Cyclic Computation
See also:
The
Mathematical Analysis,
In this discussion I use finite discrete notation. If you haven't already familiarised yourself with this then refer to here.
Consider a fundamental cyclic process represented by 
which
is iterated according to 
thus producing subsequent values of z raised to successively
higher powers, hence 
.
Say it takes A steps to complete one full cycle so in each
iteration the phase evolves by discrete steps of 
which
is the phase quantisation and dt is the simulation time step.
When 
then
one full cycle has been completed. If 
then
after A iterations the phase has moved through 
,
so there is a slight phase offset. However we still treat A
iterations as one full cycle.
The primitive cycle is called a (0)cycle and is treated as one
primitive iteration within a higher level iterative cyclic process,
so the (0)cycles are themselves (1)iterations that step the phase
around a (1)cycle where 
.
In this manner we build a model of cycles within cycles where each
(n)cycle requires A n iterations
to complete and each iteration is in fact a whole lower level cycle
(except for the primitive cycles). As the primitive cycle rotates
these later cycles emanate or unfold in sequence.
If 
then
all higher level cycles have 
and
no dynamics can be simulated via these cycles. It is only when 
that there is an inherent phase eccentricity that propagates
throughout the cycles causing the phase to evolve within the context
of each cycle.
We now have multiple nested temporal contexts so we will use
temporal notation, but express
all times in their native units for now. Let each primitive
iteration have a period of T0 = 1
(0)iteration then 
which
is equal to A0 (0)iterations or one
(1)iteration and in general 
and 
if
A n = A for all n,
i.e. if all cycles require the same number of iterations. In this
discourse we will consider this latter case.
The phase step for each cycle varies throughout the cycles as 
.
Now that we have initiated this cascade of cyclic processes, they unfold with successively longer periods. However each of these cycles is considered to represent a plane-wave or non-localised quanta with well defined momentum within a simulation universe, so the period of the cycle gives a frequency f and an energy E. The (0)cycle defines a maximum energy ceiling out of which unfolds lower energy cycles or quanta.
These cycles unfold until beyond some point the period becomes so
long that the corresponding energy is less than the minimum energy
representable within the simulation space dE (see discussion
on finite discrete information) and quantisation entropy makes it
impossible for the lower cycles to represent information within the
simulation space. So the last cycle that can take part in an
empirical simulation has 
(expressed
as a number of (0)iterations) where Ln is
the largest possible value of n. So 
so 
is
the requisite variety of the cycle-period state variable.
Energy is a measure of activity or change within the dynamical
simulation space and this is represented by the phase component.
Consider a single cycle; 
so
in each iteration there is only dφ amount of dynamical change
that can occur within the period of that iteration. The highest
energy is 
when
the phase displacement occurs in a single primitive iteration and the
lowest energy is 
when
the phase displacement takes an entire (Ln)cycle.
So 
and so
we get 
and 
as
the requisite variety of the energy state variable, which is equal to
K t.
For each cycle there is a period and an energy so there are equal numbers of these values so the requisite variety of each state variable is equal. This is the case for all dynamical variables where there is one value associated with each cycle. Hence Kn = K t = KE = K f = K p... etc.
Now consider the case where A = 2π which is a 'canonical'
case where 
and each cycle takes 2π iterations to complete, thus 
and
the cycles form within a base 2π information space which is
fitting for a cyclic process. Furthermore, let us define a unit of
time within the simulation space where one sim-second corresponds to
one radian of phase evolution within the (Ln)cycle.
Hence, by definition, one (Ln)cycle takes 2π
sim-seconds (or s-sec's) to complete. Therefore 
Now that we have unfolded these cycles out to some well defined
limit, we know the extent of the cyclic simulation process and can
effectively compress this whole process of cycles within
cycles down into a single cycle by setting 
(0)iterations
instead of 
(where
primes indicate old values) so we have taken the longest period cycle
and compressed it into one cycle, thereby compressing all the other
cycles as well. This also brings the numerical value of 
,
expressed in terms of (0)iterations into line with 
expressed
in terms of sim-seconds. We redefine the value of the fundamental
period as 
so
there are effectively now a large number of short period iterations
in place of a single previous iteration, and instead of 2π
iterations to complete approximately 2π radians, it now
takes 
iterations.
So dφ has changed from 
to 
which
is exact because there is no more need of Δ since now
everything has been compressed into one cycle and the whole cycle
must be complete in itself so there is no phase eccentricity (it has
been built-in).
Let us consider this briefly and check that it is still equivalent with the original cyclic process.
After 2π iterations of this new process the simulation
time has advanced by 
where 
is
the conversion factor from the old time (primed) to the new time (see
Conversion
Factors for more details) so this is equivalent to 2π
iterations of the old regime.
In this new regime the highest energy is 
and
the lowest energy is 
,
note that we still have 
.
In this compressed regime we have dφ = T0
and this defines the maximum energy value as one, however if the
inhabitants of the simulation universe were to use their own units of
energy then the numerical values of these need to be converted. Say
that the inhabitants define their maximum energy value as Ep
(they call it the Planck Energy) then 
and 
since 
and 
which
is the Planck Time and dφ = h, which is Planck's Constant.
Now we have 
,
,
dφ = h and T0 = Tp
so this brings us into a dynamical context with properties much like
that of our physical universe.
Furthermore, here lies the reason why the equations of quantum physics transform into classical equations as h → 0. Since h is the quanta of empirical action for each iteration, as h → 0 whilst the simulation dynamics remain unchanged Δt → 0 and Δφ → 0 and the iteration frequency becomes infinite. Hence all empirical contexts become infinitely detailed and empirical processes become perfectly smooth.
Furthermore, energy can be calculated using the various cycle's dφ
n over a full cycle. So we get 
and 
.
So far we have cycles with lengthening periods so descending
frequency and energy. When interpreted as quanta they are conceived
of as waves within a simulation space and therefore have a wave
length λ0 that is defined in terms of
spatial units within the simulation space. This quantity λ0
is constant for all cycles and is the means by which one translates
between transcendent computational quantities and empirical spatial
quantities. The various frequencies fn
produce various velocities vn where 
so
these plane waves have a phase velocity less than c. There is
also a λ n associated with each cycle and
it represents the wavelength of a photon associated with that
cycle 
so
this is equivalent to a photon with energy En
< Ep and velocity equal to c.
The cycles represent decreasing phase velocities
from 
to 
and 
as expected.
The velocities defined above are the phase velocities (redefine as
vφ) for plane-wave quanta associated with
an empirical particle, however the velocity of the particle vp
is measured by inhabitants of the empirical universe as twice this
velocity. This is because here we are dealing with non-localised
plane-wave quanta; their energy or momentum is well defined but their
location in space is undefined. To transform this momentum
representation into a position representation we must localise the
quanta in space. This requires the superposition of a large number of
plane-waves with subtly different frequencies and phases. We build up
a superposition of these waves using a fourier integral and they
interfere and produce a well defined wave packet or localised quanta,
which has a group velocity equal to the particle velocity. However
now the momentum is undefined. These issues have been discussed at
length elsewhere in regards to the general properties of all waves
and in particular the Heisenberg Uncertainty Principle where 
is
a fundamental constraint on our knowledge regarding the empirical
properties of these quanta.
This process is reminiscent of a hologram. The particle perspective is a bit like a holographic image that is constructed from the interference of countless waves, and the individual rays associated with individual state vector elements in the FDIS (Finite Discrete Information System) are like elements in a holographic medium. This is discussed further a little later in regards to holographic simulation.
At the level of the ceiling c is the maximum velocity within the empirical context. This (0)context defines a dynamical computational regime in which the empirical velocity is v = c so only maximum energy quanta or photons can exist in this context. This context, when considered in conjunction with the SMN implementation of this cyclic model, produces something remarkably like an Akashic Field, which provides a low level inter-connective infrastructure that permeates the whole simulation cosmos.
As one considers multiple iterations one moves into a context in which photons may have different frequencies and particles may have v < c and mass m > 0. Thus we enter the empirical simulation universe proper.
The first fully empirical cycle is associated with the largest
empirical energy so it represents the first phase velocity less than
c, 
but
the particle velocity is twice the phase velocity so this cannot
alone be associated with a physical particle. However it can
represent a high energy photon with 
and 
or
it can be a component in a superposition that produces a wave packet
associated with a particle but it represents the ultra high frequency
aspects of the particle's evolution and the particle's velocity is
dominated by the many lower frequency, lower velocity plane-wave
components.
When 
this
is the phase velocity associated with a quanta that represents the
lightest possible particle accelerated as close to the speed of light
as it can get and thereby possessing maximum particle momentum.
At the level of the floor, the (L n)cycle
represents the lowest empirical energy so it represents either 
,
which is the minimum phase velocity associated with a particle or the
longest wavelength 
for
the lowest energy empirical photon.
Note that 
and 
.
The above model defines aspects of the programming that simulates this empirical universe, however the type of universe that one perceives within the simulation depends upon how one interacts with the underlying cyclic processes and this depends upon the nature of ones perceptual apparatus. The cyclic process forms a common information engine but empirical systems may perceive and interpret the information differently.
At the lowest level a system's perceptual apparatus is
characterised by a perceptual frequency where 
is
the Planck frequency which is the maximum perceptual frequency for
any empirical system, this is constrained by the quantisation within
the TP itself otherwise we could have 
and 
for
a smoothly varying classical universe with infinite resolution
and infinite energy.
The minimum perceptual frequency is 
or
once per complete cycle.
When pf = Lpf the system can perceive or
sample every single iteration so there are 
possible
values that can be perceived, and the values vary with every
primitive iteration so the perceptual system needs to be able to
distinguish this many unique values. When pf = dpf the system
can only sample once per cycle and the cycles are not eccentric so
the value will remain constant and no variation will be perceived.
Given a particular empirical system with a particular pf, let us determine the nature of the empirical world that they will perceive. First let us consider a minimum capacity perceptual system with pf = dpf, then we'll consider pf = 1 and then the maximum capacity perceptual system.
If 
then
gives
a value of 
so
there is only a single constant cycle or a single type of quanta or a
single frequency that is perceptible since 
hence
constant values. (note: here T0 represents
the smallest cycle period that the system can perceive, which is not
necessarily the absolute primitive cycle.)
If pf = 1 then 
(this
is the native T0 unit in this compressed
regime) gives a value of 
so
there is more than one cycle and 
.
But what is the value of dn?
so 
but
.
Combining these two equations we get 
where 
,
hence 
.
Now Kφ = Kn and
Lφ = 2π so 
so
it takes approximately 4g iterations to complete a full cycle
and there are 4g distinct virtual cycles each representing a
particular type of empirical quanta.
Also 
and 
and
where and 
.
When 
then
gives
a value of 
and 
(too
small for my calculator to compute in this way)
But KE = Kn so
consider 
and 
and
so 
and 
and
so 
and 
however
we can rescale these to fit any arbitrary energy scale so we multiply
through by Ep to align it with the standard
units for empirical energy used by the simulation inhabitants,
so 
and 
.
In the above calculation, when we defined the empirical unit of
time we considered only the transcendent computational process and
defined it such that 
,
i.e. one radian of transcendent phase evolution corresponds to one
second of empirical time.
The exact numerical value of the Planck Time Tp depends on our arbitrary units for energy, length and so on but the actual unit of time that we use is identical to the unit defined in terms of the transcendent computational process. Is this an artefact of some oversight within the calculation or is this indicating something fundamental?
The two underlying principles responsible for our particular numerical measure for the unit of seconds is the system of 24/60/60 time keeping inherited from the Babylonians (I think) and the angular momentum of the planet Earth. The tilt of the axis and the orbit around the Sun combine to modulate this fundamental frequency and produce the seasonal and yearly cycles, however on average the length of a day is determined by the angular momentum of the planet.
Does this indicate that the Babylonians understood much of this, or that our very planet is somehow aligned with a harmonic of the fundamental frequency of the universe? Could these correspondences be related to the fact of our existence here at all? Could sentient life have formed here if these harmonic relations were not satisfied? This is deeply puzzling! Indeed our own heart beats are approximately 60 beats per minute or one second per beat.
So far we have considered only equations of the type 
where 
and
f varies for each cycle. However if we consider this in terms
of a simple harmonic oscillator where 
and 
where
p indicates Planck values, then 
and
the iterative equation we are dealing with is 
.
So as the energy increases 
and
since t is the iteration time step 
we
get 
and
the phase component is zero so there is no dynamical computation or
one may say the dynamics are timeless. Hence as the energy of
a quanta increases, either due to greater mass-energy or kinetic
energy, the rate at which it's dynamics proceeds tends to zero. So
the fastest rate of time occurs for the lightest and slowest quanta.
This produces the phenomenon of relativistic time dilation.
The above discussion describes how a large energy computation
results in a very small dφ but recall that 
so
many iterations are required to effect the appropriate phase
displacement associated with that energy so more computational
iterations are required to simulate the quanta, hence more
computational resources are drawn from the transcendent computer.
A system within a simulation world can draw upon unlimited (but not infinite) resources from the transcendent computer; it may conceivably require a computation that pushes the computer to the limit. External observers of the computer will see it slow down due to congestion, but from within the empirical universe the being can only perceive within the context of particular simulation moments. So even if the computer becomes so loaded that the time to compute one simulation moment changes from milliseconds to days, within the simulation the moments keep flowing in sequence and no difference can be discerned. So different systems in different contexts such as states of motion are observed to exhibit different time dilations due to their different computational loads, however from within their own context things seem unaffected.
Furthermore, one can go through a similar argument as above but keeping the ω.t term constant and using dx as one's simulation spatial step (analogous to a simulation time step) and λ n as a variable wavelength that translates the transcendent computational quantities into equivalent physical distances. One then finds that as the energy increases λ n → xp and all empirical systems experience the phenomenon of length contraction.
This then gives us a quantised relativistic empirical context or a physical universe with properties much like our own, both in terms of quantum phenomena and relativistic phenomena.
How does this analysis integrate with SMN and FDIS's? In this analysis I have essentially been exploring the properties of a single primitive datum as it flows through the computational process or the computational stream (or causal 'string') associated with a single primitive system. This is just one element in the state vector (SV) and one element of the system matrix (SM) as they interact. But in the wider context of SMN the state vector contains countless such datums and is operated on by the whole system matrix, indeed each row of the SM processes the entire SV and produces a new state for a particular SV element. Thus we have many primitive systems (SV elements) which are processed in a massively parallel manner.
Throughout most of the development of the concept of an FDIS I have focused on system models where the systems and subsystems are actual objects in a world or where they are abstract variables or pure states within a computation such as cases where they are components of a quantum probability distribution or dynamical variable associated with a Newtonian particle. But from the cyclic computational analysis we get causal strings (virtual system processes) that have 'vibratory' properties much like light, hence I call them 'rays' in this context. Thus when a SM row processes the SV and merges all of these rays into a single resultant state, these rays interfere just like light. There are many subtle phase variations and these create interference patterns.
Hence we have a field of primitive elements in parallel and also phase interference. This suggests to me that holographic concepts may be involved here. Could it be that an FDIS combined with cyclic computational elements produces a holographic computational context that is massively parallel and that creates an empirical simulation universe in the form of a hologram? Thinking of the primitive systems as explicit objects in a world out of which higher level systems are composed is analogous to pixels in a photograph and thinking of them as interfering rays that form a collective composite pattern is analogous to holographic elements within a hologram.
This physical universe and also consciousness itself exhibits some holographic properties that have been discussed at length in recent years. Thus it is possible that these mathematical models will lead in that direction.
But are there two levels of parallelism or are they equivalent and we have conceptualised them differently? The finite state automata (FSA) provides a massively parallel computational framework that allows every point to interact with every point in every moment of existence without any explicitly holographic concepts being required. But are these interacting points primitive systems within an empirical simulation or are they elements within a holographic medium?
If it is the prior case then the empirical context is directly simulated with the primitive datums within the computation being the actual primitive systems within the universe and the FSA computes the causal connectivity in a massively parallel manner. This is analogous to a photographic analogy where the primitive systems are pixels.
If it is the latter then the medium contains the empirical information distributed in a holographic manner, thus there are two levels of parallelism. In this holographic case the underlying computational framework processes the medium in parallel where each holographic element interacts with every other element in each moment and this dynamic holographic medium encodes the empirical simulation as phase interference patterns that are distributed throughout the entire medium. Only through a process of superposition and interference does the hologram of the simulation context (or the empirical universe) arise. This is possibly related to the wave particle duality where the particle picture is the holographic image and the many wave components that superpose to form that image are associated with elements within the holographic medium. These elements interact via the FSA and this implements the dynamics of the whole context. This was briefly mentioned earlier in regards to Phase Velocity and Particle Velocity.
In most respects the first level of interconnection within the FSA can explain all of the interconnectivity within the empirical universe and allow for quantum non-locality, but can it also explain the parallelism within systems such as the human brain and collective consciousness or perhaps the subtle field effects of the Akashic Field? Can it explain underlying phenomena that are not simply interconnected but are uniformly distributed such as the fields?
However, unless there are indeed explicit holographic effects, like fragments producing low resolution images of the whole, I see no need to include holographic phenomena. Is our image of a particle really a low resolution image of the entire cosmos? I think not, but a particle is a fragment of the holographic image and not necessarily of the holographic medium.
It could be that the massive parallelism and distributed nature of the FSA computational process led people to liken it to holograms when it is in fact a FSA rather than a hologram. But what of consciousness and memory, there are some rather holographic phenomena there I believe, and consciousness, in its most primal form, is the essence of the computation that underlies the cosmic simulation process.
Perhaps there are parallels between finite state automata and holograms that have yet to be explicitly realised, but more likely there are indeed two levels of parallelism, one within the FSA computation that implements the dynamics and one within the holographic medium that encodes a holographic empirical universe.
I do not as yet understand all of the details of how this Cyclic Computational Model fits into the general scheme of Finite Discrete Closed Information Systems so I am still a long way from being able to computationally simulate a realistic physical universe. However the confluence of intricate connections between this cyclic model, FDCIS's, string theory, the Akashic field, the holographic universe, quantum theory and general relativity suggests that these ideas may lead us in a direction wherein we may eventually unify quantum physics and general relativity and we may fully understand the meaning of string theory and thereby develop a “Theory of Everything”.
Furthermore, due to the intimate links between system theory and string theory this may provide us with a truly “Universal Theory of Everything” that is expressed not only at the lowest possible level of strings, quanta and particles but which may also be extended to all systems on all scales and in all contexts. It would also encompass all perceptual phenomena, such as our experience of the duality of subject and object and their unification within the non-dual transcendent information space thereby providing not only an outward science of reality but also a fully transcendent science that encompasses both the empirical perspective (traditional science – outward looking) and the subjective perspective (spiritual science – inward looking).
Nothing less than this deserves the name “Theory of Everything”.
For the broader context in which the above analysis took place see: The Mathematical Analysis.