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](../research-planning/), [idea-generation](../idea-generation/)
• Downstream: [experiment-code](../experiment-code/), [algorithm-design](../algorithm-design/)
• See also: [math-reasoning](../math-reasoning/)