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, NM87508www.sheliaksystem.com
Initially this was to be my response to a query about a novelty perspective for our looming ‘fiscal cliff, since defining novelty or ‘habit’ (entropy) for specific processes or events has largely been subjective and somewhat speculative, This unfortunate state of affairs is largely due to 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 arbitrary classification, I thought that some theoretical and mathematical clarification might be in order concerning the nature and implications of Novelty Theory and the Timewave as currently formulated and mathematically constructed.  
First of all, there are neither embedded ‘infinities’ nor singularities intrinsic to the formulation of Novelty Theory or to the mathematical construction of the current Timewave, since the mathematics contains no features that include a division by zero.  In fact, all Timewave fractal features are produced and ‘modulated’ by an infinite series expansion - or fractal transformation – which is a mathematical operation that generates waveform data values that decay to zero magnitude based on time remaining to an established zero date/time – i.e. 6 a.m. on 12/21/2012.Although the rationale for aligning this fractal transformed waveform zero value to this zero date is somewhat vague, it was apparently done with some thought given to the ‘novelty fit’ of the Timewave time axis to other sections of the waveform.So it appears as though both the established waveform zero value, and its’ zero date temporal alignment are at the very least somewhat arbitrary, and certainly much more theoretical than empirical in nature.Nevertheless, in order to examine Novelty features as expressed in the vicinity of the zero date, and potentially explore alternative interpretation or such features, it could be revealing to fit the Timewave fractal waveform function with a decaying exponential function.
It is important to note that the operative feature of this exponential decay function[1]is its’ convergence to a zero-valued y-asymptote, meaning that the ‘zero point’ is not an actual point but an asymptote that is approached only as elapsed time approaches an extremely large value, or as t ®¥, which is something not achieved in the real world.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 side to the other with each step.”The curve-fitting candidate for the aggregate Timewave graph would be a decaying exponential function expressed as, 
y = b*exp (-t/t)   ⁽¹⁾
Where: y is the Timewave entropy state in nats; t is the Timewave elapsed time in Julian days; b is the initial Timewave entropy state, y_b, in nats at time t=0, defining the Timewave entropy state initial condition[2]; and t 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’[3]asymptote approached by the Timewave.Note that if we define novelty as inverse entropy using the information theory notation, I (h_i)= (y^-1)= (1/y)[4], then a zero entropy is equivalent to a novelty/information singularity at y_¥ = 0, as t®¥.
So let's examine the consequential results from the proposed exponential fit to the fractal expansion Timewave graph.  The information of interest is how fast an established set of events changes with established measures of time.The mathematical operation for expressing the 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 – an operation that is clearly defined mathematically by the following differential expression,
dy/dt = -(b/t)* exp(-t/t)   (2)
Where: the derivative, dy/dt, expresses the rate at which a set of unspecified, but established events unfolds, changes, or advances with established measures of elapsed time.This differential shows that the time rate at which an established set of events unfolds is diminishing over time (expansion of time between events) by the factor, (-b/t).This result implies that the time it takes to experience an established sequence of events is expanding; accompanied by the perception that time has passed more quickly – accelerated.This could mean that the normal sequence of all the established set of events, activities, etc. in one’s life would appear to be occurring over increasingly longer time intervals; and the derivative dy/dt would define the time rate for such an unfolding event set mathematically.  In the limit where the derivative dy/dt ® 0 as t®¥, the number of events unfolding during an elapsed time measured by our local clocks would approach zero, along with the perception that the passage of time had accelerated to infinity – or a temporal collapse entirely.This concept is rather tricky to envision, since the experience, and the measure of time is in fact relative.Additionally, if we experience an expanding time interval between events, as measured by our local clocks, then and actual time acceleration or compression means that we experience a contracting time interval between events with our internal clocks. 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 = (-t/y), or alternately,   dt = (-t)* (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, is being compressed by the product of a increasing novelty magnitude (y^-1), a diminishing entropy change (∆y), and the system time constant (t).A time span that is undergoing temporal compression over time is a time span that is accelerating over time.
In order to tie 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, nearly impossible to predict (probability distribution of potential emergent states 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 the Timewave as established by Novelty Theory is demonstrated to be a fractal temporal waveform that is converging to an asymptotic zero entropy state during any given planetary evolutionary epoch, then a plausible argument can be made that some type of phase transition rather than ‘end date’ is occurring.Such a phase transition could include a chaotic phase transition leading to an unknowable emergent state or emergent paradigm that can be associated with complex or non-linear systems principles.  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 state turns out to be one stage of a multi-stage global evolutionary metamorphosis, then perhaps complex systems theory can give us insights into the process that would help light the way.
In summary, Novelty theory and its’ Timewave expression 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.Moreover, Novelty Theory as proposed by McKenna contains no features that imply or state an historical ‘termination date’.   In view of these facts, I would argue that the Timewave is an incomplete expression of an immature theory of Novelty or Information.I would also argue that this particular type of ‘phase transition’ is most likely limited to our local star system in general, and perhaps even ‘collective subsets’ of our planetary system specifically.After all, any ‘evolutionary metamorphosis’ can only take place when the caterpillar ceases its’ environmental rampage, and falls into a pupation process of enzymatic dissolution. There is nothing in Novelty Theory that suggest or implies a termination singularity.In fact, Novelty Theory actually suggests otherwise, and that we may very well live in a cosmos where the creation and conservation of increasingly higher ordered states of complex form proceeds indefinitely through a expanding series of phase transitions.Here’s hoping..

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[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]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.[3]Zero entropy, or infinite novelty/information would imply that from an infinite number of possible ‘emergent’ novelty states, h_i where i ®¥, each having a probability approaching zero, P (h_i) ® 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.[4]Define novelty as self-information, I (h_i), which is associated with the novelty outcome, h_i, having a probability P (h_i) which is then expressed: I (h_i) = log [1/ P (h_i)] = - log [P (h_i)].Information Entropy can then be defined mathematically as:H (W) = -∑P (h_i)*log [P (h_i)]