| Implement the following workflow in great looking practical UI: # AI Model Collaboration Cookbook for Math Research Development A sophisticated workflow utilizing multiple AI models in a collaborative process to develop, refine, and formalize a mathematical research idea. This cookbook provides developers with practical API call examples for each AI service mentioned in the workflow. ## GPT-3.5/4 (OpenAI) GPT models from OpenAI initiate the research process by generating initial ideas and responding to feedback. ### Sample cURL Command ```bash curl https://api.openai.com/v1/chat/completions \ -H "Content-Type: application/json" \ -H "Authorization: Bearer YOUR_OPENAI_API_KEY" \ -d '{ "model": "gpt-4o", "messages": [ { "role": "system", "content": "You are a helpful mathematics research assistant." }, { "role": "user", "content": "I'm exploring new approaches to mathematical optimization in topological spaces. Can you suggest 7 potentially novel research directions?" } ], "temperature": 0.7 }' ``` ### Sample Response ```json { "id": "chatcmpl-123ABC", "object": "chat.completion", "created": 1714675020, "model": "gpt-4o", "choices": [ { "index": 0, "message": { "role": "assistant", "content": "Here are 7 potentially novel research directions for mathematical optimization in topological spaces:\n\n1. Persistent Homology for Optimization Landscapes: Applying persistent homology to characterize the topological structure of non-convex optimization landscapes...\n\n2. Sheaf-Theoretic Approaches to Distributed Optimization: Developing frameworks where optimization variables are sections of sheaves...\n\n[...additional ideas...]" }, "finish_reason": "stop" } ], "usage": { "prompt_tokens": 65, "completion_tokens": 420, "total_tokens": 485 } } ``` ## Gemini API (Google) Gemini serves as a critical evaluation step, identifying errors and problems in the developing research. ### Sample cURL Command ```bash curl -X POST \ https://generativelanguage.googleapis.com/v1beta/models/gemini-pro:generateContent \ -H "Content-Type: application/json" \ -H "Authorization: Bearer YOUR_GEMINI_API_KEY" \ -d '{ "contents": [ { "parts": [ { "text": "Review the following mathematical approach for errors and conceptual problems:\n\nTopological Optimization Through Sheaf Cohomology\n\nWe propose a novel optimization framework for non-convex functions on manifolds using sheaf cohomology. The key insight is that optimization trajectories can be represented as sections of sheaves..." } ] } ], "generationConfig": { "temperature": 0.2, "topK": 40, "topP": 0.95, "maxOutputTokens": 8192 } }' ``` ### Sample Response ```json { "candidates": [ { "content": { "parts": [ { "text": "I've reviewed your mathematical approach and identified several issues:\n\n1. Conceptual Error: The proposed sheaf cohomology framework assumes that optimization trajectories form global sections, but this isn't generally true for non-convex functions on arbitrary manifolds.\n\n2. Mathematical Inconsistency: Your definition of the obstruction cocycle doesn't account for singularities in the gradient field...\n\n[...additional feedback...]" } ], "role": "model" }, "finishReason": "STOP", "index": 0 } ], "promptFeedback": { "safetyRatings": [ { "category": "HARM_CATEGORY_DANGEROUS_CONTENT", "probability": "NEGLIGIBLE" } ] } } ``` ## Grok-3 API (xAI) Grok-3 provides a fresh perspective on the research, identifying fundamental errors and suggesting alternative approaches. ### Sample cURL Command ```bash curl -X POST \ https://api.grok.x/v1/chat/completions \ -H "Content-Type: application/json" \ -H "Authorization: Bearer YOUR_GROK_API_KEY" \ -H "Cookie: x-anonuserid=YOUR_ANON_USER_ID; x-challenge=YOUR_CHALLENGE; x-signature=YOUR_SIGNATURE; sso=YOUR_SSO; sso-rw=YOUR_SSO_RW" \ -d '{ "messages": [ { "role": "system", "content": "You are a critical mathematical researcher who can identify fundamental issues in approaches." }, { "role": "user", "content": "Here is a detailed report on a mathematical research approach. Please identify any fundamental errors and suggest alternative directions if needed:\n\nConsensus Approach: Topological Optimization Through Sheaf Cohomology with Persistence Diagram Analysis...[detailed mathematical content]" } ], "model": "grok-3", "temperature": 0.7 }' ``` ### Sample Response ```json { "id": "chatcmpl-XYZ789", "object": "chat.completion", "created": 1714675100, "model": "grok-3", "choices": [ { "index": 0, "message": { "role": "assistant", "content": "I've identified a fundamental flaw in your approach that completely undermines the central thesis:\n\nThe primary error lies in the assumption that sheaf cohomology obstructions can be systematically minimized using gradient descent. This overlooks a critical issue: the obstruction classes themselves form a discrete invariant, not a continuous one amenable to gradient-based optimization.\n\nInstead, I suggest pivoting to a completely different direction: consider using discrete Morse theory combined with stratified spaces. This approach would allow you to...\n\n[...detailed alternative approach...]" }, "finish_reason": "stop" } ] } ``` ## Claude API (Anthropic) Claude finalizes the process by outlining and drafting a formal academic paper with extensive LaTeX code. ### Sample cURL Command ```bash curl -X POST \ https://api.anthropic.com/v1/messages \ -H "Content-Type: application/json" \ -H "x-api-key: YOUR_CLAUDE_API_KEY" \ -H "anthropic-version: 2023-06-01" \ -d '{ "model": "claude-3-7-sonnet", "max_tokens": 4000, "messages": [ { "role": "user", "content": "Based on the following consensus research report, please outline a formal academic paper following standard mathematics journal conventions. After I approve the outline, I will ask you to generate LaTeX for each section according to my style/notation conventions.\n\nConsensus Report: Discrete Morse Theory Approaches to Non-Convex Optimization on Stratified Spaces...[detailed research content]" } ] }' ``` ### Sample Response ```json { "id": "msg_01ABCxyz", "type": "message", "role": "assistant", "content": [ { "type": "text", "text": "# Proposed Paper Outline: \"Discrete Morse Theory for Non-Convex Optimization on Stratified Spaces\"\n\n## 1. Introduction\n- Motivation from optimization challenges on manifolds\n- Limitations of current approaches\n- Brief overview of stratified spaces and discrete Morse theory\n- Statement of main contributions\n\n## 2. Preliminaries\n- 2.1 Stratified Spaces\n- 2.2 Discrete Morse Theory\n- 2.3 Optimization on Manifolds\n\n## 3. Discrete Morse Optimization Framework\n- 3.1 Discretization of the Domain\n- 3.2 Morse Function Construction\n- 3.3 Critical Point Analysis\n\n[...additional sections...]\n\n## 7. Conclusion and Future Work\n\nDoes this outline meet your expectations? Once approved, I can proceed with generating LaTeX for each section according to your style conventions." } ], "model": "claude-3-7-sonnet", "stop_reason": "end_turn" } ``` ## Iterative Development Flow The complete workflow involves iterative refinement across these models: 1. **Initial Research Ideas (GPT)**: Generate potential research directions 2. **First Evaluation (Gemini)**: Identify errors and provide feedback 3. **Refinement (GPT)**: Process feedback and revise approach 4. **Second Evaluation (Gemini)**: Find additional problems 5. **Iterative Convergence**: Repeat steps 3-4 until reaching consensus 6. **Fresh Perspective (Grok-3)**: Identify fundamental issues and redirect if needed 7. **Final Consensus**: Synthesize insights from all models 8. **Academic Formalization (Claude)**: Outline and draft comprehensive LaTeX paper ## Implementation Notes 1. **Authentication**: Each service requires different authentication methods: - OpenAI: Bearer token authentication[^7][^8] - Gemini: API key authentication via Bearer token[^4] - Grok-3: Cookie-based authentication (unofficial API)[^5] - Claude: API key via x-api-key header 2. **Error Handling**: Implement robust error handling for each API call, as reliability may vary across services. 3. **Rate Limiting**: All services implement rate limits. Monitor the response headers like `x-ratelimit-remaining-requests` to avoid exceeding limits[^8]. 4. **Context Management**: For complex mathematical research, models may need longer context windows. Select appropriate model versions that support longer contexts. 5. **Response Processing**: Parse and extract key insights from each model response to feed into the next stage of the workflow. ## Conclusion This multi-model approach leverages the unique strengths of each AI system to create a robust research development pipeline. By validating ideas across different models, researchers can identify and address conceptual flaws more effectively than with any single model. The workflow represents an emerging pattern of AI collaboration that might become standard practice for complex intellectual work. Developers implementing this workflow should monitor the evolution of these APIs, as capabilities and endpoints may change as the underlying models are updated. |