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

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.

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

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?

Pillar 2: Model — What the model actually provides

Read the model definition (usually Section 2 or "Setup") and inventory:

• What distributions / processes does the model generate?
• What structural properties does the model have? (linearity, sparsity, convexity, Markov, etc.)
• What regularity does the model provide? (smoothness, tail behavior, mixing, etc.)
• Are there properties the model has that the theory does NOT exploit?
• Are there properties the theory assumes that the model does NOT guarantee?

Pillar 3: Experiments — What the empirical evaluation actually tests

Read the experiment section and inventory:

• What data distributions / models are tested?
• What sample sizes and dimensions are used?
• What metrics are reported?
• Which theoretical predictions are actually verified empirically?
• Which theoretical conditions are violated in practice?

0C: Build the Alignment Map

Produce a Theory-Model-Experiment Alignment Matrix:

## Alignment Matrix

| Assumption / Claim | Theory requires | Model provides | Experiments test | Alignment |
|--------------------|----------------|----------------|-----------------|-----------|
| i.i.d. data | Yes (Assumption 1) | Yes (by construction) | Yes | ALIGNED |
| Sub-Gaussian tails | Yes (Assumption 3) | Gaussian (stronger) | Heavy-tailed also tested | THEORY-EXP GAP |
| Dimension fixed | Yes (implicit in rate) | Not restricted | d=50,100,500 tested | THEORY-EXP GAP |
| Strong convexity | Yes (Assumption 2) | Only local convexity | Non-convex also tested | THEORY-MODEL GAP |
| Rate O(n^{-1/2}) | Theorem 1 | — | Matches empirically | ALIGNED |
| Uniform over Θ | Theorem 2 claims | Model has compact Θ | Fixed θ only | THEORY-EXP GAP |

Legend:
  ALIGNED         — theory, model, and experiments all agree
  THEORY-MODEL GAP — theory assumes something the model doesn't guarantee (or vice versa)
  THEORY-EXP GAP  — theory doesn't cover what experiments test (or vice versa)
  MODEL-EXP GAP   — model definition differs from experimental setup
  EXPLOITABLE     — model provides something stronger that theory doesn't use

Step 0.5: Framework Classification (MANDATORY — must complete before any relaxation analysis)

Relaxation pathways are framework-dependent. Trying to "relax i.i.d. to mixing" is meaningless for a paper on cross-sectional regression. Classify the paper on three axes BEFORE filtering the pathway library.

Auto-Detection + Literature-Anchored Confirmation

The classification proceeds in three sub-steps:

0.5A: Paper-internal inference (agent) Read paper sections (intro, model setup, assumptions, theorems) and infer a draft classification on three axes from the paper's own statements.

0.5B: Literature-anchored validation (agent — parallel literature search) Search for RECENT T1-venue papers in the same topic, and check whether the classification is consistent with how the field currently frames similar problems.

0.5C: Forced user confirmation (mandatory) Present BOTH the paper-internal classification AND the literature evidence to the user. Do not proceed until user confirms.

0.5B: Literature-Anchored Validation (details)

After drafting the classification in 0.5A, build a topic signature and run a targeted multi-source literature search to validate.

Topic signature components:

• Subject domain (e.g., "treatment effect estimation", "stochastic optimization", "high-dim regression", "Markov chain Monte Carlo", "online learning")
• Main technique keywords (e.g., "M-estimation", "lasso", "kernel regression", "Poisson equation", "doubly robust")
• Data structure (from Axis 1)
• Framework (from Axis 2)
• Regime (from Axis 3)

Search strategy (parallel agents, T1 priority, recent first):

Agent 1: Recent T1 journal papers (last 3 years preferred, 5 years max)
  Search: [topic] + [technique] + [framework keyword]
  Venues filter: AoS, Biometrika, JASA, JRSS-B, Econometrica, JOE, JBES, RESS,
                  JMLR, Annals of Probability, Bernoulli, EJS, Stat Sinica
  Source: WebFetch on Semantic Scholar API with venue + year filter
  
Agent 2: Recent T1 conference papers (last 3 years preferred)
  Search: same topic signature
  Venues filter: NeurIPS, ICML, ICLR, COLT, AISTATS, AAAI, KDD, UAI
  Source: WebSearch with site filter + arXiv with venue annotation

Agent 3: Most cited recent papers in this framework (last 5 years, ≥30 citations)
  Search: broader topic
  Source: Semantic Scholar sorted by citation count
  Purpose: identify the "current consensus" framework

For each found paper, extract:

• Title, authors, year, venue, citation count
• How THEY classified their problem on the three axes (data type, framework, regime)
• What assumptions they use (compare with our paper's assumptions)
• What rates they prove
• What techniques they use

Build the Literature Anchor Table:

## Recent T1 Papers in This Framework (top 5-10)

| # | Paper | Venue (Year) | Cite | Their Axis 1 | Their Axis 2 | Their Axis 3 | Key technique |
|---|-------|-------------|------|--------------|--------------|--------------|---------------|
| 1 | Chen & Li (2024) | AoS | 25 | mixing TS | SEMI | CLA | Orthogonal scores + HAC |
| 2 | Zhang et al. (2023) | Econometrica | 80 | mixing TS | SEMI | CLA + FS | Cross-fitting + DML |
| 3 | Wang & Park (2024) | JASA | 12 | mixing TS | NONPAR | CLA | Sieve + peeling |
| 4 | Kim (2023) | NeurIPS | 50 | i.i.d. | SEMI | HD | RSC + LASSO |
| 5 | ... |

Classification cross-check:

If the literature evidence DISAGREES with the paper-internal inference, flag this explicitly:

## Classification Validation Report

### Axis 1 (Data): paper says i.i.d.
Literature check: Of 10 recent papers on the same topic ("dynamic treatment effect 
under adaptive randomization"), 7 frame it as sequential-adaptive, 2 as i.i.d.-with-
conditioning, 1 as Markov.
Verdict: ⚠ MISMATCH — paper's i.i.d. framing may be unconventional for this topic.
Recommendation: Reclassify as `SEQ` and re-examine i.i.d. assumption.

### Axis 2 (Framework): paper says parametric
Literature check: 8/10 recent papers use semiparametric framework with nuisance.
Verdict: ⚠ POSSIBLE MISMATCH — modern treatment may have shifted to SEMI.
Note: If paper genuinely is PAR, this is fine but flag as a positioning issue.

### Axis 3 (Regime): paper says classical asymptotic
Literature check: 6/10 recent papers prove non-asymptotic bounds.
Verdict: 🟡 TREND — field is moving toward finite-sample; classical is acceptable
but consider adding finite-sample version.

Output to user (Step 0.5C — forced confirmation):

## Draft Framework Classification (please confirm or correct)

### Axis 1: Data Structure
- Paper-internal inference: [class] (confidence: HIGH/MEDIUM/LOW)
- Paper evidence: "Section 2 line 47 defines X_i as i.i.d. samples from P"
- Literature check (last 3-5 years, T1 only): X of N recent papers in this topic
  use the same framing; Y use [alternative]
- Verdict: ✅ CONSISTENT / ⚠ MISMATCH / 🟡 TREND
- Sample recent T1 papers in same framing:
  • Chen & Li (2024, AoS, 25 cites)
  • Zhang et al. (2023, Ectrica, 80 cites)

### Axis 2: Modeling Framework
- Paper-internal inference: [parametric / semiparametric / nonparametric / structured-nonpar]
- Paper evidence: "Parameter θ ∈ R^d with d fixed" / "Target β with nuisance η(·)"
- Literature check: X of N recent T1 papers use [same framework]; trend = [observation]
- Verdict: ✅ / ⚠ / 🟡
- Sample papers: ...

### Axis 3: Asymptotic Regime
- Paper-internal inference: [classical / proportional / high-d sparse / non-asymptotic / online]
- Paper evidence: "Theorem 1 states 'as n→∞' with no d dependence"
- Literature check: X of N recent papers prove [same regime]; field trend = ...
- Verdict: ✅ / ⚠ / 🟡
- Sample papers: ...

### Cross-axis consistency
- Does the (Data, Framework, Regime) triple match common combinations in literature?
- Are there standard "default" choices being deviated from? (e.g., NEMS is usually
  framed as SEMI + finite-sample, but this paper uses PAR + classical)

### Multi-regime papers
If the paper proves results in MULTIPLE regimes (common!), list all that apply.

### Literature-recommended pathway hints (preview)
Based on recent T1 papers in your framework, these relaxation directions are
TRENDING and likely to be reviewer-asked:
- [pathway 1 — supported by N recent T1 papers]
- [pathway 2 — supported by M recent T1 papers]
- [pathway 3 — emerging direction in last 12 months]

User actions after seeing this report:

1. CONFIRM the classification → proceed to Step 1 with filtered pathways
2. CORRECT one or more axes → agent updates and reruns the filter
3. ASK for more detail on a specific paper from the literature anchor table
4. REQUEST classification under a DIFFERENT venue's framing (e.g., "show me how an Econometrica reviewer would classify this vs a NeurIPS reviewer")

Literature search for classification and pathways

Query templates by framework axis, the recency and venue gating rules (prefer T1 within the last ~5 years; an older paper needs a reason), and worked pathway-relevance examples: ../stat-shared-references/theory-search-templates.md. Venue tiers are rule data in ../stat-shared-references/scripts/venue_tiers.py.

Gate: a classification or pathway asserted without literature support is a draft, not a finding. Confirm the framework classification against recent T1 papers before running any relaxation analysis on it.

Step 1: Assumption Relaxation Analysis

For EACH assumption in the theory, systematically assess whether it can be weakened within the confirmed framework from Step 0.5.

1A: Assumption Relaxation Table

| Assumption | Current form | Relaxation candidate | Feasibility | Literature support | Priority |
|-----------|-------------|---------------------|-------------|-------------------|----------|
| A1: i.i.d. | Samples are i.i.d. | β-mixing with polynomial decay | LIKELY | Doukhan (1994), Rio (2017) | HIGH if time-series app |
| A2: Strong convexity (μ>0) | ∇²f ≽ μI globally | Local strong convexity + growth | POSSIBLE | Mei et al. (2018, AoS) | MEDIUM |
| A3: Sub-Gaussian | ψ₂-norm bounded | Sub-exponential (ψ₁) | LIKELY | Vershynin (2018, Cambridge) | HIGH |
| A3: Sub-Gaussian | ψ₂-norm bounded | Bounded 4th moment only | HARD | Catoni (2012, AoS) — different technique | LOW |
| A4: Compact Θ | Θ is compact | Locally compact + growth at ∞ | POSSIBLE | van der Vaart (1998), Ch. 5 | MEDIUM |

1B: Relaxation Feasibility Assessment

For each relaxation candidate, answer these questions:

1. Which proof step breaks?

   • Trace through the proof: where is the strong assumption actually used?
   • Is it used once (easy to patch) or pervasively (hard to patch)?

2. What replacement technique exists?

   • i.i.d. → mixing: blocking technique, coupling, Berbee's lemma
   • Sub-Gaussian → heavy-tailed: truncation, median-of-means, Catoni's estimator
   • Strong convexity → local: restricted strong convexity, local Rademacher complexity
   • Compact → non-compact: localization, peeling, sieve

3. Does the rate change?

   • Some relaxations preserve the rate (e.g., i.i.d. → mixing with fast decay)
   • Some worsen the rate (e.g., sub-Gaussian → bounded moment adds log factors)
   • Record the rate change explicitly

4. Is this aligned with the model and experiments?

   • If the model is actually Gaussian, relaxing to heavy-tailed is theoretically nice but practically unnecessary → PRIORITY: LOW
   • If experiments test heavy-tailed data but theory assumes sub-Gaussian → PRIORITY: HIGH

1C: Choose relaxation pathways from the catalogue

For each assumption marked feasible in 1B, select a candidate pathway from ../stat-shared-references/relaxation-pathways.md, which tags every standard route by framework axis (data structure, modelling framework, asymptotic regime).

The selection rule, which matters more than the catalogue: filter by the Step 0.5 classification first. A pathway developed for i.i.d. parametric classical-asymptotic work usually does not transfer to a dependent-data high-dimensional setting, and proposing it wastes the author's time and signals the analysis did not read the paper's own regime. A pathway that survives the filter still needs literature support (Step 5) and a proof sketch before it is called feasible.

If no catalogued pathway survives the filter, say so: "no standard pathway applies in this framework" is a real finding, and it usually means the relaxation is research-level rather than a known technique.

Step 2: Rate Sharpness Analysis

For EACH rate claimed in the paper, assess whether it can be improved.

2A: Rate Inventory & Optimality Check

| Result | Claimed rate | Best known rate (literature) | Minimax lower bound | Gap | Sharpening possible? |
|--------|-------------|---------------------------|-------------------|-----|---------------------|
| Thm 1: LLN | O(n^{-1/2}) | n^{-1/2} (sharp for CLT) | n^{-1/2} (Cramer-Rao) | None | NO — already optimal |
| Thm 2: Estimation | O(n^{-2/5}) | n^{-2/(2+d)} (Stone 1982) | n^{-2/(2+d)} (Stone) | Yes if d>1 | POSSIBLE — check technique |
| Thm 3: Uniform bound | O(√(d log n / n)) | √(d/n) (no log) | √(d/n) (Rademacher) | log n factor | LIKELY removable |
| Cor 1: Prediction | O(n^{-1/3}) | n^{-1/2} (parametric) | Depends on model | Large gap | CHECK model assumptions |

2B: Sharpening Directions

For each improvable rate, identify the specific source of suboptimality:

Common sources of loose rates and their fixes (venue-verified with Codex):

2C: Minimax Lower Bound Cross-Check

For the main theorem, check if a matching lower bound exists:

1. Search for minimax lower bounds in the same problem class

   • Use venue-tier system from /proof-repair (T1 priority)
   • Focus: AoS, JRSS-B, JMLR, Econometrica, COLT, information-theoretic bounds

2. If lower bound matches → rate is minimax optimal; note this as a strength

3. If lower bound is better → gap exists; check if it's:

   • A fundamental gap (technique limitation) → suggest new technique from literature
   • An artifact of loose analysis → suggest tightening specific steps
   • A different model class → clarify which lower bound applies

4. If no lower bound exists → consider whether constructing one would strengthen the paper (and cite relevant methodology: Fano, Le Cam, Assouad)

2D: Rate-Under-Relaxed-Assumptions

Cross-reference with Step 1: if assumptions are relaxed, how do rates change?

| Result | Current rate | Under relaxed assumption | Rate change | Worth it? |
|--------|-------------|------------------------|-------------|-----------|
| Thm 1 | n^{-1/2} (i.i.d.) | n^{-1/2} (β-mixing, poly decay) | Same | YES — free generalization |
| Thm 2 | n^{-2/5} (sub-Gaussian) | n^{-2/5} · (log n) (sub-exp) | Extra log | MAYBE — depends on application |
| Thm 3 | √(d/n) (strong convex) | (d/n)^{1/3} (local SC) | Worse | NO — rate degradation too large |

Step 2E: Reviewer-Critical Dimensions Audit

Identified via Codex cross-review. These are dimensions that top-venue reviewers (AoS, Biometrika, Econometrica, NeurIPS/ICML/COLT) routinely demand but that assumption-relaxation and rate-sharpening alone do not cover.

For EACH dimension, check whether the paper addresses it. If not, flag as an improvement opportunity.

Audit template per dimension

For each dimension, produce:

| Dimension | Paper addresses it? | How? | Gap? | Improvement suggestion |
|-----------|-------------------|------|------|----------------------|
| Lower bounds | No | — | YES | Prove Fano lower bound for the estimation rate |
| Inference/UQ | Partially (CLT only) | Thm 4 | Partial: no bootstrap validity | Add bootstrap consistency theorem |
| Identification | Yes | Assumption 1 | None | — |
| Adaptivity | No — tuning parameter λ is oracle | — | YES | Lepski-type or cross-validation analysis |
| Computation | No — NP-hard in general | — | YES | Show benign landscape under conditions or use convex relaxation |

Step 3: Theory-Model Alignment Audit

Check whether the theoretical framework actually matches the model.

3A: Assumption-Model Match

For EACH assumption the theory uses, check:

| Assumption | Theory requires | Model provides | Match? | Issue |
|-----------|----------------|----------------|--------|-------|
| A1: i.i.d. | Independent samples | Sequential adaptive design | MISMATCH | Samples depend on past allocations |
| A2: Bounded gradient | ‖∇f‖ ≤ B | Neural net — unbounded | MISMATCH | Needs truncation or clipping argument |
| A3: Lipschitz loss | L-Lipschitz | Logistic loss — yes, L=1 | MATCH | — |
| A4: Sub-Gaussian noise | ψ₂ ≤ σ | Gaussian noise — yes | OVER-SPECIFIED | Model gives Gaussian, only need sub-G |

Match types:

• MATCH: Theory and model agree exactly
• OVER-SPECIFIED: Theory assumes more than needed; model gives more than theory uses → Opportunity: exploit the extra structure for sharper results
• MISMATCH: Theory assumes something the model doesn't provide → Problem: theorem may not apply to the paper's own model
• IMPLICIT: Assumption is used in the proof but not stated in the model → Problem: hidden assumption gap

3B: Exploitable Model Properties

When the model provides MORE than the theory uses (OVER-SPECIFIED), list improvements:

## Exploitable Properties

| Model property | Theory uses only | Could exploit for | Potential improvement |
|---------------|-----------------|-------------------|---------------------|
| Gaussian noise | Sub-Gaussian bound | Exact Gaussian tail | Remove log factors, exact constants |
| Linear model | Generic Lipschitz | Linear structure | Parametric rate n^{-1/2} instead of nonparametric |
| Sparse signal | Dense estimation | Sparsity + LASSO theory | Rate s·log(d)/n instead of d/n |
| Known covariance | Unknown Σ bounds | Plug-in Σ | Remove condition number dependence |

3C: Theory-Model Gap Resolution Strategies

For each MISMATCH, propose a resolution:

1. Weaken the assumption to match what the model actually provides (→ Step 1)
2. Strengthen the model description if the model does satisfy the condition but the paper didn't state it explicitly
3. Add a bridging lemma that derives the theoretical condition from model properties
4. Acknowledge the gap as a limitation and suggest it for future work

Step 4: Theory-Experiment Alignment Audit

Check whether the theoretical predictions are actually testable and tested.

4A: Coverage Check

| Theoretical result | Experimentally verified? | How? | Discrepancy? |
|-------------------|------------------------|------|-------------|
| Thm 1: √n-consistency | Yes — Fig 3 shows convergence | MSE vs n plot, slope ≈ -1 | ALIGNED |
| Thm 2: Uniform over Θ | No — only fixed θ tested | — | NOT TESTED |
| Thm 3: d can grow with n | Partially — d=50,100,500 but n fixed | Only 3 dimension values | WEAK EVIDENCE |
| Cor 1: Asymptotic normality | No — no QQ plots or coverage | — | NOT TESTED |
| Rate: O(n^{-1/2}) | Contradicted — empirical rate ≈ n^{-1/3} | Log-log regression in Fig 4 | CONTRADICTION ⚠ |

4B: Experimental Gap Analysis

Theory predicts but experiments don't test:

• Missing experiments → suggest what to add
• Untestable predictions → note as limitation

Experiments show but theory doesn't cover:

• Empirical success beyond theoretical regime → opportunity to strengthen theory
• Empirical failure where theory predicts success → possible theoretical error

Theory and experiments contradict:

• Empirical rate worse than theoretical rate → check:
  • Is sample size large enough for asymptotics to kick in?
  • Is a hidden constant very large?
  • Is there an error in the proof?
• Empirical rate better than theoretical rate → check:
  • Is the theoretical bound loose?
  • Is the model providing extra structure not captured by theory?

4C: Regime Relevance Check

Do the theoretical conditions match realistic experimental settings?

| Condition | Theory requires | Experiments use | Practice needs | Relevance |
|-----------|----------------|----------------|----------------|-----------|
| n ≥ C·d² | n > d² | n=1000, d=50 (n=0.4d²) | n ≈ d typical | IMPRACTICAL — threshold too high |
| ε ≤ 1/d | vanishing ε | ε = 0.1 | small but fixed ε | POSSIBLY OK |
| T → ∞ | asymptotic | T = 100 iterations | fixed budget | FINITE-SAMPLE NEEDED |
| σ known | known noise level | estimated σ̂ | unknown σ | PLUG-IN ANALYSIS NEEDED |

If theoretical conditions are impractical, suggest:

• Finite-sample versions of asymptotic results
• Conditions in terms of observable quantities (not unknown parameters)
• Adaptive procedures that don't require knowing constants

Step 5: Literature Benchmarking

Compare the paper's results against the current state of the art.

5.cache: Cache-consult first (mandatory)

Before any web search, consult the durable literature cache. Protocol in ../stat-shared-references/literature-cache-protocol.md. For Step 5 (benchmarking) the typical loads are:

• literature-cache-protocol.md (router).
• citation-purpose-protocol.md — benchmarking citations are typically benchmark_claim (rate / constant comparison) or lineage_positioning (placing the work in a methodological line). independently_checked floor for both.
• applicability-axes.md — to verify the cached competitor's axes match the paper's setting.

Read the INDEX and the domain shards relevant to the paper's primary_line (in the theoretical_lineage declared earlier). Cache hits at independently_checked cover direct competitors and assumption frontier without a fresh fetch; misses go to the three search agents below.

5A: Venue-Tiered Literature Search

Apply the venue credibility tiers from /proof-repair:

T1 Priority: AoS, JASA, JRSS-B, Biometrika, Econometrica, JOE, NeurIPS, ICML, ICLR, COLT, JMLR, Math Programming, SIAM Opt

Search strategy (parallel agents):

Agent 1: Direct Competitors

Search for papers solving the SAME problem with DIFFERENT techniques.
Queries:
- "[problem name] convergence rate" + venue filter
- "[problem name] optimal rate minimax"
- "[model class] estimation [loss function]"
Focus: results from last 5 years in T1 venues
Extract: their assumptions, their rates, their technique

Agent 2: Assumption Frontier

Search for papers solving SIMILAR problems under WEAKER assumptions.
Queries:
- "[problem name] without [strong assumption]"
- "[problem name] heavy-tailed" / "dependent data" / "high-dimensional"
- "[technique name] relaxed conditions"
Focus: which assumptions have been successfully relaxed in related work?

Agent 3: Rate Frontier

Search for the BEST KNOWN rates for this problem class.
Queries:
- "[problem class] minimax rate"
- "[problem class] optimal estimation"
- "[problem class] lower bound information-theoretic"
Focus: minimax lower bounds, matching upper bounds, rate-optimal procedures

5B: Competitive Positioning Table

## Competitive Positioning

| Paper | Venue (Tier) | Year | Assumptions | Rate | Technique | vs. Ours |
|-------|-------------|------|-------------|------|-----------|----------|
| This paper | — | — | A1-A4 | n^{-1/2} | M-estimation + Poisson eq | BASELINE |
| Chen & Li | AoS (T1) | 2022 | A1-A3 (no A4) | n^{-1/2} | Empirical process | WEAKER ASSUMPTIONS, SAME RATE |
| Zhang et al. | NeurIPS (T1) | 2023 | A1-A4 + sparsity | s·log(d)/n | LASSO-type | SHARPER UNDER SPARSITY |
| Wang | JMLR (T1) | 2021 | A1-A2 only | n^{-1/3} | Robust estimation | WEAKER ASSUMPTIONS, WORSE RATE |
| Kim & Park | Econometrica (T1) | 2023 | A1-A4 | n^{-1/2} | GMM | SAME RESULT, DIFFERENT TECHNIQUE |
| Lower bound: Tsybakov | AoS (T1) | 2009 | Nonparametric class | n^{-2/(2+d)} | Le Cam / Fano | OUR RATE IS OPTIMAL IN FIXED-d |

## Key Findings
1. Chen & Li (2022) achieved the same rate without A4 → our A4 may be removable
2. Under sparsity (Zhang 2023), sharper rate is possible → extension opportunity
3. Our rate matches the minimax lower bound → rate is tight (strength to highlight)
4. No competitor handles the adaptive design setting → our contribution is unique here

5C: Gap-to-Frontier Analysis

For each gap between this paper and the frontier:

| Gap | Frontier paper | What they achieved | What we'd need to match | Feasibility | Priority |
|-----|---------------|-------------------|------------------------|-------------|----------|
| A4 removable | Chen & Li (2022) | Same rate without A4 | Replace Lemma C.3 technique | MEDIUM | HIGH |
| Log factor | Zhang (2023) | No log in uniform bound | Use chaining instead of net | LIKELY | HIGH |
| Heavy tails | Lugosi & Mendelson (2019) | Bounded 2nd moment only | Median-of-means wrapper | HARD | MEDIUM |

Step 5B: Codex Independent Assessment (if Codex MCP available)

Send the sharpening analysis to Codex per ../stat-shared-references/codex-protocol.md for an independent read on three questions: which relaxations are genuinely feasible, whether the rates are actually sharp, and which theory-practice gaps a referee will raise first.

Reconcile per finding with an explicit disposition and reasoning. Codex disagreeing does not make a relaxation infeasible, and Codex agreeing does not make one feasible — the literature support and the proof sketch decide that. Record both positions where they differ.

Reconciliation shape and a worked exchange: ../stat-shared-references/examples/theory-sharpen-codex-example.md.

Step 6: Improvement Roadmap

Synthesize all findings (Claude + Codex) into a prioritized improvement plan.

6A: Priority Scoring

Score each potential improvement on five dimensions:

Priority = 3×Impact + 2×Feasibility + 2×Literature + 1×Alignment + 1.5×Reviewer

The "Reviewer demand" dimension addresses what top-venue reviewers actually ask for (Step 2E). Weight is 1.5× (intermediate) — not so high that it dominates Impact, but high enough to push reviewer-targeted improvements up the priority list.

Tuning tip: For a paper aimed at a SPECIFIC venue, customize the Reviewer-demand scoring by reviewer profile:

• AoS/Biometrika: weight lower bounds, inference/UQ, adaptivity higher
• Econometrica/JOE: weight identification, robustness, uniformity higher
• NeurIPS/ICML/COLT: weight computational attainability, structural guarantees higher

6B: Improvement Roadmap Table

| Rank | Improvement | Type | Impact | Feas | Lit | Align | Reviewer | Score | Key reference |
|------|------------|------|--------|------|-----|-------|----------|-------|---------------|
| 1 | Remove log factor in Thm 3 | Rate-Sharpen | 4 | 5 | 5 | 3 | 4 | 41.0 | Talagrand (2014) |
| 2 | Remove Assumption A4 | Assumption-Relax | 5 | 3 | 4 | 4 | 5 | 40.5 | Chen & Li (2022, AoS) |
| 3 | Heavy-tail extension | Assumption-Relax | 3 | 3 | 5 | 5 | 4 | 36.0 | Lugosi & Mendelson (2019, AoS) |
| 4 | Finite-sample version of Thm 1 | Regime-Extend | 4 | 4 | 3 | 5 | 3 | 35.5 | — (needs new analysis) |
| 5 | Exploit Gaussian structure | Model-Exploit | 2 | 5 | 5 | 2 | 2 | 31.0 | — (standard) |

Every row carries all five 6A dimensions, including Reviewer demand; a table that drops
a dimension silently changes the ranking the formula produces.

6C: Per-Improvement Specification

For each top-ranked improvement, write:

## Improvement I-1: Remove log factor in Theorem 3

### Current state
Theorem 3 claims ‖θ̂ − θ*‖ = O(√(d log n / n)) uniformly over Θ.
The log n factor comes from a union bound over an ε-net in the proof of Lemma D.4.

### Target state
Replace with ‖θ̂ − θ*‖ = O(√(d / n)), matching the minimax lower bound.

### Technique
Replace ε-net + union bound (Lemma D.4, lines 842-867) with Dudley's entropy
integral or generic chaining (Talagrand 2014).

### Which proof steps change
- Lemma D.4: replace entirely with chaining-based uniform bound
- Theorem 3 proof: update the line citing Lemma D.4's bound
- No other lemmas affected (D.4 is used only by Theorem 3)

### Literature support
| Reference | Venue (Tier) | Credibility | What it provides |
|-----------|-------------|-------------|------------------|
| Talagrand (2014), Upper and Lower Bounds for Stochastic Processes | Springer (T1) | GOLD | Generic chaining removes log factors |
| van der Vaart & Wellner (1996), Weak Convergence, Ch. 2.5 | Springer (T1) | GOLD | Dudley entropy integral |
| Wainwright (2019), High-Dimensional Statistics, Ch. 5 | Cambridge (T1) | GOLD | Localized Rademacher complexity |

### Alignment impact
- Resolves log n discrepancy between theoretical rate and empirical rate in Fig 4
- Makes the bound match the information-theoretic lower bound exactly

### Downstream effects
- Theorem 3 rate improves: this propagates to Corollary 3.1 and Section 5 applications
- No assumptions change → all existing results remain valid

### Estimated effort
MEDIUM — requires rewriting one lemma proof (Lemma D.4) using chaining technique.
The technique is standard in empirical process theory.

### Connection to pipeline
- If accepted: feed to `/proof-repair` as a voluntary improvement (not a bug fix)
- The new Lemma D.4 proof should be written via `/proof-writer` with full rigor

Step 7: Write SHARPEN_REPORT.md

Write papers/<paper-name>/SHARPEN_REPORT.md:

# Theory Sharpening Report: [Paper Title]

## Executive Summary
- Assumptions analyzed: N
- Relaxable assumptions: K (M with T1 literature support)
- Rates analyzed: R
- Sharpenable rates: S
- Theory-model gaps: G_M
- Theory-experiment gaps: G_E
- Top priority improvements: [list top 3]

## Theory-Model-Experiment Alignment Matrix
[From Step 0C]

## Assumption Relaxation Opportunities
### Feasible (T1 literature support)
[table]
### Possible (T2/T3 support or new technique needed)
[table]
### Infeasible (fundamental barriers)
[table]

## Rate Sharpening Opportunities
### Achievable (known techniques)
[table]
### Research-level (requires new ideas)
[table]

## Minimax Optimality Status
| Result | Current rate | Minimax lower bound | Optimal? | Gap source |

## Theory-Model Gaps
[From Step 3A-3C, with resolution strategies]

## Theory-Experiment Gaps
[From Step 4A-4C, including contradictions]

## Competitive Positioning
[From Step 5B — how this paper compares to state of the art]

## Codex Cross-Assessment (independent second opinion)
### Agreement summary
- Findings where Claude + Codex agree: X (HIGH confidence)
- Findings where they disagree: Y (needs human review ⚠)
- Findings only Codex found: Z (added to roadmap)
[From Step 5B Codex reconciliation table]

## Improvement Roadmap (prioritized)
[From Step 6B — ranked improvements with scores, integrating Codex findings]

## Detailed Improvement Specifications
[From Step 6C — one section per top improvement]

## New References
| # | Key | Full citation | Venue (Tier) | Credibility | Supports |
[All references found, venue-tiered]

## Recommended Actions for Authors
### Quick wins (1-2 days each)
1. [improvement]
### Medium effort (1-2 weeks each)
1. [improvement]
### Future work suggestions
1. [improvement]

Also write supporting files:

• papers/<paper-name>/sharpen_references.bib — BibTeX for all new references
• `papers/<pap

…(truncated)