A question of discrete space-time, part 3

Dear members,

Part 1 ended by proposing four postulates and an emergent property of a
discrete space-time model: (1) A maximum speed that cannot be
exceeded.

Part 2 ended by proposing two more emergent properties of a discrete
space-time model: (2) The speed of a particle at constant motion would
not vary with the speed of its source, and consequently this maximum
speed would be a constant from any frame of reference, and (3) an
equation for adding velocity, with increased mass at increased
velocity.

Unfortunately, none of these three emergent properties of a discrete
space-time model are sufficiently different from continuum physics to
allow for a concrete test that will distinguish between the two.

Notwithstanding these annoying similarities to current continuum
physics, let's continue searching for an emergent property of a
discrete space-time model that does not violate known continuum
physics, but is different enough to be used as a basis for a concrete
test that can distinguish between the two (if this is possible - to be
the same, but different).

If apparent motion really were a series of probabilities of motion
between 0% and 100% within a discrete space-time lattice, then there
should be an inherent uncertainty in the measured speed of any object.


To illustrate the inherent uncertainty in the measurement of
probability, consider the example of a bent coin. Since the coin is
bent, and no longer lies flat, it is probably no longer "fair" with a
50% chance of landing heads. If you want to determine the bent
coin's true probability of landing heads, you would likely begin by
flipping the coin 100 times. Such an experiment gives you two facts to
work with: the actual number of heads obtained, somewhere between 0 and
100 (its past performance), as well as a rough estimate of the coin's
true probability of landing heads (its future performance).

In considering whether you might want to flip the same bent coin
another 1000 or 10,000 times, you could predict two things with some
certainty. First, flipping the coin 10,000 more times would yield a
narrower estimate of the coin's true probability of landing heads,
than flipping it another 1000 times. Second, flipping the coin 10,000
more times would yield a wider range in the actual number of heads
obtained, than flipping it another 1000 times.

This tradeoff between a narrower estimate of the coin's true
probability of landing heads, and a wider range in the actual number of
heads obtained, seems trivial when dealing with a bent coin, but
consider the motion of a unit particle with an unknown probability of
motion. If you want to determine the unit particle's true
probability of motion, you would likely begin by observing the unit
particle for 100 moments. Such an experiment gives you two facts to
work with: the actual number of motions obtained, somewhere between 0
and 100, as well as a rough estimate of the particle's true
probability of motion.

In considering whether you might want to observe the same unit particle
another 1000 or 10,000 moments, you could predict two things with some
certainty. First, observing the particle 10,000 more moments would
yield a narrower estimate of the particle's true probability of
motion, than observing it another 1000 moments. Second, observing the
particle 10,000 more moments would yield a wider range in the actual
number of motions obtained, than observing it another 1000 moments.

On the one hand, a narrower estimate of the unit particle's true
probability of motion is good (in terms of precise measurements). On
the other hand, a wider range in the actual number of motions obtained,
i.e., a wider range in the position of the unit particle, is bad (in
terms of precise measurements).

To quantify these probability/uncertainty ranges, we could use the
classical textbook statistical formulas from your nearest spread***.
I have MS Excel at hand with the built-in "confidence" function, which
requires three inputs: an alpha significance level, the population
standard deviation, and a sample size. (Although motion within a
discrete space-time lattice would be a binomial event, the central
limit theorem allows the use of the normal distribution for this
analysis.) For illustration purposes, let's use an arbitrary 99%
significance level (a 1% alpha), and the limiting case of a 0.5
population standard deviation - the maximum possible value (and also
the central value) for a binomial event. For sample sizes of 100,
1000, and 10000, the spread*** function gives the following results:

CONFIDENCE(0.01,0.5,100) = 0.128792
I take this to mean that the estimated probability of motion, as
measured during 100 moments, is plus or minus 0.128792 from its "true"
value (a range of 0.257583), and that the actual number of motions
obtained is plus or minus 12.8792 from its "normal" value (a range of
25.7583), with a confidence level of 99%.

CONFIDENCE(0.01,0.5,1000) = 0.040728
I take this to mean that the estimated probability of motion, as
measured during 1000 moments, is plus or minus 0.040728 from its "true"
value (a range of 0.081455), and that the actual number of motions
obtained is plus or minus 40.728 from its "normal" value (a range of
81.4550), with a confidence level of 99%.

CONFIDENCE(0.01,0.5,10000) = 0.012879
I take this to mean that the estimated probability of motion, as
measured during 10000 moments, is plus or minus 0.012879 from its
"true" value (a range of 0.025758), and that the actual number of
motions obtained is plus or minus 128.79 from its "normal" value (a
range of 257.583), with a confidence level of 99%.

The three examples above illustrate the textbook probability
calculations applicable to three different sample sizes in the
measurement of a hypothetical unit particle's probabilistic velocity
and position within a discrete space-time model. The larger the sample
size, the narrower the range of the velocity being measured (delta-V),
and the wider the range of the position being measured (delta-P). The
resulting values for (delta-V * delta-P) are a constant regardless of
sample size!

To summarize these three spread*** results:

sample
size delta-V delta-P delta-V*delta-P
100 0.257583 25.7583 6.635
1000 0.081455 81.4550 6.635
10000 0.025758 257.583 6.635

This fourth emergent property of a discrete space-time model (a
tradeoff between measuring a unit particle's probability of motion,
i.e., its velocity, and its position, where the product of the two is a
constant) seems to correspond to Hiesenberg's uncertainty principle
in conventional continuum physics. Unfortunately, once again, this
emergent property is not sufficiently different from continuum physics
to allow for a concrete test that will distinguish between the two.

I don't really know if this makes any sense, or if it is just some sort
of bizarre coincidence?

Regards,
Ed Hanna

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