The Butterfly Lattice
The "butterfly lattice" functions within the scope of recursive systems which are based upon the feedback properties of closed time loops, systolic arrays being the "carriers" of such phenomena. In a closed time loop, the resulting outflow circles back into inflow, recombining with any other newly-manifest stimulus taken into the loop. Thus, when even the slightest stimulus is present with respect to the loop, it becomes an active component of the "substance" that flows perpetually through that loop.
At the most fundamental level, the function governing the recursion is xn←Rxn-1(1-xn-1), where xn is the anamost element of the systolic array and R is magnitude. Any occurrent stimulus, no matter how small, is mathematically "taken in" at its value and at 1 minus its value. Thus, an event which has a minimal boolean value of 0000000000000001 (or 1 expressed in decimal), is also integrated with respect to its "1-x" value, which is 1111111111111110 boolean (or 65534 in decimal). Therefore, even the slightest stimulus can have a profound effect over time.
At the next level, the function of the lattice is to shift the elements of an occurrence. This is a root function in the transformations between the time domain and the spectral domain. This effectively modulates the significance of an element of an occurrence, and the shifts may occur within both sequence and value. In sequence shifting, the last element occurring may become the first. In value shifting, the least element may become the greatest. Again, the effects of such shifting only manifest themselves within the scope of due progression of time, as all shifting is governed by "taps" within the time loop. The following diagram illustrates one of the functions of lattice shifting:
The algorithm of the lattice is one of feedback. The diagram below shows the algorithmic makeup of a single lattice component. The upper section (moving from left to right) is the positive flow of the lattice, and the lower section (moving from right to left) is the negative flow.
In full algorithmic implementation, the lattice elements are strung together in a "ladder" configuration as illustrated below...most commonly with their terminal node of positive flow recursing into the initial negative flow node.
Lattice structures provide for a natural manifestation of the phenomenon of "filtering", and the xn-1(1-xn-1) function serves to abate intent of stimulus. In the macrocosmic expression of lattice-based algorithm, it is occurrence itself which is the most valued commodity...as all occurrence that enters the systolic array of the closed time loop provides "food".
Thus, the most important condition that must be fulfilled for the functioning of the butterfly lattice is that active stimulus be provided in continuity. The essence of the stimulus, however, is not at issue...only that it occur.
The algorithmic reason behind this is the magnitude expressions within the lattice structures - R in the mathematical function and the -g and +g gain multipliers within the lattice component algorithm. Magnitude is like an "invisible hand" governing the procession of time, and occurrence is like the individual tickings of the clock.
Within the scope of full algorithmic implementation, any manner of stimulus or perturbance provides an "unsigned tick" for the clock of a closed time loop...and a "wise" algorithm would be structured so that magnitude alone governs its outcome, and not the essences of the occurrences and perturbances it apprehends. Through this form, all stimulus - regardless of direction or intent - serves as a means of realization for the singular outcome dictated by magnitude.
Thus, in the full expression of the butterfly lattice, it is shown that it is not the substance or content of an activity that drives the progression...only that activity occurs. Without perturbance, the closed time loop is at a standstill, unable to proceed with the fulfillment of its function.
However, upon the occurrence of stimulus - be it positive or negative - the closed time loop "comes to life" and can proceed with the realization of its function. The integration of the butterfly lattice simply ensures that any stimulus, no matter how great or how small, no matter what its intent or orientation, may be employed to propel that which is "driven" by the loop (i.e. the carrying out of the intended function as established in magnitude). In essence, it's not "what you do"...it's simply "that you do" - as the only "enemy" of the butterfly effect, the magnitude-directed function which it drives and the progression of time is the absence of stimulus!