A 6 phase clock can produce chiral time as does a 3 phase clock. If each of the 3 phases of time is associated with a CA coordinate axis, then the sequence x, y, z is chiral, (and different than the sequence z, y, x). There appears to be good reasons for assuming that discrete time goes through six phases. This added complexity to the nature of time seems a good compromise in reducing the overall complexity of the DM model. The author is ready and willing to be proven wrong. The rule, R, that governs how things change is actually best represented as 6 rules: R0, R1, R2, R3, R4, R5. Each of the Rules are the same except for orientation in the array. An example of the different orientations for each phase is the following:
R0 → (x y, –x –y), (x –y, –x y) Na Red
R1 → (y z, –y –z), (y –z, –y z) Cl Anti-Green
R2 → (x z, –x –z), (x –z, –x z) Na Blue
R3 → (x z, –x –z), (x –z, –x z) Cl Anti-Blue
R4 → (y z, –y –z), (y –z, –y z) Na Green
R5 → (x y, –x –y), (x –z, –x z) Cl Anti-Red
The Rule-Orientation map above can be understood by thinking of a cube in the xyz lattice. Every rule neighborhood is in a plane parallel to some face of the cube.
“…R1 → y z, –y –z, y –z, –y z …” means that the neighborhood pattern for Rule 1 is in the YZ plane and oriented in a direction specified by (y, z), (–y, –z), (y, –z), or (–y, z). Since even Rules govern movement in the Chloride sub-array and odd Rules govern movement in the Sodium sub-array, it must be true that some series of odd Rule applications can cause Na bits to make progress in any direction. The same is true for a series of even Rule vectors and the Cl sub-array.
In one 6 step cycle, each sub-array undergoes 3 possible steps. In the Na sub-array at time R0 a particular bit might remain in place or it might take a step in one of 4 directions. Since that makes 5 possibilities in each of 3 steps, there are 125 (53) different possible moves in that cycle. Not all 3 step moves lead to unique endpoints. There are actually 63 different endpoints arrayed symmetrically about the starting point with total distances as follows:
The colors, “Red, Anti-Green…” are relevant to QCD and will be discussed under DP Quarks and Color Charge.
It is best to think of the progress of a particular bit, even though its motion is always a consequence of 2 bits swapping places. If Rule R1 is about to cause a swap, then a particular bit of the pair of bits will take a diagonal step to its nearest spatial neighbor in 1 of 4 directions. The 4 directions for R1 are a change in position of (0, 1, 1), (0, –1, –1), (0, 1, –1), or (0, –1, 1).
The explanation for the rest of the entries in the Ri table is that they were chosen to try to simultaneously capture a number of different properties. First, for both the even subscripted rules and the odd subscripted rules, there must be sequences of steps that can make progress in ±x or in ±y or in ±z. For example, consider the Na sub-array sequence:
R0= x, y, R2 = x, z, R4 = –y, –z The total motion is 2x.
The same kind of motion is also possible in the Cl sub-array.
A bit may complete a spiral orbit while making progress in the X direction (or any of a large number of other directions) by being swapped 3 times in one 6 micro-step cycle. Other bits will end up being swapped fewer times or perhaps not be swapped for an arbitrary number of micro-time steps.
The chirality of the same motion (such as 2x) is opposite when 2x is done in the conjugate sub-array. We say “chirality” because every such path is actually a spiral that has the bit in a 2D triangular orbit perpendicular to the direction of motion. The reason that the rule orientation table is what it is that the sequence R0, R2, R4, must be able to be the same as the sequence R5, R3, R1; for CPT symmetry. Every particle that has an antiparticle is a structure keyed to either the Na or the Cl sub-array. While part of the machinery of every particle is in both sub-arrays, the difference between a positron and an electron is that every part of the electron that is in the Na sub-array corresponds exactly to the same part of the positron in the Cl sub-array, and visa versa. Note how the inherent chirality of R0, R2, R4 is exactly the same as R5, R3, R1. The chirality of the Na sub-array going forwards in time is the same as that of the Cl sub-array going backwards in time. The chirality of the Cl sub-array going forwards in time is the same as that of the Na sub-array going backwards in time. That is part of how DM achieves CPT; why an electron going backwards in time is the same as a positron going forwards in time.
There is a problem with the above example in that the YZ plane is treated differently than the XY and XZ planes, but only in terms of the order of sequencing. To the author, this appears as a possibly fatal flaw, but this model nevertheless combines many of the right features. This is stage 1 of trying to capture facts of physics, and we have to be very tolerant of the things that are simply wrong or missing.
The Rule subscripts correspond to micro-time steps taken modulo 6. We assume that the time subscript, τ, counts up every time step. At every point in time, the overall 2nd order state of the system is represented by 2 global states, the past and the present and by the time phase that will be applied. The smaller time subscript identifies the past sub-state and the larger time subscript identifies the present sub-state. One cycle consists of 6 corresponding 2nd order states of the system: (S0, S1), (S2, S1), (S2, S3), (S4, S3), (S4, S5) and (S6, S5). The first member of each pair, (S2i, S2i±1) always has an even subscript and, along with the time phase, represents the overall state of the Sodium sub-array. The second member of each pair always has an odd subscript and along with the time phase, represents the state of the Chloride sub-array. Every cell with an even time subscript must have x, y and z add up to an even number. Every cell with an odd time subscript must have x, y and z add up to an odd number. R1 is the rule that changes S0 into S2, leaving S1 unchanged. R2 is the rule that changes S1 into S3, leaving S2 unchanged. The rule that is applied to a second order step has the same time subscript, Modulo 6 as does the state of the present. The rule R must meet the following requirement:
R2t 1(S2t, S2t 1) → (S2t 2, S2t 2) and : R2t 1 (S2t 2, S2t 2) → (S2t, S2t 1). This requirement is easily met by many functions. One large class of such functions is a conditional permutation or swap of 2 elements. R1 of 2 elements of S0 produces S2; whether or not the swap takes place depending only on S1. Notice that a swap done twice is the null operation and since the question as to whether or not to do the swap depends only on S1; not at all on S0 when going forwards and not at all on S2 when going in the reverse direction. It is the same function of the same S1 going forwards in time or going backwards in time. It may be surprising, but it is easy to make such reversible systems computation universal!
When time is reversed, we have to come up with some new labels in order to avoid confusion. If going forwards, a function of the present changes the past into the future, then when time is reversed we will say that a function of the present changes the future into the past. R1(S2, S1) → (S0, S1). So, the larger subscript is the future and the smaller subscript is the present. The next step would be R0(S0, S1) → (S0, S-1).
Given an overall state and a phase, we can determine the direction of time in the following way. The phase always specifies the rule that is about to be applied. The present is always the state with the same time subscript as the phase. If the other state has a time subscript that is one less than the phase, (τ -1); the system is going forwards in time. If the other state has a time subscript that is one more than the phase, (τ 1); the system is going backwards in time. R1(S0, S1) is going forwards in time, R1(S2, S1) is going backwards in time.
