Accelerated Twisting on the Magic Paving Stones

In The Unexpected Result of Twisting on the Magic Paving Stones, I showed that the scenario behind the Magic Paving Stones puzzle/paradox – at least when twisted – correlates with the notion of the OE curve, in a FUGE universe.  The monster correlates with a photon, the magic paving stones inflate the path/universe at a rate that is proportional to the preset number, and you are an observer.  While it sounds like being reached by a photon is not quite as dire as being reached by a monster, I never did actually say what the monster would do when it reached you, perhaps all it would do is trigger one of the photoreceptor cells in your eye.

When I mentioned this at reddit, I noted that expansion of the universe is thought to be accelerating in the Standard Model.  But my thinking was that since this is equivalent to an ever-increasing number of magic paving stones being added each round (on either side of the monster, in front and behind), it would make sense that the monster would still get you – eventually.  There would still be a possibility of paving stones appearing behind the monster, meaning that it might take longer but the monster would still get you.

I modified my code to incorporate expansion such that, each round, the pathlength would increase by a truncated percentage.  For example, if the initial pathlength was 25 paving stones and the percentage selected was 4%, then in the first round the number of paving stones would be increased by 25*0.04=1, then they would increase by one each round, for 25 rounds, until there were 50 (50*0.04=2).  Then they would increase by 2 each round for 13 rounds, until there were 76 (76*0.04=3.04), then the increase would be 3 for 8 rounds, 4 four 7 rounds, 5 for 5 rounds, 6 for 4 rounds, 7 for 4 rounds, and so on.

What I found was that, for an increase value of 4% and for initial pathlengths of greater than 25 with the twisted scenario (so the monster is one step closer at the start), there was small chance that the monster would not get you.  That chance increases rapidly as you increase the initial pathlength.

The most common result with an initial pathlength of 26 was something like this:

A far less common result, for which I had to run the simulation 79 times the first time, 383 times the second time and 322 times the third time before it occurred (and the fourth time I exited after 400 refreshes without any triggering), was this:

That little kick up at the end is more extreme (occurs earlier and is more frequent) when the initial pathlength is greater, for example using 40 when it manifested immediately five times in a row:

This is interesting, but we need to be careful about these chunky initial states, by which I mean states in which the random influences are comparitively large. I wanted to look at a less chunky scenario, specifically where the initial pathlength is 801, the monster starts on 800 and the percentage increase per round in 0.125% (=1/800).  My intuition (which was wrong) was that this would also be unstable.  It’s not, I could refresh numerous times and the monster gets you every time.

It only gets unstable (for a percentage of 0.125%) at a pathlength of about 1032-ish at which point monster only gets you roughly half the time – of course it happens earlier but at a lower frequency.

A result like the above is not unusual with those values (although there’s usually either a kick at the end, or the monster has got you before 10,000 rounds).

There seems to be a pattern here, if we increase the initial pathlength and decrease the acceleration proportionately, we get this:

In this case the monster got you, but only after a very long time.  There are so many calculations with values of this size that my machine slows down too much to carry out many refreshes, and increasing the pathlength by another order of magnitude would be an overnight (or weekend) task.

Keeneyed readers will note that I increased the initial pathlength by very slightly more than a factor of 10.  This is because I was intending to test the notion that the instability kicks in when the pathlength equals the inverse of the percentage increase per round times the square root of one and two thirds.  While it does seem to really manifest at about pathlength=1.29*1/percentage, 1.29 is not the actual value (and neither is 1.29099444) – the monster seems to get you every time with the values above (usually within about 40,000-60,000 rounds).

The precisely nature of this effect, and the mechanism behind it remains unclear.  What is clear though is that the closer the monster is at the beginning, the more likely it is to get you:

A single paving stone is actually enough to massively increase the likelihood that the monster will get you, or in this case about 0.1% closer.

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The question that I have, of course, is whether this in any way correlates with the accelerated expansion of the Standard Model universe during the Dark-Energy-Dominated era.  We’d have to think about the time at which that era commenced, about 4 billion years ago, at which time the Hubble parameter value was about 66.5 km/s/Mpc (because in the Standard Model, in that era, H(t)=2/(3t) where t is the time elapsed since the Big Bang).  Assuming that the Hubble parameter has increased uniformly since then, to reach the current value (H(t)≈1/t≈70), and using a year as our unit of time, this would equate to an increase (Δ) given by:

Δ=^4000000000√(70/66.5)-1=1.28×10^-9%

This is using a 4 billionth root, but of course the granularity of the universe isn’t at the scale of years, it’s about 6×10⁵⁰ times finer, meaning that the increase per Planck time is in the order of 2×10^-60%.  (This is because x=y^1/z-1, where y-1→0, then y^1/(z*w)-1→x/w.  I asked whether there is proof for this at Reddit and was given one.  I don't doubt that proof, but it's one of those back of the napkin things that will work for a mathematician but have shades of appeal to authority for anyone else.  But that is better than the brute method that I had which was to just keep inserting values in and find that it works every time, which you can try yourself if you prefer.)

By implication from the above, it would seem therefore that a photon would reach us reliably (albeit after a long time) if it were emitted a little bit less than about 6×10⁵⁹ Planck lengths away, which is 3.5×10¹⁶ light years, or about 35 million billion light years.

If the logic above is correct, then the claim that a photon that is only 15 billion light years away would never reach us would be incorrect.  Perhaps the nature of the acceleration of expansion in the Standard Model is not as I have described.