$$ \newcommand \DeadlineTimeout {\mathrm{DeadlineTimeout}} \newcommand \Sortition {\mathrm{Sortition}} \newcommand \Prop {\mathit{propose}} \newcommand \Soft {\mathit{soft}} \newcommand \Cert {\mathit{cert}} \newcommand \Next {\mathit{next}} \newcommand \TP {\mathrm{TransactionPool}} $$

Jalapeño Run (Recovery)

Let us now assume a scenario similar to the Vanilla run, with the following difference: on round \( r = i \), when \( s = 2 \), before block commitment, the network experiences a network partitioning. The voting stake is fragmented, and no network partition has enough voting power to certify a block.

timeline
    title Jalapeño Run
    section (r = i, p = 0, 0 <= s <= 1)
        Proposal (s = 0) time = 0                       : Emits proposals for i-th round from selected accounts registered on the node
        Soft Vote (s = 1) time = DynamicTO(p)           : Filters proposals (lowest hash criteria) : Soft vote on the best proposal for round i : A Soft Bundle is observed and a pinned value is enstabilished
    section (r = i, p = 0, s >= 2) Network partition K begins
        Certification (s = 2) time = DeadlineTO(p=0)    : Certification vote fails, no Committe Threshold is reached
        Recovery (s > 3) time > DeadlineTO(p=0)         : Next and Fast Recovery votes triggered
    section (r = i, p = 0, s >= 3) Network partition K ends
        Recovery (s > 3) time > DeadlineTO(p=0)         : Next Bundle observed, new period begins
    section (r = i, p = 1, s = 2)
        Certification (s = 2) time < DeadlineTO(p=1)    : Certification vote of a pinned value
                                                        : Appended i-th block to the Ledger
                                                        : State deltas applied
                                                        : Transaction pool purged

Run

Let us assume the network conditions are those described in the initial context.

Regular Propose and Soft steps

The network starts round \( r \) performing regular \( \Prop \) and \( \Soft \) steps.

Suppose that during the \( \Soft \) step (\( s = 1 \)), a \( \Soft \)-Bundle is observed, therefore a pinned-value \( \bar{v} \) is established on the nodes, and the \( \Cert \) step begins.

Network Partitioning begins: failing Certification step

During the \( \Cert \) step, before the block commitment, the network experiences a partitioning \( K \).

For any two accounts in the certification committee, the nodes playing for them have their connected peers \( K_t \) and \( K_l \), with \( t \neq l \).

Given the proposed network graph, players reach \( \DeadlineTimeout(p = 0) \) without a committable block proposal (that is, no \( \Cert \)-Bundle supporting any proposal-value has been observed).

Recovery

The protocol enters into recovery mode, and several \( \Next \) votes and fast recovery votes sessions happen, without any of them being able to form a Bundle for a value, due to the persisting network partition.

Suppose now that after a given time, connections are restored and \( K \) is solved. For any two accounts in the certification committee, the nodes in which they are registered are part of the same (unique) network. In other words, this time \( K_t = K_l \) for all nodes whose managed accounts are chosen by \( \Sortition \) procedure to vote in a step \( s = \Next_h \), with \( 3 \leq h < 248 \).

Since during \( \Soft \) step (\( s = 1 \)), before the network partitioning occurred, a \( \Soft \)-Bundle had been observed, causing a pinned-value \( \bar{v} \) to be established on the nodes. Therefore, all selected players vote on this pinned-value, and a \( \Next_h \) Bundle is observed for \( \bar{v} \).

Network Partitioning ends: new period

The protocol agrees to move into the next period \( p = 1 \), garbage collects old period (\( p = 0 \)) data, and restarts from \( s = \Prop = 0 \).

Since there is already an agreed-upon value \( \bar{v} \), there are no proposals and the protocol moves quickly into \( \Soft \) and then \( \Cert \) votes.

This time the network is sufficiently connected to observe a \( \Cert \)-Bundle for \( \bar{v} \) and so, before \( \DeadlineTimeout(p = 1) \), the network is able to reach an agreement, and a new block is committed in the same way as in the Vanilla run; advancing the network into the new round.