Here goes: A1: Complex numbers exists. Call this C. A2: x = x A3: x + y = y + x A4: A is a subset of B if B contains A and B - A not = the empty set. Index.....Statement...…………………………………………………………………………………….. Reason 1...…...….Construct S = C x C.......................................................................................A1, A2 1.1...…….S is 4 dimensional....................................................................................…...1 1.2.........Set the components of S = S_1,2,3,4 in the following order: Re, Im, Re, Im.....1, A2 2......…...S can transform into two Riemann Spheres......................................……...A1, 1 3............Construct two Riemann Spheres in S, call it RS x RS = Pp......................…..A1, 1 4............Isolate the Riemann Circle of S_3, 4 and call it P_T.................................…A1, 3 4.1......…I'm going to use physical terminology below........................................…..Declaration 4.2...…..Construct "physical space" = S_P = CxC/S_4.......................................…...…A1, A2 5......…..Let P_T advance by one (rotate relative to S1,2,3) when encountering a space node and let the rotation be a quantum rotation. Call this "freq" = T_S......................….............A1, 4, 4.2, A2 7......…..Define "Change in freq" by T_Sf - T_Si…………………………………....5 8...........Let S_1,2 be perpendicular to S_3,4.................................................................1 11..........Construct {for all n = 1 to N: n(T_Sf - T_Si)} . Call this "Changes in freqs.".........5,7 12.........Define "basic time interval" = Delta t_B = 1/[(1/N) \sum \limits_{n=1}^N n(T_Sf - T_Si)]...........................................................................................................................1-11, A3, A2 13.......Construct MxT_S, M element of Natural Numbers subset of C..................................5, A4 14.......Define " Basic time" = t_B = {1/[(1/M) (\sum \limits_{n=1}^M n#T_S)]}*Delta t_B...............12, 5, A3 15........Couple t_B to every node of S_P and call the result "basic spacetime"= B_ST...........4.2, A2, A2 15.1......Construct S_i = CxC........................................................................................A1 16........Construct RSxRS in S_i, call it Pp..................................................................15.1, 2 17........Isolate the Riemann Circle in Pp and call it P_BT....................................................A1, 16 18........Let P_BT advance by one (rotate relative to S_i1,2,3) when encountering A B_ST node and let the rotation be a quantum rotation. Call this "freq2"= T_BST.................................17, A2 19........Construct KxT_BST, K element of Natural Numbers, subspace of C........................18, A4 20........Define "Tim1" = t_1 = 1/[(1/K)(\sum \limits_{n=1}^K n#T_BST)]..............................A3, A2, 18 21.......Pp is in every particle of the clock..........................................................................Requirement 22......."Tim1" advances like a clock, it depends on the Pp in the clock and on the route in B_ST.................................................................................................................................18, 21 23......."Tim1" = Time........................................................................................................A2, 22 Further: 24.........Define a constant c = DS_P/Dt_B........................................................................4.2, A2 24.1......Let c be the maximum speed trough B_ST i.e. the speed at which the particle sees no distance between succeeding nodes of B_ST..................................................................4.2 24.1......Define an operator: "<->" as "superposition operator".......................................1, 2 24.2......Construct S = CxC from a copy of B_ST................................................................A1 25.........From S, define a new RS <-> RS, subject to item 32.2..........................................24.2 29.........Construct S_AP = (-C)x(-C)2)...............................................................................A1 30.........Construct from S_AP a RS_AP <-> RS_AP. Call it F_1............................................29, 2 31.........Construct any given vector in a RS <-> RS set = F_1, call it "four-momentum" = p. This is done by identifying four numbers in F_1. Call such particle qFp_1.1).........................3, 18, 4.1 32..........p is 4 dimensional.............................................................................................31 33..........Construct the same vector as in 31 x (-1) in F_1. Call such particle qFp_1............31, 28 34 Identify a marker in F_1's origin and at the origin in F_1.......................................31, 33 35..........Set FP_1 = qFp_1 and leave out 4 distinguised points just below the unit circle crossing a curled up axis. Call the four points c_1, c_1, c_2, c_2.........................................24.2 36..........Set Fp_1 = qFp_1 and leave out 4 distinguised points just below the unit circle crossing a curled up axis. Call the four points c_1, c_1, c_2, c_2.........................................29 37.........Let S_1, S_2 of Fp_1 look like in Figure 1.1........................................................24.2, 35 Figure 1.1 38..........Let S_AP1, S_AP2 of Fp_1 look like in Figure 1.1, (just turned upside down)..29 -> 32.1 39..........Let the starting position (after one instance of time) of Fp_1 and Fp_1 be as drawn in figure 1.2 (only the curled up x and y-direction shown).................................................29 Figure 1.2 40..........Let the four points c_n sense the closest four nodes of B_ST in direction p and let them engage these nodes even if the whole Fp_1 needs to turn or move linearly...………………………….....35 41.........If four nodes were engaged: distinguish four new nodes and go to 40...........35 42.........Let Fp_1 move similarly to 40, just sensing nearest nodes of negative coordinates in the down direction....................................................................................................35 43.........Fp_1 and Fp_1 may be polarised: cicularly, transversely or longetudinally......37 45.........Fp_1 has spin 1...............................................................................................44 46.........The nodes of B_ST causes a force with nonzero component in the up direction. Define F = ma. With m = 0 we have infinite acceleration thus infinite speed. But infinite speed is just c. Hence Fp_1 goes upwads at the speed of light.........................................................24.2, 37 47..........That the movement of Fp_1 causes Electro magnetic waves can be seen from the following figure. The F forces have a tiny reaction force in the up direction...............figure 1.3 Figure 1.3 48.........Fp_1 is a photon ...............................................................................................43 -> 47 Comments: 1) We construct particles in theory, we construct them first as particles to see what the machinery of particles is. We will later construct only in particle-antiparticle pairs. Note that by figure 1.2 momentum of Fp_1 in down direction is prohibited. 2) This way the particle and antiparticle may look identical except for phase difference of 180 degrees.

What does 'x' mean in this context? Do you just mean that each element of S has 4 components? It seems you do. Okay, so elements of S have 4 ordered components, which are all real numbers? Can it? What do you mean by that? S is a set of 4-component elements. What is the transformation? I don't understand the construction you're proposing. What's a Riemann circle and how do you get one from the last two components of elements of S? You're losing me. You've lost me at this point. I don't know what the notation CxC/S_4 means. Here's where I stop. P_T, you told us, is a "Riemann circle". What does it mean for a Riemann circle to "advance by one"? What is a "space node"? What is a "quantum rotation"? None of these terms has been defined.

x means: Cartesian Product. Yes. No. S_2 and S_4 are pure complex numbers. Yes, because 1 complex plane and a copy of it may transform into a Riemann Sphere. On the Riemann Sphere the Real axis curls up and connects at infinity to the north pole of the Riemann Sphere, similarly for the Imaginary axis. The transformation is stereographic projection. Look at: https://en.wikipedia.org/wiki/Riemann_sphere I means: transform the two Complex plains into Riemann Spheres and superimpose the two. Riemann Circle is one axis as projected onto the Riemann Sphere. The line must read: 4............Isolate the Riemann Circle of S_4 and call it P_T.................................…A1, 3 It means form the Cartesian product of 2 C's and discard the 4th component S_4. This gives physical space just with the S_2-axis multiplied by i. It means: Rotate by an unit distance measured along the curve of S_4. A space node is a point in S_P. A quantum rotation is one that happens instantaneously (a jump).