Dear Professor Mattfolk,
I am deducing the logic leading to the sub-geometries step by step here.
1, I think the ultimate theory is hidden behind Einstein's question, "The most incomprehensible thing about the world is that it is comprehensible", since this is a question about the ultimate nature of physics. My interpretation is that the great master wouldn't believe the world is intellectual designed, but he couldn't explain why it looks as orderly designed and who designed it. Even though Einstein didn't give an answer, it still offered a direction and I don't think this is an unanswerable question.
2, In fact, Weinberg answered it half way in a prescription for the theory of everything, "... [it] has to be simple ... equations that are based on a simple physical principle ... it has to give us the feeling that it could scarcely be different from what it is..." [1] (that is, it is based on one unified principle and is non-designed, but simply cannot be otherwise, e.g. electromagnetism.)
3, Therefore, the problem is not that Einstein's question cannot be answered, but that, for strong, weak and gravitation forces, there exists no theory as un-designed as EM. (Unfortunately, neither standard model nor general relativity looks as un-designed.)
4, In order to reach un-designed theories, it's important to note that certain critical concepts must NOT be assumed, because assumption is subjective and leads to intellectual (human-, not God-) designed theories.
5, Actually, two unnoticed assumptions exist in today's physics, namely, 1. Pre-assumed plane angle scales (i.e. automatic space flatness, axes perpendicularity and existence of symmetry) without physical definition, which leads to "designed" Standard Model. When we wonder why Standard Model looks like designed. The reason is simple: because it is in fact designed, not by God, but by ourselves. 2. Preselected inertial frames in special relativity leading to "designed" GR.
6, Let's consider the first assumption. What does it mean by "pre-assumed plane angle scales"? Take 4d spacetime (and EM) as an example. Special Relativity used light speed to define the 4 linear scales. What was not mentioned is the 6 circular magnetic and electric fields running among the 4 axes which define the "equivalencies" among the 4 axes. Without this definition, light would not be measured at equal speeds in different directions, rotational symmetry would not exist and photons cannot be generated.
7, Then, what are the fields running among the 6 "planes" to define the equivalencies among the 6 "plane angle scales". Just like linear scales, these equivalencies cannot be assumed, but "must" be defined by real physical fields running among the 6 planes. These fields are conjectured to be the "classical" weak fields. We may say these fields are running in solid (3d-) angles among planes (2d-surfaces). When equivalencies among plane angle scales are defined by 3d-angle rotation fields, an SO(6)~SU(4) (or SO(10)~SU(5) for 5d spacetime) symmetry surfaces, which is just the observed particle spectrum (without quarks). The relation between weak fields and plane angle scales are exactly the same as that between EM and linear scales, making weak fields as un-designed as EM.
8, Likewise, there are two more levels of sub-geometries: fields running in 4d-angles among 3d-surfaces, which is conjectured to be CP-violation fields, and fields running in 5d-angles among 4d-surfaces, which is conjectured to be strong fields. Rotations in 5d-angles are believed to be causing baryon and various lepton numbers. The "relation between CP-violation fields and 3d-angle scales" and "the relation between strong fields and 4d-angle scales" are also the same as that between EM and linear scales, making CP-violation and strong fields as un-designed as EM. Details are in reference [2], "Theory of Fields of Unified Origin (TFUO)".
9, If the 6 (or 10) angle scales are not defined to be equivalent by weak fields, a full circle on xy-plane may be 360 degrees, while that on yz-plane may be 362 degrees, then the 4d-spacetime would be warped and perpendicularity of axes cannot exist and symmetry would not surface. To be more precise, without TFUO, or 3 levels of sub-geometries, linear spacetime on top would be warped and perpendicularity of axes cannot exist and symmetry would not surface.
10, In TFUO, strong, weak, CP-violation and EM fields are all originated from the same principle as Weinberg prescribed (each defining a critical scale). Electromagnetism would be as complicated as other forces if not for the change of geometry by Special Relativity. What sub-geometries do to other forces is exactly the same thing as what Special Relativity does to electromagnetism. At the same time, complete particle zoo is generated from all layers of geometries. This is a big achievement through removal of assumption of automatic equivalencies of plane angle scales (or space flatness, or axes perpendicularity, or symmetry presence). We see assumption often deprives us of otherwise possibility to uncover real nature of physics.
11, It is important to emphasize that, whether it's 4d, 5d, 11d, or 26d, there cannot be automatic flatness of space, automatic perpendicularity of axes and automatic symmetry, unless sub-geometries exist to support them. (It may be possible in mathematics but not in physics, because two persons could define differently, but physics will only follow what is defined by Nature). Take 11d as an example, if the (11x10/2=) 55 plane angle scales are not made equivalent to each other by fields running among them, then the space would be warped and perpendicularity of 55 axes is lost and the 11d symmetry would not surface. Simply put, the wished-for 11d symmetry wouldn't exist if sub-geometries don't exist. But if sub-geometries exist, 11d micro dimensions are no longer needed, because the sub-geometries already offer all the symmetries needed for particle spectrum. In fact, the 11 micro dimensions are never observed. (Also, the sub-geometry of 55 planes should generate SO(55) spectrum, which is not observed either.)
12, For reason of completeness, the gravitation part of TFUO is included here. Let's consider the second assumption, "preselected inertial frames". It is well known that inertial frames (uniform frames) are "preselected" before spacetime scales are defined by light waves to verify uniformity. Removal of this assumption leads to 5d spacetime [3].
To be published is the ultimate non-designed 5d gravitation, which is "linear" and quantize-able based on the 5d-spacetime. It meets all 3 famous tests just like GR. Note that, the 3 tests: bending of light, perihelion motion of Mercury and gravitational red shift, did not test GR completely, as they are all based on Schwarzschild solution with Einstein/stress-energy tensor set to 0. This means the exact "non-0 expression" of Einstein/stress-energy tensor has not been tested, since a different expression (e.g. this 5d linear gravitation) could work just as well, as long as it can be set to 0 in these situations. The 5d gravitation eventually joins TFUO to form the ultimate theory, which answers Einstein's ultimate question.
There are reasons to believe this theory is "not" just another fancy idea, but is THE long-sought-for ultimate theory, as: 1. It meets Weinberg prescription for the theory of everything, as all forces originate from the same principle (i.e. each defining a critical geometrical scale). 2. It is able to answer Einstein's ultimate question, as intellectual designer is eliminated, since this is a non-designed theory for all forces and particles, just as EM and photons. 3. While more verifications are needed, the symmetry, SO(4) or SO(5), is already met with particle spectrum without quarks. 4. Linear gravity can be quantized. 5. The most important and most strong evidence is that no micro dimensions are observed for any symmetry for standard model or string theory or whatsoever. On the other hand, the sub-geometry is the most (or the only) plausible explanation for particle symmetries.
If you have other questions, we may discuss further. If you find it makes sense, your dissemination would be appreciated.
Since the paper attached through FQXi seems not working, you or anyone can just send an email to: qchiang2@yahoo.com , I will send a copy from there.
References
[1] Steven Weinberg, "Will a theory of everything reign?", TIME April 10, 2000, p. 86.
[2] Kwan C. Chiang, "Anatomy of spacetime and possible origins of internal symmetry and all particle quantum numbers", Physics Essays, Vol. 33, N.3 p342-347, 2020.
[3] K. C. Chiang: "A Unified Gravitation and Quantum Mechanical Space-Time Structure through a Unified Origin of Inertial and Gravitational Masses and a discussion of the Foundation of Special Relativity", Il Nuovo Cimento Vol. 68B, N.2 p322, 1982.
