Equilibrium of the synchronous seven-state ring

Read the equation

At equilibrium, the constant reset fixes states five and six at one, the reciprocal fixes state seven at one, and the increment sets state one to two. The decrement and identity links then set states two through four to one. This yields the vector (2,1,1,1,1,1,1), not seven identical values.

Terms & definitions

How it works

Read the definition from its right-hand expression to the quantity on the left. Its surrounding source records the intended input, state, or parameter context. Where the source leaves an operation unspecified, this page preserves that gap.

Assumptions & limits

Source recovery establishes that the expression appears in this record. Its mathematical validity, implementation behavior, and interpretation must be assessed separately. See the linked source and research notes for the checks available for this version.

Original context

s5,t+1=1,  s6,t+1=s5,t,  s7,t+1=1/s6,t.

At a valid equilibrium these equations give

s1,s2,s3,s4,s5,s6,s7=2,1,1,1,1,1,1.

Starting all seven states at two, the implementation reaches this vector by update seven and remains there. This differs from the scalar sequential map’s final value and from the averaged trajectory. The wiring, state layout, scheduling, and coupling rule matter. The word “loop” alone does not select among these models.

7. Coupling, speed, thresholds, and persistent divergence

7.1 Proposition 4: no coupling means no center change

\left({s}_{1},{s}_{2},{s}_{3},{s}_{4},{s}_{5},{s}_{6},{s}_{7}\right)=\left(2,1,1,1,1,1,1\right).