Atomic Decomposition
Decompose research ideas into atomic concepts with math formula <-> code implementation mapping.
Input
$0— Research idea, paper, or method description
References
- Decomposition prompts and workflow:
~/.claude/skills/atomic-decomposition/references/decomposition-prompts.md
Workflow (from AI-Researcher Survey Agent)
Step 1: Break Down into Atomic Definitions
Analyze the research idea and decompose into atomic, self-contained concepts:
- Each atom should be a single concept
- Must have clear mathematical foundations
- Must be implementable in code
- Must be traceable to specific papers
Step 2: For Each Atomic Definition
A. Paper Survey (Math Formula)
- Search papers for the mathematical formulation
- Extract the exact LaTeX formula
- Note assumptions and constraints
- Record reference papers
B. Code Survey (Implementation)
- Search codebases for implementations
- Extract the corresponding code
- Note implementation details and variations
- Record reference repositories
C. Create Knowledge Entry
{
"definition": "Kernelized Gumbel-Softmax Operator",
"math_formula": "Z = \\text{softmax}((\\log \\pi + g) / \\tau), g \\sim \\text{Gumbel}(0,1)",
"code_implementation": "def gumbel_softmax(logits, tau=1.0): ...",
"reference_papers": ["Paper Title 1"],
"reference_codebases": ["github_user/repo_name"],
"assumptions": ["Differentiable relaxation of discrete sampling"],
"connections": ["Used in Component X of the proposed method"]
}Step 3: Compile Knowledge Base
- Merge all atomic definitions into a structured knowledge base
- Verify consistency: every math formula has a code implementation
- Verify completeness: every code module traces to a formal definition
- Identify any gaps (formulas without code, or code without theory)
Rules
- Each atomic definition must be specific enough to trace to concrete formulas and code
- Do not skip or combine definitions — analyze each separately
- If unsure about atomicity, err on the side of breaking down further
- Document breakdown reasoning before analysis
- Every mathematical concept in the paper must have verified code
- Every code module must trace back to a formal mathematical definition
Related Skills
- Upstream: research-planning, idea-generation
- Downstream: experiment-code, algorithm-design
- See also: math-reasoning
