theory-sharpen

Theory-Sharpen — Systematic Theoretical Improvement Assessment

🔬 Model Recommendation: Run this skill on Claude Opus for best results. Framework classification, assumption-relaxation analysis, and rate-sharpening all require deep mathematical reasoning. If your session is not on Opus, run /model opus before invoking. Literature search and Codex cross-review will use Opus sub-agents.

Go beyond "is the proof correct?" to ask "can the theory be stronger, sharper, and better aligned with the model, the literature, and the experiments?"

Pipeline position:

/proofcheck → /proof-repair → /theory-sharpen → /proof-writer
  Correct?      Fix issues       Improve theory     Write new proofs

This skill can also run standalone on any paper with theoretical results.

Context: $ARGUMENTS

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Core Philosophy

A good theory paper is evaluated on three axes:

1. Strength: Are assumptions as weak as possible? Are rates as sharp as possible?
2. Alignment: Does the theory match what the model actually provides and what the experiments actually test?
3. Positioning: How does the result compare to the best known results in the literature?

This skill systematically audits all three axes and produces an actionable improvement roadmap.

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Step 0: Ingest & Map the Theory-Model-Experiment Triangle

0A: Locate Inputs

Parse $ARGUMENTS. Accept:

• A .tex file path → read directly
• A paper directory → read paper.tex + any /proofcheck audit if it exists
• If /proofcheck audit exists, leverage assumption_ledger.md, theorem_inventory.md, dependency_graph.md for a head start

0B: Extract the Three Pillars

Read the paper and extract three structured inventories:

Pillar 1: Theory — What the theorems claim

ID	Result	Assumptions used	Rate / bound	Constants	Regime	Location

For each result, record:

• Exact assumptions (named + implicit)
• Convergence rate or bound (e.g., $O(n^{-1/2})$, $O_P(n^{-2/(2+d)})$)
• Whether rate is minimax, near-minimax, or suboptimal (if known)
• Sample size / dimension regime (e.g., $n \gg d$, $n \gg d^2$, fixed $d$)
• Constants: universal, dimension-dependent, problem-parameter-dependent?