foculus: consensus by convergence

foculus is the consensus mechanism where agreement emerges from the mathematics of convergence rather than from voting, leaders, or message rounds. the tri-kernel iterates over the cybergraph, redistributing focus until a unique stationary distribution $\phi^*$ stabilizes. that distribution IS the consensus.

the moment a signal becomes knowledge

before consensus, a cyberlink is a proposal -- a neuron's claim about the relationship between two particles. after consensus, it has finality. the cybergraph absorbs it as permanent structure. the signal becomes knowledge

every vimputer node applies the same signals in the same order, converging on identical state. safety: no two nodes disagree. liveness: the system keeps producing steps

why agreement emerges

consensus is an equilibrium, not an axiom. no rule forces neurons to agree -- incentives make disagreement costly and agreement profitable

every cyberlink costs focus -- a costly signal. lying wastes finite resources on claims the graph will eventually down-rank. bayesian truth serum extracts honest beliefs by rewarding predictions that match the crowd's private information. karma accumulates for those whose signals increase syntropy, decays for those whose signals add noise

the result: rational agents converge to agreement because cooperation dominates defection in the iterated game. consistency across the cybergraph is a nash equilibrium, not a design choice

finality: the point of no return

once a signal achieves finality, its cyberlinks are permanent in the cybergraph -- the focus is spent, the link is irreversible

in foculus, finality is deterministic. a particle is final when $\phi^*_i > \tau$, where $\tau$ is an adaptive threshold that self-adjusts to network conditions. once crossed, there are no rollbacks, no probabilistic confirmation depths, no waiting for additional blocks

finality is the moment the cybergraph commits: this signal is knowledge, this link is structure, this state is canonical

how foculus differs from BFT and Nakamoto

classical BFT consensus (Tendermint, HotStuff, PBFT) works by coordinated voting: a leader proposes, validators vote, and 2/3 agreement produces a block. the cost is coordination -- O(N) or O(N^2) messages per round, leaders, quorums, view changes, finality gadgets. validator sets are bounded to hundreds or low thousands

Nakamoto consensus (Bitcoin, Ethereum PoW) works by competitive mining: miners race to find a valid hash, the longest chain wins. finality is probabilistic -- deeper burial means lower reversal probability, but certainty is asymptotic. throughput is constrained by the orphan rate tradeoff

foculus works by convergence: neurons gossip cyberlinks, each node independently computes $\hat\phi^*$ via GPU-accelerated sparse matrix-vector multiplication, and particles finalize when their $\phi^*$ mass crosses the adaptive threshold $\tau$. there is no leader to elect, no quorum to assemble, no chain to extend

the key insight: pi is the fork choice rule

in every consensus protocol, there is a fork choice rule -- the algorithm that decides which of two conflicting histories is canonical. in Bitcoin, it is the longest chain. in Ethereum, it is LMD-GHOST. in Tendermint, it is the latest signed block

in foculus, $\phi^*$ is the fork choice rule. when two particles conflict (shared nullifier, equivocation, resource collision), the one with higher $\phi^*_i$ wins. this is the outcome of the entire network's cyberlink structure converging through the tri-kernel -- the sum of all stake-weighted attention, computed independently by every node, yielding the same result everywhere

$\phi^*$ integrates all information from all neurons, weighted by staked tokens. manipulating it requires controlling the topology of the cybergraph itself, which costs real economic resources. exclusive support ensures conflicting particles split a finite mass pool rather than duplicating it. the tri-kernel contraction property amplifies initial advantages exponentially -- a slight honest majority compounds into decisive separation

there is no voting. there is no leader election. there is no block ordering. there is a graph, an operator, and a fixed point. the fixed point is consensus.

see foculus for the complete protocol specification. see convergence for the mathematical foundations. see collective focus theorem for convergence proofs. see tri-kernel for the operators