Reappearance Rates

[Brief summary:  For those who are new to this website, a brief recapitulation. Ultimate Event Theory aims to be a description of the physical world where the event as opposed to the object (or field) is the primary item. The axioms of UET are given in an earlier post but the most important is the Axiom of Finitude which stipulates that Every event is composed of a finite number of ultimate events which are not further decomposable. Ultimate events ‘have occurrence’ on an Event Locality which exists only so as to enable ultimate events to take place somewhere  and to remain discrete. Spots on the Locality where events may and do have occurrence have three ‘spatial’ dimensions each of unit size, 1 stralda, and one temporal dimension of 1 ksana. Both the stralda and ksana are minimal and cannot be meaningfully subdivided. The physical world, or what we apprehend of it, is made up of event-chains and event-clusters which are bonded together and appear as relatively persistent objects. All so-called objects are thus, by hypothesis, discontinuous at a certain level and there are distinct gaps between the successive appearances of recurring ultimate events. These gaps, as opposed to the ‘grid-spots’, have ‘flexible’ extent with however a minimum and a maximum.]

Every repeating event, or event cluster, is in 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 0 or any rational number n/m ─ but no irrational value. The r/r is quite distinct from the space/time displacement rate, the equivalent of ‘speed’, since it concerns the number of times an ultimate event repeats in successive quite apart from how far the repeat event is displaced ‘laterally’ from its previous position.
If r/r = 0, this means that the event in question does not repeat.
But this value is to be distinguished from r/r = 0/1 which signifies that the ultimate event reappears at every ksana but does not displace itself ‘laterally’ ― it is, if you like, a ‘rest’ event-chain.
If r/r = 1/1 the ultimate event reappears at every ksana and displaces itself one stralda at every ksana, the minimal spatial displacement. (Both the stralda and the ksana, being absolute minimums, are indivisible.)
        If r/r = m/n (with m, n positive whole numbers) this signifies that the ultimate 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, as it were, when one space-time slice is replaced by the next. We generally assume that we ourselves constitute a standard  inertial system relative to which all others can be compared ─ we ‘are where we are’ at all instants and feel ourselves to be at rest except when our ‘natural state’ is manifestly disrupted, i.e. when we are accelerated by an outside force. In a similar way, in UET we conceive of ourselves as constituting a rest event-chain to which all others can be related. But we cannot see ourselves so we generally choose instead as a standard landmark event chain some (apparent) object that remains fixed at a constant distance 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 — ‘normal’ physics has the same problem . The crucial difference is, of course, not between ‘vertical’ event-paths (‘stationary’ event-chains)  and ‘slanting’ event-paths (the equivalent of straight-line constant motion), but rather between ‘straight’ paths (whether vertical or slanting) and ones that are not straight, i.e. curved. 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 constant straight-line 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 time only stands still for the space of a single ksana — ‘the moving finger writes and having writ, moves on” as the Rubaiyàt puts it. 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.
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 the r/r of an event is 4/5 stralda per ksana we cannot conclude that the event in question shifts 4/5ths of a stralda to the right at every ksana (because there is no such thing as a fifth of a stralda). 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, 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 particularly when we consider collisions. If there is any likelihood of confusion the denominator, which is the ksana value,  will be marked in bold, thus 187/187.
Also, the stralda/ksana ratio for event-chains 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 » 10⁸ metres/second as the upper limit for causal processes. 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). Since N is such an enormous number, this constraint can usually be ignored. An event or event-chain simply ceases to repeat when it reaches the limit.
These restrictions imply 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 number 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 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). (And this in turn means that much of the mathematical assumptions of Analysis and other areas of mathematics are unrealistic.)
Why is all this important or even worth stating? Because, unlike traditional physical systems, UET not only makes a distinction between constant and accelerated ‘motion’ (or rather their equivalents) but also between event-chains which have the same displacement rate but very different ‘reappearance rates’ — some repeating event-chain ‘miss out’ more ksanas than others.
A continuous function in Calculus is modelled as an unbroken line and, if we are dealing with a ‘moving object’, this object is assumed to ‘exist’ at every instant. In UET even a solid object is always discontinuous in that there is always a minute gap between consecutive appearances even in the case of the ‘densest’ event-chains. But, over and above this discontinuity which, since it is general and so minute, can usually be neglected, there remains the possibility of far more substantial discontinuities when a regularly repeating event may ‘miss out’ a large number of intermediate ksanas while nonetheless maintaining a regular rhythm. Giving the overall ‘speed’ and direction of an event-chain is not sufficient to identify it: a third property, the re-appearance rate, is required. There is all the difference in the world between an event-chain whose members (constituent ultimate events) appear at every consecutive ksana and an event-chain which only repeats at, say, every seventh or twentieth or hundredth ksana.
An important consequence is that a ‘particle’ (dense event-chain) can ‘pass through’ a solid barrier if the latter has a ‘tight’ reappearance rate while the ‘particle’ has one that is much more ‘spaced out’. Moreover, two ‘particles’ that have the same ‘speed’ (lateral displacement rate) but very different reappearance rates will behave very differently especially if their speeds are high relative to a barrier in front of them.
This feature of UET enables me to make a prediction even at this early stage. Both photons and neutrinos have speeds that are close to c, but their behaviour is remarkably different. Whilst it is very easy to block light rays, neutrinos are incredibly difficult to detect because they have no difficulty ‘passing through’ barriers as substantial as the Earth itself without leaving a trace. It has been said that a neutrino can pass through miles of solid lead without interacting with anything and indeed at this moment thousands are believed to be passing through my body and yours. On the basis of UET principles, this can only be so if the two event-chains known as ‘photon’ and ‘neutrino’ have wildly different reappearance rates, the neutrino being the most ‘spaced out’ r/r that is currently known to exist. Thus, if it should ever become possible to detect the ultimate event patterns of these event-chains, the ‘neutrino’ event-chain would be extremely ‘gapped’ while the photon would be extremely dense, i.e. apparently ‘continuous’ (Note 2). The accompanying diagram will give some idea of what I have in mind.

The existence and importance of reappearance-rates is one of the two principal innovations of Ultimate Event Theory and it may well have a bearing on the vexed question of so-called wave-particle duality. From the UET perspective, neither waves nor particles are truly fundamental  entities since both are bonded collections of ultimate events. A ‘wave’ is a ‘spaced-out’ collection of ultimate events, a ‘particle’ a dense conglomeration (although both wave and particle at sufficiently high magnification would reveal themselves to be discontinuous). Nonetheless, the perspective of UET is clearly much closer to the ‘particle’ approach to electro-magnetism (favoured by Newton) and gives rise to the following prediction. Since high frequency, short wave phenomena are clearly more ‘bunched up’ than low frequency, long-wave phenomena, it should one day, perhaps soon, be possible to detect discontinuities in very long wave radio transmissions while short wavelength phenomena would still appear to be continuous. The discontinuity would manifest itself as an irreducible ‘flicker’ like that of a light rapidly turned on and off — and may well have been already observed as a strangely persistent annoyance. Moreover, one can only suppose that there is some mechanism at work which shifts wave-particle phenomena from one mode to the other; such a mechanism simply ‘spaces out’ the constituent ultimate events of an apparent particle, or forcefully combines wave-like ultimate events into a dense bundle.
SH   16/03/20

Note 1 The statement “Between any two ultimate events, it is always possible to introduce an intermediate ultimate event” would be the equivalent in UET terms 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 there is always another point” 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, the  so-called Principle of Parmenides (who first enunciated it) slightly adapted,  “If there were no limits, nothing would persist”.
As against the above axiom of continuity of ‘Ordered Geometry’ which underlies Calculus and much else, one could, if need be,  introduce into UET the axiom, “It is not always possible to introduce a further ultimate event between two distinct ultimate events”.

Note 2. It is possible that this facility of passing through apparently impenetrable barriers is the explanation of ‘electron tunnelling’ which undoubtedly exists because a microscope has been manufactured that relies on the principle.