Theory Design

Theory-Design — Paper-Type-Aware Theoretical Framework Design

> 🔬 Model Recommendation: Run this skill on Claude Opus for best results. > Framework design requires deep reasoning across mathematical and methodological > dimensions. If your session is not on Opus, run /model opus.

Goes from "I have a new research topic" to a structured FRAMEWORK_DESIGN.md that downstream skills can consume. The skill's central insight: the logical order of theoretical-framework design depends on what kind of paper you're writing.

Context: $ARGUMENTS

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Pipeline Position

research-refine  →  /theory-design  →  /proof-writer  →  /theory-simulation  →  /proofcheck  →  /theory-sharpen  →  /proof-repair
(rough idea)        (framework)        (theorems)        (verify)              (audit)         (improve)           (fix)

This skill is the planning layer — chooses the framework, declares assumptions, sets target results — before any theorem is written or experiment run.

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Step 0: Declare paper type (MANDATORY first decision)

Statistics papers come in three types with fundamentally different logical orders. The skill REFUSES to proceed until the user declares type. If unsure, the skill helps classify based on the topic description.

The three paper types

| Type | Contribution | Theory's role | Estimator | Lower bound | Simulation | Real data | |------|-------------|---------------|-----------|-------------|-----------|-----------| | THEORY | New theoretical tool, explanation of phenomenon, impossibility result | The contribution itself | Often none/abstract | Often equally important to upper bound | Illustrative | Rarely | | METHODOLOGY | A new method (estimator/test/algorithm) | Guarantees statistical correctness of method | The centerpiece | Sometimes | Validating | Usually demonstrating | | APPLICATION | New empirical insight on scientific data | Supports / justifies method choice | Uses existing | Almost never | Almost never | The main event |

Decision aid (if user is unsure)

Ask in order:

1. Is your contribution primarily:

• a new mathematical insight / theorem / theoretical tool → THEORY
• a new procedure / algorithm / estimator → METHODOLOGY
• a new finding from analyzing real data → APPLICATION

1. What would the "main result" section show?

• A theorem (rate, distribution, lower bound, characterization) → THEORY
• Properties of a procedure (consistency, rate, coverage of the procedure) → METHODOLOGY
• Tables/figures of empirical estimates and p-values from real data → APPLICATION

1. What does the reviewer most care about?

• Mathematical novelty and proof rigor → THEORY
• Whether the method works better than competitors + has theoretical backing → METHODOLOGY
• Substantive scientific conclusion + valid inference → APPLICATION

If still unclear: the topic might be a hybrid; force a choice and revisit.

Mode routing

Based on declared type, the skill runs ONE of three workflows:

• Theory mode → Step T1-T7
• Methodology mode → Step M1-M7
• Application mode → Step A1-A7

But FIRST, regardless of mode, run Step 0.5.

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Step 0.5: MANDATORY Literature Anchoring

This step is NON-NEGOTIABLE. A theoretical framework designed without literature context is almost certainly either:

• Reinventing what's already done
• Deviating from field conventions for no reason
• Missing the "theoretical inertia" — how the community frames this problem
• Unpositioned — making it hard to assess the contribution

Build the literature anchor BEFORE any phase decision. Every subsequent phase must reference back to this anchor when making a choice.

0.5.cache: Cache-consult first (mandatory)

Before invoking any web search, consult the durable literature cache at ~/.claude/literature_cache/. The protocol lives in stat-shared-references/literature-cache-protocol.md (router with Minimum Load Map). For theory-design Step 0.5 the typical loads are:

• literature-cache-protocol.md (router) — always.
• citation-purpose-protocol.md — for lineage_positioning and benchmark_claim role identification of the anchor papers.
• applicability-axes.md — when matching cached papers to the topic signature's axis declarations (data structure, regime, framework, etc.).

Procedure: read the INDEX (and any relevant domain shards under ~/.claude/literature_cache/domains/) to identify which papers in the topic's theoretical line are already cached at source_checked or higher. Result-scoped loads only. The cache hits become the starting set; the four parallel-agent searches in 0.5B target only the cache misses.

Cache hits at independently_checked or higher are admissible as lineage anchors and benchmarks without a fresh fetch. Cache hits at source_checked are admissible for background and prose-only positioning; promotion to independently_checked is required if the paper enters the positioning argument as a benchmark.

0.5A: Topic signature for search

Build a structured topic signature for the search queries:

Topic signature:
- Primary subject:     [1-3 word phrase, e.g., "treatment effect estimation"]
- Method/technique:    [e.g., "doubly robust", "Poisson equation", "lasso", "M-estimation"]
- Data structure:      [iid / TS / mixing / Markov / panel / spatial / sequential / network]
- Modeling framework:  [parametric / semiparametric / nonparametric]
- Regime:              [classical / proportional / high-d / non-asymptotic / online]
- Application area (if applicable): [causal inference / genomics / finance / etc.]

0.5B: Multi-source T1 literature search (parallel)

Run in parallel — use Agent tool for each search agent:

Agent 1: T1 statistics journals (last 5 years preferred)

Search Semantic Scholar API:
  query = [topic signature components]
  year = current_year - 5 .. current_year
  venue ∈ {Annals of Statistics, JASA T&M, JASA ACS, AOAS, JRSS-B, Biometrika,
            Bernoulli, EJS, Statistica Sinica, Biostatistics, JCGS}
  fields = title, authors, year, abstract, venue, citationCount, externalIds

Apply filters:
  - Sort by recency; prefer last 3 years
  - Drop low-citation older papers
  - Keep top 10-15 most-relevant

Agent 2: T1 ML/AI conferences (last 5 years preferred)

WebSearch + Semantic Scholar:
  query = [topic signature components]
  venue ∈ {NeurIPS, ICML, ICLR, COLT, AISTATS, JMLR, UAI}
  
Same filtering rules.

Agent 3: T1 econometrics journals (if applicable)

Search:
  venue ∈ {Econometrica, Journal of Econometrics (JOE), Review of Economic Studies,
            Quantitative Economics, JBES, Econometric Theory, RFS, JF}

Agent 4: Highly-cited "consensus" papers (last 10 years)

Search:
  query = broader topic signature
  sort = citationCount desc
  year = current_year - 10 .. current_year
  limit = 10

Purpose: identify what the field considers CANONICAL recent work
(papers that define the current framework).

0.5C: Extract from each found paper

For each paper, extract structured information:

## [Paper N] Author (Year, Venue, citations)

### Problem framing
- How is the problem stated?
- What gap does it address?
- One-sentence contribution

### Theoretical anchor
- Data structure used
- Modeling framework
- Asymptotic regime
- Target estimand/object

### Assumption profile
- Key assumptions (≤5)
- Anything unusual or contested

### Result type
- Rate / asymptotic distribution / coverage / lower bound / structural recovery?

### Proof technique
- Main tool used

### Position in literature
- Direct predecessor it extends
- Alternative approach it competes with

Compile into papers//design/LITERATURE_ANCHOR.md.

0.5D: Identify the "theoretical inertia"

From the extracted papers, identify the current consensus framework:

## Theoretical Inertia of the Field

### Default data structure: [most common across recent T1 papers]
Example: "Most recent CATE papers use i.i.d. observations even when
applications are clustered."

### Default modeling framework: [most common]
Example: "Semiparametric framework with infinite-dim nuisance is dominant
for treatment effect since Robins-Rotnitzky-Zhao (1994); pure parametric
is now rare."

### Default asymptotic regime: [most common]
Example: "Non-asymptotic high-d bounds with sparsity have become standard
in the last 5 years; classical asymptotic n→∞ with d fixed is now seen
as a special case to confirm."

### Default proof technique: [most common]
Example: "Cross-fitting + orthogonal scores is now the dominant proof
technique in this subfield (Chernozhukov et al. 2018)."

### Default contribution shape
Example: "Recent papers tend to: (a) propose a new method, (b) prove
n^{-1/2} rate under semiparametric assumptions, (c) demonstrate finite-sample
performance via simulation."

This is the inertia — the path of least resistance for the field. You can either follow it (lower friction in review) or deviate from it (higher reward but must justify the deviation).

0.5E: Identify positioning options

For your contribution, where does it sit relative to the inertia?

## Positioning Options

### Option 1: INCREMENTAL — refine within the inertia
- Adopts default data structure, framework, regime
- Provides a sharper rate, weaker assumption, OR new estimator in the standard frame
- Easier to review and publish (referees see a familiar landscape)
- Lower-novelty perception unless the refinement is technically substantial

### Option 2: LATERAL — same problem, different angle
- Same problem, but pick an alternative framework or regime
- Example: most CATE papers use cross-fitting; you might use posterior contraction
- Must justify why your angle reveals something the standard angle misses
- Higher review difficulty (referee needs to be familiar with your alternative)

### Option 3: DISRUPTIVE — challenges the inertia
- Argues the standard framework is wrong / suboptimal / mis-applied here
- Requires either (a) a counterexample showing standard framework fails, or
  (b) a new framework that supersedes the standard
- Highest reward, highest risk; usually requires a paper-length argument for the
  reframing itself

For each option, also identify:

• Which T1 venues are most receptive
• Which 3-5 reference papers should be cited for positioning

0.5F: Anchor → design constraints

The literature anchor feeds into every subsequent phase as constraints:

## Constraints derived from anchor

For Step 1 (problem framing):
- The motivation must distinguish from [list 3 most similar papers]
- The gap must be precisely articulated; vague gaps will be attacked

For Step 2-3 (model/framework choice):
- If you adopt the inertia: cite [canonical papers]
- If you deviate: justify deviation with [specific reasoning]

For Step 5-6 (target results / proof):
- Your rate must beat / match / explicitly differ from [best known: list]
- Your proof technique should either use [dominant tool] or justify why not

For Step 7 (downstream connections):
- Specify which existing papers your work supersedes or complements

0.5G: Mandatory user confirmation

Present the LITERATURE_ANCHOR.md to the user. Force confirmation:

"Here is the literature anchor for your topic.

  - X recent T1 papers identified
  - Theoretical inertia: [summary]
  - Recommended positioning: [option]

Do you confirm this anchor before proceeding to framework design?"

User can: confirm / correct misreadings / add papers / change positioning.

Without explicit confirmation, the skill REFUSES to proceed to Step T1/M1/A1.

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THEORY MODE (T1-T7)

Centerpiece: the theorem itself. Estimator often degenerate or absent.

T1: Phenomenon / object identification

Q1.1: What mathematical phenomenon, object, or impossibility are you trying to
      characterize, explain, or formalize?
      (Examples: BBP transition in high-dim regression; new concentration
       inequality for dependent data; minimax lower bound for a new class)

Q1.2: Is this an explanation of an observed effect, the introduction of a new
      abstraction, or both?

Q1.3: Why is the current theoretical toolkit insufficient?
      State precisely the technical step that cannot be handled by existing tools.

T2: Mathematical landscape mapping

Q2.1: What existing theoretical frameworks are most relevant?
      (Empirical process / RKHS / random matrix / convex analysis / information
       theory / Markov chains / ...)

Q2.2: Which tools from each framework can be reused?
      Which need extension or replacement?

Q2.3: Are there sibling results in adjacent fields (probability, optimization,
      info theory) that solved analogous structures?

Q2.4: Do a quick T1 literature scan for related results in the last 5 years.
      Save findings to FRAMEWORK_DESIGN.md as the "landscape" section.

T3: Conceptual framework & notation

Q3.1: At what level of abstraction does the theory live?
      (Pointwise theorem / parameter class / function class / operator level)

Q3.2: What is the right notation system?
      Compatible with which existing literature?
      Where to introduce new notation? (do so sparingly)

Q3.3: What is the underlying probability space / σ-algebra / filtration setup
      if dependence or sequential structure is involved?

T4: Formal problem setup

Q4.1: Define the spaces, parameter classes Θ, function classes F, etc.
      Be explicit about: domain, range, topology, measure.

Q4.2: Regularity on the PROBLEM (not on methods):
      - Smoothness of underlying object?
      - Identifiability conditions?
      - Boundedness / tail behavior?

Q4.3: Asymptotic regime (which path is taken to ∞?)
      Same axes as theory-sharpen: classical / proportional / sparse / non-asymptotic / online.

T5: Target theorems (conjectures + impossibility)

This is where theory papers differ most from methodology: both upper and lower bounds are equally first-class citizens.

Q5.1: Upper-bound conjecture: what positive result do you AIM to prove?
      Write a placeholder theorem statement.

Q5.2: Lower-bound conjecture: what impossibility result is needed to certify the
      rate / characterization is sharp?
      (Minimax via Fano / Le Cam / Assouad; or worst-case lower bound)

Q5.3: Are upper and lower bounds going to match? (If not, the gap is itself a
      contribution or a future-work problem.)

Q5.4: Auxiliary theorems: are there secondary results (consistency, structural
      characterization, phase transitions) that strengthen the paper?

Q5.5: For each target theorem: sketch the regime / parameter range where it should hold.

T6: Proof strategy & lemma scaffold

Q6.1: High-level path: direct / contradiction / induction / coupling /
      reduction / construction / variational?

Q6.2: Major decomposition (if any): e.g., bias + variance, oracle + remainder,
      martingale + drift, signal + noise, low-rank + sparse.

Q6.3: Key tools needed: chaining / localization / Bernstein / Poisson equation /
      decoupling / spike-and-slab / random matrix concentration / etc.

Q6.4: Anticipated hardest step: where do you expect the proof to fight back?
      Have related papers handled similar difficulties?

Q6.5: Lemma scaffold: list 3-8 auxiliary lemmas you anticipate needing.
      (proof-writer can later be invoked to draft each)

Q6.6: Sketch a proof outline (≤ 1 page) BEFORE detailed proving.

T7: Connections (open to downstream)

Q7.1: What method designs does this theory enable / inform?
      (Theory paper authors should NOT design the method here; they should
       enumerate what becomes possible.)

Q7.2: Who will cite this paper? (theorists / methodologists / applied users)
      What is the chain of impact?

Q7.3: Open problems your theory raises but doesn't solve.
      These go in the Di

…

## Source & license

This open-source skill is cataloged on AgentStack and links to its original source — we do not rehost the code.

- **Author:** [gyf9712](https://github.com/gyf9712)
- **Source:** [gyf9712/stat-theory-skills](https://github.com/gyf9712/stat-theory-skills)
- **License:** MIT

Install and usage instructions live in the source repository linked above.