"Cosmic foam bubbles? Can they be measured, observed, investigated empirically in any way? What is the foam and where does it come from?"
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The Sloan Digital Sky Survey has mapped over half a million galaxies in 3D. Of course, there are more gaps than mapped regions. Our view is blocked by the disk of our own galaxy, and we can't see what lies beyond other galaxies. The task of mapping all the visible galaxies may take decades.
I haven't actually seen the map in stereo vision, but those who have describe it as a giant bubble bath with walls of galaxies surrounding the voids. A void is the interior of a bubble. They believe dark matter completes the fabric of the bubble walls. This is what I am calling the cosmic foam.
We are just beginning to get statistical measurements, and it appears that the median bubble size in the cosmic foam is roughly 10^24 meters across. That being the case, a median-size bubble occupies roughly one millionth of the volume of the visible universe. (I am saying that, by definition, half the volume of a region is contained in bubbles larger than the median size.)
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"This looks like another one of those "math creating reality" claims instead of math reflecting testable reality."
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I am no mathematician. Numbers are a weakness of my model---not a strength. I use very rough approximations to provide a starting point from which others may someday refine the model. There is no insubstantial mathematical space in my model. Everything consists of waves in a hard, massive medium; and that medium consists of particles which are made of waves in a finer medium, and so on ad infinitum. There are no finite empty spaces to be bridged by insubstantial mathematical forces.
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"How do you know the sub-universe is running backwards if all of your tests run forward, which they must?"
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We cannot subject the sub-universe to tests; if we could it would be part of our universe. We can only infer that it must run backwards because of the way the model explains the expansion of space. If a cubic meter is 10^105 median-size bubbles, and that number is constant, then the expansion of space means the number of bubbles in a region of space must be increasing. When a foam fizzes, bubbles are popping, which decreases the number of bubbles. A bubble wall pops, and two bubbles merge into one. For the number of bubbles to increase, they must be un-popping. New bubble walls must be appearing, dividing one bubble into two. The second law of thermodynamics prevents bubbles from un-popping in forward time. (Note: I am assuming the number of bubbles per cubic meter is constant. More generally, it makes sense that the number of bubbles increases as space expands. In keeping with Occam's razor, I choose constant as the simplest explanation until such time as it leads to a contradiction.)
What makes the bubbles pop is the expansion of space in the sub-universe. The cosmic-foam bubbles of the sub-universe are stretched to their breaking point by expansion of sub-universe space. So that is a forward time cause of popping bubbles, from a sub-universe perspective. From our perspective, those same bubbles are un-popping. The cause of the popping is expansion of sub-universe space, which occurred before the popping from a sub-universe perspective. From our perspective, the cause is after the bubble un-pops. So the effect precedes the cause from our perspective.
Also, a popping bubble generates pressure waves which radiate outward. From our perspective, those pressure waves converge to a point where they seem to cause a new bubble wall to appear. The pressure waves are dark energy, and they are converted to new space.
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"Don't forget common sense just because it's not fashionable in the non-scientific quasi-physics world."
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I'm sure Einstein's heard many similar admonitions from his contemporaries. "Common sense" is a euphemism for thinking well inside the box. Paint by the numbers; don't cross the lines. If we never think outside the box, we'll be condemned to add new patches to the same old flawed concepts for ever.