Log-space index paper

A storage-efficient cumulative interest index using log-space arithmetic.

The core result

Standard lending-protocol designs accumulate compound interest as a multiplicative index:

$$I_{t+1}=I_t\cdot (1+r\cdot \mathrm{\Delta }t)$$

This is conceptually simple but suffers two problems:

1. Overflow. At default rates, the index overflows uint256 in ~29 years. A protocol expected to last decades hits this limit.
2. Precision loss. Each multiplication truncates fractional bits. Across many compoundings, this accumulates.

XPower Banq uses log-space accumulation instead:

$$L_{t+1}=L_t+r\cdot \mathrm{\Delta }t$$

The actual index is recovered as I = exp(L) only when needed for balance computations.

Properties

• Overflow horizon ≈ 10⁵⁸ years (effectively unlimited).
• Tighter precision bound than the multiplicative approach (2 ULP worst-case per the log-space paper, vs. 2N ULP after N multiplicative compoundings — empirically +10–20 wei closer to the mathematical ideal).
• Gas cost ~1,200 less per accrual (addition is cheaper than multiplication).

Why this works

Adding logs is mathematically equivalent to multiplying linears. The log-space sum is exact (no per-step truncation), and the exponentiation at read time is a single operation with bounded error.

Implementation

The protocol uses a high-precision fixed-point representation for L. The exponentiation function is taken from the OpenZeppelin Math library or equivalent.

Where this matters

For users: invisibly. Your balance is computed correctly across decades.

For developers: you need to be aware that L is the stored value, not I. SDK calls handle the conversion.

Where to find the paper

• PDF: banq-log.pdf (latest release)
• Unified bundle: banq-all.pdf — this paper is one of two engineering primitives in Part II.

Where to go next

• Interest rates — the rate model
• Architecture overview — where this lives