Every event or event cluster is in Ultimate Event Theory (UET) attributed a **recurrence rate **(**r/r**) given in absolute units *stralda/ksana *where the *stralda *is the minimal spatial interval and the *ksana *the minimal temporal interval. **r/r **can in principle take the value of any rational number *n/m *or *zero *─ but no irrational value. The **r/r **of an event is roughly the equivalent of its *speed* in traditional physics, i.e. it is a distance/time ratio.

If **r/r **= 0, this means that the event in question does not repeat.

If **r/r **= *m/n *this signifies that the event repeats *m *positions to the right every *n *ksanas and if **r/r **= −*m/n *it repeats *m *positions to the left.

But right or left relative to what? It is necessary to assume a **landmark event-chain **where successive ultimate events lie exactly above (or underneath) each other when one space-time ‘slice’ is replaced by the next. Such an event-chain is roughly the equivalent of an *inertial system* in normal physics. We generally assume that we ourselves constitute a *standard* inertial system relative to which all other inertial systems can be compared ─ we ‘are where we are’ at all instants and so, in a certain sense, are always at rest. In a similar way we constitute a sort of *standard landmark event-chain *to which all other event-chains can be related. But we cannot see ourselves so we choose instead as standard landmark event chain some object (=repeating event-cluster) that remains at a constant distance from us as far as we can tell. Such a choice is clearly relative, but we have to choose *some *repeating event chain as standard in order to get going at all. The crucial difference is, of course, not between ‘*vertical*’ event-paths and ‘*slanting*’ event-paths but between ‘*straight’ *paths, whether vertical or not, and ones that are jagged or curved, i.e. not straight (assuming these terms are appropriate in this context). As we know, dynamics only really took off when Galileo, as compared to Aristotle, realized that it was the distinction between *accelerated *and *non-accelerated *motion that was fundamental, not that between *rest* and *motion*.

So, the positive or negative (right or left) *m *variable in *m/n *assumes some convenient ‘vertical’ landmark sequence.

The denominator *n *of the stralda/ksana ratio cannot ever be zero ─ not so much because ‘division by zero is not allowed’ as because ‘*the moving finger writes and having writ, moves on” *as the Rubaiyàt puts it, i.e. time only stands still for the space of a single ksana. So, an **r/r **where an event repeats but ‘stays where it is’ at each appearance, takes the value *0/n *which we need to distinguish from *0*.

Thus *0/n ≠** 0 *

*m/n *is a ratio but, since the numerator is in the absolute unit of distance, the *stralda*, *m:n *is not the same as *(m/n) : 1 *unless *m = n*. To say a particle’s speed is *4/5ths *of a metre per second is meaningful, but if **r/r = ***4/5 stralda per ksana* we cannot conclude that the event in question shifts *4/5ths *of a *stralda *to the right every *ksana (*because the *stralda *is indivisible). All we can conclude is that the event in question repeats every fifth ksana at a position four spaces to the right relative to its original position.

We thus need to distinguish between recurrence rates which appear to be the same because of cancelling. The denominator will thus, unless stipulated otherwise, always refer to the next appearance of an event. *187/187 s/k *is for example very different from *1/1 s/k *since in the first case the event only repeats every 187^{th} ksana while in the second case it repeats every ksana. This distinction is important when we consider collisions. If there is any likelihood of confusion the denominator will be marked in bold, thus *187/ 187*.

Also, the *stralda/ksana ratio* for event-chains always has an upper limit. That is, it is not possible for a given ultimate event to reappear more than *M *stralda to the right or left of its original position at the next ksana ─ this is more or less equivalent to setting *c **»** 10 ^{8} metres/second *as the upper limit for causal processes according to Special Relativity. There is also an absolute limit

*N*for the denominator irrespective of the value of the numerator, i.e. the event-chain with r/r = m/n terminates after

*n = (N−1)*— or at the

*N*th ksana if it is allowed to attain the limit.

These restrictions mean that the Locality, even when completely void of events, has certain inbuilt constraints. Given any two positions *A *and *B *occupied by ultimate events at *ksana k*, there is an upper limit to the amount of ultimate events that can be fitted into the interval *AB *at the next or any subsequent ksana. This means that, although the Locality is certainly not metrical in the way ordinary spatial expanses are, it is not true in UET that “*Between any two ultimate events, it is always possible to introduce an intermediate ultimate event*”(**Note 1**). *SH 11/09/19*

**Note 1 **The statement *“Between any two ultimate events, it is always possible to introduce an intermediate ultimate event*” is the equivalent in UET of the axiom *“Between any two points there is always another point”* which underlies both classical Calculus and modern number theory. Coxeter (*Introduction to Geometry* p. 178) introduces *“Between any two points….” *as a *theorem *derived from the axioms of ‘Ordered Geometry’, an extremely basic form of geometry that takes as ‘primitive concepts’ only *points *and *betweenness*. The proof only works because the geometrical space in question entirely lacks the concept of distance whereas in UET the Locality, although in general non-metrical and thus distance-less, does have the concept of a *minimum *separation between positions where ultimate events can have occurrence. This follows from the general principle of UET based on a maxim of the great ancient Greek philosopher Parmenides:

*“If there were no limits, nothing would persist **─** except the limitless itself”. *