Random Quadratic Form with random forcing: Metastable synchronization by noise

We study the Random Quadratic Form (RQF) on a sphere in the presence of random Brownian forcing. We show that the forcing does not effectively change the law of the process but affects the synchronization properties of the system. While the RQF without forcing exhibits partial synchronization due to the intrinsic symmetries, the introduction of an arbitrarily small forcing results in long-term symmetry breaking and leads to full synchronization. In this work we focus on the small forcing regime and recover the multiscale behavior of the two-point process. We show that in the first stage the model converges to an anti-polar configuration due to the symmetries of the RQF and in the second stage the two clusters meet due to the symmetry breaking phenomenon. The model is motivated by continuous-time machine learning models such as Neural ODEs and continuous-time formulations of transformers. In particular, the results of this work explain the role of the bias and the scale of its initialization.

Andy: A Mathematical Agent for Rigorous Proof and Autonomous Research

Andy is an autonomous mathematical research agent that turns a mathematical problem into a traceable proof. It solves or verifies a submitted problem, formulates a literature-grounded new problem through a research-value gate, and carries it through proof construction and final verification. It organizes proof steps in an executable DAG, verifies each step independently and binds the result to a certificate, retains verified work whose interfaces remain unchanged during local repair, and records the full path from problem formulation to final proof. The system separates proof generation from correctness evaluation and can acquire, retain, retrieve, and reuse knowledge from existing results. Starting from a self-triggered impulsive consensus result, Andy formulates a global exponential leader-follower synchronization problem for delayed heterogeneous networks with switching communication topologies. The proposed hybrid control combines self-triggered impulses with execution delay and continuous feedback over a recovery window. After each delayed impulse, the feedback cancels the delayed error channel until the pre-impulse history leaves the active delay interval. Sufficient conditions for global exponential synchronization are established, Zeno behavior is excluded for both timing sequences, and a numerical example illustrates the result.

Same physical state, different collective dynamics: state encodings select synchronization outcomes in language-model agents

Language-model agents act on state encodings of their environment, yet these are treated as interchangeable interfaces. Using pretrained language models, we designed a circular-synchronization experiment applying a state-encoding intervention while holding the physical system fixed: each agent sees only a summary of its neighbours' relative phases and chooses to advance, stay or retard. Encoding that state as low-order circular moments rather than as a histogram selected different collective outcomes. In GPT the moment encoding synchronized the population in 6/6 seeds and the histogram encodings in 0/6; the effect replicated in Claude but reversed direction. Replaying identical fields shifted each agent's advance/stay/retard probabilities far beyond within-encoding repeat variation, in GPT, Claude and Gemini; in GPT, presentation alone shifted the operator with the moment values fixed. State encodings therefore form part of a model-dependent effective interaction law, not a neutral interface.

TwistedMerge: Certified Higher-Order Diagnostics and Abstention for Model Merging

Model merging combines independently trained or fine-tuned models, but pairwise alignability does not imply globally consistent alignment. We formulate merging as a finite descent problem in which checkpoints are local objects, alignment maps are transitions, and cycle products are residuals. TwistedMerge is a conservative certification pipeline that separates fixed-chart averaging, synchronization-removable gauge inconsistency, a certified central obstruction on a specified comparison complex, and nonabelian holonomy. A residual is promoted to a cohomology class only after inverse-consistency, coefficient-identification, centrality, and closure tests; otherwise the method abstains and returns an ordinary or synchronized fallback. We prove a constant-edge no-go result, frozen-complex three-way and predeclared-family error-control theorems, and a refinement test for comparison-complex sensitivity. A planted neural alignment defect is removed by cycle-consistent synchronization, showing that a nonzero cycle score alone is not a higher obstruction. Controlled central systems recover the predicted non-coboundary and projective-rank behavior, while noisy estimates move from certification to abstention without false lifts on the tested controls. A trained low-rank-adapter audit shows that naive factor averaging depends on the chosen GLr representative, whereas global factor synchronization and dense-delta SVD are stable. On natural checkpoint collections, cycle residuals do not predict merge degradation and no natural central or period-index class is certified. The results position descent theory as a falsifiable certification and abstention framework.

Temporal-Causal Unity as an Operational Framework for Collective Dynamics: Causal-Progress Clocks, Synchronization, and Polarization

This paper develops temporal-causal unity (TCU), a framework connecting a process-philosophical thesis -- time is the ordered unfolding of causal change -- to an operational model of cognitive and social dynamics. The framework deliberately separates three claims: an interpretive thesis about becoming, a measurable causal-progress coordinate, and a stochastic network model. Causal progress is defined by $τ(t)=\int_0^tλ(s\mid\mathcal H_s)\,{\rm d}s$, where the nonnegative event intensity $λ$ must be specified independently of the outcome. Agents carry an orientation phase and an activation amplitude; weighted interaction, heterogeneous drift, external input, anchoring, and diffusion govern their evolution in $τ$. First- and second-harmonic order parameters separate consensus from bipolar polarization. For the all-to-all noisy Kuramoto special case with Lorentzian drift width $Δ$, synchronization begins at the conditional threshold $K_c = 2(Δ+ D)$, not at a universal constant. Reproducible numerical illustrations illustrate (not empirically demonstrate) this threshold, causal-clock curve collapse, and the consensus-polarization distinction. Six historical episodes are treated as scope probes rather than validation data. The paper derives falsifiable hypotheses and an out-of-sample protocol for comparing causal-progress and chronological-time models. TCU is therefore offered as a disciplined bridge between process ontology and complex-systems modeling, not as a replacement for spacetime physics or as an empirically established identity between time and causation.

Swarm-Inspired Generation of Collective Behaviors in Graph Dynamical Systems

Collective behavior arises when locally interacting units produce coordinated global organization, from synchronization in dynamical systems to task-relevant information flow on graphs. The central challenge is not only to explain how collective behavior emerges, but to design local interaction rules that can produce desired global organization and generalize across graphs, dynamics and tasks.To address this challenge, we introduce the Swarm-Inspired Emergent Synchronizer (SIES), a graph-dynamical framework that learns generalizable local-interaction laws for controllable collective organization. Each node is an agent-like dynamical unit with a state and task cue, and signed source-target-conditioned attention acts as an adaptive coupling term inside an explicit evolution model. Therefore, SIES combines an explicit dynamical engine with local agent intelligence, similar to biological swarms. For synchronization control, SIES learns a generalizable coupling operator that produces prescribed synchronization patterns for CDSs across untrained network scales, target phase relations, and intrinsic node dynamics without retraining. The learned operator also reaches gait-related modes faster than three oscillator baselines and generalizes synchronization-driven locomotion to simulated multi-legged robots of different scales and a physical hexapod after leg disablement. For graph representation learning, SIES applies the same signed interaction principle to message passing and achieves the highest performance among the compared methods on heterophilous node-classification benchmarks. Together, these results position SIES as a generalizable and learnable graph-dynamical interaction framework with promise for synchronization control, adaptive robot coordination, and heterophilous graph representation learning.

Stochastic Scaling Limits and Synchronization by Noise in Deep Transformer Models

We prove pathwise convergence of the layerwise evolution of tokens in a finite-depth, finite-width transformer model with MultiLayer Perceptron (MLP) blocks to a continuous-time stochastic interacting particle system. We also identify the stochastic partial differential equation describing the evolution of the tokens' distribution in this limit and prove propagation of chaos when the number of such tokens is large. The bounds we establish are quantitative and the limits we consider commute. We further prove that the limiting stochastic model displays synchronization by noise and establish exponential dissipation of the interaction energy on average, provided that the common noise is sufficiently coercive relative to the deterministic self-attention drift. We finally characterize the activation functions satisfying the former condition.