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The Precision Challenge

Cosmos SDK and Ethereum use different decimal precisions for their native tokens:
  • Cosmos SDK: 6 decimals (1 ATOM = 10^6 uatom)
  • Ethereum: 18 decimals (1 ETH = 10^18 wei)
This 12-decimal difference creates challenges when bridging the two ecosystems. Simply scaling values would lose precision or create rounding errors.

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Mathematical Foundation

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Balance Representation

Any account balance can be decomposed into integer and fractional components: 
Total Balance = Integer Part × Conversion Factor + Fractional Part
For an account with balance a(n) in smallest units (18 decimals): 
a(n) = b(n) × C + f(n)
Where:
  • a(n) = Total balance in 18-decimal units (aatom)
  • b(n) = Integer balance in 6-decimal units (uatom)
  • f(n) = Fractional balance (remainder)
  • C = Conversion factor (10^12)
  • Constraint: 0 ≤ f(n) < C

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Example Decomposition

Consider a balance of 1,234,567,890,123,456,789 aatom: 
1,234,567,890,123,456,789 = 1,234,567 × 10^12 + 890,123,456,789

Where:
- b(n) = 1,234,567 uatom (stored in bank)
- f(n) = 890,123,456,789 aatom (stored in precisebank)
- Total = 1.234567890123456789 ATOM

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The Reserve Account

To maintain supply consistency, a reserve account holds backing for all fractional balances:

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Reserve Equation

Reserve Balance × C = Sum of All Fractional Balances + Remainder
Formally: 
b(R) × C = Σf(n) + r
Where:
  • b(R) = Reserve balance in uatom
  • Σf(n) = Sum of all account fractional balances
  • r = Remainder (fractional amount not in circulation)
  • Constraint: 0 ≤ r < C

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Supply Invariant

This ensures the fundamental invariant: 
Total_aatom = Total_uatom × 10^12 - remainder
The remainder represents sub-uatom amounts that exist in the system but aren’t assigned to any specific account.

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Operation Algorithms

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Transfer Algorithm

Transferring between accounts requires careful handling of carries and borrows: From account 1 to account 2, amount a:
  1. Calculate new fractional balances: 
    f'(1) = (f(1) - a) mod C
f'(2) = (f(2) + a) mod C
  2. Handle integer updates with carry/borrow: 
    b'(1) = b(1) - ⌊a/C⌋ - (f'(1) > f(1) ? 1 : 0)
b'(2) = b(2) + ⌊a/C⌋ + (f'(2) < f(2) ? 1 : 0)
  3. Update reserve based on carry conditions:
    • Both carry/borrow: No reserve change
    • Only sender borrows: Reserve decreases by 1
    • Only receiver carries: Reserve increases by 1

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Mint Algorithm

Creating new tokens while maintaining backing:
  1. Update account: 
    f'(n) = (f(n) + a) mod C
b'(n) = b(n) + ⌊a/C⌋ + (f'(n) < f(n) ? 1 : 0)
  2. Update remainder: 
    r' = (r - a) mod C
  3. Adjust reserve for consistency

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Burn Algorithm

Removing tokens from circulation:
  1. Update account: 
    f'(n) = (f(n) - a) mod C
b'(n) = b(n) - ⌊a/C⌋ - (f'(n) > f(n) ? 1 : 0)
  2. Update remainder: 
    r' = (r + a) mod C
  3. Adjust reserve for consistency

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Proof: Remainder Unchanged in Transfers

A critical property is that transfers don’t change the global remainder: Proof:
  1. Start with transfer affecting fractional balances
  2. Take modulo C of the balance equation
  3. Since reserve changes are multiples of C, they vanish mod C
  4. Fractional changes sum to zero (amount subtracted equals amount added)
  5. Therefore: r' = r
This elegant property means transfers are purely redistributive - they don’t create or destroy fractional amounts.

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Precision Hierarchy

The system manages three precision levels:
UnitPrecisionDecimalsUsage
ATOM10^00Human display
uatom10^-66Cosmos native
aatom10^-1818EVM native

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Conversion Examples

1 ATOM = 1,000,000 uatom = 1,000,000,000,000,000,000 aatom
0.000001 ATOM = 1 uatom = 1,000,000,000,000 aatom
0.000000000000000001 ATOM = 0.000000000001 uatom = 1 aatom

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Edge Cases and Limits

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Minimum Transferable Amount

  • In Cosmos: 1 uatom (0.000001 ATOM)
  • In EVM: 1 aatom (0.000000000000000001 ATOM)

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Maximum Precision

  • Cosmos operations: Limited to uatom granularity
  • EVM operations: Full aatom precision
  • Cross-system: Automatic precision handling

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Dust Amounts

Amounts smaller than 1 uatom but larger than 0 aatom:
  • Tracked in fractional balances
  • Accumulate until reaching 1 uatom
  • Never lost or rounded away

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Implementation Strategy

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Storage Optimization

Rather than storing 18-decimal balances directly:
  1. Store 6-decimal amounts in existing bank module
  2. Store only the fractional remainder separately
  3. Reconstruct full precision on demand
This approach:
  • Maintains backward compatibility
  • Minimizes storage overhead
  • Preserves full precision

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Query Performance

When querying balances: 
fullBalance = bankBalance * 10^12 + fractionalBalance
This single multiplication and addition reconstructs the full 18-decimal balance efficiently.

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Why This Matters

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For Users

  • No precision loss: Every aatom is accounted for
  • Seamless experience: Automatic handling across systems
  • Fair transactions: No rounding advantages or disadvantages

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For Developers

  • Standard interfaces: Use familiar decimals for each system
  • Automatic conversion: No manual precision management
  • Predictable behavior: Mathematical guarantees on operations

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For the Ecosystem

  • True interoperability: Native precision for both ecosystems
  • Future proof: Extensible to other precision requirements
  • Efficient design: Minimal overhead for maximum capability

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Comparison with Alternatives

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Simple Scaling

// Naive approach - loses precision
evmAmount = cosmosAmount * 10^12
Problems: Loses sub-uatom amounts, rounding errors accumulate

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Fixed-Point Arithmetic

// Complex fixed-point math
amount = FixedPoint{mantissa: 123456, exponent: -18}
Problems: Complex implementation, performance overhead, compatibility issues

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Separate Balances

// Maintain two separate balance systems
cosmosBalance: 1000000 uatom
evmBalance: 1000000000000000000 aatom
Problems: Synchronization issues, double accounting, complexity

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PreciseBank Solution

// Elegant decomposition
totalBalance = integerPart * 10^12 + fractionalPart
Advantages: Simple, efficient, precise, compatible

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Future Extensions

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Multi-Precision Support

The mathematical framework extends to any precision:
  • 8 decimals for certain tokens
  • 27 decimals for high-precision applications
  • Variable precision based on token type

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Cross-Chain Precision

Standardized precision handling for:
  • IBC transfers with different precisions
  • Bridge protocols with varying decimals
  • Universal precision abstraction layer