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### Ten

One of the hardest concepts for humans to grasp is infinity. Normally when I think of infinity I think of some indeterminate end a long, long ways away, but every so often I am reminded that infinity also occurs between discrete numbers, and of the infinite number of discrete numbers, what happens between zero and one is often incredibly interesting. Consider one concept fundamental to real-time operating systems - achieving the highest possible probability of repeatable success. When the number one is defined as one hundred percent success rate, attempts to achieve ever greater success asymptotically approach one but never reach it. Consider another concept fundamental to real-time operating systems - the time slice that bounds the event. Ever smaller and smaller time slices asymptotically approach zero, but again, never reach it.

The number system Nature obeys is not positive and negative numbers on a continuous line, but a logorithmic sectioning where we define our location by what power of ten we exist in. It does not matter whether we are talking about time, temperature, or reliability. Stars live and die at scales of 10^6 degrees K, humans live and die over scales at 10^1 K, molecules live and die at scales of 10^-6 K. The half-life of a free neutron dwarfs that of a muon, tectonic plate migration dwarfs the lifespans of humans, and rotations of galaxies can be measured in thousands if not millions of rotations of solar systems.

Every day scientists achieve ever-closer temperatures to the posited absolute zero but it should be obvious that temperature operates on the same powers of ten scaling at the low end as it does at the high end. Today maybe it's 1 x 10^-7 K, tomorrow it'll be 1 x 10^-8 K. Sure, there's new and interesting things going on like Bose-Einstein Condensates, but in a scale that goes to infinity, this is no closer to absolute zero than the three million degree corona of our Sun. It's just a different segment.

In RTOS and uptimes, success rates are measured in ever-increasing N nines. Success can be shown to be identical to other aspects of nature by looking at the delta: 99.999% (also known as "five nines") is equal to 1 - .99999 = 1 x 10^-5. Six nines is 1 x 10^-6, and so on. In physics, a calculated value that is merely two or three times the measured value is irrelevant, but an order of magnitude difference, that's a mistake. Is RTOS any different? The competitive edge is in "The Nines," that area where one vendor can claim reliability to five decimal places, only to be trumped by a vendor that claims six. Nobody cares about the numbers between 10^-5 and 10^-6, like 3.14159 x 10^-6 and 7 x 10^-6, but there are more numbers between them than stars in this galaxy, or grains of sand on a beach, grains that look like mountains to paramecium.

In trying to define a narrow position within a power of ten, the landscape becomes fractal, where scale is irrelevant because it all looks the same. If the time slice bounding an event is one millisecond, attempting to determine when the event occurs exactly creates more time slices within the time slice, so that microseconds divide milliseconds, nanoseconds divide microseconds, picoseconds divide nanoseconds, ad infinitum. Yet, without the precise knowledge of when the event occurs and what time slice bounds it, claiming success to "n" decimal places is a statistical artifact borne of repeated testing instead of hard timing.

In the interview with Paul McKenney, three time scales emerged as regularly used time metrics in RTOS. I'll define them to be Human (medical research), Digital (heating systems), and Gaming (real-time physics). To this I'll add two more time scales - Geologic and Financial. Geologic is simple enough to comprehend. If the four and a half billion years the Earth has existed is represented on a time line where one millimeter equals one million years, the time line would be 4.5 meters long, or roughly fifteen feet. The time humans have been on the Earth would be less than the period at the end of this sentence.

Much as to the Geologic Time metric is to Human, so is Financial to Human. Why "Financial?" Because the edge that ever-smaller time slices offer an electronic trader is the force behind the Holy Grail of the electronic stock market - option spread trading. Option spread is simple in concept - buy one option at one price, sell it at another, pocket the difference. Instead of betting on the option, the trader is betting on the difference in value at two separate oints in time. It's a second order derivative of the stock market, where options are the rate of change of the stock, and the spread is the rate of change of the option. Just like the formulation of calculus rose from taking smaller and smaller deltas, electronic option spread trading focuses on ever smaller time-slices that bound two prices. If I'm a trader and my system can calculate the value of an option using Black-Scholes faster than my competitor across the street, maybe I can buy the option from a seller and turn and sell it before my competitor catches up. If fiber optic cable can transmit information one mile every microsecond, in that microsecond someone has a price that is out-of-date and that's just enough time to do an infinite number of trades. In the end, Black-Scholes becomes a measurement of The Nines, where regardless of the time slice, the mathematics predict the fractal landscape of trading on nanotrends. Dust to dust, ashes to ashes, the lifespan of option spreads is the same to us as our lives are to the Universe, scale-defined self-similar analogues of virtual particles existing in the vacuum of quantized space.

The definition of a time slice is inherently linked to the definition of success. To what degree of nines do you want to guarantee success? Given enough time, it is possible to achieve twenty, fifty, or a thousand nines. There's also failure - what time slice is failure measured by? If failure means the complete lack of success, what are the consequences? Is failure the difference of one microsecond between five nines and six, as in if an event takes one extra microsecond to complete, does this reduce the success probability one order of magnitude so that 99.999% of the events will have a variance of less than one microsecond but 99.9999% of the same events will likely contain at least one event with a variance greater than that one microsecond? How much more than one microsecond? A tenth of a microsecond?

What emerges is a duality, where time-slices and success rates are digital particle-wave equivalents, complete with a corrollary to the Heisenberg Uncertainty Principle, where the degree to which the time slice is accurately measured blurs the success rate and the more precise the success rate the wider the range of time slice. Even in the hardest metal, most deterministic system, there is some power of ten definition of time-slice or success rate that makes the system non-deterministic. Moore's Law may move that power of ten segment, but the landscape is the same.

In theory, space and time become quantized at small enough powers of ten, where the passage of time becomes like a movie reel, frame by frame, and movement in space is the distance light travels from frame to frame. At the other end of limits of powers of ten in Nature is the distance across the universe, and the temperature inside the Big Bang. Personally, I just think these are anthropomorphic idealizations of zero and one, really interesting but not all that there is. There is, after all, the number two....