a state the system maps to itself: $T(x^*) = x^*$. apply the rule again — nothing moves. that is the mathematical root. under a contraction (banach fixed-point theorem), every starting point is pulled to the same unique $x^*$
Schelling named the social twin: a focal point — the place minds meet without a meeting. two people lose each other in a city; they both go to the clock tower. no message, no vote — shared structure picks the rendezvous
in cyber these are the same object. the tri-kernel is $T$; its unique fixed point is collective focus $\phi^*$. every node that runs the same arithmetic on the same cyberlinks lands on the same root — a mathematical Schelling point of attention. cyberank is that root per particle; foculus finality is $\phi^*_i > \tau$: enough of the focal mass has gathered. the collective focus theorem is the proof that the rendezvous exists and is unique
see rank, rational neuron, tri-kernel
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