**Finite
Discrete Closed Information Systems**

See also:

The
Mathematical Analysis,

Elsewhere I have develop the concept of an FDIS,
which is a particular type of Finite State Automata (FSA) interpreted
as a system model. Here we perform a group theoretic analysis of an
FDIS as a model of a dynamical system or as a *dynamical group*.
In order to form a group it must meet certain conditions that impose
further constraints on the nature of the empirical dynamics that can
arise within the simulation space. These constraints result in a
dynamical regime that is in many ways remarkably like our own
physical universe.

There are intricate and subtle connections between the concepts of groups, algebra, information spaces, system state spaces, and so on. Here we map out the connection between system models and groups, thereby between causal programming and algebra.

Here we only consider finite groups, i.e. the set of integers is
an infinite group but the set of all possible states within a finite
discrete state space is a finite group. Thus the maximum size of the
state space depends upon the underlying primitive state variables and
their combined requisite variety. This then gives a finite bound on
the number of possible states, which also constrains the largest
possible state transition in a single iteration and hence the largest
exponent of the system matrix required to implement that transition.
These constrain the duration of the longest period cyclic process
that can be simulated and also the temporal resolution of the
simulation. In general the longest cycle in a particular system is
less than the absolute limit and this defines a tighter upper limit
on the total number of states **N** and the largest matrix
exponent, thus on the size of the group required to represent that
system.

Consider the set where
**M**^{0} is the identity matrix and **M**^{1}
is an ergodic system matrix that corresponds to a particular type of
simulation system or empirical cosmos, defining the types of
behaviours that are possible therein. The higher powers are simply
the SM raised to higher powers, thus representing all possible state
transitions; this represents the *transcendent context* of the
computational process.

Also consider the sequence where
each **v**_{t} is a state vector that represents
the entire state of a simulation universe frozen in a single moment
of time. The vector **v**_{0} defines a particular
starting condition which then defines all later states. These form a
time sequence that represents the *empirical context* of the
computational process or the space within which empirical states
transform over time thus producing the empirical world. It is a
sequence of column vectors thus forming a two dimensional matrix that
is indexed by state or by time. A column vector is a state vector,
which is a cosmos frozen in a single moment in time, it is like a
frame in a movie. A row is a time series for a particular state, thus
it is a signal with amplitude as a function of time, it is like a
beam of light from the movie projector.

This set is conceptually equivalent to the “Akashic Record” from esoteric metaphysics, it is a representation of an entire universe from beginning to end. Whilst the FDIS or finite state automata operates only within a single transcendent computational moment with no past or future, the virtual world that it 'simulates' must be a whole and complete process that is 'closed' in a group theoretic sense from beginning to end. Thus all past, present and future states are represented within this set even though in the absolute reality in which the information is actually being computed there is only the ever changing NOW. This explains how it could be that in some cases phenomena seem to be “spread out” in time with past and future interacting, whilst in other ways there is only the present moment that exists.

The group formed by contains
an FDIS, its higher powers and the set of all associated state
vectors. These are considered as a *dynamical group*; i.e. as a
class of mathematical entities and relations that produces an algebra
that implements the causal structure of a dynamical system, and also
an information space that is the whole computational space that
embodies both the transcendent and empirical contexts (or SM's and
SV's).

The empirical context is otherwise known as the physical universe,
it is the domain of traditional physics; it is the domain of
relations between empirical states as functions of time, thus there
are patterns within the matrix of the empirical space. However, this
empirical space is only where the phenomena manifest, it is not the
cause of the phenomena. SMN takes the causal structure deeper, where
the empirical space is a projection and the casual structure is
explicitly represented within an information space manifested by a
system matrix. Thus, from an empirical perspective, the causal
structure of the physical universe is represented transcendently, as
underlying causal programming. Thus the empirical and transcendent
contexts are defined, in terms of the SV and SM spaces respectively.
Our senses respond only to the empirical states, but sayings such as
“look within” or “look behind the veil” or “penetrate the
outer form and see into the essence” or “see God / Brahman within
all things” these all mean to use the intellect and the intuition
to see into the SM side of the group and comprehend the causal
structures behind and within all *things* or all empirical
phenomena or all patterns within the empirical space.

Consider now the set of all possible state vectors, i.e. all
conceivable permutations of the underlying state variables. This is
the set which
is the complete state space. The set (**M**, **N**)is a set of
operators that transform any system state to any system state within
**S** by traversing the causal pathways of the system as defined
by **M**, thus the system may move about in the state space.
However, by using a different matrix **M** a different set of
causal pathways form within the state space hence there is a
different *phase portrait*, describing a different behavioural
topology and thereby implementing a different system.

The group must be closed in every sense, mathematically as well as
in the context of the system's causal structure and behaviour, i.e.
not just in the flow of symbols but also in the logical meaning of
their flow. Any deviation from closure and the system can *crash*,
i.e. it can enter a *non-computable* state and the next state of
the simulation is then *undefined*, this would bring the
empirical universe to an unexpected end. If the group is closed in a
*group* sense then every state transition results in a valid
state so one moment can continue to follow another indefinitely. The
causal algebra must be invested with symbolic meaning and woven into
a simulator that is behaviourally closed, in that it does not
simulate paradoxical behaviour but instead “the simulation cosmos
rests comfortably within the fundamental constraints without *touching*
them”. For example, rather than loop a spatial coordinate so that
one jumps from one side to another as one crosses the limit, like in
many computer games, one can use relativistic constraints programmed
into the universe at a low level, thus producing a finite spatial
universe within which one can never directly experience the finite
limit. Thus it is finite but unlimited.

Whilst the cosmic FDIS programming must be closed in order to function as a group or an information space the whole cosmic process may or may not be closed, there may be a higher world that we can interact with but nothing can be said about that from the perspective of this theory, other than that, there is no way of knowing unless we are informed about it by that higher world.

The way in which dynamical groups remain closed with a finite number of empirical states is that the dynamics must all be cyclic thus any SM can be applied to any SV thereby producing a valid SV. Thus and . Even a chaotic system orbiting a strange attractor will be cyclic within a finite discrete regime because at a given resolution there will at some stage be two points that become indistinguishable due to quantisation entropy. Since the future state of an FSA depends solely on the present state and this present state has been experienced before, this loops back thus forming a cycle within state space.

So long as and the process is cyclic and only valid states arise, thus the group is closed.

and and

**M**^{0}** = I** the unit matrix hence

Due to the cyclic nature of the process so which
moves the simulation **j** steps backwards. Furthermore, these
matrices are generally hermitian so their transpose conjugate is also
their inverse; the transpose makes every input an output and every
output an input, thus reversing the flow of information, then the
conjugation reverses the sign of the phase component thus reversing
the direction of phase evolution or dynamical simulation. Thus the
signals flow along reversed channels and the signals themselves are
reversed in time, thus the entire scenario is reversed.

so
(**M**, **v**, **N**) is a non-abelian group,

but so
(**M**, **v**, **N**) is an abelian group under operator
composition. Thus operators can be combined in any order thereby
producing another operator within that group.

A finite discrete FSA in a purely deterministic regime would remain trapped in particular cycles and would not explore or utilise most of its state space or its behavioural potential. For engineered systems this is preferred so that neatly defined cycles can be formed into networks of integrated cycles, and noise is minimised to keep the systems bound within their deterministic cycles. For general systems a degree of noise is optimal, to allow the system to skip between nearby state space trajectories and thereby fully occupy its whole state space and explore its full range of potential behaviour. A zero noise condition is a sub-optimal operating condition for any complex dynamical system. For example, the experience of modern sanitised lifestyles and immune system dysfunctions indicates that a little bacteria would be good noise. Furthermore, a certain degree of noise is required at the low level to allow for the propagation of subtle signals via the phenomenon of stochastic resonance.

The effects on a system due to noise are analogous to states of mind, the broad, open, brainstorming and diverging state of mind is a noisy, fuzzy FSA that rambles through a broad state space and the narrow, focused, converging state of mind is a precise and finely tuned FSA that follows well defined tracks. Indeed consciousness can be defined as the ability to recognise and break out of loops, thus a purely noiseless deterministic system would remain trapped in loops and could not manifest the potential for consciousness, whereas a system with optimal noise would be able to traverse any loops and retain its freedom of movement within state space or its consciousness.

Here we are discussing the details of the simulation *programming*
that underlies an empirical context or phenomenal world. We must
ensure numerical closure of the data types within the programming,
this is different to the group closure discussed earlier. This
produces a dynamical simulation program that is completely fault
tolerant and cannot *crash* for any internal reason.

In a variable dynamical regime closure is ensured by the absence of underflow or overflow in any of the dynamical equations. Thus for each equation and each variable one must check all extremum cases to ensure that no data loss or corruption occurs.

Below I use finite discrete notation. If you haven't already familiarised yourself with this then refer to here.

Thus if, for example **x = v.t** then we must ensure that
either **dx ≤ dv.dt**. Or another method is to ensure both **dx
≤ dv.D**_{t} and **dx ≤ D**_{v}**.dt**
where the variables **v** and **t** are interrelated via the
programming so that values less than **dx** do not arise, thus
making values such as **dv.dt** impossible.

See here
for a discussion of *numerical closure* and non-entropic
computation that loses no information and that cannot crash. The
basic principles are equal requisite variety, no underflow and no
overflow.

These last two requirements place dynamical constraints on the
range and resolution of empirical quantities and thereby on the
nature of the empirical universe that is being computed. A constraint
being broken is equivalent to a computer program incurring a “divide
by zero” error and crashing or the simulation data becoming
corrupted over time and nonsensical phenomena occurring. But this
universe cannot *crash* and it remains coherent throughout time.

These considerations apply to the efficient and harmonious long
term functioning of any complex dynamical system and thus to the
issue of *systemic health*. If all information channels are
coherent and no information is lost then the processes can persist
indefinitely.

A simple way for the physical space to be closed is for it to loop around on itself but this is not the case, as has been recently discovered by scientists (this universe is spatially flat). Besides, one cannot loop energy or velocity values, however phase is looped into a circle of 2π radians.

Instead, the constraints are built into the programming of the (Simulator), just as in a computer it is up to the program not to ask for an illegal memory address so too is it up to the Simulator to not generate empirical values that cannot be represented by the transcendent computational infrastructure (the TC).

For instance, gravitational potential energy and underflow place constraints on the largest physical distance between any two objects, whereas overflow places constraints on their smallest separation.

so thus defines a maximum separation for the lightest particles but in general the maximum separation depends on the two masses multiplied by . These constraints imply that the empirical universe is topologically flat and finite in extent, and even has a center or origin defined as the center of mass of the physical universe, thus there is a fundamental inertial reference frame. Furthermore, the distance that one can travel from the center is proportional to ones mass with only the lightest particles out near the edge of the universe.

Also so and
since we
find that hence and
this is related to the Planck mass that is which
is different by a factor of **2π**. I'm not yet sure what the **2π**
factor means, it is likely to be related to the correlation between
the planck context and the (0)context of the cyclic
model. Note, in the cyclic model **L**_{φ}**
= 2π **.

All of the Planck level quantities can be now be derived; for example, by substituting into we get and this is related to the planck energy and so on for the other planck constants such as and and and .

Each iteration is a whole computation, it takes in the state of
the entire cosmos and from that computes the next state. Therefore
quantities that apply over multiple steps need not be represented or
even representable. For example, consider a particular velocity ,
there may be another velocity
where .
However if **Δt** is the simulation time step for (**0**)cycle
computations, then **L**_{t} cannot occur within
one single computation so a FD variable capable of storing **v'**
need not be used. Hence the FD constraints are modified by these
*iterative constraints*.

A further iterative constraint is that if **x' = x.y** is
iterated and **x** is to remain bounded, then **x.x**^{*}**
≤ 1** and **y.y**^{*}** ≤ 1**. So **x**
and **y** are complex *probabilities*.

Below is a calculation for quanta or photons that maps out all of the limits imposed by numerical closure. It is easiest to represent as a conceptual network for clarity. The calculation begins at the top, maps out the structure of the dynamical algebra and at the bottom we see that there is only one velocity, i.e. the speed of light.

Furthermore, if and then (equal requisite variety, thus equivalent information spaces).

If then
we get a set of dynamical *constants* that correspond with the
Planck scale of our physical universe, this is elaborated on shortly.
If there are more energies or frequencies allowed then the quanta can
have varying energies and and
all the dynamics occur safely within the underlying constraints.

The Planck level of our physical universe provides a full set of dynamical quantities with just the right relations to produce a simulator that meets the requirements of a finite discrete closed information system.

The actual state data that is being
computed by SMN is of the form and
it has a modulus and 1, 3 or 7 phase components see here
for more detail. These are the most low level components of the
simulation programming and this low level of implementation is
extremely fine tuned and cut down to the simplest possible structure.
The values **Δφ**, **Δω** and **Δt** are all constants
so
and
so zero bits are required to represent the different values because
there is only one value, therefore the corresponding FD data
structure is simply **{t**_{p}**}** instead of
**{n**_{t}** , dt}** or **{t**_{min}**
, n**_{t}** , dt}**, and so on for the other
variables; this is far more compact and simple. In this context we
will dispense with the **dt** and **L**_{t}
notation and simply use **t**_{p} or **x**_{p}
and so on, where the **p** subscript indicates that these are in
fact Planck values.

In the discussion on “cyclic
computational processes” it turned out that the primitive
period **T**_{0}** = t**_{p} so
the Planck context is equivalent to the (**0**)context of the
cyclic computational model. So one should refer to this for further
details regarding many aspects of the Simulator's programming.

Within this context all dynamical quantities have constant values. This is as well as the constant speed of light , which in this context is the only allowable speed. Furthermore, there is the usual range of dynamical variables and equations such as all of which involve the Planck constants woven together into a dynamical algebra or program.

These various dynamical quantities collectively define the
dynamical properties of the (**0**)context and since they are all
constants they are trivially finite and discrete so there can be no
entropy in this context.

Fundamental iteration time step.

Fundamental cycle frequency.

Fundamental spatial displacement.

Momentum of fundamental quanta.

Energy of fundamental quanta.

Effective mass of fundamental quanta.

So the fundamental iterative cycle has a period **t**_{p}
which represents a quanta with maximum energy and all the above
dynamical properties. On the level of the simulation program there is
just an iterative complex equation with evolving phase but from
within the simulation there is a quanta with a range of dynamical
properties.

In this sense the (**0**)context is the inner face of the
simulation program and is the furthest or deepest that we may
perceive in any empirical sense; it is the quantum vacuum. It is the
lowest level of the transcendent computational machinery, which is
related to the Akashic Field and it forms a high energy ceiling to
our empirical universe. In this context there are no variations of
any of these values, this forms a static framework from which the
causal structure of the empirical universe depends.

Next we explore the domain of multiple iterations and of variable
valued dynamical quantities, where velocity can be less than **c**
and the energy less than **E**_{p} and
displacements less than **x**_{p} and so on. This
is the context where ,
for details, see the cyclic
computational model. Within that context we descend into the
series of cycles from the (**dn**)cycle to the (**L**_{n})cycle
and explore the dynamics therein. But more generally we consider an
arbitrary dynamical context with variable valued dynamical quantities
where all equations are bound by fundamental constraints as
illustrated in the conceptual network calculation above and there are
many distinct energy states.

There is much more variety here that must be represented in some way. This can be represented explicitly using FD variables with more requisite variety since within the dynamical group there are SM's that correspond to multiple iteration processes and these can be used to perform calculations of explicit values that can only arise or be computed over multiple iterations. This is the conceptually simpler method and is used, for example, in the preceding analysis.

Or conversely one could retain only the fundamental cyclic processes, this is the structurally simpler method and strictly more correct. Each FSA computation is a whole universal moment so only the constants and the phases are stored and all other values must arise from virtual computations that are implicit within the cyclic processes. These are essentially virtual computational spaces that arise within the cyclic computational process that is driven by the fundamental FSA iterations. Hence their computational values need not be explicitly represented, they are encoded into the underlying framework of the dynamical ceiling using minute phase differences.

An indication of multiple iteration computation is related to the solution of the Schroedinger wave equation where is the Hamiltonian, which is the key dynamical energy equation for the system that steps the system forward in time. But is a matrix, so to calculate an equation with a matrix exponent one uses the taylor series expansion of the exponential function, which is where the higher powered matrices involve interactions over multiple iterations. Thus, the dynamical group must be considered in regards to all of its SM's, or all possible transitions between system states to determine a particular state. Thus all the SM's in the dynamical group are required to compute this summation, but the factorials in the denominators soon make the higher powered terms negligible so only a few terms need be computed.

The above connection with the fundamental constants and the requirements of a FDCIS dynamical regime implies that the fundamental constants of this universe are tuned precisely so that they may provide such a dynamical regime within which this empirical universe could arise. Or more likely, they are thus tuned because they arise out of such a dynamical regime and could not manifest a coherent universe otherwise. The connection adds credence to the underlying proposition of the computational paradigm; that this physical universe is indeed a simulation being computed by a finite discrete closed information process.

The dynamical ceiling or the Simulator/EC membrane or the Zero
Point Field or the Planck scale of our physical universe is as far
into the underlying computing machinery of our virtual reality that we can
explore using purely empiricist techniques. We cannot perceive and
measure any deeper; to go deeper requires intellect and inference.
However the effects of the deeper computing machinery can be felt
since it is that machinery which computes every aspect of our virtual reality
so its effects are seen throughout the whole of the empirical
universe on all levels. These effects give rise to the idea of an
Akashic Field, but this is a way of conceptually grasping the whole
of the underlying computational process, it is not one thing but it
has a coherent influence so it seems to be one thing from our
perspective. Thus the phrase *Akashic Field* refers to
everything *above* the Simulator/EC membrane, i.e. the
Simulator, Transcendent Context (TC) and transcendent process (TP);
whilst the phrase physical universe refers solely to the empirical
context (EC). For more information regarding these terms refer to the
detailed
VR analogy.

The most general FDCIS implementation would not use complex data but octonions, which are similar to complex data except that they have one real and seven imaginary components. These are the most general form of data since all other forms can be implemented using octonions (real, imaginary, complex and quaternion). As one goes from real to complex to quaternion to octonion the amount of coherent causal structure within the mathematics diminishes so beyond octonions there is simply not enough structure to produce a coherent group or information space; so these are the most primitive and low level form of information.

As described above, the *empirical context* is a two
dimensional structure indexed by state and time (this is the SV
accumulated over all iterations), this is now filled with eight
dimensional data giving a ten dimensional empirical universe. The
seven imaginary components produce seven independent phase components
or seven fundamental dynamical quantities, which could be interpreted
as three spatial dimensions and four forces. It is no surprise that
String Theory suggests a ten dimensional universe and also relies
heavily on the use of octonion data. From within a simulated
empirical universe the FDCIS interaction loops could be conceived of
as one dimensional filaments (causal strings) that vibrate through
ten dimensions and thereby manifest empirical phenomena. However it
is far clearer and more intuitive when one conceives of the context
as being that of a FDCIS that is structured according to SMN and
system theory.

It seems highly likely that these correspondences between FDCIS's and String Theory are not just an accident. If this universe was a product of an FDCIS then the system theoretic principles apply on all levels from the lowest to the highest. Whilst String Theory has been searching for a Theory of Everything, it was really only expected to apply to the lowest possible level and then we would have to extrapolate this enormously to get any sensible understanding of everyday phenomena. However it seems possible that String Theory and System Theory are in fact the same theory, one applied to the lowest level, the other applied to all levels. If this is the case then any progress on the lowest level of String Theory could be translated into high level understanding via System Theory and conversely advancements in System Theory may provide deeper insights into String Theory. In fact, because of the universality of System Theory, all particular scientific domains are sub domains of System Theory so System Theory can act in this way as a cross fertilisation and communication facilitator between all domains and eventually all may be seamlessly unified into a single coherent body of human collective knowledge.