## Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?

The jury is still out on whether Einstein was right (QM cannot be considered complete) or Bohr (yes it can). There is a remarkable proof by Bell that no local hidden variables theory will be able to reproduce the predictions by Quantum Mechanics, and this proof seems to favour Bohr's point of view. After Bell, others, such as Clauser et al. have made similar claims that to a lesser degree rely on the assumptions that underlie Bell's proof, such as the soundness of counterfactuals.

Bell's and others' writings about hidden variable theories almost always are rather unpronounced as to what space-time, the backcloth of Bell's experiment, or rather of the Bohm-Aharonov experiment, is. This is surprising from a General Relativistic point of view. Read for example Hans Reichenbach's The Philosophy of Space and Time and then try to apply what you have learned as you read about the EPR experiment. But notice: the Special Theory of Relativity is already taken into account in all discussions about the EPR experiment, and is even seen as a necessary prerequisite. It is only as you step up to studying the foundations of General Relativity that new and relevant considerations come into view.

With the following story I want to illustrate my point. The story, although very unlikely to happen, is realistic. Care has been taken to let nothing a-causal slip in. Although the characters may seem to act in ways that are more based on superstition than reason, the fiction is not in conflict with known physical laws. As a special case in point, the validity of the quantum formalism is not disputed. This is all old school physics and not about yet another loophole.

While writing this story, I discovered that I had been wrong thinking that the problem addressed by Bell is about reproducing the predictions of QM. I had also been wrong thinking that counterfactual statements played an essential role in all proofs of the non-existence of HV-theories that reproduce QM.

My new insight was that Bell and Clauser, Horne, Shimony and Holt prove that

• it is impossible to construct a HV-theory that reproduces QM
• and that presupposes a three- or higher dimensional space as the backcloth against which measurements take place, a measurement being defined as the combination of a direction in 3D space and a binary outcome,
• and that assumes that each of the detector settings and the hidden variable have all N-1 degrees of freedom of a unit vector in an N-dimensional space.

By removing one degree of freedom from either one of the detector settings or from the hidden variable, it is perfectly possible to construct a local HV-theory that models the outcomes and statistics of a series of pairs of spin component measurements on an ensemble of systems in the singlet state, allowing the experimenters to make last-minute decisions as to which component to measure.

If a problem can be solved by removing a degree of freedom, one way to explain that is to say that there exists an unanticipated dependency. Another explanation, however, is that the problem was stated against a backcloth with more degrees of freedom than can be accounted for. This might be the case if for example our macroscopic three-dimensional space is an illusion created by interactions taking place in two dimensions.

A 2-D image of the 3-D steel sculpture by Marco Cianfanelli, taken at the unique vantage point where a 2-D picture of a 3-D subject can be seen: the point where Nelson Mandela was arrested in 1962, just outside Howick, SA.

Yet another explanation is that there may be degenerated phenomena in space for which one degree of freedom is collapsed. An example from General Relativity is given.

So, the reason why we have difficulty understanding QM can be the failure to let Hidden Variables turn QM into a fully deterministic theory (yes, I know, QM is deterministic as far as probabilities of outcomes of future experiments go - it just doesn't say anything about the next outcome of my experiment for sure), but it can also be the failure to appreciate the true nature of space and time. Or both.

# A Cosmic EPR Experiment

Bart Jongejan

## A catastrophe

Long, long time ago and now almost forgotten, there were two planets that went by the names of Bryiks and Heolaam. They were populated by intelligent species of which very little is known. These peoples' true names have been lost and therefore we call them Bryiksians and Heolaamites.

Bryiksians were making a living of agriculture, cultivating their small fields to make the best out of it. Bryiksians did not have much success planning the proper time for seeding and harvesting, and therefore they sought advice from wise men.

Many light years from Bryiks, the Heolaamites spent there days as hunters of the shy animals as they went to their drinking place during twilight. The life of the Heolaamite hunters was just as miserable as that of the Bryiksians. It was never to say whether the animals came to the drinking place at dawn or at dusk. The hunters' life style was an impediment for lying in wait at both times, so they often went to sleep with a hollow feeling in their belly. To improve their chances, the Heolaamites had subjected themselves to a cast of priests.

One day, an enormous explosion blew apart the far away star Nektuura Alpha. Nektuura Alpha was one of the two stars controlling the Nektuura solar system. Nektuura's inner planets turned into dead planets, but against all odds the people on the fifth planet, Cricsis, survived. These people were highly developed, the result of their willingness to bet and take risks. There were rumours that the explosion was the result of a nuclear experiment that had run out of control, but nobody was sure. After the explosion the complete scientific staff of the Campana Institute for Quantum Experiments was missing and the Campana laboratory fell into a deplorable state. Its walls begun to crumble and, without anybody taking notice of it, puffs of smoke and malodorous gas leaked away through holes in the roof. Nobody did really care, because taking risks was in high vogue on Cricsis.

By the time the intense radiation from Nektuura Alpha had traveled many light years through outer space and reached Bryiks, the explosion was seen by the wise men. Their power over the Bryiksians increased overnight when they told the farmers that the heavens had given a sign, because many simple minds also had seen something very extraordinary in the skies.

The Bryiksian wise men built a temple. Only a small group of wise men was allowed access to the temple. Nothing ever became known of what was inside the temple to anybody outside this small circle of wise men.

The temple was not a place for contemplation but rather a place where the farmers could receive advice from the wise men - rumours went that there was an oracle inside the temple. An applicant for advice had to give up an agricultural tool that was dear to him and leave it behind on the doorsteps to the temple. The wise men kept all those utensils in a safe place inside the temple, and quite likely they became part of the oracle. All those who had sacrificed a utensil would receive an engraved seal as proof that they were entitled to receive advice in all future ceremonies of the Oracle.

The paper roll started with a sign that symbolised the vision of an exploding star. After this symbol followed a list with two columns. Each ceremony of the oracle concluded with the addition of a new line to the list, the first column containing the seal of the farmer who had won the advice contest and the other column containing the wise men's advice, a V for "do it" and an O for "don't do it".

Some seals occurred considerably more often on the paper roll than average, and that was because not all farmers were in a like position to work for the wise men, and some might not always be willing to spend the whole night outside. As to the given advises, the farmers knew well that the chances of getting a V or a O were the same and that to predict the answer was just as hopeless as predicting 'head' or 'tail' when tossing a coin. Some farmers went one step further and said that the oracle was no better at giving usefull advise than a coin, but such heretics stood quite isolated from the other farmers.

We must ensure that decisions taken on Heolaam do not influence things on Bryiks and that decisions taken on Bryiks do not influence things on Heolaam. This is commonly done by making sure that the two places are well separated, especially in such a way that no signal conceivably can travel from one place to the other in a short enough time. Also, free will - especially combined with ignorance - can be a player, although it is difficult to incorporate free will into microscopic physics. Free will and the asymmetry of time seem intricately interwoven, but the only theories that have at least some success explaining time asymmetry are far from microscopic: thermodynamics and cosmology.

On Heolaam, far, far away from the Nektuura solar system, and even farther away from Bryiks, the explosion was also seen. The priests, like the wise men on Bryiks, were inspired by the celestial event to build a temple. The priests let nobody enter and leave the temple. They told the hunters that the temple concealed a secret power that could answer questions of a practical nature, and they urged the hunters to bring a personal trophy to the temple if they wanted to enjoy the oracle power of the temple. The priests would keep the trophies in a safe place inside the temple. As a proof of ownership, each Heolaamite hunter who had brought a trophy to the temple, received an iron ring with a distinctive inlaid design of ivory.

Like the Bryiksian wise men, the Heolaamite priests held ceremonies at times that only they knew about in advance. The oracle spoke for only one hunter at a time, so to find out who was going to receive advice, the priests released the Aima, a snow rat (rattus nix) that the priests had raised especially for this occasion. The hunter who brought back the Aima was the winner. The hunters were mostly interested in knowing whether it was best to hunt just after dawn or to hunt before dusk, so the priests' answer was either "dawn" or "dusk". To put down the verdicts for eternity, the priests used animal skins that they stitched together to form a journal. The design on the elected hunter's ring was carefully copied to the parchment. The verdicts were represented by holes cut in distinctive patterns next to the elected hunter's ring design. A "dawn" verdict was depicted by five holes in the corners of a regular pentangle and a "dusk" verdict by three equidistant holes on a row. Perhaps some hunters were more successful at bringing back the Aima than others, but each hunter was just as likely to get the "dawn" answer as the "dusk" answer, and there were no tendencies in the sequence of answers that allowed prediction of the next answer any better than by flipping a coin.

These temple ceremonies went on for generations. The Bryiksian seals and Heolaamite rings were handed from generation to generation, and generation after generation followed the advice given by the wise men and the priests. The list on the Bryiksian paper roll contained more entries than most inhabitants on the planet Bryiks were able to count. And the Heolaamite priests had to stitch ever more parchments together so that all ceremonies could be properly performed with the required punching of ever more holes, until ...

## Visitors

One day, captain Wejurch of the Alic Wee flipped the deceleration switch. The ionic front thrusters spew a blinding plasma and the Cricsian flying saucer coasted into orbit around Heolaam. Wejurch and his crew looked at the world below and then descended to a place that had drawn their interest. They landed a few hill tops away from the temple, or rather, what had been the temple. Just at that moment a revolt was taking place against the priests. In their anger, the hunters had completely shattered the building. All that was left were rubbles. Silence fell when the foreigners came nearer. A priest held the journal of parchments in the air and with a blank gaze stared at Wejurch, who seemed to be the leader of the visitors. Wejurch understood why the priest looked at him and he nodded to indicate his consent. While the priest put the parchments in Wejurch's custody, his crew tried to find out what had been happening. They understood that the priests as of late had been unable to give advice, as though whatever was in the temple was on strike. The hunters explained by gesticulating and pointing at their engraved rings that the priests had broken a deal. Wejurch thought he knew why the ceremonies didn't occur anymore, but there was nothing he could do to reconcile the parties. The visitors returned to their space ship and begun a long journey back to Cricsis, the parchments in a safe place aboard.

At about the same time when the Alic Wee arrived at Cricsis, another space ship, the Bo Cowzay, returned from Bryiks. Its crew told about the turmoil they had met when they visited the Bryiksian farmers. Other than descriptions from farmers, no trace of the wise men was left. The wise men's temple was burnt to ashes. By chance, the Cricsian space farers noticed how the wind was playing with what looked like a fluffy stick. It was the paper roll, partly burried under the loose sand of the temple's parvis and already disintegrating at the unprotected free end. Since the Bryksians clearly had lost all interest in the document, the Cricsians picked it up and brought it to their own planet for a closer analysis.

## Cause and effect

Apart from the scant accounts that the farmers and hunters had been able to give the visitors during their short visits, the only information about what had happened during the temple seances had to be found in the paper roll and the journal of parchments. But nothing did the Cricsians know about what had been happening inside the temples. Or did they? How could it be that both space ship crews were witnessing the strikingly similar aftermaths of religious upheaval? Had the timing of their missions anything to do with the timing of those tragic events? Was there a common cause for all this? And had the common cause to be found on Cricsis?

Since the explosion of Nektuura Alpha the planet Cricsis had orbited around the remaining central star, Nektuura Beta, in a trajectory much more stable and predictable than previously. Reflecting the changed circumstances, the people of Cricsis had evolved into a race seeking certainty and predictability, forsaking excursions into the unknown and speculative. As the ages went by the Cricsian power agency, during a rigorous check of the finances, discovered that part of the energy production at the electricity utility seemed to disappear in the blue. There was a leak. It turned out that delivery of electricity to the Campana laboratory, although it had been unmanned since that day when Cricsians woke up under one, instead of two suns, had not been stopped. The electricity utility sent inspectors to the laboratory and they found two robotic instruments that seemed to be aimed at two distant constellations that we know as the Bryiks and Heolaam solar systems. On the floor lay a piece of paper, almost crumbled beyond recognition by the forces of time and an oxidizing atmosphere, with a handwritten notice saying

spukhafte Fernwirkung?

## Spooky action at a distance?

The note with the mysterious words was later recognised as the start of a new scientific endeavour in the strange world of causation where no causation should exist. At that time, the inspectors did not know that, and they summarily pulled the chord to the robotic machine. For completing the financial statement of the utility with a full account of the lost energy, space ships were sent off to Bryiks and Heolaam to find out what the energy had been used for. Would the traveleres be able to find any evidence of events that could be related to the activities of the robotic instruments under the roof of the Campana laboratory? Since the instruments had been silenced before the space ships lifted off, the crews knew that good places to look for such evidence were areas with visible traces of the recent sudden termination of old traditions.

When the space ships had returned, the secretary general of the Cricsian Academy of Science, Tuelleor Raxep, decided to do a comparative study of the of the book and the paper roll. Strikingly, the number of ceremonies in the journal and in the paper roll were the same. Obviously at least something had been coordinated between the two planets. Was this the spooky action at a distance that the note hinted at? Or was there a more natural explanation, namely that the robotic machine in all those years had dictated when a session, involving ceremonies on each of the two planets Bryiks and Heolaam, had to take place? But why would the scientists in the Campana laboratory in their days have been interested in dictating ceremonious events on distant planets, like otherworldly toastmasters? Tuelleor Raxep decided to compile a new data set, consisting of all the data in the journal, ceremony-wise paired with all the data in the paper roll.

Seeing that there were two possible verdicts in the journal as well as in the paper roll, he choose to denote them with '+' and ''. The V's in the paper roll and the five hole patterns in the journal were transcribed as '+' and the O's and three hole patterns were transcribed as ''. He also gave all farmers and hunters unique names by inventing names for each seal and for each ring design.

This is how a tiny part of the data set looked like. A thin line separated a pair of ceremonies - a session - from the next and a thicker line separated the ceremonies from each other, a vague reminiscence of the enormous spatial separation at which the ceremonies had taken place.

 Bryiks Heolaam farmer verdict hunter verdict Abrafo − Jun + Dede + Gina − Paki + Hiro − Nsia + Kaori − Kgosi − Mika + Udo − Rika − Yaw + Shig + Sizwe − Yuko + Olumide − Ume + Obi + Kei + Nkruma − Mari − Sipho + Fuji + Gugu − Cho + Gazini + Kaede − Dube + Emiko − Tafari − Akira −

Tuelleor Raxep asked three members of the Fig clan - the brothers Sorbil and Yrfes and their cousin Oeg - to assist him with the analysis of this data set. The Figs were known for having diverging views on almost everything and Tuelleor Raxep hoped that their discussions would result in a well-balanced report. A report from the crews of the space ships describing the circumstances under which the data were created, in essence the introduction to this story, completed the material that the four-man committee had to work with.

The secretary general instructed the Figs to look for regularities in the data and especially to look for correlations between ceremonies that could not be explained without assuming spooky action at a distance - presumably an action that transgressed the boundaries restricting normal actions.

## Correlations over a cosmic distance

To start with, the committee looked for regularities in the data pertaining to each planet in isolation, but they found no correlations whatsoever. Given any amount of data from either the columns containing the Bryiks data or the Heolaam data, the outcome of a session was completely unpredictable, whether you looked at the available data for all farmers or hunters or only at the data for a single farmer or hunter. These were considered important observations, and so the committee wrote indifference and blindness on the interactive wall in the meeting room.

SINGLE PLANET REGULARITIES:

indifference
Within small margins, there are as many +'es as 'es.
blindness
It is even the case that every single farmer or hunter has about as many +'es as 'es.

(blindness implies indifference)

Whatever decided the outcome of a verdict, it behaved like a coin that had been flipped.

The committee continued their analysis, now concentrating on the combination of the data from the two planets. This was much tougher work, because the committee had to delve through enormous amounts of data to make statistically significant observations. They found four related regularities, correlation, spreading, twinning, inversion and locality.

TWO PLANETS REGULARITIES:

correlation
If you pick out one farmer and one hunter, you will normally see that their verdicts are correlated: From one verdict you can predict the other better (more often correctly) than by tossing a coin or any other method.
The verdicts of different farmer-hunter pairs are correlated to different degrees. A few pairs may seem not to be correlated at all.
twinning
Seemingly almost each farmer on Bryiks was paired with a hunter on Heolaam in the sense that they almost always received the same verdict if they happened to be elected for the same pair of ceremonies. The converse is also true: almost each hunter had a twin farmer. It was often, but not always the case that the twin of a twin was the same person.
inversion
There were farmer-hunter couples that almost always got opposite verdicts. They were not real twins, but inverse twins, so to speak.
locality
There were no farmer-hunter combinations for which either the farmer's or the hunter's outcomes were skewed. In other words: if, from the set of all verdicts given to a certain farmer, you selected only those verdicts that were given in ceremonies that were paired with hunter ceremonies for which a given hunter was elected, you would still see that these farmer verdicts exhibited blindness, and the same was true for the hunter verdicts in this subset of sessions. Thus, all verdicts in any farmer-hunter combination were blind.
antipode
Every hunter had a hunter colleague who would have got the opposite verdict, and every farmer had a farmer colleague who would have got the opposite verdict.

The data very strongly suggested that the wise men and the priests were able to accurately coordinate their answers for certain combinations of farmers and hunters, whereas other combinations seemed to have been coordinated to a much lesser degree.

Oeg Fig, who taught 3-D modelling at the prestigious Local University and tended to see shapes and forms in every list of numbers, and who once had been criticised for teaching the view that the High One more fittingly could be called "Zero" because of certain parallels - omnipresence, invisibility and the ability to reproduce itself by multiplication with anything else - walked to the interactive wall and briskly added antipode.

When asked to explain what he meant, Oeg Fig remarked that if you combined twinning and inversion you had to conclude that the twin farmer and the inverse farmer of a given hunter must be two farmers who always have opposite verdicts in the same ceremony. The other members scoffed at Oeg Fig for his audaceous remark, and added that he was alluding to a hidden property, the existence of which was unprovable, because there were never verdicts for two farmers during the same ceremony. There was nothing in the data from Bryiks supporting Oeg Fig's proposition, and combining data from Heolaam with data from Bryiks seemed an unlikely way to show that two farmers somehow were related. Oeg Fig's antipode was a fiction. Oeg Fig was set back by being ridiculed by his colleagues, and he did not understand how it was possible not to think that antipodes were real. Perhaps, he thought, the antipode might become useful in a later stage of the committee's work. But for now, the antipode was removed from the interactive wall.

After having analysed even more data, new regular patterns became quite clear. On an unused corner of the interactive wall the committee wrote isotropy and homogeneity.

BIPLANETARY CORRELATIONS IN MORE DETAIL:

isotropy
Each degree of correlation, expressed as the number of equal verdicts in a run of, say, 16 sessions (16 being the base of the Cricsian number system), occurred about as often as any other degree of correlation. It was not the case that some degrees of correlation were dramatically underrepresented and others overrepresented.
homogeneity
Looking at the data pertaining to one farmer (or hunter) at a time, one can see that all degrees of correlation are equally represented in the set of pairs that combine this farmer (hunter) with all hunters (farmers).

(homogeneity implies isotropy)

Even at cosmic distances, farmers and hunters formed a community based on everybody being equal, it seemed.

## Missives

Pondering about the seeming coordination between the activities of the wise men and those of the priests, the committee deduced that some common cause was the only sensible explanation. Such a common cause could be that packets of information had been sent from a third place to each of the planets Bryiks and Heolaam. This third place might very well be the committee's own planet, Cricsis, because the robotic instrument at the Campana laboratory was directed towards both planets.

Sorbil Fig, who had a flair for numbers, came up with a scheme that could do the trick. Suppose, he said, that you have some sort of missive that predicts each verdict. The first line in the missive must contain the verdict to be given if the first farmer or hunter is elected, the second the verdict if the second farmer or hunter is elected, and so on. The number of lines in the missive must of course be at least as big as the number of farmers and the number of hunters, so that everybody can receive a personalized advice. In this scheme, all that is needed as a preparatory measure is that the wise men and priests, when they register the tributes from the farmers and hunters, give the farmers and hunters unique numbers, starting with 1 and then counting upwards: 2,3,... . In the end, in the bookkeeping of the temples each farmer and each hunter would have such a personal number. From then on, when copies of a missive had been sent to each temple, upon arrival ceremonies were held and two people, a farmer and a hunter singled out by chance, received advice.

Table 2 shows how such a missive might look like if there are no more than 16 farmers and no more than 16 hunters.

 per- son ver- dict 1 + 2 + 3 − 4 − 5 + 6 + 7 − 8 − 9 + 10 + 11 − 12 − 13 + 14 + 15 − 16 −

A new set of such missives had to be sent whenever a couple of ceremonies were to be held. Whenever the elected farmer and the elected hunter happened to have the same personal number, they would also obtain the same verdict, because the wise men and the priests always received identical missives. So that would explain the existence of twins, and that would also explain that not every farmer was a twin of every hunter.

Tuelleor Raxep found it implausible that the machine in the Campana laboratory had been printing missives and sending them to the two planets Bryiks and Heolaam. The inventor of the scheme, Sorbil Fig, retorted that of course "missive" was just a way of saying. In practice "sending a missive" could be realised by sending a sequence of bits over a radio or optical channel. And if that scheme also seemed too unlikely, one could also devise a scheme in which the wise men and the priests had access to copies of a thick book that contained all missives for all sessions, one column per missive. Copies of such a book could have been sent by ordinary space mail shortly after the stellar explosion. After having received their copies of the book, the wise men and the priests only had to work through the books, for each ceremony selecting a column in some prescribed order, for example from begin to end in serial order, and pick the verdict from that column. The wise men and priests would not even have to wait for a sign from heaven, but could, for example, plan their ceremonies at the solstice of their suns, at times that were completely unrelated to the other planet's ceremony. The book could look like shown in Table 3 (abridged to only 16 person numbers):

 session⇒ person⇓ 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1 + + + + + + + + + + + + + + + 2 − + + + + + + + + + + + + + + 3 + − − + + + + + + + + + + + + 4 − − − + + + + + + + + + + + + 5 + + + − − − − + + + + + + + + 6 − + + − − − − + + + + + + + + 7 + − − − − − − + + + + + + + + 8 − − − − − − − + + + + + + + + 9 + + + + + + + − − − − − − − − 10 − + + + + + + − − − − − − − − 11 + − − + + + + − − − − − − − − 12 − − − + + + + − − − − − − − − 13 + + + − − − − − − − − − − − − 14 − + + − − − − − − − − − − − − 15 + − − − − − − − − − − − − − − 16 − − − − − − − − − − − − − − −

Apparently, the single missive shown in Table 2 is equivalent to the second or third column in the big book shown in Table 3.

It was quickly confirmed that this scheme shows
indifference: There are as many +'es as −'es,
correlation: For example, farmer 3 and hunter 4 had the same verdict in 14 out of 15 sessions. Only in session no. 1 they received different verdicts.
spreading: For example, farmer 3 and hunter 5 had only the same verdict in nine out of 15 sessions, as opposed to farmer 3 and hunter 4.
twinning: Equal numbered farmers and hunters always got the same verdict and therefore were twins.
inversion: Farmer 1 and hunter 16 always got opposite verdicts and therefore were inverse twins, and likewise farmer 2 and hunter 15, farmer 3 and hunter 14, and so on.

A minor defect with this scheme was that farmer 1 and hunter 1 always received the same verdict (+), and likewise farmer 16 and hunter 16 (−). All other farmers and hunters also received answers that were biased, only to a lesser degree. This was not in agreement with the observed blindness regularity, but the lack of blindness was very easily remedied by simply adding another fifteen columns to the book that are the inverse of the first fifteen columns of +'es and −'ses. See Table 4.

Does such a Book of Missives automatically guarantee locality? Although the verdict for farmer X in no way depends on whether hunter Y or hunter Z was elected, it was easy to construct a sequence of pairs of verdicts that defied locality, without assuming any kind of orchestrated coordination between the planets' respective election processes. If, on Bryiks, farmer with number 1 always won the competition and if at the same time hunters with, say, numbers 3 and 8 took turns winning the competition, the verdicts for farmer 1 would be extremely strongly correlated with who was elected on Heolaam, because the verdicts for farmer 1 did take turns as well, as can be seen in Table 4, verdict + always coinciding with one hunter and verdict − with the other. By randomizing the order of columns in Table 4 one might evade this kind of non-locality, but the problem does not go away. One could imagine a scenario where the competition on Bryiks and Heolaam is corrupted and the order in which each of the farmers and hunters are elected is completely predictable. In that case there is a non-negligible chance that the same column is used again and again when farmer X and hunter Y are elected, repeating the same verdict for farmer X again and again, suggesting that hunter Y has something to do with that. On the other hand, if the Book of Missives was thick enough to make it unnecessary to ever start over at the first column after reaching the last column, and if the columns in the book were well-shuffled, all thinkable corrupted election processes were deemed to be without effect on the statistics.

 session⇒ person⇓ 1 2 3 4 5 6 7 8 9 1 0 1 1 1 2 1 3 1 4 1 5 1 6 1 7 1 8 1 9 2 0 2 1 2 2 2 3 2 4 2 5 2 6 2 7 2 8 2 9 3 0 1 + − + − + − + − + − + − + − + − + − + − + − + − + − + − + − 2 − + + − + − + − + − + − + − + − + − + − + − + − + − + − + − 3 + − − + − + + − + − + − + − + − + − + − + − + − + − + − + − 4 − + − + − + + − + − + − + − + − + − + − + − + − + − + − + − 5 + − + − + − − + − + − + − + + − + − + − + − + − + − + − + − 6 − + + − + − − + − + − + − + + − + − + − + − + − + − + − + − 7 + − − + − + − + − + − + − + + − + − + − + − + − + − + − + − 8 − + − + − + − + − + − + − + + − + − + − + − + − + − + − + − 9 + − + − + − + − + − + − + − − + − + − + − + − + − + − + − + 10 − + + − + − + − + − + − + − − + − + − + − + − + − + − + − + 11 + − − + − + + − + − + − + − − + − + − + − + − + − + − + − + 12 − + − + − + + − + − + − + − − + − + − + − + − + − + − + − + 13 + − + − + − − + − + − + − + − + − + − + − + − + − + − + − + 14 − + + − + − − + − + − + − + − + − + − + − + − + − + − + − + 15 + − − + − + − + − + − + − + − + − + − + − + − + − + − + − + 16 − + − + − + − + − + − + − + − + − + − + − + − + − + − + − +

How did Table 4 give insight in the data returned from Bryiks and Heolaam? The committee had observed the regularities that they had coined isotropy and homogeneity. If the table was a model for how the real data had arisen, the same isotropy and homogeneity properties had to be inherent to Table 4. To check this, the committee needed to look at the number of times that farmer k and hunter l had received different verdicts. Table 5 shows the result of this assessment for each farmer-hunter combination. Each cell in the table contains a number. A cell was given the value 0 if no verdicts were dissimilar; the value 1 was given if 1 out of 15 verdicts were dissimilar, and so on. The degree of correlation is maximal if the cell value is 0, whereas if the cell value is 15, the verdicts are always opposite, indicating a perfect anti-correlation. Verdicts were more or less unrelated if the cell value was close to seven or eight. The committee called the numbers in the cells 'correlation separations', because the greater the number, the further away from perfect correlation.

The other committee members were utterly bored by the many numbers and had started to look out of the window. In a moment of pedagogical afterthought Sorbil Fig decided to make Table 5 more visually attractive and elucidating. He added a background to each cell in Table 5, using white for perfect correlation, black for perfect anti-correlation and shades of grey for all other correlation values.

The number in cellkl in Table 5 is the Hamming distance between the rows k and l in Table 4.

By definition, a farmer and a hunter with the same person numbers have maximal correlation, so all cells on the diagonal from the upper left to the lower right of Table 5 have zeros and are coloured white. The other diagonal has all black cells. These cells denote the correlation between the farmers and their inverse twin hunters.

 farmer⇒ hunter⇓ 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 2 1 0 3 2 5 4 7 6 9 8 11 10 13 12 15 14 3 2 3 0 1 6 7 4 5 10 11 8 9 14 15 12 13 4 3 2 1 0 7 6 5 4 11 10 9 8 15 14 13 12 5 4 5 6 7 0 1 2 3 12 13 14 15 8 9 10 11 6 5 4 7 6 1 0 3 2 13 12 15 14 9 8 11 10 7 6 7 4 5 2 3 0 1 14 15 12 13 10 11 8 9 8 7 6 5 4 3 2 1 0 15 14 13 12 11 10 9 8 9 8 9 10 11 12 13 14 15 0 1 2 3 4 5 6 7 10 9 8 11 10 13 12 15 14 1 0 3 2 5 4 7 6 11 10 11 8 9 14 15 12 13 2 3 0 1 6 7 4 5 12 11 10 9 8 15 14 13 12 3 2 1 0 7 6 5 4 13 12 13 14 15 8 9 10 11 4 5 6 7 0 1 2 3 14 13 12 15 14 9 8 11 10 5 4 7 6 1 0 3 2 15 14 15 12 13 10 11 8 9 6 7 4 5 2 3 0 1 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0

By inspection of Table 5 the committee was able to confirm that the two remaining conditions also were fulfilled:
isotropy: All degrees of correlation were equally represented in Table 5, because each value 0..15 occurs 16 times
homogeneity: Even if you looked at a single farmer or hunter, all correlations were equally represented, because in each column or row each value occurs once.

Sorbil Fig drew attention to the fractal appearance of Table 5. Each of the four quadrants in Table 5 looked like a copy of the whole picture - the only difference being an overall adjustment of the blackness level. To demonstrate this, Sorbil Fig ordered the interactive wall to construct a table for 256 farmers hand 256 hunters, this time without showing the numbers. Apart from the absence of numbers, the 256x256 table looked the same as the much smaller 16x16 table. In both illustrations, white corresponded with the cell value 0, maximum correlation, but black corresponded to the cell value 15 in the small illustration and to 255 in the big illustration, while a light grey had replaced black to denote the value 15.

An easy way to compute the correlation separation between farmer k and hunter m according to the Book of Missives is as follows: subtract 1 from both k and m (so that the counting starts at 0 instead of 1) and perform an exclusive OR operation (XOR) on the binary representations of these numbers. The resulting sequence of bits is the binary representation of the Hamming distance:

`Hamming distance = XOR(k-1, m-1)`

The reason why this "trick" works is that one line in the Book of Missives is a representation of the the farmer or hunter number. For example, the number 11 is represented as follows: subtract 1 from the number, so we get 10. Write this in a binary representation, this is 1010 (because 10 is 2 + 8 and 2 is represented 10 in binary and 8 is represented as 1000 in binary, their sum being 10+1000=1010. Now 1010 is a positional representation system: the first "1" is worth four times as much as the second "1", because each extra position leftwards doubles the value of the digit. Turning our back to the positional system, we just repeat each digit the number of times it has gained significance by being shifted leftwards in the positional notation. So the first digit, 1, is repeated eight times, the second digit, 0, is repeated four times, the third digit, 1, is repeated two times and the last digit is just written once. This gives the sequence 111111110000110. Now replace each '1' by '+ −' and replace each '0' by '− +' to obtain + − + − + − + − + − + − + − + −  − + − + − + − + + − + −  − +. Turn this string of +es and  − ses backwards to obtain the row numbered 11 in Table 4: + −  − + − + + − + − + − + −  − + − + − + − + − + − + − + − +.

The committee adopted Sorbil Fig's Book of Missives as a valid scheme to explain how the combination of the book of parchment and the paper roll could display correlations between non-interacting civilisations. This solution gave a completely causal explanation of the correlations between the verdicts on the two planets. The committee agreed that no spooky action at a distance had been in play. But to be completely sure, they had to check that the real data could be fit into a correlation separation table along the lines of the scheme. Yrfes Fig was appointed to this task.

## The failure of the Book of Missives

Yrfes Fig decided to make a correlation separation table for 16 real farmers and 16 real hunters. He selected the names of 16 Bryiksian farmers and also the names of 16 Heolaamite hunters. From the big book Yrfes Fig extracted the data that only involved farmers and hunters from these selections, while disregarding all other data. Then he picked out one farmer, Buru. From the now much reduced amount of data he took aside all data in which this farmer was one of the two participants. He put the correlation separations derived from these data into the first column of his correlation separation table, arranging the values in increasing order, so that the cell with the background shading closest to white was in the uppermost cell of the first column.

This was how Yrfes Fig computed the correlation separation from the data pertaining to all sessions in which a specific farmer and a specific hunter both were elected: $\mathrm{correlation separation}=15\frac{N\left(\mathrm{sessions with dissimilar outcomes}\right)}{N\left(\mathrm{all sessions}\right)}$ The number 15 is a "normalisation factor" which makes the resulting table comparable with the results obtained from the Book of Missives for 16 person numbers. For the same reason of comparability, all correlation numbers were rounded to the nearest natural number.

From top to bottom the first cell contained the correlation separation pertaining to the farmer and the farmer's hunter twin, Emi, the next cell contained the correlation separation pertaining to the farmer and the hunter with the next best correlation, Aki, and so on. The number in the last cell of the first column pertained to the farmer and the hunter who most often had received verdicts opposite to the farmer - the farmer's inverse twin, Jun (see Table 7a). The procedure was repeated to fill the first row of cells, taking all data in which the farmer's hunter twin, Emi, was one of the participants and computing the correlation separations pertaining to this particular hunter and all farmers (see Table 7b). When the first column and the first row were filled and sorted according to increasing blackness level, each hunter and each farmer in the selection had fixed positions along the table's axes: each row corresponded to a hunter and each farmer had his own column. Up to now the result looked very much the same as the left column and the top row in Table 5, so the expectations were high that the book-of-missives hypothesis was sound. Yrfes Fig filled the still empty cells with the correlation data pertaining to the remaining farmers and hunters. The result looked like Table 7c.

 farmer⇒ hunter⇓ Bu ru Emi 0 Aki 1 Nao 2 Ren 3 Ryo 4 Yuu 4 Aya 6 Cho 7 Ume 9 Ken 9 Eri 10 Ran 11 Kyo 13 Aoi 13 Rei 14 Jun 15
 farmer⇒ hunter⇓ Bu ru Da yo A yo E nu Ne o O bi Ta u U do Y aw A ma E si I fe I ge Ni a O ni Ya a Emi 0 1 2 3 4 6 6 7 8 9 10 10 12 12 14 15
 farmer⇒ hunter⇓ Bu ru Da yo A yo E nu Ne o O bi Ta u U do Y aw A ma E si I fe I ge Ni a O ni Ya a Emi 0 1 2 3 4 6 6 7 8 9 10 10 12 12 14 15 Aki 1 7 4 1 1 10 12 6 2 7 11 15 7 7 15 12 Nao 2 5 8 0 8 11 6 1 6 13 5 12 13 7 13 10 Ren 3 1 0 9 4 1 7 13 11 4 13 5 9 15 10 14 Ryo 4 0 4 8 10 2 1 9 14 10 7 3 14 15 8 12 Yuu 4 6 1 7 0 5 13 12 4 1 15 10 4 10 13 13 Aya 6 12 9 3 3 14 14 5 0 6 9 15 4 2 12 7 Cho 7 4 11 6 14 8 0 3 13 15 1 5 14 9 6 6 Ume 9 11 4 9 1 7 15 12 2 0 14 10 0 6 9 8 Ken 9 12 14 3 10 15 9 0 3 12 2 12 8 1 8 3 Eri 10 4 6 14 11 1 3 12 15 7 7 0 9 13 2 8 Ran 11 8 4 15 5 2 10 15 9 1 12 3 3 10 4 8 Kyo 13 12 15 8 14 12 5 3 8 12 1 6 8 2 2 0 Aoi 13 15 13 9 7 13 12 6 2 6 6 10 2 0 4 1 Rei 14 9 12 13 14 6 3 8 12 9 3 1 7 7 0 2 Jun 15 13 11 14 9 7 9 11 7 4 7 4 2 4 1 2

After filling out all cells the table looked markedly different from the table constructed by using Sorbil Fig's book-of-missives scheme, Table 5. It was immediately clear that homogeneity was not exact: in some columns and rows some values were lacking, while others occurred more than once. But it seemed unlikely that this was an essential difference. More disturbing was that the fractal appearance of Table 5 was completely absent in Table 7c. If you looked closely at Table 7c you saw an irregular and grainy pattern of dark and light cells that was lost at larger scales. Yrfes demonstrated this for himself by selecting a lot more farmers and hunters, 1024 of each, and plotting their correlations with just one pixel for each farmer-hunter pair on the interactive wall in the meeting room. (See Table 8)

The resulting square had two light and two dark corners and mostly gradual transitions from dark to light. Could the difference be ascribed to errors in the data or in the transcription from the journals to the combined table? A wrong assignment of a correlation in the first step could easily mean that a hunter ended in the wrong position in the column, and that would mean that the correlation separation table had both a column and a row in the wrong position. Might the table of Sorbil Fig that was full of regularities become something that looked like the irregular real-world table constructed by Yrfes Fig if some more of these errors were made? Yrfes Fig soon realised that the noise level in the data and the error rate in the transcription process had to be quite low, because otherwise the good correlations between some of the hunters and farmers would be statistically very improbable and close to miraculous. So Yrfes Fig reported that the real data could not be explained by the book-of-missives scheme.

## A game of Chapshap

The four brave committee members, their high expectations being shattered by Yrfes Fig's conclusion, took a break to play a card game called Chapshap.

In Chapshap each of the four players has a hand of four cards with ranks 2, 3, 4 and 5. In each round each player in turn lays down one card. Each player subtracts the value of his own card from the sum of the other three cards. If his score is less than 4 (two times the value of the lowest ranked card, 2) or if his score is more than 10 (two times the value of the highest ranked card, 5), he shouts "Campana!", hits a table bell and earns one point for each notch on the score scale past the boundaries of 4 or 10, so a score of 12 earns the winner 2 points (12 − 10) and a score of 3 earns the winner 1 point (4 − 3). The highest possible score is equal to three times the highest ranking card minus the lowest ranking card (3 × 5 − 2 = 13), and the lowest possible score is three times the lowest ranking card minus the highest ranking card (3 × 2 − 5 = 1). A player earning the highest possible or lowest possible score yells "POIROT!". In respect for obtaining a poirot score, the other three players shout a threefold Hooray and throw their remaining cards (if they have any) in the air. The game ends after four rounds. The player who has earned the most points wins.

Table 9 shows how the cards landed on the table in the first round of Chapshap, Table 10 shows how all four rounds went, and Table 11 shows how the points were divided.

 Player A Player D Player B Player C
 player⇒round⇓ A B C D Score Comment 1 2 3 4 4 A: − 2 + 3 + 4 + 4 = 9 B: + 2 − 3 + 4 + 4 = 7 C: + 2 + 3 − 4 + 4 = 5 D: + 2 + 3 + 4 − 4 = 5 no winner 2 5 2 2 2 A: − 5 + 2 + 2 + 2 = 1 B: + 5 − 2 + 2 + 2 = 7 C: + 5 + 2 − 2 + 2 = 7 D: + 5 + 2 + 2 − 2 = 7 A wins Poirot 3 3 4 3 5 A: − 3 + 4 + 3 + 5 = 9 B: + 3 − 4 + 3 + 5 = 7 C: + 3 + 4 − 3 + 5 = 9 D: + 3 + 4 + 3 − 5 = 5 no winner 4 4 5 5 3 A: − 4 + 5 + 5 + 3 = 9 B: + 4 − 5 + 5 + 3 = 7 C: + 4 + 5 − 5 + 3 = 7 D: + 4 + 5 + 5 − 3 = 11 D wins
 player⇒round⇓ A B C D 1 0 0 0 0 2 3 0 0 0 3 0 0 0 0 4 0 0 0 1

The winner was player A with 3 points.

The scientists joked that the table with correlations between farmers and hunters was merely a listing of the outcomes of a galactic game of Chapshap. Each rectangle spanning the correlation separations between two farmers and two hunters could be seen as a playing table with the cards of one round lying in the corners. (See Table 13.) The values of the cards went from 0 to 15. Scores below 0 ( = 2 × 0) and above 30 ( = 2 × 15) gave the winning 'player' - in reality a couple consisting of a farmer and a hunter - as many points as the difference between the score and 0 or 30, whichever was nearest. Although it was not possible to see where a round had started, it was possible to see whether or not any farmer-hunter pair had won the round. For example, the square of cells formed by the farmers Enu and Neo and the hunters Aki and Nao contained the correlation separations 0, 1, 1 and 8. In this "correlation round" scores of 10, 8 and − 6 were possible, the last score (Neo and Nao) being a winning score. There were also non-winning rectangles, such as the one spanned by farmers Buru and Dayo and the hunters Ume and Ken, with cards 9, 11, 12 and 9. The possible scores were 23, 19 and 17, none of which was winning.

The card game Chapshap gave the winner 3 points because of a Poirot in the second round. In contrast, a remarkable observation was done on the data in Table 12: there seemed to be an upper limit to the number of earned points that was lower than what would have been the case if the numbers reflected a Chapshap game. One would expect an upper score of 15 + 15 + 15 − 0 = 45, giving 45 − 30 = 15 points, but the data did not get further than a score of 13 + 13 + 13 − 1 = 38 (the rectangle with vertical sides Ama and Oni and with horizontal sides Nao and Yuu), giving only 38 − 30 = 8 points. Similarly, one would expect a lower score of 0 + 0 + 0 − 15 = − 15, giving 15 − 0 = 15 points, but in reality Table 12 has a lowest score of only 2 + 3 + 2 − 14 = − 7 (Obi, Ige, Ryo and Ran), giving 7 − 0 = 7 points (See Table 12). The committee was puzzled by these constrained data. By what were the correlation separations constrained to never produce a Poirot? Why weren't there any rounds that even came close to a Poirot in all of the data?

 farmer⇒ hunter⇓ Bu ru Da yo A yo E nu Ne o O bi Ta u U do Y aw A ma E si I fe I ge Ni a O ni Ya a Emi 0 1 2 3 4 6 6 7 8 9 10 10 12 12 14 15 Aki 1 7 4 1 1 10 12 6 2 7 11 15 7 7 15 12 Nao 2 5 8 0 8 11 6 1 6 13 5 12 13 7 13 10 Ren 3 1 0 9 4 1 7 13 11 4 13 5 9 15 10 14 Ryo 4 0 4 8 10 2 1 9 14 10 7 3 14 15 8 12 Yuu 4 6 1 7 0 5 13 12 4 1 15 10 4 10 13 13 Aya 6 12 9 3 3 14 14 5 0 6 9 15 4 2 12 7 Cho 7 4 11 6 14 8 0 3 13 15 1 5 14 9 6 6 Ume 9 11 4 9 1 7 15 12 2 0 14 10 0 6 9 8 Ken 9 12 14 3 10 15 9 0 3 12 2 12 8 1 8 3 Eri 10 4 6 14 11 1 3 12 15 7 7 0 9 13 2 8 Ran 11 8 4 15 5 2 10 15 9 1 12 3 3 10 4 8 Kyo 13 12 15 8 14 12 5 3 8 12 1 6 8 2 2 0 Aoi 13 15 13 9 7 13 12 6 2 6 6 10 2 0 4 1 Rei 14 9 12 13 14 6 3 8 12 9 3 1 7 7 0 2 Jun 15 13 11 14 9 7 9 11 7 4 7 4 2 4 1 2
 Nao Ama Oni Yuu

The committee also scrutinized the table that was the result of the Book of Missives. If the committee had been astonished by the seeming constrainedness of the 'Chapshap numbers' in the real data, an even bigger surprise waited them in the Book of Missives: there were no Campana Wins, let alone any Poirots! Chapshap scores in the Book of Missives never got less than 0 or more than twice the maximum correlation number, so the game could not be won! This gave the committee a clear hint of how fundamentally different the Book of Missives and the real data were. What explained the lack of winning rounds in the Book of Missives scheme and what could be learned?

Oeg fig, realizing they just had discovered a regularity in the statistics of the combined ceremony log books, added a long and winding text to the whiteboard, and named it after the laboratory to commemorate that the discovery was made there.

Regularity involving SOME Quartets of TWO farmers and TWO hunters:

Campala Wins
There are quartets consisting of two farmers and two hunters exhibiting correlations with a remarkable property: for those quartets, adding three of the four correlations and subtracting the fourth results in a number that cannot be reproduced by a scheme that explains the correlations by way of a Book of Missives.

## Exit Book of Missives

To get a feel of why the Book of Missives did not produce any Poirot, the committee decided to work backwards from hypothetical Poirot data to a Book of Missives that produces such data, and to see where the attempt got off the track.

To start, the committee made up a set of four correlations that would guarantee the upper Poirot value.

They decided that Abrafo and Jun always would receive the same verdict (+ + or − −), but that Abrafo and Gina, Dede and Jun, and Dede and Gina always would receive opposite verdicts (+ − or − +). These correlations are summarized in Table 14.

From Table 14 one immediately sees that Abrafo and Jun must have the same number in a fictitious Book of Missives: they are each other's perfect twins. One also sees immediately that Abrafo and Gina and Jun and Dede are each other's inverse twins. However, because Abrafo and Jun have the same number, their inverse twins also must have the same number, which is the same as to say that these inverse twins are twins for each other. But this contradicts the assumption that Dede and Gina always should obtain opposite verdicts!

 farmer⇒ hunter⇓ Abrafo Dede Jun 0 2 Gina 2 2

Can every theory that fulfills Bell's inequalities be underpinned by a deterministic nontextual hidden variable theory? Is it enough for a theory to be nondeterministic or contextual to break Bell's inequalities, or are there such theories that stay within Bell's bounds? If the latter is the case, then what makes Quantum Theory break Bell's inequalities? What, apart from nondeterminism and contextuality, is needed to construct a theory that breaks Bell's inequalities? If a theory turns up that claims to be nondeterministic but is agnostic as to whether it is contextual or noncontextual, and that claims to reproduce the statistical predictions made by quantum mechanics, what test will decide whether this theory breaks Bell's inequalities?

Sorbil tried to defend the Book of Missives by pointing out that the real data were not quite like the hypothesised Poirot producing Table 14. The real data contained "Campana Wins", but no Poirots were ever found in the real data either. The other three members of the committee, reminding Sorbil of the fact that the Book of Missives not even could generate Campana Wins, almost fell over each other to explain to Sorbil that the Book of Missives was a lost case.

## Beyond the Book of Missives

The Book of Missives was clearly not the way by which the outcomes of the ceremonies on Bryiks and Heolaam had been causally coordinated, but there might be another way to causally coordinate spatially widely separated events. Only if it could be proven that there are no such ways, the committee would have to conclude that the existence of Campana Wins in the real data, as opposed to their absence in the data produced by the Book of Missives mechanism, had no causal explanation and that they therefore had to be ascribed spooky action at a distance.

Tuelleor Raxep noted that the Book of Missives did not distinguish between the farmer or hunter that is elected for receiving a verdict and all other farmers and hunters. If the wise men and the priests really produced their verdicts from a book of missives, they were free to elect any number of farmers and hunters in the same session and give them all their own personalised verdict. But neither the Bryiksian paper roll nor the Heolaamite parchments gave any hint that more than one farmer or one hunter was ever elected for any ceremony.

Oeg Fig wondered whether the elected farmer and hunter could be incorporated into a causal explanation without compromising the freedom of the farmers and hunters to decide how much they wanted to invest into a coming competition. The nice thing about the Book of Missives hypothesis was that it allowed a clear separation between the common cause on the one hand - the production of the Book of Missives at some time preceding the sessions - and the election of the farmers and hunters at later times and in spatially separated regions. Any alternative solution should exhibit the same separation of roles, because otherwise the election of a farmer or a hunter, instead of playing an independent role in the determination of a verdict in a far away part of the universe, would itself be determined by a common cause in the common past of the elections.

Oeg Fig proposed a direction in which to look for a more sparse solution than the Book of Missives that might incorporate the good quality of the Book of Missives without producing a plethora of verdicts. His idea was to replace the book containing verdicts with something that only was a seed for giving a verdict. The determination of the verdict would be postponed until the point in space and time where a farmer or hunter was elected. When a farmer or a hunter was elected, a verdict for the elected person would materialise, while all other farmers and hunters only would have been allowed to speculate about what verdict they would have received, had they been elected. They would have been free to imagine that the verdicts they didn't receive were real and "out there", but not known or even knowable, not even to their spiritual leaders. They might also nurture the opinion that the not received verdicts had no existence. It wouldn't make any difference which opinion they held.

This sounded all very vague, but Oeg Fig presented an analogy that at least partly illustrated his point: the measurement of the size of an object. The size of an object is something inherent to the object and yet the outcome of a measurement cannot be determined unless a unit length is chosen. So in a sense, the object is or carries the seed for the outcome of a measurement, but it does not determine the outcome on its own. However, this example does not preclude the simultaneous or successive selection of different units of lengths and the measurement of the object's size in terms of each of these units.

The problem with being able to do many measurements on the same seed is that the wise men and priests might lie several times about whom was elected, combine each counterfactually elected person with the seed and in that way derive a table that, like a book of missives, prescribes the verdict for any farmer or hunter before any farmer or hunter was elected, and then we would again be unable to explain the Campana Wins in the data. Therefore Oeg Fig added that the solution, if it existed, had to preclude this possibility.

We normally assume that measurements can be repeated, unless a measurement does something irreversible to an object. But if you think about it, it is clear that all measurements that leave a trace in a memory are irreversible. Reverting a measurement would also involve wiping the trace in the memory, which is utterly contraproductive. So if the wise men or priests had plans to secretly create a Book of Missives, they would necessarily change the seed every time a new verdict was written in the book. In the end, the verdict given to the really elected farmer or hunter would be seeded by a combination of the original seed and all the measurements that the wise men or priests had performed. Conversely, if the wise man and priests were aware of this conundrum and did revert each measurement so as to undo the change inflicted to the seed, the last created verdict in the book would have to be wiped out before a new verdict could be measured, making it impossible to fill out a complete column in the book. And if the wise man and priests had peeked at some of the data, they would have to forget these results before performing the next measurement. So it seems natural that the most basical way to think of measurements is as firework crackers that consume the stuff that causes the visual display and to think of repeatable measurements as not quite precise and in need of an explanation.

## A distanced look

The correlation separation table constructed by means of the scheme proposed by Sorbil Fig - the Book of Missives - reminded Yrfes Fig of a distance table, for example a table from which you can read all the walking distances between Cricsis' main cities. Yrfes wanted to know whether the superficial similarity was incidental, or whether the correlation separations had anything whatsoever to do with walking distances.

Yrfes wrote down what he knew for sure about distances:

Neighbourhood
If you walk a small distance away from home, you are still in your neighbourhood. Everybody in your neighbourhood has only a small walking distance to everybody else in your neighbourhood.
Triangle Inequality
If you live in Ab and you travel a long distance to your friend in Tuv, you may want to sleep in a hostel somewhere on your way, lets say in Nop, before you reach Tuv. The next day you can travel the other part of the journey from Nop to Tuv. Now suppose that the next time you visit your friend in Tuv, you manage to do the journey from Ab to Tuv in one day, because you need fewer stops for looking at the map or taking pictures, say. Under the second journey you have travelled the same distance as under the first journey, or perhaps less, because the first time you probably had to deviate from the planned route to get to the hostel in Nop. So whereas the night at the hostel may have lengthened the trip, you would not expect ever to be able to shorten your trip by staying at hostels.
Degrees of Freedom
If you can travel in all directions, as long as you stay to the ground, you can enlarge your visitable world about fourfold by traveling twice the distance. If you can dig into the ground, dive into the see, and fly in the air, your world will even be enlarged eightfold. In the other extreme, if you are confined to travelling in only one direction (and back again), your known world grows at the same rate as the distance you travel.

If your world is finite but endless, such as the surface of a planet, there is a maximum walking distance you can get away from home. You will also notice that as you are making trips further and further away from home, the part of the world that is knowable to you will grow slower and slower if your world is of a finite size. When all of the world is knownable to you, you are able to walk the maximum distance from home.

Not all numbers that grow as you move away from home are good walking distance measures. For example, during the first day of your first journey from Ab to Tuv with a stop at Nop you travelled a distance corresponding to crossing the size of a city with 9 000 000 people and the second day you travelled the distance corresponding to crossing a city with 4 000 000 inhabitants. The sum of these numbers is 13 000 000, so 13 000 000 would be the maximum population size of the city that you would have crossed in both days together if population size was a good distance measure. However, If you had travelled from Ab to your destination Tuv in one day, you would have crossed a distance corresponding to a city with many more than 13 000 000 inhabitants, namely an enormous city with a population size of about 25 000 000. Because the Triangle Inequality says that you cannot shorten the total walking distance by staying at hostels, the population size of a city is not a good measure of walking distance. However, it can still be an indicator that only needs a translation step to turn into a measure of walking distance. For example, the square root of the city population size would be a fine walking distance measure, because

$\sqrt{9000000}+\sqrt{4000000}=3000+2000=5000=\sqrt{25000000}$

Why would it be interesting to use population size as a measure of distance? As we saw earlier, the correlation separation table that was built from the true observations (Table 7c) exposed a property we called homogeneity: in each column and each row, most correlation separation values occurred once. The left column and the top row show that as you let the correlation separation grow from 0 to the maximum value, you add, apart from only a few exceptions, one person for each notch up the separation scale. That is, in the real data a given correlation separation is proportional to the size of the population consisting of the persons with that correlation separation or a correlation separation that is less than the given correlation separation:

$\mathrm{homogeneity condition}⇒\mathrm{population size}\propto \mathrm{correlation separation}$

Because the committee's assignment was to explain the relation between the data sets that were brought back from Bryiks and Heolaam, the committee decided to reject any explanation that did not fulfil the homogeneity condition, and thus accepted that correlation separation had to be proportional to population size in every proposal for a solution they would come up with.

## A walk trough the Book of Missives

To see how distances could play a role in the correlation separations brought about by the Book of Missives, Yrfes Fig first tried to simply equate the correlation separations implied by the Book of Missives with walking distances. All three properties of distance mentioned above seemed to be fulfilled. The farmers and hunters with numbers 5, 6, 7 and 8, according to Table 5, seemed to form a neighbourhood with a greatest walking distance of 3 from one member in the neighbourhood to any other member. Also, it never was the case that the sum of two correlation separations with a stop at a hostel was less than the correlation separation between the start and the end of the journey. For example, the correlation separations between number 1 (home) and number 5 (the hostel) is equal to 5 and the correlation separation between 5 (the hostel) and your friend at number 16 is equal to 11, giving a total correlation separation of 15, which so happens to be the correlation separation between 1 (home) and 16 (your friend).

The example showed that 5 (hostel) seemed to be on the route from 1 to 16. Was that a coincidence? Yrfes tried other hostels. A hostel further from home was number 12 with a correlation separation of 11 to home. But again, the total separation added up to 15, because the separation between 12 and 16 was 4.

This suggested that home was at one end of the world and 16 at the other, and that all other numbers lined up in between at regular correlation separations. That would mean that there was just one degree of freedom, the number of visitable persons growing in proportion to the distance from home. But there was something that didn't fit in this picture: the correlation separation between number 2 and number 15 was also 15 and the correlation separation between number 3 and number 14 likewise, so there were many pairs of numbers that seemed to occupy the two ends of the world, while the other numbers lined up between each of these pairs. The configuration that came to mind was that all numbers were arranged on a circle in such a way that 1 and 16 were opposite each other and likewise 2 and 15, 3 and 14, and so on.

This idea at first seemed to work right. Yrfes selected the numbers 1, 2, 3 and 4 and he arranged them on a circle in such a way that places were separated further if the correlation separations were greater. He did the same with other quartets of person numbers. See Fig. 1.

Yrfes realised that all persons not only were at one of the endpoints of a diagonal through the circle, but that they also were at the top of a right triangle, and that therefore the distance along one of the sides always was dictated by the distances along the other two sides by the relation: ${\left(\mathrm{distance along short side}\right)}^{2}+{\left(\mathrm{distance along other short side}\right)}^{2}={\left(\mathrm{distance along long side}\right)}^{2}$

This could only be the case if not the correlation separations themselves, but their square roots were equated to distances, and two (adjacent) trajectories always were perpendicular. With such a definition of walking distance, doubling your walking distance quadruples the number of people you can reach, as the following example shows. Let us assume that you are number 1 and that you are going to visit number four. That is a walking distance √3. Within that distance you can visit the three persons numbered 2, 3 and 4. Doubling this distance, √3 + √3 = √12, you have found the walking distance to person with number 13, and indeed, as you can test in Fig. 1, you can reach four times as many, 12, different persons at shorter walking distances than person 13.

But that should not let us conclude that there is a two-dimensional world underlying the Book of Missives. Interestingly, if the visitable population size is proportional to the correlation separation, you can, at least in the case of the correlations inherent to the Book of Missives, argue for any number of dimensions, because setting the walking distance equal to the correlation separation to the inverse power of the number of dimensions is tautologically equivalent to computing the population size by taking the walking distance to the power of the number of dimensions, for any non-zero number of dimensions. $\begin{array}{ccc}\begin{array}{c}\mathrm{walking distance}=\sqrt[\mathrm{dimensions}]{\mathrm{correlation separation}}\\ \mathrm{population size}\propto \mathrm{correlation separation}\end{array}}& ⇒& \mathrm{walking distance}\propto \sqrt[\mathrm{dimensions}]{\mathrm{population size}}\\ & ⇒& \mathrm{population size}\propto {\left(\mathrm{walking distance}\right)}^{\mathrm{dimensions}}\end{array}$

There was however one more aspect to the Book of Missives that seemed to single out an underlying world where distance was defined as the square root of the correlation separation. Just as a neighbourhood of four persons could fit on a rectangle with sides √1 and √2 (see Fig. 1), twice that number of persons could be arranged at the vertices of cuboid with edges √1, √2 and √4. Likewise, 16 persons could be positioned on the vertices of a hypercuboid with edges √1, √2, √4 and √8. And so we can continue, adding a dimension for every doubling of persons to allocate. If you wonder where the 'missing' distances √3, √5 and √6, etc. are, then, well, according to Pythagoras, they are precisely the distances along the diagonals connecting the vertices of a cuboid.)

A distance defined as the square root of the correlation separation allows us to construct a world where alle persons are located at uniquely defined positions. It is a strange world that offers the degrees of freedom of a two dimensional plane, but which has the structure of an N-dimensional cuboid, where N is more than two if there are more than four persons.

## A spherical take on correlation numbers

Because we saw that homogeneity, in the case of the Book of Missives, implied that we cannot deduce the dimensionality of an underlying world, we might also be led to think that we cannot say anything about the dimensionality of a world underlying the real data. Nothing could be more wrong.

Yrfes had noticed that the maximum separation having the value '15' (see Table 7a-c and Table 12) was just a consequence of him having taken a look at a sample of 15+1=16 farmers and hunters. Factoring out the sample size minus one, the normalized correlation separation would vary between 0 and 1 where the value 0 corresponded to full correlation and the value 1 to full anticorrelation.

$\mathrm{normalized correlation separation}=\frac{\mathrm{correlation separation}}{\mathrm{sample size}-1}$

Even if all data were taken into account, the value of the normalized correlation separation would always be somewhere between 0 and 1. Furthermore, the normalized correlation separation that followed from the data that were brought back from Bryiks and Heolaam pertaining to two specific persons, was not influenced by how many other persons were taken into account.

Because the correlation separation between two persons seemed to be a property of some underlying world and not of the number of inhabitants in the world, Yrfes's idea was to evenly spread a two planets' worth of people over a closed, finite sized area, a spherical surface for simplicity. He had not decided whether that should be a surface with 1, 2, 3 or even more dimensions - in other words whether the people would be confined to a circle, a globe or a higher dimensional surface. Of course, he himself didn't have the power to shuffle people around, but he could do some modelling in a computer.

First Yrfes tried a very simple model of the world: a circle. The populations were evenly spread out over the circle. Yrfes defined "home" as the position of one of these people on one of the planets. Then Yrfes divided this world in concentric "rings" (which become pairs of points if the world is a circle) centred on "home" in such a way that between a ring and the next ring there are always the same number of people. If he had 1000 people in all, 500 'Bryiksian farmers' and 500 'Heolaam hunters', and between each neighbouring pair of rings there would be 20 people, about ten from each planet, he would create 49 rings, the last one of which would encircle a 'polar cap' around the point opposite "home".

 0 ? 0.5*Math.PI : -0.5*Math.PI; return phi; } //http://www.filosophy.org/post/35/normaldistributed_random_values_in_javascript_using_the_ziggurat_algorithm/ function Ziggurat(){ var jsr = 123456789; var wn = Array(128); var fn = Array(128); var kn = Array(128); function RNOR(){ var hz = SHR3(); var iz = hz & 127; return (Math.abs(hz) < kn[iz]) ? hz * wn[iz] : nfix(hz, iz); } this.nextGaussian = function(){ return RNOR(); } function nfix(hz, iz){ var r = 3.442619855899; var r1 = 1.0 / r; var x; var y; while(true){ x = hz * wn[iz]; if( iz == 0 ){ x = (-Math.log(UNI()) * r1); y = -Math.log(UNI()); while( y + y < x * x){ x = (-Math.log(UNI()) * r1); y = -Math.log(UNI()); } return ( hz > 0 ) ? r+x : -r-x; } if( fn[iz] + UNI() * (fn[iz-1] - fn[iz]) < Math.exp(-0.5 * x * x) ){ return x; } hz = SHR3(); iz = hz & 127; if( Math.abs(hz) < kn[iz]){ return (hz * wn[iz]); } } } function SHR3(){ var jz = jsr; var jzr = jsr; jzr ^= (jzr << 13); jzr ^= (jzr >>> 17); jzr ^= (jzr << 5); jsr = jzr; return (jz+jzr) | 0; } function UNI(){ return 0.5 * (1 + SHR3() / -Math.pow(2,31)); } function zigset(){ // seed generator based on current time jsr ^= new Date().getTime(); var m1 = 2147483648.0; var dn = 3.442619855899; var tn = dn; var vn = 9.91256303526217e-3; var q = vn / Math.exp(-0.5 * dn * dn); kn[0] = Math.floor((dn/q)*m1); kn[1] = 0; wn[0] = q / m1; wn[127] = dn / m1; fn[0] = 1.0; fn[127] = Math.exp(-0.5 * dn * dn); for(var i = 126; i >= 1; i--){ dn = Math.sqrt(-2.0 * Math.log( vn / dn + Math.exp( -0.5 * dn * dn))); kn[i+1] = Math.floor((dn/tn)*m1); tn = dn; fn[i] = Math.exp(-0.5 * dn * dn); wn[i] = dn / m1; } } zigset(); } var z = new Ziggurat(); function initRandomUnitVector(dim,oldvector,olddim) { var sum2 = 0.0; var e; if(olddim == 0) e = []; else e = oldvector; var k; for(k = olddim;k < dim;++k) { /* var d = Math.pow(2, -32); var U = Math.random()+d; var V = Math.random()+d; var root = Math.sqrt(-2.0 * Math.log(U)); var XXd = root * Math.cos(2.0*Math.PI*V); e[k] = XXd; */ e[k] = z.nextGaussian(); } if(olddim == 0) { var a = 0.8 * e[0] - 0.6 * e[1]; var b = 0.6 * e[0] + 0.8 * e[1]; e[0] = a; e[1] = b; } return e; } function normalised(e,dim) { var sum2 = 0.0; var k; var ret = []; for(k = 0;k < dim;++k) { var X = e[k]; sum2 += X * X; } var f = Math.sqrt(sum2); if(f > 0.0) { for(k = 0;k < dim;++k) { ret[k] = e[k]/f; } } return ret; } function planetNo(p) { return p - 2 * Math.floor(p/2); } function Accent(p) { var planet = planetNo(p); var plan = plnt[planet]; return (plan[0] == p || plan[1] == p) ? 1 : 0; } function ei(dim,cx,cy,r) { this.dim = dim; var svg = this; var g = document.createElementNS(svgns,"g"); var homenode = document.createElementNS(svgns,"circle"); this.homenode = homenode; homenode.setAttribute("cx",850); homenode.setAttribute("cy",100); homenode.setAttribute("r",8); homenode.setAttribute("fill","red"); g.appendChild(homenode); var moredimensions = document.createElementNS(svgns,"circle"); this.moredimensions = moredimensions; moredimensions.setAttribute("cx",400); moredimensions.setAttribute("cy",150); moredimensions.setAttribute("r",20); moredimensions.setAttribute("fill","red"); g.appendChild(moredimensions); var fewerdimensions = document.createElementNS(svgns,"circle"); this.fewerdimensions = fewerdimensions; fewerdimensions.setAttribute("cx",400); fewerdimensions.setAttribute("cy",250); fewerdimensions.setAttribute("r",20); fewerdimensions.setAttribute("fill","blue"); g.appendChild(fewerdimensions); this.svg = g; this.globes = null; this.root2 = document.getElementById("svgwortel"); this.egg = function(r,x,y,angle,Z,stroke,fill) { var R = Math.sqrt(1.0 - Z*Z); // circle diameter (long axis of ellipse when circle is projected) var rx = Math.floor(r * R); var ry = Math.floor(r * R * Math.sin(angle)); var cx = Math.floor(x); var cy = Math.floor(y - r * Z * Math.cos(angle)); var ellipse = document.createElementNS(svgns,"ellipse"); ellipse.setAttribute("cx",cx); ellipse.setAttribute("cy",cy); ellipse.setAttribute("rx",rx); ellipse.setAttribute("ry",ry); var hstroke = ("000000" + Math.floor(stroke).toString(16)).substr(-6); var hfill = ("000000" + Math.floor( fill).toString(16)).substr(-6); ellipse.setAttribute("style","stroke:#"+hstroke+"; fill:#"+hfill); this.globes.appendChild(ellipse); this.ellipse = ellipse; } this.spot = function(r, x,y,theta,phi,R,stroke,fill,p,rl) { // theta == 0.0 : top // theta == pi : bodem // phi == 0.0 : midden // phi < 0.0 : links // phi > 0.0 : rechts /* A/a (elliptical arc) rx, ry x-axis-rotation large-arc-flag sweep-flag x, y Draws an elliptical arc from the current point to (x, y). The size and orientation of the ellipse are defined by two radii (rx, ry) and an x-axis-rotation, which indicates how the ellipse as a whole is rotated relative to the current coordinate system. The center (cx, cy) of the ellipse is calculated automatically to satisfy the constraints imposed by the other parameters. Large-arc-flag and sweep-flag contribute to the automatic calculations and help determine how the arc is drawn. the arc starts at the current point the arc ends at point (x, y) the ellipse has the two radii (rx, ry) the x-axis of the ellipse is rotated by x-axis-rotation relative to the x-axis of the current coordinate system. */ var X, Y, RX, RY; var large_arc_flag = 1; var sweep_flag = 1; var dx = r*Math.sin(theta)*Math.sin(phi); var dy = -r*Math.cos(theta); var cx = x + dx; var cy = y + dy; var R2 = dx*dx + dy*dy; var R1 = Math.sqrt(R2); var x_axis_rotation = 180.0/Math.PI*Math.asin(dx/R1); // positief: draaiing met de klok mee var flat = Math.sqrt(r*r - R2)/r;// cosine of angle away from direction out of the screen. if(Math.cos(theta) < 0) x_axis_rotation = -x_axis_rotation; X = Math.floor(cx - (flat*Math.abs(dx)-dy)*R/R1); Y = Math.floor(cy); RX = Math.floor(R); RY = Math.round(R*flat); var path = document.createElementNS(svgns,"path"); //--X; path.setAttribute("d","M"+X +","+Y +" A"+RX +","+RY +" "+Math.floor(x_axis_rotation) +" "+large_arc_flag +" "+sweep_flag +","+X +","+(Y+1) ); var tjek = path.getAttribute("d"); var hstroke = ("000000" + Math.floor(stroke).toString(16)).substr(-6); var hfill = ("000000" + Math.floor( fill).toString(16)).substr(-6); path.setAttribute("style","stroke:#"+hstroke+"; fill:#"+hfill); path.setAttribute("id",p+"."+rl); this.globes.appendChild(path); } this.slice = function(r,x,y,Zd,Zu,stroke,fill) { var Rd = Math.sqrt(1.0 - Zd*Zd); var xdL = Math.floor(x - r * Rd); var ydL = Math.floor(y - r * Zd); var xdR = Math.floor(x + r * Rd); var ydR = Math.floor(y - r * Zd); var R = Math.floor(r); var path = document.createElementNS(svgns,"path"); if(Zu <= 1.0) { var Ru = Math.sqrt(1.0 - Zu*Zu); var xuL = Math.floor(x - r * Ru); var yuL = Math.floor(y - r * Zu); var xuR = Math.floor(x + r * Ru); var yuR = Math.floor(y - r * Zu); path.setAttribute("d","M"+xdL+","+ydL +" A"+R+","+R+" 0 0,1 "+xuL+","+yuL +" H"+xuR +" A"+R+","+R+" 0 0,1 "+xdR+","+ydR +" H"+xdL ); } else { path.setAttribute("d","M"+xdL+","+ydL +" A"+R+","+R+" 0 0,1 "+xdR+","+ydR +" H"+xdL ); } var tjek = path.getAttribute("d"); var hstroke = ("000000" + Math.floor(stroke).toString(16)).substr(-6); var hfill = ("000000" + Math.floor( fill).toString(16)).substr(-6); path.setAttribute("style","stroke:#"+hstroke+"; fill:#"+hfill); this.globes.appendChild(path); } this.integral = function(x,n) { if(n > 1) return -1.0/n * Math.pow(Math.sin(x),n-1) * Math.cos(x) + (n-1)/n * this.integral(x,n-2); else if(n == 1) return 1 - Math.cos(x); else return x; } this.setMinMaxIntegral = function(dimensions) { this.maxintegral = this.integral(Math.PI,dimensions - 2); this.minintegral = this.integral(0.0,dimensions - 2); } this.expectationValueA = function(angle,dimensions) { var ret; if(dimensions == 3) ret = Math.cos(angle)/*inp*/; else if(dimensions == 2) { ret = 1.0 - (angle / Math.PI*0.5); } else { var inp = this.integral(angle,dimensions - 2); ret = 1.0 - 2.0*(inp - this.minintegral)/(this.maxintegral - this.minintegral); } return ret; } this.expectationValue = function(p1,p2,dimensions) { console.log('p1:'+p1); console.log('p2:'+p2); var A = points[p1]; var B = points[p2]; console.log('A:'+A); console.log('B:'+B); var inner = 0; var normA2 = 0; var normB2 = 0; console.log('dimensions:'+dimensions); for(i = 0;i < this.dim;++i) { normA2 += A[i]*A[i]; normB2 += B[i]*B[i]; inner += A[i]*B[i]; } inner /= Math.sqrt(normA2*normB2); console.log('inner:'+inner); var angle = Math.acos(inner); console.log('angle:'+angle); exp = this.expectationValueA(angle,dimensions); console.log('exp:'+exp); return exp; } this.angle = function(expectectationvalue) { var mina = 0.0; var maxa = Math.PI; var mine = this.expectationValueA(mina,this.dim); var maxe = this.expectationValueA(maxa,this.dim); if(expectectationvalue == mine) return mina; else if(expectectationvalue == maxe) return maxa; else { var h = 0.0; var i; for(i = 0;i < 100 && maxa - mina > 0.0000000001;++i) { h = (mina+maxa)/2.0; var e = this.expectationValueA(h,this.dim); if(e > expectectationvalue) { mine = e; mina = h; } else if(e < expectectationvalue) { maxe = e; maxa = h; } else break; } return h; } } this.fz = function(Z) { var ret; if(Z <= -0.999) { return -1; } if(Z >= 0.999) { return 1; } switch(this.dim) { case 2: ret = Math.cos((1.0 - Z) * Math.PI*0.5); break; default: var a = this.angle(Z); ret = Math.cos(a); } return ret; } this.density = function(Z) { var FZ = this.fz(Z) if(FZ != 0) return Z / FZ; else return 1; } this.col = function(Z,maxdensity,mindensity) { var dens; if(maxdensity > mindensity) dens = (this.density(Z) - mindensity) / (maxdensity - mindensity); else dens = 0.5; var allbits = 0xffffff; var idens = ((1.0 - (dens)) * 255.0); return (idens << 16) + (idens << 8) + idens; } this.setDim = function(dim) { this.setMinMaxIntegral(dim); this.dim = dim; var p; for(p = 0;p < 1000;++p) { var e = initRandomUnitVector(this.dim,points[p],DIMENSIONS); points[p] = e; } } this.setTop = function(top) { console.log("setTop:"+top); var planet = planetNo(top); console.log("planet:"+planet); var plan = plnt[planet]; if(plan[0] != top) { plan[1] = plan[0]; plan[0] = top; console.log("planN:"+plan[0]+" "+plan[1]); plnt[planet] = plan; } /* for(p = 0;p < 1000;++p) { rot[p] = points[p]; } */ var d; for(d = this.dim-1;d > 0;--d) { ptop = points[top]; var a = ptop[0]; var b = ptop[d]; var Determinant_ = Math.sqrt(1.0/(a*a + b*b)); for(p = 0;p < 1000;++p) { var e = points[p]; var X = Determinant_ * ( a*e[0] + b*e[d]); var Y = Determinant_ * (-b*e[0] + a*e[d]); e[0] = X; e[d] = Y; points[p] = e; } } var pl,pt; for(pl = 0;pl < 2;++pl) for(pt = 0;pt < 2;++pt) { if(plnt[pl][pt] == -1) return; } var separations = []; var sumseparations = 0; var biggestseparation = 0; var dBiggest; for(d = 0;d < 4;++d) { separations[d] = 1 - this.expectationValue(plnt[0][parseInt(d/2)],plnt[1][parseInt(d%2)],this.dim); console.log('d:'+d+' : '+separations[d]); sumseparations += separations[d]; } var minchsh = sumseparations - 2*separations[0]; var maxchsh = minchsh; var mind = 0; var maxd = 0; for(d = 0;d < 4;++d) { var chsh = sumseparations - 2*separations[d]; if(chsh < minchsh) { minchsh = chsh; mind = d; } if(chsh > maxchsh) { maxchsh = chsh; maxd = d; } } console.log('chsh: '+minchsh+' '+maxchsh); var difmin = -minchsh; var difmax = maxchsh - 4; dBiggest = difmin > difmax ? mind : maxd; var iA = parseInt(dBiggest/2); var iB = parseInt(dBiggest%2); var A = plnt[0][iA]; var B = plnt[1][1-iB]; var C = plnt[0][1-iA]; var D = plnt[1][iB]; console.log('AB:'+separations[2* iA +1-iB]); console.log('CB:'+separations[2*(1-iA)+1-iB]); console.log('CD:'+separations[2*(1-iA)+ iB]); console.log('AD:'+separations[2* iA + iB]); } this.Egg = function() { if(this.globes != null) { var y = this.root2.getElementById("globes"); y.parentNode.removeChild(y); } var globes = document.createElementNS(svgns,"g") globes.setAttribute("id","globes") this.globes = globes; this.svg.appendChild(this.globes) var yy = this.root2.getElementById("globes"); var desc; switch(this.dim) { case 2: desc = "Circle mapped onto 3-D sphere"; break; case 3: desc = "Normal 3-D sphere"; break; default: desc = "Higher dimensional sphere mapped onto 3-D sphere"; } var Z = 0.0; var inc = 0.04; var maxdensity = 0.0; var mindensity = 1.79769e+308; var rings = 0; for(Z = -1.0;Z < 1.0; Z += inc) { var temp = this.density(Z+0.5*inc); if(temp > maxdensity) maxdensity = temp; if(temp < mindensity) mindensity = temp; } if(maxdensity < 1.001 * mindensity) maxdensity = mindensity; var tilt = 1.0; for(Z = -1.0;Z <= 1.0; Z += inc) { // bin[Slice++] = 0; this.egg(160.0, 200.0, 200.0, tilt, this.fz(Z), 0, this.col(Z+0.5*inc,maxdensity,mindensity)); } /* var MaxSlice = Slice; */ for(Z = -1.0;Z < 1.0; Z += inc) { ++rings; this.slice(160.0, 600.0, 200.0, this.fz(Z), this.fz(Z+inc), 0, this.col(Z+0.5*inc,maxdensity,mindensity)); } var sintilt = Math.sin(tilt); var costilt = Math.cos(tilt); var colours = [0xff8000,0x00ff00]; for(p = 0;p < 1000;++p) { // var e = points[p]; planet = planetNo(p); var e = points[p]; var n = normalised(e,this.dim); // Normalized in dim dimensions! Not in three! var Y = n[0]; var Z = n[1]; var X = 0; /* var Slice = 0; var ZZ = -1.0; while(Slice < MaxSlice) { if(this.fz(ZZ) >= Y) { bin[Slice - 1] += 1; break; } ++Slice; ZZ += inc; } */ var theta = Math.acos(Y); //This ensures that the apparent // angle from a pole to another point is the true angle. // All other angles are distorted! // So we take the Y from the n-dimensional sphere as the Y // of the three-dimensional sphere. var phi; if(this.dim > 2) { X = n[2]; phi = Atan(X,Z); } else if(Z > 0) phi = -0.5; else phi = Math.PI - 0.5; if(Z > 0) { this.spot(160.0, 600.0, 200.0, theta, phi, 4*(1+Accent(p)), 2, colours[planet],p,0); } else if(Accent(p) == 1) { this.spot(160.0, 600.0, 200.0, theta, phi, 4*(1+Accent(p)), 2, (colours[planet]|0xaaaaaa),p,0); } var sintheta = Math.sin(theta); var X1 = Math.sin(phi)*sintheta; // NOT the same as X! var Z1 = Math.cos(phi)*sintheta; // NOT the same as Z! var Y2 = costilt*Y - sintilt*Z1; var Z2 = sintilt*Y + costilt*Z1; if(Z2 > 0) { var thetatilt = Math.acos(Y2); var phitilt = Atan(X1,Z2); this.spot(160.0, 200.0, 200.0, thetatilt, phitilt, 4*(1+Accent(p)), 0, colours[planet],p,1); } else if(Accent(p) == 1) { this.spot(160.0, 200.0, 200.0, thetatilt, phitilt, 4*(1+Accent(p)), 0, (colours[planet]|0xaaaaaa),p,1); } } // this.spot(160.0, 600.0, 200.0, 0, 0, 4.0, 0, 0xff8000,0,2); // this.spot(160.0, 200.0, 200.0, tilt, 0, 4.0, 0, 0xff8000,0,3); DIMENSIONS = this.dim; return 0; } homenode.addEventListener("click",clickhome,false); moredimensions.addEventListener("click",clickmore,false); fewerdimensions.addEventListener("click",clickfewer,false); this.setDim(this.dim); this.setTop(theTop); this.Egg(); var ip2 = document.getElementById("ip2"); ip2.appendChild(this.svg); document.querySelector('#svgwortel').addEventListener('click', function(event) { if (event.target.tagName.toLowerCase() === 'path') { console.log("klik:"+event.target.id.split('.')[0]); var the = parseInt(event.target.id.split('.')[0]); if(!isNaN(the)) { theTop = the; Ei.setTop(theTop); Ei.Egg(); } } }); }; var Ei; var dim = 2; function clickhome(evt) { Ei.setTop(theTop+1); Ei.Egg(); } function clickmore(evt) { document.getElementById("dim").firstChild.nodeValue = dim + ' dimension' + ((dim > 1) ? 's' : ''); dim += 1; Ei.setDim(dim); Ei.setTop(theTop); Ei.Egg(); } function clickfewer(evt) { if(dim > 2) { dim -= 1; Ei.setDim(dim); Ei.setTop(theTop); Ei.Egg(); } document.getElementById("dim").firstChild.nodeValue = (dim-1) + ' dimension' + ((dim > 2) ? 's' : ''); } function main(evt) { Ei = new ei(dim,100,100,45); } ]]> 1 dimension

Yrfes defined the correlation separation between the person at "home" and another person as the number of rings someone would have to cross to walk from "home" to the other person, divided by the total number of rings, thus: $\mathrm{correlation separation}=\frac{N\left(\mathrm{rings crossed}\right)}{N\left(\mathrm{all rings}\right)}$ Each person could only visit people with the other colour, so each person could make 500 different visits. This was how Yrfes modelled the circumstance that the bare data exhibited correlations between persons from different planets, but not between persons from the same planet.

Yrfes computed the correlation separation for each farmer-hunter pair in which "home" was a specific farmer, and built the the first column of a new correlation separation table in the same way as described for Table 7a, taking care to sort the rows in order of increasing correlation separation (decreasing correlation), or increasing number of rings to cross. This procedure was repeated for all farmers inside the first ring, but without sorting the rows. From now on, each person had its own row, defined by the sorting order of the very first iteration. When all persons inside the first ring had been "home", Yrfes picked a person outside the first ring, but inside the second ring and declared him to be the new "home". Yrfes repeated the computation for this second person and filled in the second column. And so he went on until all persons had been "home". This came out of it if the world was a circle:

This correlation separation table looked remarkably like the correlation separation table constructed from the real data, so Yrfes set out to check whether there were any Campana Wins to be found, as had been the case for the real data. But alas, such did not occur. But Yrfes had only tried a very simplistic world, where everybody lived on a circle. So, next, Yrfes evenly populated the surface of a globe, which was not very different from how people were spread over his own planet, Cricsis.

A similar table was the result:

To his delight, Yrfes found many combinations of "farmers" and "hunters" in his model world that created Campana Wins. The reason why there were Campana Wins was the following:

If person A counts N rings between him and another person B, and if A counts M rings between person B and a third person C, then person A counts either |N-M| or N+M rings between him and person C, depending on whether C is "between" A and B or B is "between" A and C. However, if the sphere has more than one dimensions it may be true that B, from his perspective, counts fewer than M rings between him and person C, although he would of necessity agree with A that the number of rings between B and A is N. So the "outsider" A, when counting the number of rings between two other persons B and C, counts a greater number of rings than any of the two "insider" persons B and C, in the same way that a journey from Nop to Tuv in an earlier example corresponds to crossing a city with fewer people than the difference between the city population corresponding to a walk from Ab all the way to Tuv and the city population corresponding to the shorter walk from Ab to Nop.

If the dimensionality of the sphere is greater than 1, there will always be "outsider" persons who count a greater number of rings between two persons than any of the two involved persons is able to count. This puts the Book of Missives in another light. We saw that a convenient way to visualize the correlations was to take the square root of the correlation separation as a walking distance. With this definition of walking distance, all persons can be arranged on a cuboid, the edges and diagonals of which have lengths equal to these walking distances. This would give the correlation separations the status of population size in a two-dimensional world. However, the correlation separations themselves could also have been taken as walking distances, because they obey the triangle inequality. This sets the Book of Missives apart from the real data, which shows signs of correlation separations that do not obey the triangle inequality, and therefore cannot be interpreted as walking distances.

Yrfes created worlds with even more dimensions and got these results:

## Spin

In Newtonian physics everything happens at definite locations in absolute space at an absolute time, where "absolute" indicates that there is no room for disagreement about these locations. Still today, in physical theories we often assume the existence of a frame of reference encompassing everything, or, if that sounds too ambitious, at least the laboratory.

In every global (or laboratory-sized) frame of reference you can define a global system of coordinates. To do this you do not need to go out and put poles in the earth to indicate the mesh points of your system of coordinates. Because you know (or at least assume) that space and time are absolute, it is sufficient to use a handful of physical gauge points, and the rest follows by interpolation. With a global system of coordinates every location can be represented by a set of four numbers, and every set of four numbers represents a location.

In Einstein's General Relativity Theory you can not take the existence of a global system of coordinates for granted. That is not to say that you cannot speak of coordinates. On the contrary, you are free to define any system of coordinates you want, and nothing will change very much in your physical expressions, but the price you have to pay is that you must have thought about how every point in your system of coordinates corresponds with physical locations out there in the real world. If you cannot operationally define this correspondence, your system of coordinates is pretty much useless as a vehicle for physical theories.

General Relativity is about geometry, literaly "measuring the world", but it is also about gravitation. The latter is distinctly more sexy than the first, and that may explain why we seldom see the need for operational definitions of frames of reference play a major role in discussions about fundamental questions in physics.

An important part of operationally defining a system of coordinates is defining distances. A value for a distance has two components. On the one hand you have the measuring stick that defines the unit length and on the other hand you have the coordinate distance, which counts the number of unit measuring sticks needed to span the distance. In General Relativity both components occupy prominent positions, whereas in the rest of physics the unit measuring stick is regarded as somewhat dull, and only seldom attracts attention, such as in 1998 when the Mars Climate Orbiter crashed and the distinction between imperial and metric units made headlines.

In discussions about the conceptual difficulties of Quantum Mechanics the general trend is to solve these problems within the "easier" conceptual framework of Galilean or Special Relativistic geometry. It is seen as outraOegus to mention General Relativity as a relevant source of useful concepts, because if we cannot assume that space time at the scale of the laboratory is flat and very well described by Special Relativity, then how do we explain that we got so far using this assumption? Giving General Relativity a prominent place in solving the conceptual issues of Quantum Mechanics may also seem to turn things on its head: the generally accepted idea is that General Relativity has to give in to Quantum Mechanics in order to arrive at a theory that encompasses both.

Yrfes Fig endorsed his uncle's idea that something that was not expressible in numbers was sent from Cricsis to the planets. If these things were feeble enough, the wise man and priests would have only one chance to let these things "talk" and decide the verdict, because any attempt to listen to the message would change or destroy it. Examples of such things could be pairs of tops spinning in opposite directions, which would be able to operationally define parallelism at a distance, without the need to invoke coordinates (which are numbers) and even without the need to worry about the space-time structure. The real data apparently allowed one to find out which farmer somehow was parallel with which hunter, the angle between a farmer and a hunter being defined by the angle between the points on the sphere that represented them. It was difficult to envisage other operational definitions of parallelism at a distance than by way of the constancy of the angular momentum of a material object.

Several committee members thought this would not work for the following reason. Once it had been established how the temples on Bryiks and Heolaam were spatially aligned with respect to each other, using the aforementioned pairs of spinning tops, it was possible to define a system of coordinates encompassing both temples. Knowing that angular momentum can be denoted by an axial vector, it would seem natural to imagine that during every session, the mediating spins could be decomposed in components, using the coordinate system as the framework for defining a set of basis vectors along which to take the components. But then, these committee members said, we would be back to the missives-in-a-book model. If an angular momentum can be fully described by its components (three numbers), then we can do without the angular momenta and just send the numbers. On these grounds they did not think that the angular momentum was any improvement over the book and they rejected Yrfes Fig's idea.

Yrfes Fig countered this "reductio ad absurdum" by saying that you cannot say much about the individual angular moments' components if you do not know the metric structure of space in detail. If you assume a flat or otherwise nice structure, then you can indeed use the parallelism of the temples to define components at all times, but how can we know that space time has such a simple structure? Especially if the angular momenta themselves and individually influence the metric structure, arguing like these committee members becomes similar to arguing that all particles in a gas have the same velocity, as defined by the temperature of the gas.

Yrfes Fig also said that his opponents' reasoning would lead to the absurd conclusion that sending numbers could help establish and maintain a system of coordinates incorporating both temples. Did his opponents think that numbers alone could define a coordinate system? If not, what else was needed and how much of that?

To explain the data, the physical objects that had been used for coordinating the verdicts on the two planets should somehow resist decomposition, because such decomposition would make it possible to create a Book of Missives. One way (perhaps the only way!) to understand how a physical object can escape decomposition is by assuming that there is no frame of reference to decompose into.

Remains the question how a model that is agnostic as to counterfactual verdicts can be shown to produce Campana Wins. Indeed, not assigning verdicts to persons that did not win the competition is a prerequisite, as the failure of the Book of Missives has shown, but not sufficient.

# Think now, imagine later

1. Think of the farmers being represented by the item each of them brought to the temple. Likewise think of the hunters being represented by the items they brought to their temple.
2. Instead of manipulating symbols, the farmers' wise men and hunters' priests undertook something that cannot be replaced by consultation of a book of prescripts: they let something point at an item, the item belonging to the elected farmer or hunter, establishing a direction in doing so.
• If you think that the pointing at an item can be fully described by symbols, then think again: Can you ensure that an alien intelligent race can construct a copy of an object that is sitting in front of you by following ever so detailed instructions that you sent to them? Wouldn't the reconstruction have a 50% chance of being mirrored? Wouldn't it be necessary to also send them a physical reference?
3. Instead of a book with prescriptions of how to decide the outcome of ceremonies, think of directions being sent, one direction for each session. During each session, the two temples receive the same direction. For the moment, do not try to picture a direction. Especially, don't make a projection of a direction on the model of physical space that you carry around in your mind. Also, the 'sameness' of directions is not meant in the usual sense in which two lines can be parallel in Euclidean space.
• Do not think of the received direction as pointing at something or other, for example the item a fellow hunter or farmer brought to the temple. This explanation requires that you accept that the 3D world we perceive is a construction based on innumerable measurements of the kind involved in the cosmic experiment.
• The sameness of the directions received at the temples is an after-the-fact attribute, given because there were correlations between hunters and farmers.
4. Between the direction received at a temple and the direction of the elected hunter's/farmer's item is an angle. You may think of that angle as something real, 'out there', but not measurable in practice, except for a single bit of information, telling whether the angle is acute or obtuse. There exists machines that can elicit this single bit of information. To obtain the bit of information that you want, as part of the preparation of the machine you have to turn in way so that it points in the direction of the elected person's item. More precise information about this angle, as everything else about the received direction, is hidden.
5. The answer to the hunter or farmer would be an arbitrary (but consistently adhered-to) mapping to the 'yes/no' like answers they receive from the priests/wise men.
6. Although distributions of angles at the temples seem to indicate complete randomness, the three directions involved in one session - the received direction, the direction of the farmer and the direction of the hunter - are not independent. That is, whereas any two of the involved angles are uncorrelated, the third angle cannot be chosen completely free.
• This is easy to see with an example: if the farmer and the hunter have almost the same direction, then the angle between the hunter and the received direction and the angle between the farmer and the received direction must be almost the same.
7. Free will is allowed to decide the hunter's and farmer's directions. Free will has nothing to do with the received direction.
8. There are innumerable distributions of three vectors, two of which can be chosen at random, while the third is more or less restricted, all distributions having marginal distributions of angles that indicate randomness.
9. Find the distribution that reproduces the CHSH-statistics.
10. Even though it is possible to construct a complete model for what was going on in the temples it is not possible to do a replay of the cosmic experiment in a virtual world, involving real players with free will and real big distances between them to ensure that one ceremony's outcome cannot be used to make a last moment adjustment of the other ceremony's outcome. All virtual worlds consist of manipulations of dimensionless symbols, and as we have seen, the messages that have to be sent cannot be embodied by symbols.
11. Explanatory models that can reproduce the statistical correlations of the cosmic experiment must themselves utilize the kind of messages that were employed in the cosmic experiment. In other words: such models are not really explanatory, but pedagogical at best.

# Definitions

ceremony
An event localised in space to a planet (Bryiks or Heolaam) and in time to about one morning.
correlation
If expressed numerically, a correlation of 1 means perfect agreement between the verdicts of a hunter and a farmer over a long run of ceremonies. A correlation of -1 also means a perfect agreement, but verdicts are always found to be the opposites of each other. A correlation of 0 indicates that a farmer verdict corresponds as many times with the same and the opposite hunter verdict.
correlation separation
An alternative way to numerically express a correlation. A correlation separation of 0 is the lowest possible correlation separation and corresponds to perfect correlation (1). If normalized, the maximum correlation separation is 1, corresponding to a correlation of -1. In some contexts correlation separation is expressed as a natural number.
session
A pair of ceremonies, one taking place on Bryiks, the other taking place on Heolaam. Pairing is done by combining the n-th ceremony in the paper roll with the n-th ceremony in the journal of parchments.
farmer
An animate agent with free will, inhabiting the planet Bryiks and acting independently of far away events on other planets, such as decisions taken by hunters (on Heolaam) and laboratory experiments (on Cricsis).
hunter
An animate agent with free will, inhabiting the planet Heolaam and acting independently of far away events on other planets, such as decisions taken by farmers (on Bryics) and laboratory experiments (on Cricsis).
item
An inanimate object representing a farmer or hunter. Items are, in their turn, uniquely represented by symbols.
priest
An agent following strict procedures, with no free will. Local of the planet Heolaam.
wise man
An agent following strict procedures, with no free will. Local of the planet Bryiks.
selection
Procedure by which, during a ceremony, a farmer or hunter, and therefore an item, is singled out. No two items can be selected during the same ceremony. Selection is decided by those who are nearby and have free will.
symbol
An element of meaning that an animate agent can be reminded of when observing a shape, rythm or item. The same symbol can have more than one physical representation, thus establishing symbolic links between physical objects.
verdict
The outcome of a ceremony: one of two symbols
identity
An emergent property of things physical. When an animate agent is reminded of the same symbol on different occasions, the mind tends to ascribe unique identity to the physical causes of the aroused memory.
laws
A sine qua non for the existence of symbols and identity. And without symbols and identity, no laws.
log book
A memory of all ceremonies. Contains, for each ceremony, a symbol denoting the selected item and a symbol denoting the verdict.

# Epilogue

Space-time may be emergent, an epiphenomenon. The kind of experiment that Bell envisaged is one way to get a closer view at what is underlying. This closer view is perceived as entanglement. As an analogue, spreading rings on a surface of water are clearly coordinated if seen as a deviation from quietness, but the dynamical laws of water molecules are the same. The rings are not any more correlated than the quietness. It is characteristic of the smallest constituent of a thing that it is behaving in a different way than the thing. A water molecule does not behave like a volume of water, a letter does not behave like a text, a corn of sand does not behave like a dune and a coin does not behave like a budget. Spin half particles are not oddly behaving guests in a normal world. They are the essence of the world.

## YOU CANNOT GET SUCKED INTO A TRON WORLD AND GET BACK AGAIN

### Why?

As you enter a tron world inside a computer, you loose any sense of orientation, for good. Left and right cannot be expressed in bits. Even if there were a system of coordinates, in terms of which you always had x, y, z, and t coordinates and you were heading in direction theta, phi, with your feet pointing to xi zeta, you would not be able to use your coordinates when you wanted to return. A computer is a Turing machine, a long tape and a machine moving back and forth, with no concept of orientation.

## The CHSH inequality

It is fairly easy to understand that the sum of the shortest separations between A and B, between B and C and between C and D cannot be shorter than the separation between A and D. $\stackrel{‾}{\mathrm{AD}}\le \stackrel{‾}{\mathrm{AB}}+\stackrel{‾}{\mathrm{BC}}+\stackrel{‾}{\mathrm{CD}}$ That is the triangle inequality applied twice. But the CHSH inequality is about two, not one, limit. What is the other limit? The separations CHSH look at have a maximum. While the smallest separation corresponds to the highest correlation, the largest separation corresponds to the highest anti-correlation. But we could just as well have defined separation in terms of anti-correlation, the smallest separation corresponding to the highest anti-correlation and the largest separation to the highest correlation. If separation has a range from 0 to 2, we have the simple conversion rule $\mathrm{anticorrelation separation}=2-\mathrm{correlation separation}$ By writing down the CHSH inequality for the anticorrelation separation and then substituting the above formula, we get $\left(2-\stackrel{‾}{\mathrm{AD}}\right)\le \left(2-\stackrel{‾}{\mathrm{AB}}\right)+\left(2-\stackrel{‾}{\mathrm{BC}}\right)+\left(2-\stackrel{‾}{\mathrm{CD}}\right)$ or $\stackrel{‾}{\mathrm{AB}}+\stackrel{‾}{\mathrm{BC}}+\stackrel{‾}{\mathrm{CD}}-4\le \stackrel{‾}{\mathrm{AD}}$ or, combined $\stackrel{‾}{\mathrm{AB}}+\stackrel{‾}{\mathrm{BC}}+\stackrel{‾}{\mathrm{CD}}-4\le \stackrel{‾}{\mathrm{AD}}\le \stackrel{‾}{\mathrm{AB}}+\stackrel{‾}{\mathrm{BC}}+\stackrel{‾}{\mathrm{CD}}$

## Events, labels, regions, manifolds

TODO: formulate a la: "There are events e(r,n,l,o) where region r in {H,B}, number n in {0...N}, label l in {L}, outcome o in {yes,no}, instructions lambda(n) such that ..."

## Two dimensional analogon

Draw two concentric circles of different radius. The centre of the circle is the parameter lambda. each circle is a measurement, which can be binary, conditioned on the radius of the circle. The distance betwen the circles is the angle between the measuring instruments. Two of the three numbers can be chosen freely. Once two numbers are chosen, the third is limited to two possible values.

(Will be edited and continued)