Ampcode: Research Paper to Mathematica Notebook

The Challenge

What GPT-5 Couldn't Do

Professor Gilo's paper on insurance collusion contains Lemma 1 — a nuanced mathematical proof involving implicit differentiation, Arrow-Pratt risk aversion, and limit arguments. GPT-5 kept making algebraic errors when trying to verify it and apply it to CRRA utility.

❌ GPT-5 Failures

Incorrect implicit differentiation of FOC
Wrong Arrow-Pratt substitution steps
Missed limit behavior at P→0 and P→1
Could not produce executable code
Repeated mistakes across multiple attempts

✅ What Was Needed

Verify every equation (14–19) step by step
Compute optimal α* numerically via FOC
Apply Lemma 1(i) to CRRA utility U=A^(1-γ)/(1-γ)
Find critical γ threshold analytically
Interactive notebook for Professor Gilo

The Magic Moment

One Prompt. Complete Solution.

The entire task was accomplished with a single natural-language request:

User → Use Wolfram Mathematica CK skill. Attached is the research paper "Avraham_and_Gilo_Insurance_collusion". Start from Lemma 1 in the paper. I want to verify that the lemma and its proof (which is in the appendix) are correct, and then show that Lemma 1(i) can apply to consumers with CRRA utility. Result should be interactive Wolfram Mathematica .nb where user can insert inputs and use sliders, documented in detail for a scientific presentation to Professor Gilo.

That's it. No manual equation entry. No debugging. No iteration.

Under The Hood

What Happens Automatically

📄 Step 1

AI reads the full PDF and extracts every equation

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📚 Step 2

CandleKeep library provides Wolfram Language syntax reference

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⚙️ Step 3

Skill generates validated .nb with 12 sections

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✅ Step 4

Syntax validation: brackets, characters, structure

🔧 Infrastructure Stack

Ampcode Agent — orchestrates the entire pipeline
Wolfram-Mathematica Skill — paper-to-notebook generation rules
CandleKeep Library — 60+ Wolfram Language reference docs
PDF Extraction — equations, definitions, proof steps
6-Point Syntax Validator — ensures error-free .nb files

The Paper

Avraham & Gilo (2025)

Insurance Collusion and Imperfect Competition when Insurers Increase Risk

📋 Model Setup

Wealth W, Harm H, probability P
Loading factor q ≥ 1 (supra-competitive)
Risk-averse consumers: U''(A) < 0
Arrow-Pratt: Ra(A) = −U''(A)/U'(A)
Coverage α* ∈ [0,1] maximizes EU

📐 Lemma 1

(i) Single-peaked: If Ra decreases fast enough, ∃ P̄ maximizing demand
(ii) Monotone: If Ra is not too decreasing, demand falls with P
Proof: Implicit differentiation → Eq.(17)
Limit arguments at P→0 and P→1

Mathematics

Key Equations Verified

FOC (Eq. 1)

−(1−P)·q·U'(W₁) + (1−qP)·U'(W₂) = 0

Derivative (Eq. 17)

dα*/dP = [(1−P)αqH(Ra₁−Ra₂) + (q−1)/(1−qP)] / [−(1−P)H(qP·Ra₁ + (1−qP)·Ra₂)]

CRRA Specialization

U(A) = A^(1−γ) / (1−γ) Ra(A) = γ / A (DARA)

Critical γ Threshold

γ_crit = (q−1)·W·(W−H(1−α)) / (α·q·H²·(1−α))

All 6 equations (14–19) from the Appendix verified computationally in the notebook.

Deliverable

Interactive Mathematica Notebook

📊 3 Manipulate Panels

Coverage α*(P) with peak detection
Arrow-Pratt decomposition of Eq.(17)
Aggregate demand with heterogeneous consumers

🔬 6 Validation Tests

FOC consistency
Wealth ordering W₁ > W₂
Arrow-Pratt identity
Denominator sign < 0
P→1 limit behavior
Reproducibility (fixed seed)

🎛️ Interactive Sliders

Wealth W: 50–500
Harm H: 5–80
Loading q: 1.01–3.0
CRRA γ: 0.1–15.0
Consumer heterogeneity range

Results

Key Findings

Finding	Detail
✅ Lemma 1 Proof Correct	All algebraic steps (Eq. 14→19) verified computationally
✅ CRRA Satisfies 1(i)	For γ > γ_crit, insurance demand is single-peaked in P
📊 Critical γ ≈ 1–3	For typical parameters (W=100, H=40, q=1.3)
🎯 Empirically Relevant	Standard CRRA estimates γ=2–5 consistently satisfy Lemma 1(i)
📈 Phase Diagram	Complete (γ, q) map showing Lemma 1(i) vs 1(ii) regions
✅ P→1 Limit	dα*/dP proven negative — concavity argument confirmed

Comparison

Ampcode + Wolfram vs. GPT-5

❌ GPT-5 (ChatGPT Plus)

Multiple attempts, repeated algebraic errors
Cannot execute or validate equations
No interactive visualizations
No numerical root-finding
Cannot produce runnable .nb files
No syntax validation pipeline
Hours of manual correction needed

✅ Ampcode + Wolfram Skill

Single prompt → complete solution
Executable Mathematica notebook
3 interactive Manipulate panels
Numerical solver with fallback methods
Production-ready .nb with 12 sections
Automated 6-point syntax validation
Minutes, not hours

Why It Works

The Infrastructure Stack

🤖 Ampcode Agent

AI orchestrator that reads papers, plans the approach, writes code, and validates output

📚 CandleKeep Library (60+ docs)

RAG-backed Wolfram Language reference — correct syntax for Manipulate, FindRoot, Plot, etc.

⚙️ Wolfram-Mathematica Skill

Custom skill with notebook generation rules, .nb format constraints, and validation checklist

🔒 6-Point Syntax Validator

Invalid characters, BoxData check, bracket balance, notebook structure, string escaping — all automated

One prompt. Verified proof. Interactive notebook.

ESL • ⚡ Ampcode • ◆ Wolfram Mathematica

Ampcode + Wolfram

The Ampcode Workflow