Precision and Mathematical Conventions

Consistent Token, native18, external6, bps, and fixed-point conventions distinguish continuous models from contract integer execution.

Distinguish Amounts from Raw Integers First

Contracts do not store floating-point USDC or Token amounts. Human-readable examples in this manual use expressions such as 1,000 USDC; formulas labeled integer implementation use raw on-chain integers.

Notation	Unit in integer formulas	Example
`T`, `a`, `b`	Token18	1 Token = `10¹⁸`
`Q`, `N`, `C`, `A`	native18 USDC	1 USDC = `10¹⁸`
`E`, `R`	external6 USDC	1 USDC = `10⁶`
`σ`	Precision conversion factor `10¹²`	Not a currency exchange rate
`BPS`	Ratio denominator `10,000`	100 bps = 1%
`PPM`	Ratio denominator `1,000,000`	10 ppm = 0.001%

In the source, ether is Solidity's numeric suffix for 10¹⁸ ; it does not mean that this protocol uses ETH as its quote asset. The current adaptation treats native Arc USDC and the ERC-20 USDC interface with 18/6 decimals; this property cannot be generalized directly to every EVM network.

Converting between USDC Precisions

From external6 to native18:

Q = E σ, σ = 10^{12}

Rounding down from native18:

E = ⌊ \frac{Q}{σ} ⌋, ε = Q mod σ

ε is native18 dust. Exact conversion requires:

Q mod σ = 0

For example, rounding down 1.234567890123456789 USDC yields 1.234567 USDC, with a remainder of 0.000000890123456789 USDC. An exact-conversion path rejects this input instead of silently rounding it.

Rounding Notation

Symbol	Meaning	Common uses
`floor(x)` / `⌊x⌋`	Round down	Ratio progress, partial allocation, index accumulation
`sqrtFloor(x)`	Integer square root rounded down	Theoretical reserves in the risk model
`sqrtCeil(x)`	Integer square root rounded up	Conservative remaining reserves in risk capacity

feeUp (A, f) = ⌈ \frac{A f}{10,000} ⌉

Standard V2 exact-output input calculation uses floor(ratio) + 1. It adds 1 even when ratio is already an integer, so it cannot always be replaced by ceil(ratio). The Quoter's maximum executable capacity also preserves a strict boundary for this reason.

Operations Whose Order Cannot Be Exchanged

In general:

⌊ \frac{a}{b} ⌋ c \neq ⌊ \frac{a c}{b} ⌋

Fixed-Point Indices

TWAP cumulative prices use 2¹¹² scaling; Funding and Legacy use 2¹²⁸ . Taking Funding as an example:

Δ I = ⌊ \frac{r Δ t 2^{128}}{10^{6} \cdot 86,400} ⌋, F = ⌊ \frac{N Δ I}{2^{128}} ⌋

The index separates the globally accumulated rate from individual position shares, avoiding a storage write for every position every second. Two downward rounding steps may produce tiny differences from floating-point approximations; Legacy separately retains a high-precision remainder as well.

Fixed Supply Basis for Market Cap

The protocol's internal market cap is based on Curve supply S = 1,000,000,000 Token:

M = ⌊ \frac{Q S}{T} ⌋

It is neither FDV calculated using the total supply of 1,199,861,071 Token nor a circulating-supply measure. Public displays should distinguish raw spot market cap from clamped risk market cap and avoid substituting one metric for another.