq 2$, we prove that dead-end densities exist and are given by a closed-form expression (as a finite alternating sum of Euler products). The argument is fully formalized in Lean/Mathlib, and was produced automatically by AxiomProver from a natural-language statement of the problem.

math.CO↗

arXiv2026

Fel's Conjecture on Syzygies of Numerical Semigroups

Let $S=\langle d_1,\dots,d_m\rangle$ be a numerical semigroup and $k[S]$ its semigroup ring. The Hilbert numerator of $k[S]$ determines normalized alternating syzygy power sums $K_p(S)$ encoding alternating power sums of syzygy degrees. Fel conjectured an explicit formula for $K_p(S)$, for all $p\ge 0$, in terms of the gap power sums $G_r(S)=\sum_{g\notin S} g^r$ and universal symmetric polynomials $T_n$ evaluated at the generator power sums $\sigma_k=\sum_i d_i^k$ (and $\delta_k=(\sigma_k-1)/2^k$). We prove Fel's conjecture via exponential generating functions and coefficient extraction, solating the universal identities for $T_n$ needed for the derivation. The argument is fully formalized in Lean/Mathlib, and was produced automatically by AxiomProver from a natural-language statement of the conjecture.

math.CO↗

arXiv2026

Parity of $k$-differentials in genus zero and one

Here we completely determine the spin parity of $k$-differentials with prescribed zero and pole orders on Riemann surfaces of genus zero and one. This result was previously obtained conditionally by the first author and Quentin Gendron assuming the truth of a number-theoretic hypothesis Conjecture A.10. We prove this hypothesis by reformulating it in terms of Jacobi symbols, reducing the proof to a combinatorial identity and standard facts about Jacobi symbols. The proof was obtained by AxiomProver and the system formalized the proof of the combinatorial identity in Lean/Mathlib (see the Appendix).

math.NT↗

arXiv2026

Joint Continual Learning of Local Language Models and Cloud Offloading Decisions with Budget Constraints

Locally deployed Small Language Models (SLMs) must continually support diverse tasks under strict memory and computation constraints, making selective reliance on cloud Large Language Models (LLMs) unavoidable. Regulating cloud assistance during continual learning is challenging, as naive reward-based reinforcement learning often yields unstable offloading behavior and exacerbates catastrophic forgetting as task distributions shift. We propose DA-GRPO, a dual-advantage extension of Group Relative Policy Optimization that incorporates cloud-usage constraints directly into advantage computation, avoiding fixed reward shaping and external routing models. This design enables the local model to jointly learn task competence and collaboration behavior, allowing cloud requests to emerge naturally during post-training while respecting a prescribed assistance budget. Experiments on mathematical reasoning and code generation benchmarks show that DA-GRPO improves post-switch accuracy, substantially reduces forgetting, and maintains stable cloud usage compared to prior collaborative and routing-based approaches.

cs.LG↗

arXiv2025

Towards Straggler-Resilient Split Federated Learning: An Unbalanced Update Approach

Split Federated Learning (SFL) enables scalable training on edge devices by combining the parallelism of Federated Learning (FL) with the computational offloading of Split Learning (SL). Despite its great success, SFL suffers significantly from the well-known straggler issue in distributed learning systems. This problem is exacerbated by the dependency between Split Server and clients: the Split Server side model update relies on receiving activations from clients. Such synchronization requirement introduces significant time latency, making straggler a critical bottleneck to the scalability and efficiency of the system. To mitigate this problem, we propose MU-SplitFed, a straggler-resilient SFL algorithm in zeroth-order optimization that decouples training progress from straggler delays via a simple yet effective unbalanced update mechanism. By enabling the server to perform $\tau$ local updates per client round, MU-SplitFed achieves a convergence rate of $O(\sqrt{d/(\tau T)})$ for non-convex objectives, demonstrating a linear speedup of $\tau$ in communication rounds. Experiments demonstrate that MU-SplitFed consistently outperforms baseline methods with the presence of stragglers and effectively mitigates their impact through adaptive tuning of $\tau$. The code for this project is available at https://github.com/Johnny-Zip/MU-SplitFed.

cs.DC↗

arXiv2025

Bridging On-Device and Cloud LLMs for Collaborative Reasoning: A Unified Methodology for Local Routing and Post-Training

Device-cloud collaboration holds promise for deploying large language models (LLMs), leveraging lightweight on-device models for efficiency while relying on powerful cloud models for superior reasoning. A central challenge in this setting is determining, for each incoming query, whether it should be processed locally or offloaded to the cloud. Existing approaches typically rely on external routers, which often struggle to determine difficulty from the prompt itself, especially for tasks involving complex reasoning. Motivated by this limitation, we propose enabling on-device LLMs to decide internally whether to invoke cloud assistance at inference time, with this capability instilled through reinforcement learning based post-training. Casting on-device LLM post-training as a reward maximization problem, we design hierarchical rewards to encourage local problem solving and judicious cloud offloading. To solve the resulting problem, we develop an algorithm featuring a group-level policy gradient that stabilizes optimization, together with adaptive prompt filtering that provides complementary learning signals to mitigate policy collapse (i.e., exclusive local execution or exclusive cloud offloading). Extensive experiments on on-device-scale LLaMA and Qwen models across multiple reasoning benchmarks show that our method consistently outperforms baselines and significantly narrows the gap to full cloud LLMs.

cs.LG↗

arXiv2025

Federated Foundation Models in Harsh Wireless Environments: Prospects, Challenges, and Future Directions

Foundation models (FMs) have shown remarkable capabilities in generalized intelligence, multimodal understanding, and adaptive learning across a wide range of domains. However, their deployment in harsh or austere environments -- characterized by intermittent connectivity, limited computation, noisy data, and dynamically changing network topologies -- remains an open challenge. Existing distributed learning methods such as federated learning (FL) struggle to adapt in such settings due to their reliance on stable infrastructure, synchronized updates, and resource-intensive training. In this work, we explore the potential of Federated Foundation Models (FFMs) as a promising paradigm to address these limitations. By integrating the scalability and generalization power of FMs with novel decentralized, communication-aware FL frameworks, we aim to enable robust, energy-efficient, and adaptive intelligence in extreme and adversarial conditions. We present a detailed breakdown of system-level constraints in harsh environments, and discuss the open research challenges in communication design, model robustness, and energy-efficient personalization for these unique settings.

cs.NI↗

arXiv2025

More Women, Same Stereotypes: Unpacking the Gender Bias Paradox in Large Language Models

Large Language Models (LLMs) have revolutionized natural language processing, yet concerns persist regarding their tendency to reflect or amplify social biases. This study introduces a novel evaluation framework to uncover gender biases in LLMs: using free-form storytelling to surface biases embedded within the models. A systematic analysis of ten prominent LLMs shows a consistent pattern of overrepresenting female characters across occupations, likely due to supervised fine-tuning (SFT) and reinforcement learning from human feedback (RLHF). Paradoxically, despite this overrepresentation, the occupational gender distributions produced by these LLMs align more closely with human stereotypes than with real-world labor data. This highlights the challenge and importance of implementing balanced mitigation measures to promote fairness and prevent the establishment of potentially new biases. We release the prompts and LLM-generated stories at GitHub.

cs.CL↗

arXiv2025

Communication-Efficient and Differentially Private Vertical Federated Learning with Zeroth-Order Optimization

Vertical Federated Learning (VFL) enables collaborative model training across feature-partitioned devices, yet its reliance on device-server information exchange introduces significant communication overhead and privacy risks. Downlink communication from the server to devices in VFL exposes gradient-related signals of the global loss that can be leveraged in inference attacks. Existing privacy-preserving VFL approaches that inject differential privacy (DP) noise on the downlink have the natural repercussion of degraded gradient quality, slowed convergence, and excessive communication rounds. In this work, we propose DPZV, a communication-efficient and differentially private ZO-VFL framework with tunable privacy guarantees. Based on zeroth-order (ZO) optimization, DPZV injects calibrated scalar-valued DP noise on the downlink, significantly reducing variance amplification while providing equivalent protection against targeted inference attacks. Through rigorous theoretical analysis, we establish convergence guarantees comparable to first-order DP-SGD, despite relying solely on ZO estimators, and prove that DPZV satisfies $(\epsilon, \delta)$-DP. Extensive experiments demonstrate that DPZV consistently achieves a superior privacy-utility tradeoff and requires fewer communication rounds than existing DP-VFL baselines under strict privacy constraints ($\epsilon \leq 10$).

cs.LG↗