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Andy: A Mathematical Agent for Rigorous Proof and Autonomous Research
Zi'an Wang
Intelligence
Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 93%
Last extracted: 8/18/2026, 5:18:04 AM
Summary
The paper introduces Andy, an autonomous mathematical research agent that solves problems, verifies proofs, and formulates new research problems. It utilizes a dual-model architecture separating a solver (DeepSeek V4 Pro) from an evaluator (Claude Sonnet 5). The system employs a Directed Acyclic Graph (DAG) for proof construction to allow localized verification and repair. A case study demonstrates Andy's ability to derive a global exponential leader-follower synchronization theorem for delayed heterogeneous networks with switching topologies, including a rigorous proof and numerical validation.
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Andy → employsmethod → Directed Acyclic Graph (DAG)
confidence 95% · Andy adapts this dependency structure to proof construction. The rigorous DAG proof route...
Andy → usesevaluator → Claude Sonnet 5
confidence 95% · The mathematical correctness evaluator uses Claude Sonnet 5.
Andy → usessolver → DeepSeek-v4-pro
confidence 95% · The main solver uses DeepSeek V4 Pro.
Zi’an Wang → affiliatedwith → Tongji University
confidence 90% · Address: School of Mathematical Sciences, Tongji University...
Andy → formulatesproblem → Global Exponential Leader-Follower Synchronization
confidence 90% · Andy formulates a global exponential leader-follower synchronization problem...
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Abstract
Abstract:Andy is an autonomous mathematical research agent that solves and verifies submitted problems, formulates new research problems, and constructs rigorous proofs. It separates proof generation from correctness evaluation and supports knowledge acquisition, targeted revision, and multistage verification. This paper illustrates the workflow using a published result on self-triggered impulsive consensus as a starting point. 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 recovery-phase continuous feedback. After each delayed impulse, this feedback cancels the delayed error channel during a recovery window. Sufficient conditions for global exponential synchronization are established, and Zeno behavior is excluded for both the sampling and impulse sequences. A numerical example confirms the result. This case demonstrates Andy's ability to learn from existing results, formulate meaningful research problems, and develop and verify rigorous proofs.
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- Source: https://arxiv.org/abs/2608.15052v1
- Canonical: https://arxiv.org/abs/2608.15052v1
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Global Exponential Leader-Follower Synchronization of Delayed Heterogeneous Networks via Switching Hybrid Control Zi’an Wang wza1012@tongji.edu.cn Address: School of Mathematical Sciences, Tongji University, Shanghai 200092, China Address: Key Laboratory of Intelligent Computing and Applications (Tongji University), Ministry of Education, Shanghai 200092, China Abstract Andy is an autonomous mathematical research agent that solves and verifies submitted problems, formulates new research problems, and constructs rigorous proofs. It separates proof generation from correctness evaluation and supports knowledge acquisition, targeted revision, and multistage verification. This paper illustrates the workflow using self-triggered impulsive consensus as a starting point [1]. 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 recovery-phase continuous feedback. After each delayed impulse, this feedback cancels the delayed error channel during a recovery window. Sufficient conditions for global exponential synchronization are established, and Zeno behavior is excluded for both the sampling and impulse sequences. A numerical example confirms the result. This case demonstrates Andy’s ability to learn from existing results, formulate meaningful research problems, and develop and verify rigorous proofs. keywords Autonomous mathematical agent ,Heterogeneous networks ,Switching topology ,Exponential synchronization 1 Introduction Recent systems illustrate complementary routes toward AI-assisted mathematical research. FunSearch pairs programs proposed by a pretrained large language model with a problem-specific systematic evaluator. AlphaGeometry uses a neural model to generate auxiliary constructions and a symbolic engine to complete geometry proofs [2, 3]. The Moonshine preprint does not treat solving a fixed proposition as the endpoint. It extracts structure from classical problems, formulates conjectures, builds connections, and identifies obstacles [4]. These developments motivate mathematical agents whose research objects, proof steps, and evaluations remain explicit and auditable. Toward the goal of making mathematical problem solving, problem formulation, and proof verification explicit, revisable, and auditable, we developed Andy, a verification-centered autonomous mathematical research agent. It uses a dual-model architecture that separates the roles of solver and evaluator. The main solver uses DeepSeek V4 Pro. It is responsible for solving the original problem, generating proofs, generating new problems, and revising proofs. The mathematical correctness evaluator uses Claude Sonnet 5. It reviews answers and proofs along three dimensions: use of premises, correctness of calculations, and logical completeness. The evaluator cannot directly modify a proof. It can only provide structured verification conclusions, error locations, and revision suggestions. The main solver then completes the revisions. This separation of roles prevents the solver model from approving its own answer without external review. To use Andy, the user must first provide a problem. If no answer is provided, Andy independently constructs a solution or proof and submits it to the correctness evaluator for review. If an answer is provided, Andy directly verifies that answer without solving the problem independently again. After the original problem and its proof pass verification, Andy proposes a new research problem. The basic evaluation of a new problem covers four dimensions: importance, originality, feasibility, and coherence. Importance measures whether solving the problem could advance a theory, method, or application. Originality checks whether the problem only renames, rewrites, or directly restates an existing problem. Feasibility examines whether there is a concrete proof strategy, usable tools, and a reasonable point of entry for the research. Coherence examines whether the new problem follows naturally from the original problem or the references and whether it has one clear research objective. It also checks that the problem is not a simple combination of several small problems. If the user provides reference papers, the evaluation system adds a literature support dimension. This dimension checks whether the new problem has traceable links to the papers and contains substantive modifications. The user can specify the target difficulty of the new problem. Every dimension is scored on a ten-point scale. An easy problem requires a total score of at least 6.06.0 and a score of at least 4.54.5 in every dimension. A standard problem requires a total score of at least 7.07.0 and a score of at least 5.55.5 in every dimension. A research-level problem requires a total score of at least 7.57.5 and a score of at least 6.06.0 in every dimension. A pure rewriting of an existing problem triggers a hard gate. A simple combination of several conclusions, the lack of a concrete feasible method, or the inclusion of several parallel subproblems also triggers a hard gate. Such a problem does not pass verification. The user’s choice of difficulty changes only the threshold for research value and the research cost. It does not reduce the correctness requirements for the final proof. If a problem does not pass the evaluation, the evaluation stage explains the reason for each score deduction. It also gives revision instructions that include the specific defect, the mathematical modification, and the acceptance criterion after modification. The generator then revises the same problem draft in a targeted manner and returns it to the evaluation stage for another review. By default, the system automatically enters the proof stage after the problem reaches the threshold. The user can also choose to pause after the evaluation and intervene manually. Before a task begins, the user can separately specify the direction for generating the problem or upload reference papers. This guidance is used only during the generation, evaluation, and revision of the new problem. It does not affect the solution and verification of the original problem. Andy analyzes the common features, differences, complementary relations, and research gaps among the papers. In the new problem, it explains which papers were used, which conditions or methods were inherited, and which substantive modifications were made. When the available materials are insufficient to assess originality, related work, or proof tools, Andy can use Tavily to search for related literature as needed. For the proof of a new problem, Andy provides two routes: a rigorous DAG proof route and a deep research route. A directed acyclic graph (DAG) is a directed graph with no directed cycles and therefore admits a topological ordering. In artificial intelligence, DAGs are widely used to encode conditional-dependence structures and organize probabilistic inference [5]. Andy adapts this dependency structure to proof construction. The rigorous DAG proof route first generates only a proof outline, necessary definitions, and lemmas. Andy then represents the proof as a directed acyclic graph. Each node corresponds to a lemma or proof step with explicit inputs, assumptions, and a conclusion. Directed edges represent logical dependencies between nodes. The system first checks node identifiers, dependency relations, and reachability of the target. It then executes the nodes one by one in topological order. When a node is executed, the solver can use only the available materials and dependency nodes that have passed verification. After each new node is completed, it undergoes a separate local verification of premises, calculations, and logic. A node is frozen after it passes verification. If a node fails, Andy repairs only that node and the descendants affected by it. Unrelated sibling nodes are not regenerated. If a revision changes only the proof text and leaves the proposition, assumptions, and dependency interface of the node unchanged, verified downstream results can be retained. After all necessary nodes pass, the system assembles the complete proof and performs a unified final verification. If the assembly stage requires a new bridging result, that result must first become a new node and pass a separate verification. DAG mode starts a main proof branch. If the main branch cannot be completed mathematically, Andy activates an alternative branch. If the new DAG executor and the alternative branch both fail, the system can also use a multibranch proof chain as a fallback and try several proof routes. For a research-level problem, the deep research route can be selected. This route first explores several proof strategies at low cost. It compares only the key construction, required lemmas, main risks, and likelihood of completion for each route. After selecting the most promising route, the system converts it into a DAG. It then completes the proof rigorously through the node-by-node proof, local verification, freezing, and final overall verification described above. For a difficult and open-ended problem, this route first obtains the necessary breadth of exploration at low cost. It then concentrates most of the computational effort on the most promising route. Figure 1 summarizes this verification-centered workflow. Submitted problem and optional solution Solve the problem or verify the supplied solution Independent correctness check: premises, calculations, and logic Literature-grounded new-problem formulation Evaluate importance, originality, feasibility, and coherence Targeted revision until the selected threshold is met Explore proof routes and select a promising strategy Build a lemma DAG with explicit dependencies Execute, verify, freeze, or locally repair each node Assemble the complete proof and perform final verification Deliver the proof, certificates, literature links, and research log repair if neededre-evaluaterepair affected node and descendants Figure 1: The research workflow of Andy. Remark 1.1. The main purpose of the DAG is to freeze the parts that have been confirmed as correct and confine errors to local regions. This reduces repeated generation and repeated verification of the entire proof. For example, we used Andy twice to process the problem “Study the zeros and monotonicity of the function x3−x^3-x.” With an ordinary proof attempt and verification process, the total time was 11 hour, 2222 minutes, and 1212 seconds. With our DAG, the total time was approximately 3737 minutes and 59.759.7 seconds. The time decreased by approximately 53.8%53.8\%. The new research problems generated by Andy in the two runs were not identical because of the randomness of the LLM. The experiment shows that the DAG avoided unnecessary execution of complete branches and substantially reduced the running time in this case. The entire time difference cannot be attributed to the DAG. Andy is designed for long-term use and debugging by researchers. It therefore also has a memory function. Andy manages logs and long-term memory separately. The logs provide complete records of problem versions, model rounds, reasoning content returned by model interfaces, tool calls, search results, evaluation reasons, verification certificates, DAG node revisions, frozen and invalidated states, causes of errors, and task checkpoints. These materials are saved as session records and written to the project-level research log. A failed or interrupted task can therefore resume from the corresponding stage. Long-term memory contains selected reusable research information. Andy can store proof-method preferences and evaluation preferences explicitly stated by the researcher. Examples include “Prefer the Lyapunov method for problems of this type in the future” and “A rewriting that only changes notation should not receive a high originality score.” It also stores verified conclusions, successful proof patterns, failed routes, key difficulties, and connections among papers. A new task retrieves only a small amount of memory relevant to the current problem. Failure records help avoid repeating an unsuccessful route, but they are not used as mathematical evidence. Andy can ultimately deliver the original problem and its verification conclusion or complete solution, the new problem generated by Andy, itemized scores and revision records, the complete proof of the new problem, and connections to reference papers and retrieved literature. If the proof fails, it returns the reason for failure and subsequent reduced-scope problems. The system can export Markdown, structured JSON, and both complete and concise PDF reports. The complete report retains scores, tools, certificates, versions, and the research process for review and auditing. The concise report retains only the original problem and its complete solution, together with the problem generated by Andy and its complete proof. It is suitable for direct reading and communication. Andy organizes its data around a runtime home directory. This directory contains configurations and credentials, project workspaces, session logs, a session database, a knowledge base, long-term memory, Agent definitions, Skills, Tools, runtime logs, and MCP server description files. During a research task, especially during the generation and proof of a research-level problem, Andy can independently select memory retrieval, knowledge-base queries, literature searches, and verification tools according to the current stage. It updates the research state through a cycle of planning, execution, verification, and local revision. We also designed a research front end to make the system easier to use and review. The front end integrates the creation of a research task, a real-time research workspace, and the final report into a complete workflow. Users can upload materials for the main problem and reference papers. They can separately enter the research requirements for the original problem and the direction for generating a new problem. They can also select the target difficulty and proof mode. During execution, users can view stable stage progress, total running time, the actual reasoning_content returned by the model interface, the verification process, evaluation reasons, literature connections, and the dependencies, revisions, frozen states, and certificate states of DAG nodes. If users find an error in the model’s reasoning, they can easily append a prompt to correct the research direction. This provides a convenient interface for human intervention. It greatly improves the accuracy of the results and helps the results better satisfy user requirements. After the task is complete, users can directly download research outputs at different levels from the front end. The main contributions of Andy are summarized as follows. (1) Andy provides an end-to-end, verification-centered research workflow. The system does more than generate a solution or imitate the form of a reference theorem. It links literature-grounded problem formulation, research-value evaluation, proof-route selection, DAG-based decomposition, independent local verification, targeted repair, and final proof assembly in one auditable process. (2) The case study in this paper tests a capability that is rarely evaluated as a whole. It examines whether a mathematical agent can start from a verified result, formulate a technically meaningful control problem with interacting delay, switching, and hybrid mechanisms, and then produce a theorem, a proof, and a numerical certificate whose dependencies remain traceable. (3) The significance lies in turning mathematical generation into a revisable research process. Failed proof steps can be localized, verified steps can be retained, and the distinction between a plausible argument and a certified proof is made explicit. The remainder of this paper is organized as follows. Section 2 presents the new control problem proposed by Andy, including the delayed heterogeneous network, the switching hybrid controller, the self-triggering mechanism, and the main synchronization theorem with its proof. Section 3 gives a numerical simulation. Section 4 concludes the paper. 2 The New Problem Proposed by Andy We made only minor formatting adjustments to the model and theorem proposed by Andy. The result is presented below. The complete research log and report for this Andy run are available at https://github.com/mowaiwaim/Andy. During this study, Codex running GPT-5.6 Sol was placed outside Andy and used as a simulated human monitor. It inspected the exposed intermediate reasoning, identified mathematical and strategic errors, and returned targeted guidance. It did not directly rewrite Andy’s result; Andy’s solver remained responsible for each revision. This section studies a class of leader-follower heterogeneous networks with switching communication topologies, time-varying delays, and impulse execution delays. The aim is to establish sufficient conditions for synchronization under the joint action of continuous and impulsive control. Andy adopts the following hybrid control idea. The network error is sampled, and impulsive control is applied after a prescribed execution delay. A recovery window whose length equals the upper bound on the time-varying state delay is then introduced. During this window, the continuous controller temporarily cancels the delay channel in the error system. When the window ends, the pre-impulse history has left the active delay interval. The delayed error channel is then reactivated, and the stability analysis resumes with the complete Lyapunov-Krasovskii functional. 2.1 System Description Let N and n be positive integers. Throughout this section, ImI_m denotes the m×m× m identity matrix, ⊗ denotes the Kronecker product, and ∥⋅∥\|·\| denotes the Euclidean vector norm or its induced matrix norm. For symmetric matrices, X⪯YX Y means that Y−XY-X is positive semidefinite. Let skk≥0\s_k\_k≥ 0 be the sampling instants determined by the self-triggered algorithm introduced below. Let τ>0τ>0 be the known impulse execution delay, and define the corresponding impulse execution instants by tk:=sk+τt_k:=s_k+τ. All state trajectories are taken to be right-continuous at the impulse execution instants. Let h¯>0 h>0, let h:[s0,∞)→ℝh:[s_0,∞) be a time-varying delay, and let f,g:ℝn→ℝnf,g:R^n ^n be nonlinear functions. Consider the following n-dimensional leader: s˙(t)=−Cs(t)+Af(s(t))+Bg(s(t−h(t))), s(t)=-Cs(t)+Af(s(t))+Bg(s(t-h(t))), (1) where C,A,B∈ℝn×nC,A,B ^n× n. The iith follower, i=1,…,Ni=1,…,N, satisfies the hybrid dynamics x˙i(t)=−Cixi(t)+Aif(xi(t))+Big(xi(t−h(t)))+uic(t),t≠tk,Δxi(tk):=xi(tk+)−xi(tk−)=uip(tk),t=tk, \ aligned x_i(t)&=-C_ix_i(t)+A_if(x_i(t))+B_ig (x_i(t-h(t)) )+u_i^c(t),&&t≠ t_k,\\ x_i(t_k)&:=x_i(t_k^+)-x_i(t_k^-)=u_i^p(t_k),&&t=t_k, aligned . (2) where Ci,Ai,Bi∈ℝn×nC_i,A_i,B_i ^n× n may differ among the nodes. The terms uic(t)u_i^c(t) and uip(tk)u_i^p(t_k) denote the continuous and impulsive control inputs, respectively. The communication topology among the network nodes can switch and will be introduced below. Assumption 2.1. The time-varying delay h is absolutely continuous and satisfies 0≤h(t)≤h¯,h˙(t)≤δ<1a.e.0≤ h(t)≤ h, h(t)≤δ<1\,a.e. The nonlinear functions f and g are globally Lipschitz continuous. Namely, there exist lf,lg>0l_f,l_g>0 such that, for any x,y∈ℝnx,y ^n, ‖f(x)−f(y)‖≤lf‖x−y‖,‖g(x)−g(y)‖≤lg‖x−y‖.\|f(x)-f(y)\|≤ l_f\|x-y\|, \|g(x)-g(y)\|≤ l_g\|x-y\|. (3) The continuous controller is designed as uic(t)= u_i^c(t)= (Ci−C)s(t)+(A−Ai)f(s(t))+(B−Bi)g(s(t−h(t))) (C_i-C)s(t)+(A-A_i)f(s(t))+(B-B_i)g(s(t-h(t))) −Ki(xi(t)−s(t))−(1−q(t))Bi[g(xi(t−h(t)))−g(s(t−h(t)))], -K_i (x_i(t)-s(t) )-(1-q(t))B_i [g(x_i(t-h(t)))-g(s(t-h(t))) ], (4) where Ki∈ℝn×nK_i ^n× n is the feedback gain. The phase indicator q(t)∈0,1q(t)∈\0,1\ is right-continuous, distinguishes the recovery phase from the normal-flow phase, and is coordinated with the impulse instants. Specifically, (i) choose the initial sampling instant s0s_0 such that q(s0−)=1q(s_0^-)=1, and set q(t)=1q(t)=1 on the initial interval [s0,t0)[s_0,t_0). (i) Set q(tk)=q(tk+)=0q(t_k)=q(t_k^+)=0 whenever t=tkt=t_k. (i) Keep q=0q=0 in the recovery window [tk,tk+h¯)[t_k,t_k+ h), and set q=1q=1 on [tk+h¯,tk+1)[t_k+ h,t_k+1). (iv) A topology switch that does not coincide with an impulse does not reset q. We next introduce impulsive control with a switching communication topology. First, consider the switching graphs. Let l be a positive integer, and let σ(t)∈:=1,…,lσ(t) :=\1,…,l\ be a left-continuous, piecewise constant switching signal. In mode r∈r , the communication graph rG_r is a connected undirected graph. Its Laplacian matrix is denoted by Lr=(ℓijr)N×N.L_r=( _ij^r)_N× N. Let aijra_ij^r denote the communication weight between nodes i and j in mode r. Then ℓijr=−aijr,i≠j,∑m≠iaimr,i=j. _ij^r= cases-a_ij^r,&i≠ j,\\[2.84526pt] _m≠ ia_im^r,&i=j. cases Let Nσ(t,u)N_σ(t,u) denote the number of topology switches in the interval [u,t)[u,t), and assume that Nσ(t,u)≤t−uTa+N0,t≥u≥s0,N_σ(t,u)≤ t-uT_a+N_0, t≥ u≥ s_0, (5) where Ta>0T_a>0 is the average dwell time and N0≥0N_0≥ 0 is the chatter bound. Assumption 2.2. The mode schedule satisfies σ(t)=rk:=σ(sk−),sk≤t≤tk+h¯,k≥0.σ(t)=r_k:=σ(s_k^-), s_k≤ t≤ t_k+ h, k≥ 0. (6) It also satisfies σ((tk+h¯)+)=rk,k≥0.σ ((t_k+ h)^+ )=r_k, k≥ 0. (7) Thus, the topology mode remains rkr_k from each sampling instant sks_k to the end of the corresponding recovery window. Its right limit at the endpoint of the recovery window is also rkr_k. In particular, no topology switching is allowed in [sk,tk+h¯][s_k,t_k+ h], including at tkt_k. An immediate switch to the right of the endpoint of the recovery window is also excluded. Choose the pinning matrix D=diag(d1,…,dN),di≥0,D=diag(d_1,…,d_N),d_i≥ 0, where di>0d_i>0 means that the iith follower has an additional leader-pinning channel in the impulsive coupling. When di=0d_i=0, the node receives the leader’s influence only indirectly through its neighbors in the impulsive topology. Given the impulse gain μk>0 _k>0, define the topology-dependent impulse mapping matrix Mk:=μk(Lσ(sk−)+D),M_k:= _k (L_σ(s_k^-)+D ), where the active topology immediately before the sampling instant sks_k is used. The mode-scheduling assumption in (6) and (7) avoids a mismatch between the sampling topology and the impulse execution topology. Remark 2.3. The impulsive control law selects the corresponding Laplacian matrix according to the active communication topology at the sampling instant. It is therefore a topology-dependent switching impulsive controller. This design can describe practical situations such as changes in adjacency relations caused by moving nodes, failures and recovery of communication links, and time-sharing schedules for wireless channels. The controller can then implement impulsive corrections according to the communication relations available at the sampling instant. Define the synchronization errors by ei(t):=xi(t)−s(t),e(t):=col(e1(t),…,eN(t)).e_i(t):=x_i(t)-s(t),e(t):=col(e_1(t),…,e_N(t)). Assume that the leader and follower state histories are continuous on [s0−h¯,s0][s_0- h,s_0]. Define the initial error segment by es0(θ):=e(s0+θ),θ∈[−h¯,0],e_s_0(θ):=e(s_0+θ), θ∈[- h,0], so es0∈([−h¯,0],ℝNn)e_s_0 ([- h,0],R^Nn), where C denotes the space of continuous functions. Define its norm by ‖es0‖h¯:=sup−h¯≤θ≤0‖e(s0+θ)‖.\|e_s_0\|_ h:= _- h≤θ≤ 0\|e(s_0+θ)\|. Let [z]i[z]_i denote the iith n-dimensional block of the stacked vector z∈ℝNnz ^Nn. Since the leader state s(t)s(t) is continuous at the impulse execution instants, the impulsive control input is designed as uip(tk) u_i^p(t_k) :=[(Mk⊗In)e(sk−)]i−ei(tk−) := [(M_k I_n)e(s_k^-) ]_i-e_i(t_k^-) (8) =−ei(tk−)+μk[∑j=1Naijσ(sk−)(ei(sk−)−ej(sk−))+diei(sk−)]. =-e_i(t_k^-)+ _k [ _j=1^Na_ij^σ(s_k^-) (e_i(s_k^-)-e_j(s_k^-) )+d_ie_i(s_k^-) ]. Since xi(tk)=xi(tk−)+uip(tk)x_i(t_k)=x_i(t_k^-)+u_i^p(t_k), the stacked error after impulse execution is e(tk)=(Mk⊗In)e(sk−).e(t_k)=(M_k I_n)e(s_k^-). (9) Substituting (4) into the leader and follower systems in (1) and (2) gives the unified closed-loop error system e˙i(t)=−(Ci+Ki)ei(t)+AiΔfi(t)+q(t)BiΔgi(t−h(t)),t≠tk,ei(tk)=[(Mk⊗In)e(sk−)]i,t=tk, \ aligned e_i(t)&=-(C_i+K_i)e_i(t)+A_i f_i(t)+q(t)B_i g_i(t-h(t)),&&t≠ t_k,\\ e_i(t_k)&= [(M_k I_n)e(s_k^-) ]_i,&&t=t_k, aligned . (10) where Δfi(t):=f(xi(t))−f(s(t)),Δgi(t−h(t)):=g(xi(t−h(t)))−g(s(t−h(t))). f_i(t):=f(x_i(t))-f(s(t)),\, g_i(t-h(t)):=g(x_i(t-h(t)))-g(s(t-h(t))). It follows from (10) that the delay channel in the error system is exactly canceled when q(t)=0q(t)=0. This delay channel is reactivated when q(t)=1q(t)=1. Furthermore, suppose that the error is identically zero at a given instant and throughout the required delay-history interval. Then Δfi(t)=0 f_i(t)=0 and Δgi(t−h(t))=0 g_i(t-h(t))=0, so the continuous flow satisfies e˙i(t)=0 e_i(t)=0. If e(sk−)=0e(s_k^-)=0, the impulse mapping gives e(tk)=(Mk⊗In)e(sk−)=0.e(t_k)=(M_k I_n)e(s_k^-)=0. Therefore, the continuous flow and the impulsive jumps both preserve the zero-error state. Once the system reaches synchronization, it remains synchronized. We next introduce self-triggered impulsive control. Let the triggering parameters be λ¯≥0 λ≥ 0 and ρ>0ρ>0, and the parameter sequence be akk≥0\a_k\_k≥ 0, where ak>0a_k>0. Assume additionally that τ≥h¯τ≥ h. The selected triggering parameter λ¯ λ will be further constrained in the theorem below by the Lyapunov matrices and their related constants. At the computed sampling instant sks_k, set Wk2:=eλ¯τW_k^2:=e λτ and define the self-triggering function for a candidate instant t≥tkt≥ t_k as Ψk(t):=(λ¯+ρ)(t−tk)−ak−lnWk2. _k(t):=( λ+ρ)(t-t_k)-a_k- W_k^2. (11) Since λ¯+ρ>0 λ+ρ>0, the function Ψk(t) _k(t) is strictly increasing in t. Therefore, the next sampling instant is defined as the first instant at which Ψk(t) _k(t) reaches a nonnegative value. Namely, sk+1:=inft>tk:Ψk(t)≥0.s_k+1:= \t>t_k: _k(t)≥ 0 \. (12) Define the time length from the kkth actual impulse execution instant tkt_k to the next sampling instant sk+1s_k+1 by Φk:=sk+1−tk=ak+lnWk2λ¯+ρ=ak+λ¯τλ¯+ρ. _k:=s_k+1-t_k= a_k+ W_k^2 λ+ρ= a_k+ λτ λ+ρ. (13) Hence, sk+1=tk+Φk=sk+τ+Φk,s_k+1=t_k+ _k=s_k+τ+ _k, Therefore, the sampling sequence and the impulse execution sequence are interlaced as s0<t0<s1<t1<⋯,s_0<t_0<s_1<t_1<·s, and tk+1−tk=sk+1−sk=τ+Φk>0.t_k+1-t_k=s_k+1-s_k=τ+ _k>0. Both the sampling intervals and the impulse execution intervals have the uniform strict positive lower bound τ. Thus, any finite time interval contains only finitely many sampling instants and impulse execution instants. Therefore, neither the self-triggered sampling sequence nor the impulse execution sequence exhibits Zeno behavior. 2.2 Main Result For any candidate symmetric positive definite matrices Pr,Sr,Rr∈ℝn×nP_r,S_r,R_r ^n× n, constants λ1,r≥0 _1,r≥ 0, and scalars εi,r>0 _i,r>0 appearing in condition (i) below, where r∈r and i=1,…,Ni=1,…,N, define Qi,r:= Q_i,r:= Pr(Ci+Ki)+(Ci+Ki)Pr, P_r(C_i+K_i)+(C_i+K_i) TP_r, Λi,r(11):= _i,r^(11):= −Qi,r+2‖PrAi‖lfIn+εi,rIn+λmax(Sr)In+h¯λmax(Rr)In, -Q_i,r+2\|P_rA_i\|l_fI_n+ _i,rI_n+ _ (S_r)I_n+ h _ (R_r)I_n, Λi,r(22):= _i,r^(22):= εi,r−1‖PrBi‖2lg2In−(1−δ)λmin(Sr)In. _i,r^-1\|P_rB_i\|^2l_g^2I_n-(1-δ) _ (S_r)I_n. Further define βr(0) _r^(0) :=max1≤i≤N−λmin(Qi,r)+2‖PrAi‖lf,βr:=max0,βr(0)λmin(Pr), := _1≤ i≤ N \- _ (Q_i,r)+2\|P_rA_i\|l_f \, _r:= \0, _r^(0)\ _ (P_r), γr(0) _r^(0) :=max1≤i≤Nλmax(Qi,r)+2‖PrAi‖lf,γr:=max0,γr(0)λmin(Pr). := _1≤ i≤ N \ _ (Q_i,r)+2\|P_rA_i\|l_f \, _r:= \0, _r^(0)\ _ (P_r). Let χ~r χ_r :=e(βr+γr)h¯(1+h¯λmax(Sr)+h¯22λmax(Rr)λmin(Pr)), :=e^( _r+ _r) h (1+ h _ (S_r)+ h^22 _ (R_r) _ (P_r) ), χ~max χ_ :=maxr∈χ~r,β+:=maxr∈βr,λ¯1:=maxr∈λ1,r. := _r χ_r, _+:= _r _r, λ_1:= _r _1,r. For a candidate sequence ηkk≥0⊂(0,1)\ _k\_k≥ 0⊂(0,1) appearing in condition (i) below, define the effective contraction over the kkth impulsive period by c~k:=−lnηk−β+h¯−lnχ~max. c_k:=- _k- _+ h- χ_ . (14) For any α>0α>0, define αeff:=α+λ¯−λ¯1. _ eff:=α+ λ- λ_1. Theorem 2.4. Consider the systems in (1) and (2) under the continuous controller (4), the impulsive controller (8), and the self-triggering mechanism defined by (11) through (13). Assumption 2.1, the average dwell-time constraint (5), the execution-delay requirement τ≥h¯τ≥ h, and the mode-scheduling assumption in (6) and (7) are imposed throughout. If the following conditions hold: (i) For each switching mode r∈r , there exist symmetric positive definite matrices Pr,Sr,Rr∈ℝn×n,P_r,S_r,R_r ^n× n, and a constant λ1,r≥0 _1,r≥ 0 that is valid for all nodes. For every i=1,…,Ni=1,…,N, there exists εi,r>0 _i,r>0 such that [Λi,r(11)00Λi,r(22)]⪯λ1,r[Pr000], bmatrix _i,r^(11)&0\\ 0& _i,r^(22) bmatrix _1,r bmatrixP_r&0\\ 0&0 bmatrix, (15) In addition, λ¯≥maxλ¯1,β+. λ≥ \ λ_1, _+\. (i) There exists a sequence ηkk≥0⊂(0,1)\ _k\_k≥ 0⊂(0,1) such that, for every k≥0k≥ 0, the impulsive map satisfies MkMk⪯ηkIN.M_k TM_k _kI_N. (16) In addition, there exists a constant μ≥1μ≥ 1 such that, for all r,j∈r,j , Pr⪯μPj,Sr⪯μSj,Rr⪯μRj.P_r μ P_j, S_r μ S_j, R_r μ R_j. (17) (i) The self-triggering parameters satisfy ak>ρh¯,k≥0.a_k>ρ h, k≥ 0. (18) There exist constants α>0α>0 and φ>0 >0 such that, for all t≥u≥s0t≥ u≥ s_0, λ¯(t−u)+lnμTa(t−u)+∑k:u≤tk<t[−c~k+(λ¯+ρ)τ]≤−α(t−u)+φ. λ(t-u)+ μT_a(t-u)+ _k:\,u≤ t_k<t [- c_k+( λ+ρ)τ ]≤-α(t-u)+ . (19) Then αeff≥α>0 _ eff≥α>0, and the closed-loop system achieves global exponential leader-follower synchronization. Specifically, there exists a constant CGE≥1C_ GE≥ 1, independent of the initial history, such that every admissible initial history satisfies ‖e(t)‖≤CGE‖es0‖h¯e−αeff2(t−s0),t≥s0,\|e(t)\|≤ C_ GE\|e_s_0\|_ he^- _ eff2(t-s_0), t≥ s_0, (20) Proof. The stated regularity and non-Zeno timing assumptions ensure that the closed-loop system admits a unique global right-continuous solution. It remains to establish the exponential estimate. Fix matrices Pr,Sr,RrP_r,S_r,R_r and constants λ1,r,εi,r _1,r, _i,r satisfying condition (i). In each recovery window, construct the quadratic Lyapunov function r0(t):=e(t)(IN⊗Pr)e(t),V_r^0(t):=e T(t)(I_N P_r)e(t), (21) In each normal phase, construct the Lyapunov-Krasovskii functional r1(t):= _r^1(t):= e(t)(IN⊗Pr)e(t)+∫t−h(t)te(v)(IN⊗Sr)e(v)v+∫−h¯0∫t+θte(v)(IN⊗Rr)e(v)vθ. e T(t)(I_N P_r)e(t)+ _t-h(t)^te T(v)(I_N S_r)e(v)\,\,dv+ _- h^0 _t+θ^te T(v)(I_N R_r)e(v)\,\,dv\,\,dθ. (22) In the following, write (t)=σ(t)0(t) V(t)=V_σ(t)^0(t) when q(t)=0q(t)=0, and write (t)=σ(t)1(t) V(t)=V_σ(t)^1(t) when q(t)=1q(t)=1. The initialization of q gives q=1q=1 at s0−s_0^- and throughout [s0,t0)[s_0,t_0). For k≥1k≥ 1, the conditions τ≥h¯τ≥ h and (18) imply that q=1q=1 at each sampling instant sks_k and each pre-impulse instant tk−t_k^-. Thus, V is the full functional at these instants. First, consider the continuous flow in a recovery window. When q=0q=0 and the mode is fixed, (10) has no delay term. Its error trajectory is absolutely continuous. Hence, r0V_r^0 is also absolutely continuous. It follows from (21), the definition of Qi,rQ_i,r, and (3) that, for almost every t, ˙r0(t)= V_r^0(t)= ∑i=1N[−eiQi,rei+2eiPrAiΔfi]≤βr(0)∑i=1N‖ei‖2≤βrr0(t). _i=1^N [-e_i TQ_i,re_i+2e_i TP_rA_i f_i ]≤ _r^(0) _i=1^N\|e_i\|^2≤ _rV_r^0(t). (23) Similarly, ˙r0(t) V_r^0(t) ≥−γr(0)∑i=1N∥ei∥2≥−γr0(t). ≥- _r^(0) _i=1^N\|e_i\|^2≥- _rV_r^0(t). (24) Therefore, in the almost-everywhere sense, −γrr0(t)≤˙r0(t)≤βrr0(t)- _rV_r^0(t)≤ V_r^0(t)≤ _rV_r^0(t). Apply the standard Gronwall inequality to these two differential inequalities. In a recovery window with a fixed mode, for any tk≤u≤v≤tk+h¯t_k≤ u≤ v≤ t_k+ h, this gives r0(v)≤eβr(v−u)r0(u),r0(u)≤eγr(v−u)r0(v).V_r^0(v)≤ e _r(v-u)V_r^0(u), _r^0(u)≤ e _r(v-u)V_r^0(v). (25) Next, consider the continuous flow in a normal phase. Let ei,h:=ei(t−h(t))e_i,h:=e_i(t-h(t)). When q=1q=1, take the Dini derivative of the quadratic term in (22) along (10). We obtain D+[e(t)(IN⊗Pr)e(t)]=∑i=1N[−eiQi,rei+2eiPrAiΔfi+2eiPrBiΔgi(t−h(t))]. D^+\! [e T(t)(I_N P_r)e(t) ]= _i=1^N [-e_i TQ_i,re_i+2e_i TP_rA_i f_i+2e_i TP_rB_i g_i(t-h(t)) ]. The Lipschitz conditions and Young’s inequality give, respectively, 2eiPrAiΔfi≤2‖PrAi‖lf‖ei‖2, 2e_i TP_rA_i f_i≤ 2\|P_rA_i\|l_f\|e_i\|^2, 2eiPrBiΔgi(t−h(t))≤εi,r‖ei‖2+εi,r−1‖PrBi‖2lg2‖ei,h‖2. 2e_i TP_rB_i g_i(t-h(t))≤ _i,r\|e_i\|^2+ _i,r^-1\|P_rB_i\|^2l_g^2\|e_i,h\|^2. Apply the Leibniz rule to the single-integral term. Using h˙(t)≤δ h(t)≤δ, we obtain D+[∫t−h(t)te(v)(IN⊗Sr)e(v)v]≤λmax(Sr)∑i=1N‖ei‖2−(1−δ)λmin(Sr)∑i=1N‖ei,h‖2. D^+\! [ _t-h(t)^te T(v)(I_N S_r)e(v)\,\,dv ]≤ _ (S_r) _i=1^N\|e_i\|^2-(1-δ) _ (S_r) _i=1^N\|e_i,h\|^2. The Dini derivative of the double-integral term satisfies D+[∫−h¯0∫t+θte(v)(IN⊗Rr)e(v)vθ] D^+\! [ _- h^0 _t+θ^te T(v)(I_N R_r)e(v)\,\,dv\,\,dθ ] =h¯e(t)(IN⊗Rr)e(t)−∫t−h¯te(v)(IN⊗Rr)e(v)v = h\,e T(t)(I_N R_r)e(t)- _t- h^te T(v)(I_N R_r)e(v)\,\,dv ≤h¯λmax(Rr)∑i=1N‖ei‖2. ≤ h _ (R_r) _i=1^N\|e_i\|^2. Combining the above estimates gives the following bound for the Dini derivative of the full functional: D+r1(t)≤∑i=1N[eiΛi,r(11)ei+ei,hΛi,r(22)ei,h]. D^+V_r^1(t)≤ _i=1^N [e_i T _i,r^(11)e_i+e_i,h T _i,r^(22)e_i,h ]. (26) By (15), D+r1(t)≤λ1,re(t)(IN⊗Pr)e(t)≤λ1,rr1(t).D^+V_r^1(t)≤ _1,re T(t)(I_N P_r)e(t)≤ _1,rV_r^1(t). (27) Suppose that a switch from mode r to mode j occurs during a normal phase. Applying (17) term by term to (22) gives j1(t+)≤μr1(t−).V_j^1(t^+)≤ _r^1(t^-). (28) We next estimate the functional transition at the end of a recovery window. Let tk⋆:=tk+h¯t_k :=t_k+ h. Since h(tk⋆)≤h¯h(t_k )≤ h, all history involved in (22) at tk⋆t_k lies in the recovery window [tk,tk⋆][t_k,t_k ]. By (25), every v in this interval satisfies r0(v)≤e(βr+γr)h¯r0(tk⋆−).V_r^0(v)≤ e^( _r+ _r) hV_r^0(t_k -). (29) Moreover, ‖e(v)‖2≤r0(v)/λmin(Pr)\|e(v)\|^2 _r^0(v)/ _ (P_r). Hence, ∫tk⋆−h(tk⋆)tk⋆e(v)(IN⊗Sr)e(v)v≤e(βr+γr)h¯h¯λmax(Sr)λmin(Pr)r0(tk⋆−), _t_k -h(t_k )^t_k e T(v)(I_N S_r)e(v)\,\,dv≤ e^( _r+ _r) h h _ (S_r) _ (P_r)V_r^0(t_k -), ∫−h¯0∫tk⋆+θtk⋆e(v)(IN⊗Rr)e(v)vθ≤e(βr+γr)h¯h¯2λmax(Rr)2λmin(Pr)r0(tk⋆−). _- h^0 _t_k +θ^t_k e T(v)(I_N R_r)e(v)\,\,dv\,\,dθ≤ e^( _r+ _r) h h^2 _ (R_r)2 _ (P_r)V_r^0(t_k -). Therefore, the definition of χ~r χ_r and (7) yield r1((tk⋆)+)≤χ~rr0(tk⋆−).V_r^1((t_k )^+)≤ χ_rV_r^0(t_k -). (30) By (6) and (7), the mode rk=σ(sk−)=σ(tk−)r_k=σ(s_k^-)=σ(t_k^-) remains unchanged on [sk,tk+h¯][s_k,t_k+ h]. Using (9), (16), and the dominance of the full functional over its current-state quadratic term, we have rk0(tk+) _r_k^0(t_k^+) =e(sk−)(MkMk⊗Prk)e(sk−)≤ηke(sk−)(IN⊗Prk)e(sk−)≤ηkrk1(sk−). =e T(s_k^-)(M_k TM_k P_r_k)e(s_k^-)≤ _ke T(s_k^-)(I_N P_r_k)e(s_k^-)≤ _kV_r_k^1(s_k^-). (31) Successive application of (23), (30), and (14) gives rk1((tk+h¯)+) _r_k^1((t_k+ h)^+) ≤χ~maxeβ+h¯ηkrk1(sk−)=e−c~krk1(sk−). ≤ χ_ e _+ h _kV_r_k^1(s_k^-)=e^- c_kV_r_k^1(s_k^-). (32) By (18), τ≥h¯τ≥ h, and (13), Φk−h¯=ak−ρh¯+λ¯(τ−h¯)λ¯+ρ>0. _k- h= a_k-ρ h+ λ(τ- h) λ+ρ>0. (33) Thus, [tk+h¯,sk+1)[t_k+ h,s_k+1) is a nondegenerate normal phase. Let NkswN_k sw denote the number of switches in this interval. Combining (27), (28), and (32), we obtain (sk+1−)≤eλ¯1(Φk−h¯)−c~kμNksw(sk−). V(s_k+1^-)≤ e λ_1( _k- h)- c_kμ^N_k sw V(s_k^-). (34) Condition (6) excludes mode switches in all other intervals of the current cycle. Thus, NkswN_k sw is also the total number of switches in [sk,sk+1)[s_k,s_k+1). Iterate (34) and use ∑i=0k−1Φi=(sk−s0)−kτ,∑i=0k−1Nisw≤sk−s0Ta+N0, _i=0^k-1 _i=(s_k-s_0)-kτ,\, _i=0^k-1N_i sw≤ s_k-s_0T_a+N_0, to obtain ln(sk−)≤ V(s_k^-)≤ ln(s0−)+λ¯1[(sk−s0)−kτ−kh¯]−∑i=0k−1c~i+lnμTa(sk−s0)+N0lnμ. V(s_0^-)+ λ_1 [(s_k-s_0)-kτ-k h ]- _i=0^k-1 c_i+ μT_a(s_k-s_0)+N_0 μ. (35) Set u=s0u=s_0 and t=skt=s_k in (19). Then −∑i=0k−1c~i+lnμTa(sk−s0)≤−(α+λ¯)(sk−s0)−k(λ¯+ρ)τ+φ.- _i=0^k-1 c_i+ μT_a(s_k-s_0)≤-(α+ λ)(s_k-s_0)-k( λ+ρ)τ+ . (36) Substitute (36) into (35) and discard the nonpositive terms involving k. Let φeff:=φ+N0lnμ _ eff:= +N_0 μ. We obtain (sk−)≤eφeff(s0−)e−αeff(sk−s0). V(s_k^-)≤ e _ eff V(s_0^-)e^- _ eff(s_k-s_0). (37) We next estimate a uniform upper bound for the triggering intervals. Fix k and take 0<ε<Φk0< < _k. Set u=tk+εu=t_k+ and t=sk+1t=s_k+1 in (19). The interval [tk+ε,sk+1)[t_k+ ,s_k+1) contains no impulse and has length Φk−ε _k- . Hence, (α+λ¯+lnμTa)(Φk−ε)≤φ. (α+ λ+ μT_a )( _k- )≤ . Letting ε↓0 0 gives 0<Φk≤Φmax:=φα+λ¯+lnμTa<∞.0< _k≤ _ := α+ λ+ μT_a<∞. (38) Thus, Φk _k has the uniform upper bound required for the estimate over the entire time domain. We now extend the estimate at the sampling instants to all times. If t∈[sk,tk)t∈[s_k,t_k), then (27) and (6) give (t)≤eλ¯τ(sk−). V(t)≤ e λτ V(s_k^-). If t∈[tk,tk+h¯)t∈[t_k,t_k+ h), then (23) and (31) give (t)≤eβ+h¯(sk−)≤eλ¯h¯(sk−). V(t)≤ e _+ h V(s_k^-)≤ e λ h V(s_k^-). If t∈[tk+h¯,sk+1)t∈[t_k+ h,s_k+1), then (30), (27), (28), (5), and (38) give (t)≤χ~maxeβ+h¯+λ¯1ΦmaxμN0+Φmax/Ta(sk−). V(t)≤ χ_ e _+ h+ λ_1 _ μ^N_0+ _ /T_a V(s_k^-). Therefore, for every t∈[sk,sk+1)t∈[s_k,s_k+1), (t)≤Ccyc(sk−), V(t)≤ C_ cyc V(s_k^-), (39) where Ccyc:=max C_ cyc:= \ eλ¯τ,eλ¯h¯,χ~maxeβ+h¯+λ¯1ΦmaxμN0+Φmax/Ta. e λτ,e λ h, χ_ e _+ h+ λ_1 _ μ^N_0+ _ /T_a \. (40) Moreover, t−sk<τ+Φmaxt-s_k<τ+ _ . It follows from (37) and (39) that (t)≤Ccyceφeff+αeff(τ+Φmax)(s0−)e−αeff(t−s0). V(t)≤ C_ cyce _ eff+ _ eff(τ+ _ ) V(s_0^-)e^- _ eff(t-s_0). (41) Define p¯ p :=minr∈λmin(Pr), := _r _ (P_r), Ch C_h :=maxr∈λmax(Pr)+h¯maxr∈λmax(Sr)+h¯22maxr∈λmax(Rr). := _r _ (P_r)+ h _r _ (S_r)+ h^22 _r _ (R_r). By (21) and (22), (t)≥p¯‖e(t)‖2,(s0−)≤Ch‖es0‖h¯2. V(t)≥ p\|e(t)\|^2, V(s_0^-)≤ C_h\|e_s_0\|_ h^2. (42) Substituting (42) into (41) gives (20), where we can take CGE:=(CcycChp¯)1/2exp[φeff+αeff(τ+Φmax)2].C_ GE:= ( C_ cycC_h p )^1/2 \! [ _ eff+ _ eff(τ+ _ )2 ]. (43) Since ei(t)=xi(t)−s(t)e_i(t)=x_i(t)-s(t), (20) shows that every follower error ei(t)e_i(t) converges to zero at an exponential rate of at least αeff/2 _ eff/2. Therefore, the closed-loop system achieves global exponential leader-follower synchronization. ∎ Remark 2.5. When γr(0)≤0 _r^(0)≤ 0 holds for every mode, γr=0 _r=0. The phase-transition factor χ~r χ_r then automatically reduces to its original form without the bidirectional growth compensation. This reduction leaves all other conditions of the theorem and the proof structure unchanged. Remark 2.6. The recovery-window feedback in (4) reveals a useful control-design idea. It matches a controller term with a difficult term in the error dynamics and removes that term during the part of the hybrid evolution in which it is most troublesome. Exact delay-channel cancellation is demanding in practice because it requires accurate knowledge of the delayed model and online access to the complete delayed states. Mature delay-system methods can retain the delayed channel and estimate its effect through Lyapunov-Krasovskii functionals, comparison arguments, and delayed impulsive inequalities [1, 6, 7]. Andy’s construction complements these methods with a structural heuristic. It first identifies the term that makes the Lyapunov derivative difficult. The controller and the Lyapunov function are then co-designed so that this term is canceled or absorbed in a controlled phase. This principle also appears in PD-controlled multi-weighted networks. A derivative feedback term can be matched with a coupling-weighted quadratic term in the Lyapunov function so that the corresponding cross-derivative terms cancel pairwise [8, 9]. In PI designs, related weight terms can be balanced in an augmented Lyapunov functional [8]. A broader version of this idea appears in recursive adaptive control. Coordinate changes, parameter-update laws, and the feedback law are constructed step by step, and stability is established through a Lyapunov argument [10]. A related learning-based approach jointly trains a nonlinear controller and a neural Lyapunov function. An SMT-based falsification step checks the Lyapunov conditions and returns counterexamples when they fail [11]. These connections suggest that Andy’s control design can inform the construction of Lyapunov functions and PID-type controllers for systems with several coupling weights, delayed channels, or derivative couplings. When exact delayed states are unavailable, a natural extension is to use observer-based or adaptive approximate cancellation and bound the residual mismatch with standard delay-analysis tools. 3 Numerical Simulation Example 3.1. Consider one three-dimensional leader and five three-dimensional heterogeneous followers. The nonlinear functions are applied componentwise and are chosen as f(z)=g(z)=tanh(z),lf=lg=1.f(z)=g(z)= (z),\,l_f=l_g=1. The matrices of the leader system are C=diag(0.45,0.50,0.55),B=diag(0.12,0.10,0.09),C=diag(0.45,0.50,0.55),\,B=diag(0.12,0.10,0.09), and A=[1.05−0.250.150.200.95−0.20−0.150.251.00].A= bmatrix1.05&-0.25&0.15\\ 0.20&0.95&-0.20\\ -0.15&0.25&1.00 bmatrix. The follower matrices are diagonal and are chosen as C1 C_1 =diag(0.30,0.36,0.42), =diag(0.30,0.36,0.42), A1 A_1 =diag(1.20,0.95,1.05), =diag(1.20,0.95,1.05), B1 B_1 =diag(0.15,0.10,0.12), =diag(0.15,0.10,0.12), C2 C_2 =diag(0.34,0.31,0.46), =diag(0.34,0.31,0.46), A2 A_2 =diag(1.10,1.18,0.92), =diag(1.10,1.18,0.92), B2 B_2 =diag(0.11,0.14,0.09), =diag(0.11,0.14,0.09), C3 C_3 =diag(0.38,0.44,0.32), =diag(0.38,0.44,0.32), A3 A_3 =diag(0.98,1.08,1.15), =diag(0.98,1.08,1.15), B3 B_3 =diag(0.13,0.08,0.15), =diag(0.13,0.08,0.15), C4 C_4 =diag(0.33,0.48,0.37), =diag(0.33,0.48,0.37), A4 A_4 =diag(1.16,1.02,1.10), =diag(1.16,1.02,1.10), B4 B_4 =diag(0.10,0.12,0.14), =diag(0.10,0.12,0.14), C5 C_5 =diag(0.41,0.35,0.50), =diag(0.41,0.35,0.50), A5 A_5 =diag(1.05,1.12,1.19), =diag(1.05,1.12,1.19), B5 B_5 =diag(0.14,0.15,0.11). =diag(0.14,0.15,0.11). The continuous feedback gains are Ki=1.20I3K_i=1.20I_3 for i=1,…,5i=1,…,5. The time-varying delay is h(t)=0.075+0.025sin(0.8t),h(t)=0.075+0.025 (0.8t), so 0.05≤h(t)≤h¯=0.100.05≤ h(t)≤ h=0.10 and maxth˙(t)=0.02<δ=0.05<1 _t h(t)=0.02<δ=0.05<1. The two switching topologies are a five-node ring and a star centered at follower 3. Their Laplacian matrices are L1=[2−100−1−12−1000−12−1000−12−1−100−12],L2=[10−10001−100−1−14−1−100−11000−101].L_1= bmatrix2&-1&0&0&-1\\ -1&2&-1&0&0\\ 0&-1&2&-1&0\\ 0&0&-1&2&-1\\ -1&0&0&-1&2 bmatrix, L_2= bmatrix1&0&-1&0&0\\ 0&1&-1&0&0\\ -1&-1&4&-1&-1\\ 0&0&-1&1&0\\ 0&0&-1&0&1 bmatrix. Only follower 1 has a direct pinning channel, and hence D=diag(1,0,0,0,0)D=diag(1,0,0,0,0). For the impulsive controller, choose μk=0.05,ηk=0.07,τ=1.00,λ¯=0,ρ=0.10,ak=0.10 _k=0.05, _k=0.07,τ=1.00, λ=0,ρ=0.10,a_k=0.10. The switch in each cycle is scheduled at sk+1.50s_k+1.50 s, after the protected interval ends at sk+τ+h¯=sk+1.10s_k+τ+ h=s_k+1.10 s. The two impulsive topology modes alternate over successive triggering cycles. The switching parameters are Ta=2.00T_a=2.00 and N0=1N_0=1. The Lyapunov parameters are selected as Pr=I3,Sr=0.25I3,Rr=0.05I3,εi,r=0.10,P_r=I_3,S_r=0.25I_3,R_r=0.05I_3, _i,r=0.10, for both modes, with λ1,r=0,μ=1,α=0.80,φ=1.97. _1,r=0,μ=1,α=0.80, =1.97. The initial histories are constant on [−0.10,0][-0.10,0]. They are specified by s(θ)=[0.60−0.400.80],[e1(θ)e2(θ)e3(θ)e4(θ)e5(θ)]=[51−342−2−5−2010−1−7−31],−0.10≤θ≤0.s(θ)= bmatrix0.60&-0.40&0.80 bmatrix T, bmatrixe_1 T(θ)\\ e_2 T(θ)\\ e_3 T(θ)\\ e_4 T(θ)\\ e_5 T(θ) bmatrix= bmatrix5&1&-3\\ 4&2&-2\\ -5&-2&0\\ 1&0&-1\\ -7&-3&1 bmatrix, -0.10≤θ≤ 0. These values satisfy all conditions of Theorem 2.4. Therefore, the network achieves global exponential leader-follower synchronization under the proposed hybrid control. Figure 2 shows that all synchronization errors converge to zero under the hybrid control. The maximum error at t=12t=12 s is 4.3163×10−74.3163× 10^-7. For comparison, all control and inter-node coupling are removed in Figure 3. The error curves remain separated, and the maximum error at t=12t=12 s is approximately 8.45278.4527. Figure 2: Synchronization errors ‖xi(t)−s(t)‖2\|x_i(t)-s(t)\|_2 of the five followers under the proposed hybrid control. Figure 3: Synchronization errors ‖xi(t)−s(t)‖2\|x_i(t)-s(t)\|_2 of the five followers without control. 4 Conclusion Andy provides an end-to-end, verification-centered workflow for mathematical research. It connects literature-grounded problem formulation, research-value evaluation, proof-route selection, DAG-based decomposition, independent local verification, targeted repair, and final proof assembly. The case study verifies that Andy can develop a technically meaningful problem from an established result and carry it through theorem construction, proof, and numerical illustration. Its explicit dependency structure keeps the process traceable, localizes failed steps, and preserves verified steps during revision. Starting from the self-triggered impulsive consensus result in [1], Andy formulated a global exponential leader-follower synchronization problem for delayed heterogeneous networks with switching communication topologies. The control-theoretic contribution is the recovery-window design. After each delayed impulse, model-matching feedback temporarily removes the delayed error channel until the pre-impulse history leaves the active delay interval. This separation permits a quadratic Lyapunov estimate during recovery and restores the full Lyapunov-Krasovskii functional afterward. It brings switching topology, impulse execution delay, and self-triggered updates into one exponential-synchronization proof. Sufficient conditions were established, Zeno behavior was excluded for both timing sequences, and the numerical example illustrates the expected synchronization behavior. Future work will investigate CrewAI-style role-based multi-agent orchestration in which the solver, evaluator, and human-like monitor operate as distinct agents. Independent roles can reduce correlated self-evaluation errors and make disagreements explicit. Parallel exploration can increase proof-route coverage and shorten the time spent on unsuccessful branches. Role-specific messages and certificates can also improve traceability, while dynamic routing, escalation, and cross-checking can make the workflow more robust to errors from a single model or reasoning route. References [1] S. Hong, Y. Zhang, Consensus of switched delay multi-agent systems via self-triggered impulsive control, International Journal of Control 95 (12) (2022) 3252–3261. doi:10.1080/00207179.2021.1970815. [2] B. Romera-Paredes, M. Barekatain, A. Novikov, M. Balog, M. P. Kumar, E. Dupont, F. J. R. Ruiz, J. S. Ellenberg, P. Wang, O. Fawzi, P. Kohli, A. 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