Thursday, December 20, 2012

Transiting the Event Horizon - Beyond 2012 Singularities

 A Timewave Revisited


 The Zero Date ‘Singularity’, and a Novelty Perspective on 'fiscal cliffs' and Planetary Evolutionary ‘Metamorphosis’


John David Sheliak – Santa Fe, NM  87508

Initially this was to be my response to a query asking for a novelty perspective for our looming ‘fiscal cliff, since defining novelty or ‘habit’ (i.e. entropy) for specific processes and events has largely been subjective and somewhat speculative so that queries like this one arise far too often.  This confusing state of affairs is largely the result of the lack of mathematical rigor and theoretical formality concerning the Timewave and its supporting Novelty theory.  In order to reduce this kind of subjective confusion over often arbitrary speculation, I thought that additional theoretical and mathematical perspectives might provide some much needed clarification on the nature and implications of Novelty Theory and the Timewave as have been formulated and mathematically constructed.  These ideas, memes, or perspectives, have far too long languished for lack of rigorous development, theoretical formality, and unambiguous empirical data.   Consequently, composing an answer to the query concerning a real world application of the concept of novelty, habit, or entropy, has provided an opportunity to finally address, and perhaps shed some light on these long ignored issues.

It is important to understand at the outset that there are neither embedded ‘infinities’ nor transition state ‘singularities’ intrinsic to the formulation of Novelty Theory, or to the mathematical construction of the Timewave.  Such singularities normally result from a mathematical operation that includes a division of some number by zero (any number or quantity, x, when divided by zero produces an infinity or singularity, ∞.).   It is true that transition state discontinuities are encountered in fractal mathematics when differentiation is attempted on these functions.  Such fractal functions are considered not to be everywhere differentiable[1] – i.e. there are point discontinuities where the slope of the curve described by the derivative, dy/dt, is undefined or discontinuous (infinity).   However, such fractal discontinuities are not end state singularities in any sense of the defining principles.  Moreover, there is nothing that I’m aware of in ordinary natural process that would produce this type of zero-valued terminal state either, including black hole event horizons.  In my view it is a very long leap of reason or logic to first arbitrarily establish a zero end state, and then define this value as being a form of singularity without any theoretical backup or mathematical development.

[1] A differentiable function is a function whose derivative exists at each point in its domain

It is also important to note that the Timewave fractal features are produced and ‘modulated’ by an infinite series expansion (or fractal transformation) - the mathematical operation performed on the 384 number set of the tri-level wave that generates the set of Timewave data values decaying to zero as a function of the time remaining to an established zero date/time – e.g. 6 a.m. on 12/21/2012.  The most likely justification for mathematically forcing the final Timewave fractal number set to a zero endpoint was that both the first order 64 number bi-directional wave set that is generated using features of the I-Ching King Wen sequence, and the 384 number tri-level wave set that is generated from the bi-directional set using I-Ching concepts, have zero endpoints, and therefore implying zero-valued endpoints for the Timewave fractal waveform.  

Nevertheless, as I pointed out in my vector analysis of the Timewave first order number set, the ‘forward wave’ and the ‘reverse wave’ appear to be linked in a cyclical fashion – i.e. each of the zero endpoints was simply some type of vibration or oscillation node, and neither need be interpreted as a zero-valued termination point.  Moreover, the rationale for the alignment of the established Timewave zero value to an established zero date is somewhat vague, although it was apparently done with at least some thought given to the ‘novelty fit’ of the Timewave to other historical events that were classified as having had ‘significant’ impact on global development and civilization.  Since precise definitions or a rigorous formulation of the concepts of novelty, entropy, or habit were never really offered nor established, such ‘novelty-to-time alignments’ should be seen as speculative and certainly more than somewhat subjective.

The consequence of any theoretical or Timewave developmental bias born of either of an understandable ignorance, or a type of theoretical ‘incoherence’, might lead one to legitimately conclude that both the established waveform zero entropy value, and its’ temporal alignment with a singular historical event associated with a predicted galactic alignment process assumed associated with the Mayan calendar, are somewhat flawed if not inherently biased speculation.  The lack of theoretical maturity or mathematical precision is certainly understandable if not expected for any radically new theory, as I’ve stated in previous Timewave work; but results born of such immaturity must be understood to be incomplete, inaccurate, or both; and certainly to be understood as more theoretical than empirical in nature.   So in order to examine Timewave Novelty events or processes occurring in the vicinity of the zero date in terms of a more natural functional process, I propose fitting the Timewave decaying fractal function with a decaying exponential function in order to explore potential alternative interpretations of such 'terminal' features. 

Before proceeding with such an analysis, it is important to note that the operative feature of a decaying exponential function[1] is its’ convergence to a zero-valued y-asymptote.   This means that the ‘zero point’ is not an actual singular point, but an asymptote that is approached only as elapsed time approaches an extremely large value, i.e. as t → ∞.   Obviously this result is a process not achieved in a practical real world sense.  A common analogy posits the question: “How many steps would it take for one to cross a room if one were able to travel ¼ of the distance from one's current position to the opposite wall with each step.”   This well known mathematical function being proposed to fit the aggregate Timewave graph is the decaying exponential that is expressed as, 

y = b*exp (-t/τ)                                                              (1)

Where: y is the Timewave entropy state in nats[2]; t is the Timewave elapsed time in Julian days; b is the initial Timewave entropy state, yb,, in nats at time t = 0, (i.e. the Timewave entropy state initial condition)[3]; and τ is the system time constant or e-folding time (how long it takes for a novelty magnitude state (y-1) to change by a factor of 2.71828 from its original value at a time, t1,, to a subsequent value at a later time, t2.  As elapsed time becomes infinitely large, t → ∞, the entropy value, y(∞) = b*exp(-∞) = b*0 = 0, represents a ‘zero entropy’[4] asymptote approached by the Timewave.  Note that if we define novelty as inverse entropy using the information theory notation, I(hi) = (y-1) = (1/y)[5], then a zero entropy is equivalent to a novelty/information singularity at y(∞) = 0, as t → ∞.

Now let's examine the results of the curve-fitting operation used to fit our decaying exponential function to the Timewave fractal function.  The information of interest here is how fast an established set of events changes with established measures of time.  The mathematical operation for expressing this change rate is the time derivative of the decaying exponential fitting function – i.e. the time rate of change of the set of events for individual and/or collective processes at any given elapsed time, t.  This is an operation that is defined mathematically by the following differential expression,

                                            dy/dt = -(b/τ)* exp(-t/τ)                                                   (2)

Where: the derivative, dy/dt,, expresses the rate at which a set of unspecified, but well-established events unfolds, changes, or advances with established measures of elapsed time.  Equation (2) shows the result of this derivative operation or the time rate at which an established set of events unfolds is diminishing over time (expansion of time between events) by the factor, (-b/τ)..  This result implies that the time it takes to experience an established sequence of events is expanding; likely accompanied by the perception that time has passed more quickly, or accelerated.  This could mean that the normal sequence of the established set of time sequential events, activities, etc. in one’s life would appear to be occurring over increasingly longer time intervals.  The derivative dy/dt provides the mathematical form and structure for the time rate of change for such an unfolding temporal sequence of events. 

In the limit where the derivative dy/dt → 0 as t → ∞, the number of established events that are unfolding during elapsed times as measured by our local clocks, would approach zero accompanied by the perception that the passage of time had become hyper-accelerated and on threshold of temporal collapse.  This concept is rather tricky to envision or wrap one’s mind around, since the experience and the measure of time is in fact relative.  As an example, let’s say that someone observes an expanding time interval between established individual events as measured by a local clock time reference – furthermore, let’s say 8 hrs elapses for an event sequence that previously took only 4 hours to unfold.  This experience would likely be accompanied by a perception that present time intervals are passing more quickly, or accelerating relative to past time intervals.  Temporal acceleration is then seen as the perception that time intervals between established events are expanding (dilating) over time, where standard local clocks measure the passage of time.  The fact is that there is no absolute time standard that can be referred to, as there is no preferred reference, or inertial frame in the cosmos.  Consequently, the perceptions based on the local frame of our ‘internal clocks’ can arguably be taken as primary.

In order to examine the time span dependence, ∆t, of specific novelty changes ∆y-1 (inverse entropy changes, y-1), one can perform an inverse differentiation using equation (2).  If the inverse derivative operation is performed the time span, ∆t, per unit of entropy (inverse novelty) change, ∆y can be quantified as well.  The result of this inverse derivative operation is expressed simply as:

dt/dy = (-τ/y), or alternately, dt = (-τ)* (dy/y)                         (3)

This expression reveals an interesting result showing that the perceived time span, dt,, for a given change in entropy, dy is compressed by a factor of (-t * y-1 * dy), where y-1 is inverse entropy or novelty (novelty = 1/y or 1/entropy).  The solution to this differential tells us that the actual time interval, ∆t, appears to be compressed by the product of a increasing novelty magnitude (y-1), a diminishing entropy change (∆y), and the system time constant (τ).  A time span that is undergoing temporal compression over time is a time span that is accelerating over time.

In tying all of this to the original query about how one could associate novelty/entropy concepts to the upcoming 'fiscal cliff', I’ll apply principles derived from information theory. The so-called ‘fiscal cliff’ is a somewhat misleading concept foisted on the American populace by an increasing corrupt and unstable political system.  As this system becomes more unstable and chaotic, information is lost and entropy increases.   Additionally if such a system is becoming ever more calcified, very little additional information is gained, and its' state of entropy moves ever closer to a chaotic collapse, disorder, and higher entropy.   If a chaotic state does transition to some type of system collapse, then additional information (novelty) is lost and entropy again increases along with a corresponding increase in the number of possible emergent states – a probability distribution that is determined by the nature of a given collapse or disintegration.  Information (novelty) is again gained when one or more of the ‘probable states’ emerges and becomes manifest.  Emergence is a phenomenon, which is by definition, virtually impossible to predict  (probability distribution of potential emergent states – the number of possible configurations of that state’s individual structural elements - is very broad, and the emergent state is one with a low probability of occurrence) - so when 'emergence' does occur, there is a huge drop in entropy and a correspondingly large increase in Information or Novelty.  Such a 'fiscal cliff' is only one feature of a complex reality that is undergoing a type of 'evolutionary metamorphosis'.  Such a transformational process is highly unlikely to occur at a specific time on a specific date, although there may be some events (as I think there already are) that one could reasonably associate with such a 'phase transition'. 

If the expression of Novelty theory through the Timewave can be demonstrated to be a fractal temporal waveform that is converging to an asymptotic zero entropy state during any given evolutionary epoch, then a plausible argument can be made that some type of phase transition rather than some ‘end date event’ is occurring.  Such a phase transition could include a chaotic phase transit into an unknowable emergent state of existence, or some type of emergent paradigmatic state that can be associated with the evolutionary principles of complex or non-linear systems.  Complex systems emergent states, paradigms, or trajectories could conceivably include two or more 'bifurcations', or multiple distinctly separate emergent states, paradigmatic structures, or trajectories.  If our current phase space turns out to be a transition phase of a multi-phase global evolutionary metamorphosis, then perhaps complex systems theory can give us some insights into the process that would help light the way.

In summary, Novelty theory and its’ expression through the Timewave should be seen as both immature and incomplete.  Additionally, the termination of the Timewave on a somewhat arbitrary ‘zero date’ must be seen as speculative at best.  As the driving principle behind the Timewave, Novelty Theory as proposed by McKenna contains no features that would imply or claim an historical ‘termination date’.   In view of these facts, I would argue that the Timewave is an incomplete expression of an immature or incomplete theory of Novelty that could be expressed using the priciples of Information Theory.  It is clear that ‘Novelty Theory’ is a theory that has never been ‘fleshed out’ – an often thankless and usually isolated pursuit that is a path normally undertaken with new theories.  South African scientist and a String Theory physicist, Dr. Neil Turok, says it very well in clearly expressing the dilemma of all new theories that challenge the wisdom and world view of any paradigmatic status quo:

“Coming up with this rough idea for how things might work, is of course exciting; but in having an idea like that, and then deciding to really pursue it, you are condemning yourself to years of misery – because you now have to flesh this out.”   

I would also argue that the particular type of ‘phase transition’ in which we may be currently embedded, could be very likely limited to our local star system in general, and perhaps restricted more specifically to ‘collective subsets’ of our planetary system.  After all, any process of ‘evolutionary metamorphosis’ can only take place when the caterpillar ceases its’ environmental rampage, and falls into a pupation process of enzymatic dissolution that permits cells of an emergent ‘creature’ to form and interconnect.  There is nothing to be found in Novelty Theory that either suggests or implies a termination singularity; and the speculative concepts of initial and terminal singularity events that are often claimed by institutional cosmologists, are slowly being deconstructed by String Theorists who challenge the rational absurdities of the status quo.   Consequently, it is becoming increasingly likely – if not evident - that we exist in a cosmos where the creation and conservation of increasingly higher ordered states of complex form is initiated and proceeds indefinitely through an ever expanding series of cosmic processes that trigger or induce both creation events and evolutionary phase transitions.

[1] Fractal modulated sigmoid functions, or logistic functions, may be proposed as alternatives to the current ‘terminal’ infinite series fractal expansion function, using principles of complex systems and information theory to project a Timewave waveform through chaotic phase transitions into increasingly complex higher-ordered novelty states.
[2] Nats are units of entropy/information when the natural log, ln, is used to calculate either entropy or information (novelty).
[3]  Entropy is the appropriate term to be preferred when referring to the Timewave y-axis values.  Novelty can then be defined as inverse entropy, or y-1 = novelty, or information increase.
[4] Zero entropy, or infinite novelty/information would imply that from an infinite number of possible ‘emergent’ novelty states, ηi where i → ∞,, each having a probability approaching zero, P(ηi) → 0, with only one of these extremely unlikely states emergent and manifest.  Obviously there is not an infinite number of possible novelty or information states having essentially zero probability of emergence, so this zero entropy asymptote is a mathematical artifact only.
[5] Define novelty as self-information, I(ηi), which is associated with the novelty outcome, ηi, having a probability P(ηi) which is then expressed: I(ηi) = log[1/ P(ηi)] = - log[P(ηi)].  Information Entropy can then be defined mathematically as: H(Ω) = -∑ P(ηi)*log[P(ηi)].