Edge Case Analysis

Overview

This document analyzes edge cases in the Shadow Liquidity mechanism to demonstrate the mathematical guarantees of protocol solvency.

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Solvency Under Extreme Conditions

Case 1: Unilateral YES Betting

All participants bet on YES outcome.

Scenario: $10,000 total volume, 100% on YES side

Metric	Value
Total USDC deposited	$10,000
Fee collected (2%)	$200
Net to curve	$9,800
Final vYES	91,116
Final vNO	109,754
Shadow YES issued	8,884
Shadow NO issued	0
Vault balance	$10,000
Maximum liability	8,884
Solvency	10,000 / 8,884 = 112.5%

Result: System remains solvent despite extreme one-sided betting.

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Case 2: Unilateral NO Betting

All participants bet on NO outcome.

Scenario: $10,000 total volume, 100% on NO side

Metric	Value
Total USDC deposited	$10,000
Shadow NO issued	8,884
Shadow YES issued	0
Vault balance	$10,000
Maximum liability	8,884
Solvency	10,000 / 8,884 = 112.5%

Result: Symmetric to Case 1. Solvency maintained.

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Case 3: Extreme Imbalance (95% YES / 5% NO)

Scenario: $10,000 volume with 95/5 split

Metric	Value
YES purchases	$9,500
NO purchases	$500
Shadow YES issued	~8,400
Shadow NO issued	~490
Vault balance	$10,000
Maximum liability	8,400
Solvency	10,000 / 8,400 = 119.0%

Result: Imbalanced markets achieve higher solvency due to premium prices on the popular side.

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Case 4: Bank Run Scenario

All participants attempt to exit simultaneously after market becomes one-sided.

Initial State:

• 5,000 Shadow YES shares outstanding

• Vault balance: $5,500

• vYES: 95,000

• vNO: 105,263

Sequential Exit Process:

Exit Order	Shares Sold	Payout	Remaining Vault
1st 1,000	1,000	$1,020	$4,480
2nd 1,000	1,000	$980	$3,500
3rd 1,000	1,000	$940	$2,560
4th 1,000	1,000	$900	$1,660
5th 1,000	1,000	$860	$800

Analysis:

• Early exiters receive premium (~$1.02/share)

• Late exiters receive discount (~$0.86/share)

• Vault never depletes entirely

• Average recovery: ~94%

Result: First-mover advantage incentivizes orderly exit rather than panic.

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Solvency Guarantees

Mathematical Proof

The CPMM guarantees solvency through three mechanisms:

1. Slippage Premium

As users buy one side, they pay increasingly higher prices:

shares = vOLD - k/vNEW

For large purchases, vNEW >> vOLD, so shares << amount spent

2. Fee Accumulation

Each trade adds to vault while issuing fractional shares:

vault_increase = amount (100%)
shares_issued < amount (due to fee + slippage)

3. Bounded Liability

Maximum liability is bounded by shares issued, which is bounded by initial reserve:

MAX(totalYES, totalNO) < INITIAL_VIRTUAL_RESERVE

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Failure Mode Analysis

Scenario: Failed Graduation

If bonding deadline expires before 100% solvency:

Solvency at Deadline	Recovery Rate
95-99%	~96%
90-94%	~92%
80-89%	~85%
< 80%	~80%

Root Causes of Failure

Cause	Likelihood	Prevention
Insufficient interest	Medium	Market discovery, promotion
Extreme imbalance w/o arbitrage	Low	Arbitrage incentives
Rapid sell-off before graduation	Low	First-mover advantage

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Parameter Sensitivity

Virtual Reserve Impact

Initial Reserve	Max Solvency	Slippage Profile
10,000	150%+	High slippage
100,000	~110%	Moderate slippage
1,000,000	~101%	Low slippage

Current configuration (100,000) balances capital efficiency with solvency margin.

Fee Impact

Fee Rate	Solvency Impact	Trading Cost
1%	Lower margin	Lower barrier
2%	Moderate margin	Moderate cost
3%	Higher margin	Higher barrier

Current configuration (2%) provides sufficient margin without excessive trading friction.