docs: small write-up.

Author: l-monninger

protocol-units/settlement/mcr/contracts/README.md

@@ -1,66 +1,29 @@
-## Foundry
+# MRC
+- **RFC**: [RFC MCR](https://github.com/movementlabsxyz/rfcs/pulls)
 
-**Foundry is a blazing fast, portable and modular toolkit for Ethereum application development written in Rust.**
+This directory contains the implementation of the MRC settlement smart contract. To test the contract, run:
 
-Foundry consists of:
-
--   **Forge**: Ethereum testing framework (like Truffle, Hardhat and DappTools).
--   **Cast**: Swiss army knife for interacting with EVM smart contracts, sending transactions and getting chain data.
--   **Anvil**: Local Ethereum node, akin to Ganache, Hardhat Network.
--   **Chisel**: Fast, utilitarian, and verbose solidity REPL.
-
-## Documentation
-
-https://book.getfoundry.sh/
-
-## Usage
-
-### Build
-
-```shell
-$ forge build
-```
-
-### Test
-
-```shell
-$ forge test
-```
-
-### Format
-
-```shell
-$ forge fmt
-```
-
-### Gas Snapshots
-
-```shell
-$ forge snapshot
+```bash
+forge test
 ```
 
-### Anvil
+There is a long-running test cover over 50 epochs. It will take about a minute and a half to complete on most machines.
 
-```shell
-$ anvil
-```
+# Implementation 
+## Description
+For a given block height, MCR selects the earliest block commitment that matches the supermajority of stake for a given epoch by:
+1. Fixing the stake parameters for the epoch; all stake changes apply to the next epoch.
+2. Tracking commitments for each block height until one exceeds the supermajority of stake.
 
-### Deploy
+## Proof of Correctness
+The stake is fixed for an epoch, so only commitments for a specific block height are considered, allowing for a straightforward proof.
 
-```shell
-$ forge script script/Counter.s.sol:CounterScript --rpc-url <your_rpc_url> --private-key <your_private_key>
-```
+Let $C$ represent all possible commitments, and $C'$ be an ordered subset of $C$. MCR returns $c_i \in C'$, the earliest commitment matching the supermajority of stake, defined as:
 
-### Cast
+$$
+\delta(C') = \frac{2}{3} \times \sum_{c \in C'} s(v(c)),
+$$
 
-```shell
-$ cast <subcommand>
-```
+where $v: C \to V$ maps a commitment to its validator and $s: V \to \mathbb{N}$ maps a validator to their stake. Define $\sigma'(C', i) = \sum_{j = 0}^{i} s(v(c_j))$, the cumulative stake up to the $i$-th commitment. $\sigma'$ is non-decreasing as $\sigma(C', i) = \sigma(C', i - 1) + s(v(c_i))$.
 
-### Help
-
-```shell
-$ forge --help
-$ anvil --help
-$ cast --help
-```
+If $\sigma(C', i) \geq \delta(C')$, then $c_i$ is the earliest commitment where the supermajority is met, since any earlier commitment $c_j$ for $j < i$ would violate the non-decreasing nature of $\sigma'$.

protocol-units/settlement/mcr/contracts/test/MCR.t.sol

@@ -257,7 +257,7 @@ contract MCRTest is Test {
         // dishonest signers
         dishonestSigners.push(signer3);
 
-        for (uint i = 0; i < 10; i++) {
+        for (uint i = 0; i < 50; i++) {
 
             for(uint j = 0; j < 10; j++) {