On Occam’s Razor (1.2.0)

Abstract What is the basis of Occam’s Razor? Why, that is to ask, should simplicity be preferred to complexity? Here we offer an answer to that question, and give some real-world examples of the employment of Occam’s Razor in problem-solving.

 

PART I: THE MATHEMATICAL FOUNDATIONS OF OCCAM’S RAZOR

According to the widely accepted principle known as Occam’s Razor, simple explanations are to be preferred to complex ones, and the question arises as to why this is so. Broadly, the answer offered here is that an explanation is a description of a state-of-affairs, and the more orderly a state-of-affairs, the more energy-costly is that state-of-affairs, and thus the less likely it is that an orderly state-of-affairs will occur than a relatively disorderly state. A more orderly state is in some sense more complex than a less orderly state. Nature therefore prefers simple explanations because she prefers less orderly states-of-affairs because she prefers to conserve energy. But there are subtleties here that can be explored by reference to a mathematics paper published in 1859 (2). In this work, a little-known mathematician named Bernhard Riemann stated that the real part of the non-trivial zeros the zeta function is “very probably” always equal to 1/2, a statement that has come to be known as the “Riemann Hypothesis”.

Expressed in this way, the RH is incomprehensible (except to a relatively small sub-set of those with post-graduate degrees in mathematics), but beneath the esotericism it is a simple statement about the prime numbers (numbers divisible only by themselves and 1) and the rate at which these thin out as the number line continues toward infinity. It was proven in 1896 (3, 4) that the limit of number of primes not greater than x as x -> Infinity is

\frac{x}{x \log }

The RH is equivalent to the stronger claim that the number of primes not greater than x is

\sum _{n=2}^x \frac{1}{n \log }

plus or minus the square root of x.

It is to be solved by re-expressing the tradition equation for a circle of area 1 as

\lim_{x\to \infty } \left(e^{2 \gamma } \sqrt{\frac{1}{e^{2 \left(\sum _{n=1}^x \frac{1}{n}-\int_1^x \frac{1}{n} \, dn\right)}}}\right){}^2=1

which can then be extended in the following way:

\lim_{x\to \infty } \left(e^{(s+1) \left(\zeta (s)-\frac{1}{s-1}\right)} \left(\left(\frac{1}{\exp \left((s+1) \left(\sum _{n=1}^x \frac{1}{n^s}-\int_1^x \frac{1}{n^s} \, dn\right)\right)}\right){}^{\frac{1}{s+1}}\right){}^{s+1}\right)=1

This extended equation involves a significant division between s = 1 and real values of s other than 1, for if and only if s = 1 does

\sum _{n=1}^x \frac{1}{n^s}-\int_1^x \frac{1}{n^s} \, dn

not reach the limit

e^{(s+1) \left(\zeta (s)-\frac{1}{s-1}\right)}

The division allows us to distinguish between long-ranged (infinite primes) and short-ranged (finite primes) progressions. For example:

If and only if the number line is long-ranged, is it the case that number of primes not greater than x is within the bounds prescribed by the RH. Hence the error term is what it appears to be, and what the Riemann originally said it was.

The RH is more than a technicality, of interest only to pure mathematicians and those with an interest in puzzle solving, for it provides amongst other things the foundation of the distinction between order and disorder, and of the idea that the universe as we know it is governed by an arrow of time that leads from a state of prime-density (order) towards to one of prime-sparsity (disorder). Consider something that Richard Feynman once said about entropy (5). He invited us to imagine that atoms are divided into blue-coloured and white-coloured varieties, and separated into compartments. If the separation is removed, then just as blue dye turns water a luke-blue colour, the atoms form a luke-blue mixture. He observes that individual collisions provide no clue as to the irreversibility of the mixing process, but that studying a film of the mixing played in reverse reveals that

…every one of the collisions is absolutely reversible, and yet the whole moving picture shows something absurd, which is that in the reverse picture the molecules start in the mixed condition… and as time goes on, through all the collisions, the blue separates from the white…

He went on to say that

.…it is not natural that the accidents of life should be such that the blues will separate themselves from the whites…

The one-way nature of this process is reflected by from the experiment in which a gas is confined to one of two compartments. If the separation between the compartments is removed, then the gas spontaneously distributes itself in a uniform manner throughout the two compartments, but it does not spontaneously revert to the separated state. More familiar still, is the breaking of an egg. We never see a broken egg spontaneously reassemble, and there is no way to reassemble an egg after it has been broken (“All the kings horses and all the kings men couldn’t put Humpty together again.”). But although both these processes involve a one-way direction when viewed from a sufficiently global perspective, they can go either way when viewed from a sufficiently local perspective: the individual atoms comprising the gas molecules might just as well go from compartment B to compartment A as from compartment A to compartment B, and if we study the individual atoms comprising Humpty Dumpty we get no clue as to the fact that Humpty cannot be reassembled. If we play a film depicting the un-breaking of an egg or the increase of the density of the prime numbers in the number line, we see something that looks absurd, and Feynman had no better explanation for this impression of absurdity in the first case other than ‘the accidents of life’. But the impression of absurdity attached to the loss of density of the prime numbers as we count down the line isn’t an ‘accident’ at all – it is mathematically necessary that the repetition of a unit be accompanied by a global decrease in prime-density. This is the Prime Number Theorem. The Riemann Hypothesis is an extension of the Prime Number Theorem: it says, not merely that the primes thin out globally, but that local changes in prime-density are equally likely to involve an increase as a decrease in density, and that they cannot exceed the upper and lower bounds marked in red and blue in the graphs below:

\text{Density}=\frac{\pi (x)}{x}

\text{Density} \text{Min}=\frac{\sum _{n=2}^x \frac{1}{n \log }-2 \left(\text{Re} \sum _{n=1}^{\infty } \text{Ei}\left(\rho _{-n} (\log x)\right)\right)}{x}

\text{Density} \text{Max}=\frac{\sum _{n=2}^x \frac{1}{H_n}-2 \left(\text{Re} \sum _{n=1}^{\infty } \text{Ei}\left(\rho _{-n} (\log x)\right)\right)}{x}

To the naked eye, the distribution of the stars in the night sky seems to be random, but looking through a telescope we realize that galaxies have a spiral shape, light-dense toward the center of the galaxy, and increasingly dark at distances further away from the center.

Same thing with the primes in the number line:

It is only by considering a sufficiently large group of stars and primes, and a sufficiently large group of particles, that the loss of energy-density known as ‘entropy’ is found to involve a one-way direction known as the arrow time. In fact there are multiple arrows of time. Well known are the thermodynamic arrow arising from the loss of heat, the cosmological arrow arising from the expansion of the universe, the radiative arrow arising from the expansion of waves outwards from their source, the causal arrow arising form the fact that effects follows causes rather than precede them, the quantum arrow arising from the collapse of the wave-function, and the psychological arrow of consciousness arising the fact that we remember the past and the future is unknown… Less well-known is the genetic arrow, which arises from the loss of mutability of DNA with generation, a consideration that explains anomalous results such as the apparent mismatch between Y-DNA extracted in 2014 from the skeleton of the English King Richard III and his contemporary paternal relatives (6), and points to the Theory of Evolution being a special case of a larger more sophisticated theory (Darwinism depends on symmetrical DNA mutation rates). But the arrow that contains and explains all the others is the arithmetic arrow. The first person to hint at the possible unification of all of these arrows was Euler (7), who he noted that the product continued to infinity of this fraction

\frac{2\ 3\ 5\ 7\ 11\ 13\ 17\ 19\text{...}}{2\ 4\ 6\ 10\ 12\ 16\ 18\text{...}}

in which the numerators are prime numbers and the denominators are one less than the numerators, equals the sum of the infinite series

1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}\text{...}

and they are both infinite. To prove his point to Euler invites us to imagine the extraction from the second series a prime denominator and all remaining multiples of that prime denominator until everything except the first term 1 has been eliminated. Let

x=1+1/2+1/3+1/4+1/5+1/6...

Then

\frac{x}{2}=\frac{1}{2}+\frac{1}{4}+\frac{1}{6}+\frac{1}{8}\text{...}

This leaves

\frac{x}{2}=1+\frac{1}{3}+\frac{1}{5}+\frac{1}{7}\text{...}

To eliminate the denominators that are divisible by 3, we divide both sides to get

\frac{x}{2\ 3}=\frac{1}{3}+\frac{1}{9}+\frac{1}{15}+\frac{1}{21}\text{...}

Subtracting again eliminates all remaining denominators that are multiples of 3 leaving

\frac{2 x}{2\ 3}=1+\frac{1}{5}+\frac{1}{7}+\frac{1}{11}+\frac{1}{13}\text{...}

Applying this eliminatory process to all of the prime numbers leaves

\left(\frac{2\ 4\ 5\ 10\ 12\ 16\ 18}{2\ 3\ 5\ 7\ 11\ 13\ 17\ 19}\text{...}\right) x=1

This is a thought-experiment -mere imagination- but if these eliminations could be performed in the physical world, they would result in the disappearance of any distinction between the form and the content of a coordinate system, and therefore the shrinking of space and the slowing of time to a zero-dimensional point. With all of reality contracted to a zero-dimensional point, the distinction between the world and the mind that surveys it is lost. The idea we can take from Euler’s thought-experiment is that, since both prime-density and energy-density must at this point be infinite. From here we can begin to formulate a theory according to which the development of the universe from this central singular point towards an ever-increasing state of de-centralization is a process involving the distribution of the prime numbers… Once this connection is made we see that all of our arrows are subject to the same dynamics as the arithmetic arrow constituted by the number line and governed by the RH, and all exhibit the same tendency to decentralization.

 

PART II: USING OCCAM’S RAZOR TO CONSTRAIN THE COMPLEXITY OF SOME PROBLEMS

1. False Flags

Conspiracy Theories are theories according to which things aren’t as they seem. There is in every conspiracy theory a veil of illusion behind which the truth hides. The ultimate conspiracy theory was proposed by Descartes, who in the Meditations (1), raised the possibility that the world is an illusion created by an evil demon, if only to banish this demon on the grounds that a world in which things are as appear to the rational mind is a condition of rational inquiry. If the world is a systematic deception, then this is a far more complex state of affairs than a world that is as it appears, and every conspiracy theory, regardless of its extent, is more complex than its non-conspiratorial counter-parts.

“False flags” are an illuminating sub-category of conspiracy theories. According to the Wikipedia definition, these are

covert operations that are designed to deceive in such a way that activities appear as though they are being carried out by individual entities, groups, or nations other than those who actually planned and executed them.

If for example The Cannibals vandalize The Devil’s clubhouse and leave behind a jacket worn by members of the The Vikings, in order to make it appear that The Viking’s were the vandals, that’s a false flag. An early example of an alleged false flag concerns the Reichstag fire, an arson attack on the Reichstag building (home of the German parliament) in Berlin on 27 February 1933, which occurred one month after Adolf Hitler had been sworn in as Chancellor of Germany.

A young Dutchman and Communist sympathizer named Marinus van der Lubbe was found by police at the scene and blamed for the fire. The Nazi’s appealed to the fire as evidence of a Communist plot against the German government, and the result was a decree, mass arrests of the members of the Communist party, and -with their rivals gone- the attainment by the Nazi party of a majority in the Reichstag… A recent example of an alleged false flag is the attack on the twin towers known as 9/11, and used by George Bush as the basis of the US invasion of Iraq. A popular example of an alleged false flag is the assassination of President Kennedy, claimed by some conspiracy theorists not to be -as it appears- the doing of Lee Harvey Oswald alone.

Solution

In the case of the Reichstag Fire, an event occurred that was to the benefit of the Nazis and Adolf Hitler; in the case of 9/11, an event occurred that was to the benefit of the Republicans and George Bush. But the fact of such benefits doesn’t mean that Marinus van der Lubbe wasn’t solely responsible for the Reichstag Fire, nor that Al Quida weren’t solely responsible for 9/11. Marinus van der Lubbe = sole perpetrator of the Reichstag Fire, Al Quida = sole perpetrators of 9/11, Lee Harvey Oswald = sole perpetrator of the assassination of JFK: these are the explanations that are in accord with Occam’s Razor, for they are simpler than the false flag explanations, and since there is no evidence to suggest that these explanations are inadequate to the facts, we conclude -at least for now- that these operations were not false flags.

2. The Mary Celeste

The Mary Celeste was a merchant brigantine discovered adrift and deserted in the Atlantic Ocean, off the Azores Islands, on December 5, 1872.

She was seaworthy, under partial sail, and her lifeboat was missing. The captain and the crew’s belongings were undisturbed, as was the cargo of denatured alcohol. No one who had been aboard the vessel was ever seen or heard from again.

Solution

The ship was carrying denatured alcohol, highly inflammable, and capable of destroying the ship should it explode. What must have happened is this: there was an explosion from the hold, and possibly flames, which prompted the Captain -Nathan Brigg’s- to order the evacuation of the ship. In his haste to leave the vessel before she exploded, Brigg’s must failed to secure the yawl to the towline. There was no sign of any damage to the ship, and no scorch marks, but experiments conducted in a replica hold showed that these are unnecessary features of an explosion. (8) Using butane gas, a large blast, together with a ball of flame, was created in the replica hold, and yet this did not result in any damage, nor was there any trace of the explosion. The story of the Mary Celeste is broadly mirrored by Herge’s fictional story The Red Sea Sharks (9), in which the main characters are held hostage aboard a merchant vessel smuggling explosives. Fire breaks out, and the Captain and crew clamber aboard the lifeboats and flee the vessel, only to have the fire die out before the explosives can ignite. They attempt to head back to the ship, by this time under the control of the hostages (one of whom is a former sea Captain), but she sails away leaving them in her wake…

3. Jack the Ripper

In 1888 there were a series of murders -at least 5- in the Whitechapel district of London. The victims were female prostitutes, all of whose throats were slashed prior to abdominal mutilations. The name Jack the Ripper was derived from the following letter -probably a hoax- received by police at the time (10):

Dear Boss,

I keep on hearing the police have caught me but they wont fix me just yet. I have laughed when they look so clever and talk about being on the right track. That joke about Leather Apron gave me real fits. I am down on whores and I shant quit ripping them till I do get buckled. Grand work the last job was. I gave the lady no time to squeal. How can they catch me now. I love my work and want to start again. You will soon hear of me with my funny little games. I saved some of the proper red stuff in a ginger beer bottle over the last job to write with but it went thick like glue and I cant use it. Red ink is fit enough I hope ha. ha. The next job I do I shall clip the ladys ears off and send to the police officers just for jolly wouldn’t you. Keep this letter back till I do a bit more work, then give it out straight. My knife’s so nice and sharp I want to get to work right away if I get a chance. Good Luck. Yours truly
Jack the Ripper

Dont mind me giving the trade name

PS Wasnt good enough to post this before I got all the red ink off my hands curse it. No luck yet. They say I’m a doctor now. ha ha.

The murders were never officially solved.

Solution

Sir Melville Macnaghten, the Assistant Chief Constable of the London Metropolitan Police, wrote that he suspected a Polish-Jew named Kosminski, saying that he had a hatred for women and homicidal tendencies. Also, Assistant Commissioner Sir Robert Anderson wrote in his memoirs The Lighter Side of My Official Life that the Ripper was a “low-class Polish Jew”. Anderson:

Undiscovered murders are rare in London, and the Jack the Ripper crimes do not fall into this category. I am almost tempted to disclose the identity of the murderer but no public benefit would result from such a course. In saying that he was a polish jew I am merely stating a definitely ascertained fact. I will merely add that the only person who ever had a good view of the murderer unhesitatingly identified the suspect the instant he was confronted with him, but he refused to testify.

Chief Inspector Donald Swanson, who led the Ripper investigation, named the Polish Jew as “Kosminski” in notes handwritten in the margin of a copy of Anderson’s memoirs. He goes on to say that Kosminski had been watched at his brothers’s Whitechapel home by police, and that he was taken to the workhouse, and then to the Colney Hatch Asylum. (11) A search conducted in 1987 of asylum records found only one Kosminski – Aaron. (12) The case notes say that he was a paranoid schizophrenic, and a compulsive masturbator, both typical traits -especially the latter- of serial killers, whose actions are largely sadistic masturbatory fantasies enacted outside the privacy of minds obsessed by self-comfort. These case notes on Aaron Kosminski match Swansons’s margin notes – Aaron Kosminksi went to the workhouse and then to Colney Hatch. A Polish-Jewish immigrant, Aaron Kosminski , worked as a hairdresser in Whitechapel.

In 2014, British author Russell Edwards published a book in which he drew on an analysis by Dr. Jari Louhelainen, an expert in historic DNA, of a shawl allegedly found lying on the ground beside the mutilated body of canonical Ripper victim Catherine Eddowes. (13) According to Louhelainen, the analysis revealed the shawl to contain mitochondrial DNA matching, on the one hand, a female line descendant of Eddoes’ sister, and on the other, a female line descendant of Kosminski’s sister… Edward’s theory that Aaron Kosminksi was the Ripper has been disputed on a number of grounds, but this is a case where, whether the theory is right or wrong, the method underlying the theory is the right one. In a 1973 documentary for Yorkshire television (14), Fred Hoyle tells Richard Feynman a story of a man who loses a key at night, and is discovered looking for it under a street lamp by a passer-by. After assisting in the search for a few minutes, the passerby asks the man if he is sure that this is where he lost the keys. The man replies “Not at all, but unless I lost it here, I’ll never find it.” The point of the story for Hoyle was that, in science, one has to make certain assumptions that one is not sure of in order to go forward. It’s applicability to the Jack the Ripper case is that, only if there is an argument such as Edwards (one which blends history and forensics), is there any hope of discovering the identity of Jack the Ripper. “Rippologists”, who seem to take some sort of perverse pleasure in the ongoing nature of this mystery, are strongly inclined for emotional reasons to reject any theory that aims -as does Edward’s theory- to constitute a final solution, and without such a theory, the Rippologists will have their way, and there will never be a solution to the mystery. The historical data alone is insufficient to establish the Ripper’s identity, and Edward’s solution – unlike any of its rivals other than Patricia Cromwell’s version of the theory that The Ripper was painter Walter Sickert (15) – involves a forensic as well as a historical perspective. Like Sickert, many of the Ripper suspects can be eliminated on false grounds that the Ripper could be a romantic figure, in the sense of someone of talent or status but with a dual identity, ala Stephenson’s Jekyll and Hyde. Like Kosminksi, serial killers are always people without great talent or status, because their lapse into a masturbatory fantasy-world where they are powerful arises from their failure to be powerful in the real world, and the feelings of worthlessness born of the narcissist’s overwhelming sense of under privilege.

4. D.B. Cooper

On November 24, 1978, a man identifying himself as “Dan Cooper” purchased a one-way plane ticket on a 30 minute flight (305) from Portland to Seattle:

Shortly after takeoff, he handed the stewardess a note, which thinking it was a proposal, she put in her handbag. At this point, he leaned toward her, saying

Miss, you’d better look at that note. I have a bomb.

He demanded 200, 000, four parachutes (two primary and 2 reserve) and a fuel truck standing by to refuel the plane when it landed at Seattle. After the re-fueling was complete, Cooper ordered the plane to take off with the rear exit door open and the staircase extended. 20 minutes later a warning light in the cockpit indicated that the aft stair apparatus had been activated, and 13 minutes later there was an upward movement in the aircraft’s tail section. When the plane landed about 2 hours later at Reno airport, it was determined that Cooper was not on board. A search of the presumed drop-zone failed to reveal any sign of Cooper or the equipment he took with him. Ultimately, the search operation -possibly the most extensive, and intensive, in U.S. history- uncovered nothing related to the hijacking. In 1978, a placard printed with instructions for lowering the aft stairs of a 727 was found by a deer hunter within Flight 305’s flight path. In 1980 a young boy, holidaying with his family 32 k from the presumed drop-zone, uncovered from a sandy river bank 3 packets of the ransom cash. The bills were degraded, but still wrapped by rubber bands. There were 10 bills missing from one packet. To date, none of the 9,710 remaining bills have turned up anywhere in the world.

Solution

The problem with the idea that Cooper survived the jump and returned to civilization is that the money he misappropriated was never spent; the problem with the idea that he didn’t survive the jump and return to civilization is that no sign of his parachute, or his body, or any of the equipment used in the jump were ever found despite considerable searching. The bigger problem is the first. Moreover, he took with him on his jump a dummy reserve parachute that had been accidentally included amongst the 4 parachutes he asked for, something an experienced parachutist would never do. An inexperienced parachutist however could very well -as Cooper did- turn down instructions on how to use the parachutes, and put the parachute on as if he knew what he was doing based only on arrogance and research. Conclusion: Cooper was not very experienced (although he may have parachuted previously), and he didn’t come out of the escapade alive, a parachute jump undertaken in the middle of the night, into a 200 mile an hour wind, with a chill factor of \[Minus]70 \[Degree]F (\[Minus]57 \[Degree]C) into wild territory. His body and the parachute weren’t found simply because it’s impossible to say exactly where the drop zone was, and the searchers were effectively looking for needles in a haystack. Those who say otherwise are motivated by the irrational desire to see Cooper as a romantic hero. The only mystery here is that of Cooper’s history, and his reasons for the hijacking.

5. Turin Shroud

The Turin Shroud -the most studied artifact in human history- is a 14.3 foot by 3.7 ( 8 by 2 cubit) linen cloth bearing the frontal and dorsal image of a naked man whose wounds are identical to those suffered by Jesus of Nazareth. The image resides only on top-most fibrils of the threads with which the TS is woven, and it is a negative image. Although very faint when viewed as a positive, the image becomes much clearer when darks and lights are reversed:

In 1978 group of scientists known as STURP (Shroud of Turin Research Project) performed an intensive series of tests on the TS. Most were skeptical and expected that they would quickly discover it to be a fake, but after three years of analyzing the data they collected they wrote in their final report:

We can conclude for now that the Shroud image is that of a real human form of a scourged, crucified man. It is not the product of an artist. The blood stains are composed of hemoglobin and also give a positive test for serum albumin. The image is an ongoing mystery and until further chemical studies are made, perhaps by this group of scientists, or perhaps by some scientists in the future, the problem remains unsolved.

Ten years later in 1988, 3 laboratories carbon-dated samplings of a sample taken form the corner of the TS (Rae’s corner) and judged this same sample to have been produced with 95% confidence between 1260 and 1390 AD (16).

Solution

Since the possibility that TS is the authentic burial garment of Jesus of Nazareth involves all kinds of complex implications, shouldn’t we, by Occam’s razor, adopt the forgery explanation without question? Most say yes, but no we shouldn’t. Putting aside the objection that the radiocarbon dating procedure of 1988 clearly broke -nay shattered- scientific protocol by the failure to use independent samples (to ensure that the results of the radiocarbon pertained to the cloth as a whole rather than to an isolated portion of the cloth), this explanation is can’t explain the body-image, a photographic negative produced 600 years before the invention of photography, and a good explanation of the TS must, first and foremost, be able to do this. Attempts to duplicate the image on the TS – such as that by chemistry professor Luigi Garlaschelli (below right)- are, not merely unconvincing, but ridiculously crude when compared to a high resolution rendition of the image on the TS:

Consider that in the above mentioned documentary Take The World From Another Point of View, Richard Feynman and Fred Hoyle rightly agree that we should try to do as much as possible with the scientific principles we presently use rather than introduce new principles. New principles are akin to conspiracy theories- an extraordinary and energy-costly state of affairs is required to produce new principles that are true- and we introduce then only when the old one’s are blatantly inadequate to the explanatory task … When it comes to the Reichstag fire, and 9/11, the assassination of JFK- and the holocaust, the moon landings, the death of Elvis etc.- the apparent explanations are adequate, and there is no need to introduce a less orthodox explanation, but this is not the case when it comes to an array of scientific phenomena, where the orthodox explanations don’t work at all and are literally incoherent. The background of many of these unexplained phenomena is General Relativity, Einstein’s theory of gravity. At root of GR is the idea that mass curves space-time (17, 18), an idea that which implies the existence of local infinities at centers of black holes (19), an account of gravity that fails to cohere with quantum mechanics (which has its own problems) (20), and fails to explain the flat rotation curves of distant galaxies. The last failure is the motivation for the idea of dark matter (21, 22):

GR readily takes us back to an initial condition of the universe such that all the mass of the universe is compressed to a point, and this same infinite compression of mass is, by the terms of the theory, also to found at the centers of black holes. But the singular nature of the initial condition of the universe represents the beginning of the time, while the singularities at the centers of black holes in some sense represent the end of time, and these forms of curvature should therefore be quite distinct. More particularly, it should not be the case that both are attributable to the infinite action of the force of gravity. This is the theory breaking down, and a sign of a false assumption. There is talk of the big bang versus the big crunch, but since both these states are associated with infinite gravity, General Relativity paints a picture of the universe that begins and ends in an identical state when, very clearly, there is throughout the universe as we know it an arrow of time leading from an energetic contracted state to an exhausted expanded one. If we give up this idea that curvature is due to mass (which is a combination of light and space), and employ instead the idea that curvature is due to imbalances of light and space (where the classical world is balanced, the atomic world is unbalanced in the direction of light, and black holes are unbalanced in the direction of space), we will find that we can solve this and other problems. Mathematically, we capture what it is to be balanced, and what it is to depart from balance, thereby producing curvature, by appeal to the machinery of the Riemann Hypothesis. Reiterating, we re-express the tradition equation for a circle of area 1 as

\lim_{x\to \infty } \left(e^{2 \gamma } \sqrt{\frac{1}{e^{2 \left(\sum _{n=1}^x \frac{1}{n}-\int_1^x \frac{1}{n} \, dn\right)}}}\right){}^2=1

Where the traditional equation fails by implying that an energy source located at the center of this area unit-circle is undiminished from center to circumference (it has either a zero or an infinite radius), the second provides us with a potentially infinite hierarchy of energy levels that are necessarily non-infinite and non-zero. Given that gamma is a spacial case of \zeta (s)-\frac{1}{s-1} for s = 1, we can go from here to the more general

\lim_{x\to \infty } \left(e^{(s+1) \left(\zeta (s)-\frac{1}{s-1}\right)} \left(\left(\frac{1}{\exp \left((s+1) \left(\sum _{n=1}^x \frac{1}{n^s}-\int_1^x \frac{1}{n^s} \, dn\right)\right)}\right){}^{\frac{1}{s+1}}\right){}^{s+1}\right)=1

Let s be a positive integer greater than 1 and let

\left(\frac{e^{2 \gamma } \left(e^{-\left(\zeta (s)-\frac{1}{s-1}\right)}\right)^2}{\hbar =e^{2 \gamma } \left(e^{-\left(\zeta (s)-\frac{1}{s-1}\right)}\right)^2-e^{2 \gamma } \left(e^{-\left(\sum _{n=1}^7 \frac{1}{n^s}-\int_1^7 \frac{1}{n^s} \, dn\right)}\right){}^2}\right){}^{1/s} = the critical radius on the one side of which curvature in the direction of light is classical.
and on the other is non-classical

and we see something like this:

Let s be a positive real number less than 1, and let

\left(\frac{e^{2 \gamma } \left(e^{-\left(\zeta (s)-\frac{1}{s-1}\right)}\right)^2}{\hbar =e^{2 \gamma } \left(e^{-\left(\zeta (s)-\frac{1}{s-1}\right)}\right)^2-e^{2 \gamma } \left(e^{-\left(\sum _{n=1}^s \frac{1}{n^s}-\int_1^s \frac{1}{n^s} \, dn\right)}\right){}^2}\right){}^{1/s} = the critical radius on the one side of which curvature in the direction of space is classical
and on the other is non-classical

and we have the following model:

Using this model we can solve the above problems (the infinities in black holes disappear, we get coherence between gravity and the other forces, and we dispense with the need for dark matter). More importantly, we solve the central problem of the TS, which is as indicated is that the body-image is a photographic negative produced long before the invention of photography. The body-image is the result of the oxidization of the topmost micro-fibers of the cloth, a phenomenon that is too subtle to have been produced by any chemical reaction, but could have been produced by a sufficiently intense and brief burst of radiation emanating from the body in the TS. Trouble is that, as John Jackson has suggested (23), a new physics is surely required to understand how this could occur (human bodies generally can’t radiate at all, let alone in such a powerful way), and from the discussion of Occams’s Razor and Feynman and Hoyle, we know that we should not introduce new physical principles until we are the sure the old ones are insufficient. But as we have seen the old principles are insufficient, and we also know that the Feynman/Hoyle stance doesn’t apply if the new principles are better than their predecessors. Chances are of course that any proposed new set of principles will be incorrect (the energy-cost of a novel solution is greater than that of an orthodox solution, and nature therefore in some sense prefers old hat solutions), but should a true set of new principles happen for whatever reason to arise, it is an easy matter to verify- or at least to test- their truth, albeit hard to devise or discover these principles in the first place.

The first person to provide a glimpse -however obscured- of this new physics was artist and sindonologist Isabel Piczek, who wrote in language few could have understood at the time of the role of the suspension of gravity in the resurrection (24):

We have stated that there is a dividing line, a real INTERFACE between the projection of the Frontal Image and the Dorsal Image that was, no doubt, created by the Body wrapped into the Shroud.

REVEALING SIGNS:

1) The hermetic separation of the two images Frontal and Dorsal without any overlap.

2) The lack of anatomical distortion of the naked Body projected on the Shroud.
\[LongDash]These both indicate that the Shroud was forced absolutely taut and precisely parallel with some kind of horizontal entity running in the center.
\[LongDash]Also is apparent the presence of an inner Enclosure, AN ISOLATED SYSTEM, with all that this Isolated System would indicate or even enforce.

3) It is clearly visible on the Shroud Images, especially on the Dorsal Image, that the muscles of the Body are not crushed and flattened against the stone bench of the tomb.

4) The Body is hovering between the upper and the lower sheet and there is NO TRACE OF GRAVITY.

5) The lack of gravity is also further proven by the Shroud linen. The linen does not fall on top of the Body, but remains in its unnaturally stretched condition at some distance from the body.

All the above tell us that the INTERFACE indicated is not an ordinary Interface. Judging by its qualities it has to be an EVENT HORIZON that blocks every communication between the two sides of the Image.

Let us see what is usually indicated by an Event Horizon and how does that relate to the Event Horizon of the Shroud:

An Event Horizon is a critical line or a radius that divides Space-Time into two distinct regions. The exterior region one can experience, but the region beyond the critical line or radius one cannot experience. The critical line marks the path of the last light pulse that still reached the Event Horizon and Time itself slowed to a halt. Looking at the critical line from the other side Time and events gain almost infinite speed and one could see the whole history of the Universe, past present and future rapidly passing to an arbitrary end.

Ordinarily Event Horizons are tied to Gravity and Time, until they both, Time and Space cease to exist in a Black Hole, the end product of the process and gravity suffers a catastrophic collapse.

Does the Interface Event Horizon of the Shroud lead to a Black Hole?

Everything on the Shroud indicates that the answer is NO. It is here that we face the most substantial paradox of our investigation. A paradox of that magnitude one cannot solve all at once, but one can assemble everything that is known so far and get closer to the magic door opening through the Shroud into a startlingly different world.

We have stated before that the images on the Shroud firmly indicate the total absence of Gravity. Yet they also firmly indicate the presence of the Event Horizon. These two seemingly contradict each other and they necessitate the past presence of something more powerful than Gravity that had the capacity to solve the above paradox.

ENTROPY TURNED AROUND. THE SECOND LAW DEFEATED.

Entropy decreasing can produce energy levels powerful enough to replace gravity while leaving other gravity-like effects in place, as the Event Horizon. This has to be studied further. It promises some very new results.

The upper region of the Isolated System of the Shroud has one Event Horizon, H1, that serves as a boundary of the Upper Region and there has to be an Event Horizon, H2, that is boundary to the Lower Region of the closed system. There is no space region but infinite density between the two. The two Boundaries can be looked at as one. Because H1 and H2 move so close to each other that they look as one, makes them eventually disappear, causing a total collapse of the Time quantum to ABSOLUTE ZERO TIME.

The total Space and Time breakdown to zero exposes that what was in the heart of the now collapsed Event Horizon. Not a Black Hole, but a very special kind of SINGULARITY, similar to the one that once assisted the creation of a universe, our own…

Isabel herself admitted that her ideas don’t quite make sense (in the light of the General Theory of Relativity, and/or The Standard Model of Particle Physics), but then these theories fall as far short of being able to explain the TS as they fall short of being able to explain the universe- but they do make sense in the light of the model applied to the case of the TS below.

The figure shows that gravity is suspended if and only if space and time are flattened, something that occurs if and only if the arrow of time loses its directionality. That such an occurrence is according to physical laws -that it is not miracle or supernatural in the sense of something requiring that these law violated- can be shown by reference to the problem known as “P versus NP”, which can be stated in the following way:

Problems that can be solved by computers in an efficient amount of time (where efficient means that as the problem-size grows arithmetically, the number of steps required to solve the problem does not grow exponentially) are classified as “P” for Polynomial time; problems whose solution can be verified in any efficient amount of time we classify as “NP” for Non-Deterministic Polynomial time; all problems in P are also NP, but not all problems in NP are in P. NP-Complete problems are both in NP, and not known to be in P. The classic example is The Travelling Salesman problem, which involves a salesman who must, after starting from a home-city, visit a number of cities exactly once before returning the home city for a certain cost.

The solution lies in the consideration of a problem, discovered in 1936 by Alan Turing (25), known as “The Halting Problem”. This concerns the impossibility of a computer that can determine of an arbitrary computer whether it will terminate when run with an arbitrary input. The argument that establishes Turing’s result is that a computer can’t on pain of contradiction determine of itself that it won’t terminate, for if it can’t then it terminates if and only if it does not terminate… If we examine the basis of this limitation, we see that it arises from the fact that a more complex computer is required to make this determination. To see that this same limitation -that there is a hierarchy of complexity such that computers can only pronounce on those that are lower than them in the hierarchy – limits the speed with which a computer can solve a problem, let the home city in the TSP = the halt state of a computer, let every other city = an atomic instruction of a computers program. If an instruction can be executed, assign a cost of 1, if an instruction cannot be executed, assign a cost of 2. If and only if the salesman can complete a circuit that visits every city exactly once for a cost of the number of cities, there is some computer that will halt when run with some input. If the number of cities is the same as the number of instructions in the program of the machine evaluating the TSP instance, the evaluation is a self-evaluation.

And if we assign a cost of 2, rather than 1, then this machine is required to determine that an input can only be run by a more complex program than itself.

There is no such limit on any problem in P, and so this extension of the Halting Problem proves that NP-Complete problems are such there is no classical algorithm, no matter how powerful, that can solve them in an efficient amount of time. We can further this argument in two ways. Firstly we identify the computational arrow of time, which along with all of the other arrows of time, is an arithmetic arrow of time governed by the RH. Secondly, we observe that Factoring is an NP-Complete problem (a TSP problem can be transformed into a Factoring problem by considering the relative prime-density of any pair of TSP problems). From Shor’s Algorithm (26), which permits the factoring of integers on a quantum computer in polynomial time, it follows that the limits we have been discussing are limits on classical, but not on quantum computational processes. The process that formed the image on the TS, we can conclude, did not violate the laws of nature, but may give this appearance because it was produced by a quantum rather than a classical computational process.

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