Gottlob1's Theory of Everything

The four inverse electromagnetic fine-structure constants, rounded up, to seventeen significant digits, are: 137.03599915121266, and 142.73600104069440, for our set of alternate universes, and, 89.266557295047035, and 62.712232460473931, for the other. The four inverse gravitational fine-structure constants, rounded up, to seventeen significant digits, are: 197.66952228533632, and 214.02899610821660, for our set of alternate universes, and, 29.970632587726477, and 10.679967924426944, for the other, say, the true anti-matter set. Note that the gravitational set of four thus numerals are each a factor of 10^(-63) less. For example, 197.66952228533632 ---> 0.19766952228533632 X 10^(-60), but, it works better as is to work out along side the first set of four thus numerals. I prefer to thus flip, say, the 137 to 0.197. Regardless, the 137 = [1][√9][7] ---> 197, in a figurative or symbolic sense. Later on, I intend to show that there's a bit more to it, in the way that one dimension grows out of another.

The four very simple equations used to mathematically determine the fine-structure constants. By plugging the integer inverse approximation values, of the constants, into the equations in a way that produces the thus constants. Integer values, in equations with integer coefficients, to produce the thus irrational, constants to an arbitrary number of (non-repeating) decimal places. The four thus equations are really, in theory, and, actually, in practice, a set or family of equations that issues from the mathematics of the first, which differs from that of the others, in that it's as is.

Equation 1a is the graph of the y-values of the sequence of sums of the two squares that begin with (7^2 + 8^2) = 113. If 1 is added to the first base, which is 7, and, 1 is subtracted from the second base, which is 8, the sum of the two squares becomes (8^2 + 7^2) = 113. Note that the value, 113, occurs twice in the sequence at the same spot, which is denoted 113/113. Thus carrying on, the sum of (9^2 + 6^2) = 117, the sum of (10^2 + 5^2) = 125, the sum of (11^2 + 4^2) = 137, and, so on. Similarly, if 1 is subtracted from the first base, which is 7, and, 1 is added to the second base, which is 8, the sum of the two squares becomes (6^2 + 9^2) = 117, which leads to 125, 137, and so on, to form the other brach of the graph. The minimum value of the graph of equation 1a is y = 113. Note that the value, 137, which is the integer inverse approximation value of the electromagnetic fine-structure constant of our universe, is included in the range of equation 1. The four equations above necessarily involve all of the integer inverse approximation values of the fine-structure constants, and, initial digits which lead up to the other mathematical/physical dimensionless constants. Simplistically speaking, not any old set of four equations leads to the fine-structure, and similarly dimensionless, constants. Equation 1b is the graph of the respective y-values of equation 1a minus 11, which are 102, 106, 114, 126, and so on. The minimum value of the graph of equation 1b is y = 102.

Equation 2a is the graph of the respective y-values of equation 1a descending instead of ascending from 113. 113 goes to 117 by adding 4; 117 goes to 125 by adding (4 + 4) = 8; 125 goes to 137 by adding (8 + 4) = 12, and, so on, by the previous increases plus 4. Similarly, 113 goes to 109 by subtracting 4; 109 goes to 101 by subtracting (4 + 4) = 8; 101 goes to 89 by subtracting (8 + 4) = 12, and, so on, by the previous decreases minus 4. The maximum value of the graph of equation 2a is y = 113. Equation 2b is the graph of the respective y-values of equation 2a minus 11, which are 102, 98, 90, 78, and so on. The maximum value of the graph of equation 1b is y = 102.

It's important, in specific, that the respective y-values of the graphs above involve x-values that increase by 1 from x = 1. Eg, y = 113/113 occurs at x = 1, for equation 1a, and, y = 102/102 occurs at x = 1, for equation 1b. The reason that intermediate values of x, and y, where x = 0, must be defined separately from the equations 1, and 2. That is, the latter values aren't per se on the graphs of equations 1, or 2, but, fit their overall numerical pattern as expressed differently.

It's important, in general, that the pairs of the corresponding graphs face each other through the y-axis, where x = 0. Ie, that equation 1a faces equation 1b, and, similarly, for equations 2a, and 2b. If arbitrarily put the graph of equation 1a, defined as the 113-graph, to the left side, and, the graph of equation 1b, defined as the 102-graph, to the right side, then y = 115 at x = 0 for the 113-graph, and, y = 104 at x = 0 for the 102-graph. To allow for the minimum value on the left, which is 113, to go up 2, to 115, before going up another 2, to 117, and, then, carrying on from 117 as above; and, for the minimum value on the left, which is 102, to go up 2, to 104, before going up another 2, to 106, and, then, carrying on from 106 as above. Then pairs of values develop by extending the left graph to the right, and, versa. 113 goes with 106; 117 goes with 114; 125 goes with 126; 137 goes with 142, and, so on. 142 goes with 137, which is taken to mean that 142 is the integer inverse approximation of the electromagnetic constant in our alternate universe.

For example, the values of the 113-graph may advance clockwise on a circle of eight axes, starting from 115//104/104 at the bottom of the vertical axis, to 113/113//106, to 117/114, to 126/125, to 137//142/162 at the top of the verticals axis, to 153/173//186, to 197/214, to 246/225, to 257//282/322 at the bottom of the vertical axis, after one complete turn, and, so on. The values of the 102-graph may then similarly advance counter-clockwise on the same circle, starting from 104//115/115 at the bottom of the vertical axis, to 102/102//117, to 106/125, to 114/126//137, to 142//153/173 at the top of the vertical axis, to 162/197, to 186/225, to 214/246//257 at the bottom of the vertical axis, after one complete turn the other way, and, so on. The top of the vertical axis represents position-0 of a cycle of eight progressions, with position-8 superimposed on. Define going backward from position-0, as to position-7, to position-6, and so on, but, not backward from position-0, to position-(-1), to position-(-2), and so on. Laid out as above, then the axes with double numerals (with one slash to separate them) for either the 113-graph, or, the 102-graph, are the vertical, and next-to-vertical-going-clockwise, axes. 113/113 is opposite 153/173, on the next-to-vertical-going-clockwise axis, for the 113-graph; 104/104 is opposite 142/162, on the vertical-going-clockwise axis, for the 102-graph.

No diagram is included because there's only the one diagram, with the eight basic positions, and, it's best understood by drawing it out for oneself. The exception occurs to keep the positions in sync when working with a cycle of 10, 12, 14, or 16. If of 10, then former position-2 goes positions- 2, 3, and 4, so that former position-3 goes 5, former position-4 goes 6, former position-5 goes 7, former position-6 goes 8, former position-7 goes 9, and, former position-8 goes 10, which, similarly, is imposed on position-0. If a cycle of 12, then the two additional positions are added at former positions- 2, and 6. If a cycle of 14, then at former positions- 2, 4, and 6. When a cycle of 16, then at former positions- 2, 6, 4, and, 8. Either of which is beyond the scope of the fine-structure constants per se. In specific the various symmetries follow how the integers are mathematically defined, in terms of the sequence, 0 to 1; 0/1 to 2/3; 0/1 and 2/3 to 4/5/6/7. Every numeral times zero is 0; every numeral to the exponent of 0 is 1. The 2/3 has four, and six, -fold symmetries, respectively, and, form the basis of the other symmetries. The five, and six, -fold symmetries spring from the four, and six, -fold symmetries. In general, the numerals reflect, however, about the average of their symmetries. Now if properly constructed, the above circle of values, as a cycle of 8 of equations 1 led to the associated values, going clockwise on the horizontal and vertical axes, of 115/104, 117/114, 137/142, 197/214, 257/282, and so on, but, going counter-clockwse on the horizontal and vertical axes, of 104/115, 106/125, 142/153, 186/225, 282/293, and so on. But, for the equations 2, going clockwise on the horizontal and vertical axes, of 111/100, 109/90, 89/62, 29/-10, -31/-78, and so on, but, going counter-clockwise on the horizontal and vertical axes, of 100/111, 101/98, 53/62, 1/18, -67/-42, and so on. But, for now, we are concerned with 29/-10, 89/62, and, 137/142, 197/214.

Two calculators were required to determine the fine-structure constants from the four equations above. Calculator 1 allows for the calculation of decimal places in the thousands of digits. I required it to go beyond the 16-significant-digit calculations of calculator 2, which I used to emperically determine the thus limited number of decimal places of the numerals plugged into the four equations above to yield the fine-structure constants. Just input one of the given equations above, and, next, start with one of the integer values of the fine-structure constants, such as 137, to see which thus limited numeral gives rise to the best known inverse approximation of the electromagnetic fine-structure constant. Currently, its accepted value is, 0.035999084, which is about (1 / 137.035999084) ---> 137.035999084 is about (1 / 0.035999084). As far as I know, there are many ongoing experimental measurements of it, and, so, every few years, its currently accepted value is revised. The later measurement, 13 Feb, 2023, was maybe the best so far, at 0.0072973525649(8), which is about (1 / 137.035999166(15)) ---> 137.035999166 is about (1 / 0.0072973525649). The bits in brackets at the ends of the measurements are within a presumed possible range of error, but, the measurements appear to swing relatively wildly beyond the bits in the brackets at the ends. Regardless, it's a goofy numeral, at best, "Therefore, (1 / 137.03600) is the asymptotic value of the fine-structure constant at zero energy," in that it already sort of looks like an inset version of itsef, by 36 = (-1 + 37) ---> 137.

The measurement above may be compared with mine above, by 137.035999166 compared with 137.03599915121266, respectively, which then is at least a decimal place better than the latest best mathematical approximation, by an indirect means, of the inverse electromagnetic fine-structure constant, at about 137.03599978677653. "It is impossible to give an account here of all of the proposed formulas for alpha that have appeared in the literature. Some recent formulas are discussed by Gilson (1997), which presents his own formula, which is probably to date the most accurate formula and it is calculated from a model of atomic physics and relativity. This formula is difficult to interpret, however, and the numbers n1 = 137 and n2 = 29 have to be selected as the best input into the formula, alpha = {[29 * cos(pi / 137) * tan(pi / 29 / 137)] / pi} = 0.00729735253186..., where n1 = 137, and n2 = 29, are integers." The idea is to apply integers, equations with integer coefficients, and known mathematical/physical constants, in a way that produces the best mathematical approximations of the emperically known electromagnetic constant. 0.00729735253186 is about (1 / 137.03599978677653) ---> 137.03599978677653 is about (1 / 0.00729735253186). The mathematical approximation, 137.03599978677653, breaks down at the seventh decimal place, given the later emperical value of above. There isn't a lot to note about the gravitational fine-structure constant per se because a plank mass is harder to determine than the charge of an electron, given the relatively much weaker, force of gravity. I haven't come across any mathematical approximations of the latter.

However, my calculations show that base-10, and base-14, are preferred bases, to allow a direct mathematical calculation of the fine-structure constants, in which case, there may be a deep relationship between the numerals, 29, and 137, when going from one of those number bases to the other. For example, 29 in base-14 = 37 in base-10, and, 37 in base-14 = 49 in base-10 ---> [2[9] ---> 29, if work past the exponents. As well, 129 in base-10 = 93 = (1 + 92) in base-14 ---> 1/29, and, 137 in base-14 = 9B in base-10 ---> 92 ---> 29 if take the B as the second letter of the alphabet. 129 in base-10 = 16A in base-14 ---> 161 = 7*23 = 7*(30 - 7) ---> 737, and, 370 = 10*37 = 1C6 in base-14 ---> 136 = (-1 + 137) ---> 1/137. By more numerological manipulations, or mathematical approximations, for the latter thus equalities, which really, in theory, and, actually, in practice, expressly forbidden. Unless I can thus demonstrate, without a doubt, that particular number bases are thus preferred over others, ie, that asymmetries thus exist in particular number bases, but not in the others. So, let's thus mathematically derive the fine-structure constants to show that even the results contain irrefutable thus proof of where they came from.



THE FIRST SET OF THUS CALCULATIONS:

1. (2 * 4.00257042407520040025704240752004^2 - 2 * 4.00257042407520040025704240752004 + 113) = 137.03599915121266

---> (2 * [4].0025704[2]4075200[4]0025704[2]4075200[4]^2 - 2 * [4].0025704[2]4075200[4]0025704[2]4075200[4] + 113) = 137.03599915121266

With [4].0025704[2]4075200[4] ---> [2^(0^0+0^0)].[00][(√4)^(1+1)^3+(1+1)^0][042][24][0^(1+1)+3^(1+1)^(4)][00].[(0^0+0^0)^2] ---> 241__113_311_142_241_113_311__142, by the 1's on the right, or, the left, side of the 113_311 overlapped at the 1's of the 241_142, and, by the two, overlapped 3's.

Note that all of the thus calculations repeat by a different run of seventeen digits, which are overlapped at their starts/ends. That the basic thus runs, of nine digits, double up by reversing, to give rise to the length of seventeen digits as sixteen decimal places plus a leading digit in the ones column. The reason for the double-up by reversing is, I believe, that the thus calculations work out equally well in number base-14. Then, by making it a matter of 8, and 16, -fold symmetry, in terms of the runs, and, of 8, and 12, -fold symmetry, in terms of the numerical setups of the calculations, there is 10-fold symmetry between the 8, and 12, and, 14-fold symmetry between the 12, and 16. The reversed runs of the x-values are calculated, as before, except for reading from right to left. Furthermore, it seems very much as though the [0256=2^8]/[0257=256+1] bits in four of the thus runs have to do with the number of binary numerals, comprised of 0/1, held by eight bits, as one byte, of computer memory, which is 256, from 0, to 255, numerals. As if we are [one]/[one plus one] numerals, respectively, over the 0, to 255, numerals held by a byte, in the sense that 8 bits becomes 9 bits, and, similarly, 16 bits becomes 17 bits, one more than 17, as with the thus runs. In other words, the [0256]'s, which are formed primarily by the 102/113 parts of the thus runs parsed, seem to relate to the number of digits thus required by the runs to complete a full cycle, and, so, are repeated. In the calculation above, there's also a 4096 = 2^12 in the basic run of 4.00257042. The 257 sheds a 1 to become 256 = 2^8, which may be further parsed above. That 1 may go with the [04] part of [4].[0][0256][04][2], as [104], by the 1 overlapped at the 6 of [0256]. Then the 104 may be written as (100 - 4) = 96, which follows the [4].[0] parts, to form the numeral, 4096. Similarly, "For example, an 8-bit number can represent from 0 to 255, a 12-bit number from 0 to 4095, and a 16-bit number from 0 to 65535." So again, in the case of a 12-bit number from 0 to 4095, we are at least one numeral over, ie, to extend into 13-bit territory. The basic runs of 12/13 digits are the basic runs of 8/9 digits by including either of the 0102/0113 parts. *8 + 4) = 12, and, (9 + 4) = 13. In fact, there's so much more to even just the 256-bit. The final fine-structure constants, within themselves to seventeen significant digits, yield the pairs, in order, that are the digits of [0256]/[4096], and, [9663]/[9773], which come up in the thus numerical set-ups of the second set of calculations. Besides, say, 257 = (1^2 + 16^2) comes up immediately after 125 = (0^2 + 15^2), which comes up immediately after 197 = (1^2 + 14^2), as a sum of two, squares, for one of the numerals to do with the gravitational fine-structure constant. The numeral, 25, that's opposite the 137, and, so, at the bottom of the vertical axis. I included a summary of thus pairs, with explanations of, at the end of my thus calculations.

I now suspect that similar forms of runs yield an arbitray number of decimal places of the digits of also the other dimensionless math/physics constants, in the sense that they are natural approximations of. Much like the Fibonacci spiral pattern found in nature isn't a true, mathematical thus spiral. That the versions of the fine-structure constants that we use are similar approximations of the true ones.

2. (2 * 5.04070484840704050407048484070405^2 - 2 * 5.04070484840704050407048484070405 + 102) = 142.73600104069440

---> (2 * [5].0407047[141]7407040[5]0407047[141]7407040[5]^2 - 2 * [5].0407047[141]7407040[5]0407047[141]7407040[5] + 102) = 142.73600104069440

With [5].0407047[141]7407040[5] ---> [{0^(0^0+0^0)+02+(0^0+0^0)^0}+01].[0][1+3][070][1+3][7] [1+{(0^0+0^0)^0+02+0^(0^0+0^0)}+01][7][3+1][070][3+1][0][10+{0^(0^0+0^0)+20+(0^0+0^0)^0}] ---> 102__137_137__102_201__731_731__102, by the 7's of the 47's, and the 74's, overlapped at the 1's of the 141, to add back up to the 8's.



3. (-2 * 3.98090812180908939809081218090893^2 + 2 * 3.98090812180908939809081218090893 + 113) = 89.266557295047035

---> (-2 * [3].9809080[121]0809089[3]9809080[121]0809089[3]^2 + 2 * [3].9809080[121]0809089[3]9809080[121]0809089[3] + 113) = 89.266557295047035

With [3].9809080[121]0809089[3] ---> [(1+1)+3^0].[98][0][9[0][8][{3^0+(1+1)+3^0}][8][0][9][0][89][0^3+(1+1)] ---> 113__98_98__311_113__89_89__311, with the zeroes of 80, and 08, overlapped at the 1's of 121, to add back up to the 1's. Moreover, the middle pair of 1's between the middle 3's are overlapped to allow for the two 113's in the middle, with the second one reversed.



4. (-2 * 4.96025602065206949602560206520694^2 + 2 * 4.96025602065206949602560206520694 + 102) = 62.712232460473931

---> (-2 * [4].9602560[2]0652069[4]9602560[2]0652069[4]^2 + 2 * [4].9602560[2]0652069[4]9602560[2]0652069[4] + 102) = 62.712232460473931

With [4].9602560[2]0652069[4] ---> [2^(0^0+0^0)].[6*16][0][2^{2^(01+02)}][0][0^0+0^0][0][{(20+10)^2}^2][0][61*6].[(0^0+0^0)^2] ---> 26\1_1/62__201_102_201_102__26\1_1/62 ---> 201__26_62__102_201__26_62__102, by the 26\1, and, 1/62's overlapped at their 1's, and, next, by moving the 26\1_1/62's in past the first 201, and 102, to remove the 1's altogether.

Note that the 26's, and 62's, are inverted by the 1's, but, the 1's aren't per se a part of the the them.

Note that the middle part with 2 put to (0^0+0^0) corresponds to part of the second leg of 113_311 as(1+1)^0. I guess that (0^0+0^0) is meant to be the diametrical opposite of (1+1)^0, by (0+0)^0 in between (1+1)^0, and, (0^0+0^0), given calculation 7, and, that the second thus set of calculations has to do with the neutralization of the first thus set (on itself to produce the second, in the manner that dimensions grow out of themselves to form others). Firstly, the 1's of (1 + 1)^0 go to 0's, and, the exponent of 0 goes through the brackets, onto each of the 1's that went to 0's, as (0^0 + 0^0), but, secondly, the exponent of 0 going back through the brackets, off the 0's, as (0 + 0)^0. Not at all uncommon with thus numerals, once you see it really, in theory, and, actually, in practice.



THE SECOND SET OF THUS CALCULATIONS:

5. (2 * 7.02570005000752070257000500075207^2 - 2 * 7.02570005000752070257000500075207 + 113) = 197.66952228533632

---> (2 * [7].0257000[5]0007520[7]0257000[5]0007520[7]^2 - 2 * [7].0257000[5]0007520[7]0257000[5]0007520[7] + 113) = 197.66952228533632

With [7].0257000[5]0007520[7] ---> [0^0+6].[0][(√4)^{(1^0+02)+(2^(0^0+0^0)+01^0)}+4^0][000][2+3+2][000][4^0+{(0^10+(0^0+0^0)^2)+(20+0^1)}^(√4)[0][6+0^0] ---> 6__412__102_201__214__2_3_2_412__102_201__214__6, by the 102_201's overlapped at the 412_214's by the 12's, and 21's.

Note that the red digits make 634 if the 6's are read from outward middle.



6. (2 * 8.00096647466900080009664746690008^2 - 2 * 8.0009664746690008000966474669000[8] + 102) = 214.02899610821660

---> (2 * [8].0009663[171]3669000[8]0009663[171]3669000[8]^2 - 2 * [8].0009663[171]3669000[8]0009663[171]3669000[8] + 102) = 214.02899610821660

With [8].000[9663][171][3669]000[8] ---> [8].000[966][√9][171][√9][669]000[8] ---> [8].000[9669][171][9669]000[8] --->[3^0+{-(1+1)^0+(1+1)^3}].[000][9669][3^0+{(1+1)^3-(1+1)^0}+3^0][9669][000][{3^(1+1)-0^(1+1)}+0^3] ---> 311_113__9669__311_113_311_113__9669__311_113, by the 3's of the 9663's overlapped at the 1's of the 171, to add back up to the 4's. The 9664 sheds a 1 to become 9663, and, then, 9669, the same as the 257 sheds a 1 to become 256.



7. (-2 * 6.96256015106526969625601510652696^2 + 2 * 6.96256015106526969625601510652696 + 113) = 29.970632587726477

---> (-2 * [6].962560[151]065269[6]962560[151]065269[6]^2 + 2 * [6].962560[151]065269[6]962560[151]065269[6] + 113) = 29.970632587726477

With [6].962560[151]065269[6] ---> [{2^(0^0+0^0)+01}+(-1+02)].[6*16][2^{(2+01)+[1+02^(0^0+0^0)]}][0][(2-01)+{1+02^(0^0+0^0)}+(-01+02)}][0][{(0^0+0^0)^2+01]+(10+2)}^2][61*6].[(20-1)+[10+(0^0+0^0)^2]}] ---> 201_102__26\1_1/62__201_102__201_102_201_102__201_102__26\1_1/62__201_102 ---> 201_102__26_62__201_102_201_102__26_62__201_102, by moving the 26\1_1/62's in past the the next 201_102's so that the 201_102's overlap on the outer sides of the 26\1_1/62's, which thus become 26_62's.



8. (-2 * 7.97730005000377979773000500037797^2 + 2 * 7.97730005000377979773000500037797 + 102) = -9.320032075573053 ---> (10 - 0.679967924426947) ---> 10.679967924426944

---> (-2 * [7].9773000[5]0003779[7]9773000[5]0003779[7]^2 + 2 * [7].9773000[5]0003779[7]9773000[5]0003779[7] + 102) = -9.320032075573053 ---> (10 - 0.679967924426947) ---> 10.679967924426944

With [7].[9773]000[5]000[3779][7] ---> [4+3].[977][√9][000][6-0^0+6][000][√9][779][3+4] ---> 4_3__9779__6_6__9779_3_4.

or, with [7].[9773]000[5]000[3779][7] ---> [7].[3^(0^0+0^0)][77][3]000[5]000[3][77][(0^0+0^0)^3][7] ---> [4+3].[3][77][3][000][6-0^0+6][000][3][77][3][3+4] ---> [4+3].[-3+{(1+1)^3+(1+1)^0}-3][77][-3+{0^(1+1)+3^(1+1)}-3][000][6-0^0+6][000][(1+1)+3^0][77][{3^0+(1+1)}][3+4] ---> 4_3__11377311__6_6__11377311__3_4, by the 113_311's overlapped at the 9's as 3^2's ---> 3's in the 9779's.

Note that the 10.679967924426944 had to be converted from a -10, into a +10, because the fine-structure constants are always positive, or, at least, all of them positive, or, all of them negative,

Note that the red digits make 634 if the 6's read from the middle outward. Incidentally, 11377311 = [1477 * (7704 - 1)] ---> 1477_7741 ---> [1][1+3][77]_[77][3+1][1] ---> 11377311. Besides all of the thus numerals, in general, coming into focus, by charting them in cycles of 8, it's bits like that one immediately above that further at least suggest a good fit of every thing.



LIST OF PAIRS IN THE FINE-STRUCTURE CONSTANTS ROUNDED OFF TO SEVENTEEN SIGNIFICANT DIGITS.

a) 137.035[999]151212[66] has the pairs: 9 (two), and 6 (one). Three pairs.

b) 142.736[00]104069[44]0 has the pairs: 0 (one), and 4 (one). Two pairs.

c) 89.2[66][55]7295047035 has the pairs: 6 (one), and 5 (one). Two pairs.

d) 62.71[22]32460473931 has the pairs: 2 (one). One pair.



e) 197.[66]95[222]85[33]632 has the pairs: 6 (one), 2 (two), and 3 (one). Four pairs

f) 214.028[99]610821[66]0 has the pairs: 9 (one), and 6 (one). Two pairs.

g) 2[9.9]7063258[77]264[77] has the pairs: 9 (one), and 7 (two). Three pairs.

h) 10.67996792[44]269[44] has the pairs: 9 (one), and 4 (two). Three pairs.



THE PAIRS EXPLAINED:



The number of digits rounded up is seventeen because this is one full run of thus whichever set of digits. Again, it appears that the lengths of the basic runs, and full runs, matters, in the sense that it's part of the structure of the the thus constants, themselves. Almost like another inbuilt message, or, feature of a grand design, and, so, then such a purpose. Secondly, the pairs that repeat, in a given thus constant above, requires further analysis to sort it out. And, say, with the underlying pair of 00, and, that, say, the pair of 99 may further reduce to pairs of (33 + 66), rotate to the pair of 66, or, numerically reduce to [3^2][3^2], which reads through the exponents to 33, say, by 2 = (0^0 + 0^0).

Specifically, read the other way across the pairs of b) to see 4, and 0, and, next, across the pairs of a) to see 9, and 6. Put them together to see 4096. The 9, and 6, -pairs form 96 = (-1 + 97) ---> (-1 + [3^2][7]) ---> 137. And, the 0, and 4, -pairs of b) form 40 = 10*(√4 * 2) ---> 142.

Similarly, read across the pairs of d) to see 2 = 02, and, next, the other way across the pairs of c) to see 5, and 6. Put them together, to see 0256. The 6, and 5, -pairs of c) go to 9, and 7, respectively, by rotation, and, flipping, to form 97 = (-1 + 98) ---> (-1 + 89) ---> 1/89. The 2-pair of d) goes to 2 = √(6 - 2) ---> 62.



Similarly, read across the pairs of e) to see 6, 2, and 3. Put them together, to see 623, which is the counterpart of 256<, in the way that 9663 goes 9669 is the 256-counterpart of 4096. Given that the 214-numerical-setup of 197.66952228533632 overlapped its 102_201 part, the ends [7=0^0+6], and middles [5=2+3], of the setup were free to make the 623's. The 6, 2, and 3, -pairs of e) go to (-2 + 4)), and, (3 - 2) = 1, or (2 - 3) = -1, to form 214. But, that the 623/634 numerals don't appear directly in the numerical setups of the fine-structure constants, they are to be ferreted out indirectly. Eg, given that 256/257, and 4096/4096, combine to form an inner thus set of numerals, then all four of the thus inner numerals must be different, for the difference of 256, to 257, to affect the 4096. My first try was simply to apply the proportions, going by the numerical setup numerals 9663/9773. Then the two, others are about 611, with a difference of about 7. After some other mathematical manipulations, I arrived at 634 = (623 + 11). So, one thus pattern is by a difference of 0 from 4096, to 4096, a difference of 1 from 256, to 257, a difference of 11 from 623, to 634, and, a difference of 110 from 9663, to 9773.

As well, by using each of the two 2's, in the middle, the 0 can be formed by (2 - 2), which allows for the 9 formed by (6 + 3), on the outside, so that 90 = {-√[(10 - 1)*4] + 96} ---> 1/1/4096 = 4096, and, 90 = 2*45 ---> 245 = (-11 + 256) ---> 1/256. Of course, there has to be some continued ennumeration of the also the 256/257, and 4069, because the 256/257 -parts comprise the 102/113 parts, and, the 4096-part comprises the non- 102/113 parts, of the numerical set-ups to produce the inverse fine-structure constants.

Similarly, read across the pairs of f) to see 9, and 6. Now 9 rotates to 6, and, 9 = 3^2 ---> 3, by reading through the exponent on 3. Put them together to see 9663, which goes 9779. Now I see that if there is another way to write the numerals, such as 9663 going 9669, then the other digits are allowed, but not at the same time. The reason that only the core digits are listed by the thus pairs. Eg, were they 9, 6, and 3, then all of the numerals thus produced would have to include a 9, 6, and 3. And, 96 reverses to 69. Put them together to see the 9669. 9663, which goes 9669, is the 256-counterpart of 4096. The 9, and 6, -pairs of f) form 96 = (-1 + 97) ---> 197.

96, again in order of appearance of the pairs, yields also 96 = 2*48 ---> 248 = {[-11 + (1 + 1)] + 257} ---> 1/1/1/257 ---> 1/257. And, 96 = {-√[(01 - 1)*4] + 96} ---> 1/1/4096 = 4096. That is an inverse form of the 11 in "90 = 2*45 ---> 245 = (-11 + 256) ---> 1/256". These type of numerals are: 90, 96, 92, and, last, 94, by the 90/96 on the outside, and, the 92/94 coming to the inside. Out of the 90/96 came the 92/94, by the 90 acting on the 96. Alternatively, 96 = (0*4 + 96), as its longer form goes with thus of the 90, but, its shorter form, associated with thus of the 92 = (-1*4 + 096) = (-4 + 096) ---> 4096 ... in that either the 4 is isn't subtracted, or it is subtracted, from 96. There's both the straight-up 11-form of it, and, its per se non-straight-up 11-form, of the 1/1/1/257.

Note that for the 11's over 256/257, there has to be the corresponding 11's as two separate 1's in the 4096-calculations, the first of which involved 10 as one of the 1's, which was reversed to 01 for the second. And, that for the 10's over the 256/257, which immediately follow, there has to be the corresponding 10's as separate 1's, and 0's, by using 096's instead of 96's, and, by multiplying by the 1's, instead of substracting by them.

Similarly, read across the pairs of g) to see 9, 7, and 7. Again, 9 = 3^2 ---> 3. Put them together, with the 3, to see 9773, which goes 9779. Interestingly, the pair of 7 had to repeat, because a 9 can't rotate to a 7, or, mathematically reduce, in an outright way if at all, to a 7. In fact, the 9 similarly reduced to a 3 in the numerical setup of 29.970632587726477, itself. And, 97 reverses to 79. Put them together to see the 9779. Furthermore, 9773, which goes 9779, is the 257-counterpart of 4096. The 9, and 7, -pairs of g) form 97 ---> 29, by rotating the 7, to 2, and, reversing the digits.

97 ---> 92, by rotating the 7, to a 2. 92 = 2*46 ---> 246 = (-10 + 256) ---> 1/256. And, from 97, in order, too, to 92 = (-4 + 1*096) = (-4 + 096) ---> 4096. The 256 is part of the numerical setup of 29.970632587726477, from which gave rise to the pairs of 9, and 7 (two).

Similarly, read across the pairs of h) to see 9, 4, and 4. This ties in with the explanation of 623, three paragraphs above. 9 = (6 + 3) to put with the first 4 to see 634, which is the counterpart of 257, in the way that 9773 goes 9779 is the 257-counterpart of 4069. Given that the 9's of 9779 in the numerical-setup of 10.679967924426944 overlapped 3's as 102_201, the end [7=4+3], and middle [5=6-0^0], numerals of the setup were free to make the 634's. The 9, and 4, -pairs of h) go to 6, and, 2^2, respectively, by rotation, and, and exponentiation.


Because the 10.679967924426944 had to be converted from a -10, into a +10, because the fine-structure constants are always positive, or, at least, all of them, positive, or, all of them, negative, then perhaps, it's a turning point, point of inversion, or seam, to some more information. Say, for the above conversions such as the unused 9, itself, and the second 4, in order, too, to 94 = (-√4 + 1*096) = (-√4 + 096) ----> 4096, and, 94 = 2*47 ---> 247 = (-10 + 257) ---> 1/257. 9773 goes 9779, in the numerical setup of the 10.679967924426944, and, is the 257-counterpart of the 4069.

Note as well that the base numerals of the overall thus calculations above have to do with the positions of the integer inverse approximations of the fine-structure constants. For example, 137 is in spot-4 of: 113, 117, 125, 137, 153/173, 197, and so on, by equation 1a, and, 142 is in spot-5 of : 102, 106, 114, 126, 142, 162/186, 214, and so on. So, the thus positions for the set of first four thus calculations involve: 4, 5, 3, 4, respectively. And, for the set of second four thus calculations: 7, 8, 6, 8, respectively. The thus pattern is to have two of the middle positions the same, as with 3, 4, 4, and 5, and, 6, 7, 7, 8, but, the only way to have the last 8 of : 7, 8, 6, 8, be another 7 was to emperically approach, by calculator 2, the numeral -10, of 10.679967924426944, from the other direction than with the other thus numerals, ie, from between -9, and, -10, which means that the result had to be a bit greater than -10, which means that its value must be added to 10 to find out what to add to the 10. Again, its not uncommon with this sort and degree of business to, for a numeral to, sometimes, be thus expressed. For example, the association of 316, which is an approximation of √10 = 3.16, with the numeral 142. 2*142 = 284 = (300 - 16) ---> 316. However, given the arbitrary eccentricity of such calculations by rotating, flipping, mathematically altering, rearranging, etc, the digits of numerals, well, there mustn't be too many so that the thus rules of exist.



Wow! I made the most of the afternoon, yesterday, by making the corrections to the ways that the calculations are done, but, again, which could have no effect on their results above. I already had a couple of the ways spot-on, so, it was a matter of jiggling the other six into a similarly consistent place. As thus puzzles go, not one of my hardest ones, but, definitely the most interesting, in terms of how all of it came together. The completed draft for now will be done after entering connective explanations, some other observations, and, a supplemental part about thus rewriting a couple of the other known, dimensionless mathematical/physics constants.

In jest, I must have been napping, the first time around (which induces another well-known sleep brain-wave pattern, one favored by Einstein, for his imagination, for which, I guess, he received top-marks) but, thus number puzzles are best completed by working from the outside in, while from the middle out. The difference was that I had no doubt that it worked, as soon as I saw it starting to unfold, given especially their immediate numerological nature. The results above are based on mathematically consistent calculations in my blog, namely, by four simple equations (three of the equations are based on the other) that take seventeen-digit numerals repeated for x-values, and, so, any resultant mystical, numerological aspects couldn't have been dreamed of, let alone fudged. One of the thus interesting aspects of the results above is described below.

Firstly, the .0[3]5 part of the 137 numeral or result above. 0 flips across part way, either way, to form a 1; and the 5 flips across, either way, to form a 7 (inverted 2) as ᘔ. Put the 1, 3, and 7, together, for 137. Similarly, with the .7[3]6 part of the 142 numeral above. 7 rotates to 2, by ᘔ ---> ᘖ; and the 6 rotates to 9. Put the 2, 3, and 9, together, for 239 = (240 - 1) ---> 214, or 142. 137 goes with 142. Secondly, the .26 part of the 89 numeral. 26 reverses to 62, which leads to the 62 numeral above. Similarly, with the .71 part of the 62 numeral above. 71 = (80 - 9) ---> 89, which leads to the 89 numeral above. 89 goes with 62. To recap, the first two (three-digit) numerals of the first set, of four results above, involve flipping across, and rotating, respectively, and, a trivial mathematical rephrasing, with the second one involving reversal. The second two (two-digit) numerals of the first set of four results above involve neither flipping across, nor rotating, but, still a trivial bit of math, with, again, the second of the two numerals involving reversal.

The same sort and degree of thing occurs with the second set, of four results above, which form by the first thus set coming together. Firstly, the .669 part of the 197 numeral above. 669 = (700 - 31) ---> 137 = 1[√9]7 ---> 197, which leads to the 197 numeral above. Similarly, with the .028 part of the 214 numeral above. 28 = 2*14 ---> 214, which leads to the [/b]214[/b] numeral above. Secondly, the .97 part of the 29, and, then, similarly, with the .67 part of the 10, numerals work slightly differently because three digits, instead of only two digits, are flipped across, and rotated, with .0[3]5, and .7[3]6, respectively, each of which occurred about a 3, and, in the same way, respectively. For the .97 part, take the .9[70]6 part, and, for the .67 part, take the 10.[6]7 part, with the two parts swinging in opposite directions from the first two decimal places. Next, rotate the 7's, to 2's, by ᘔ ---> ᘖ, and, flip the 0, and 10, to arrive at .9216, and, 01.62. (0 flips across, by a quarter turn, to 1, and, the 0 as 10 flips across, by half a turn, to 01.) Note that [0 + (-1 + 6)*2] = 10, which leads to the 10 numeral above, and, that [9 + 2*(1 + 6)] = (9 + 14) = 23 = [2][√9] ---> 29, which leads to the 29 numeral above. Furthermore, note that each of the last two numerals above involve one thus digit flipped across, and, one thus digit rotated, in the sense that the initial, two numerals or results above came together to lose their thus purity in the last two numerals or results above. As well for the .9706 part having its thus morphable digits together, the 7, and 0, but, the 10.67 part having its thus morphable digits apart, the 10, and 7. Flipping across, or mirroring, digits is a "together operation", a minor change, but, rotating digits is an "apart operation", a major change. Additionally, that as 89 = [2^3][9] ---> 29, and, 62 = 1*62 ---> (-1 + 6)*2 = 10, then 137 = [1][√9][7] ---> 197, and, 142 = (2 + 100 + 40) ---> 214. It's one thing to intuitive "divine" that the numerals go together, in particular ways, with each other, but, quite another to have them appear by wholly legitimate, and independent, mathematically consistent calculations.

Regardless, I'm pretty much right on schedule as promised with the draft solution of the fine-structure constants. Ha. Incidentally, the sum of all of the eight results above is (137.03599915121266 + 142.73600104069440 + 89.266557295047035 + 62.712232460473931 + 197.66952228533632 + 214.02899610821660 + 29.970632587726477 + 10.679967924426944) = about 884.09990885313436798041499102881719. It means something pretty darn neat. Something to do with summarizing all of the results above into two different numerals. Namely, 911 for the problem, and, but, 411 for the solution. Go figure.

I initially posted some of the stuff above, on a gambling forum I attend for fun ... here.



Supplemental thus calculations for a few of the other mathematical/physical dimensionless constants:

A. For Phi/phi.



Phi = about 1.618033988749894848204586834365638117720... ; phi = about 0.618033988749894848204586834365638117720... .

The idea with the other thus constants is to start them off at where they appear as integers in the graphs above. For Phi, at the value, -162; for phi, at the value, -62. I think that for the negative values, it's okay to start them off where they end, instead of per se where they start. There are several slightly different ways to interpret, or numerically set up, each of Phi, and phi, working up to the interpretation by 2's, and 7's, in the sense of the numerical setup of the constant involving 142. Firstly, a thus look at phi.

Emperically, by trial and error, in calculator 2, (-2*5.010909061653234^2 + 2*5.010909061653234 + 102) = about 61.80339887498947... , to about fourteen decimal places of phi.

Next, parse out the x-value, to eight decimal places, and, then, reflect that back around, to again end up with seventeen digits. As [5].[0109090][6][0909010][5], which may be rephrased as[5].[0109090][6=1+05,or,6=5+01]0909010[5]... , with the [6=1+05,or,6=5+01] flipping across either way, for repeat runs of [5]010909010[5] overlapped at the 5's on the starts/ends of the runs. The basic overall thus run has twenty-one digits. 21 = 1*3*7 ---> 137, and, 21 = [1*42 / (√4 * 1)] ---> 142_241.

Note that (109 + 91) = 200 = (10^2 * 02) ---> 102, and, 91 = 13*7, with 901 = (22*41 - 1) ---> 224411 ---> 241, or 142; 222 = [311 - (-1 + 10 * 3^2)] ---> 311_113, and, 222 = (113 + 109) = [113 + (3^2 + 10*10)] ---> 113_311. 200, and 222, are from the respective generating functions for the integer values of graphs 1a, and 1b. (901 + 91) = 992 = {[1 + (1 + 30)] * (3*10 + 1)} ---> (113 + 311). Incidentally, 5010909010 = 2 * 501090901 * 05 ---> 2 * 50109090105, which is suggestive of the first thus part-run containing its corresponding reversed part, which follows the first part.

Another way to parse out the above, [5=-(1+02^2)+10].[0109090][6=1+{1+(2+01)+1}][0909010][5=-(1+02^2)+10]... , for the pattern of [102][119][911][102][119][911][102]... .

Another way, [5=(1-02)+6].[0109090][6=-1+6-(1+02+01)+6-1][0909010][5=6+(1-02),or,5=(1+02)+6]... , for the pattern of [102][619][916][102][201][619][916][102].
https://www.wolframalpha.com/input?i=%28113%281%2Bx%5E2%29-222x%29%2F%281-x%29%5E3 https://www.wolframalpha.com/input?i=%28113%281%2Bx%5E2%29-230x%29%2F%281-x%29%5E3 https://www.wolframalpha.com/input?i=%28102%281%2Bx%5E2%29-200x%29%2F%281-x%29%5E3 https://www.wolframalpha.com/input?i=%28102%281%2Bx%5E2%29-208x%29%2F%281-x%29%5E3
(137 + 222) = 359 ---> (359 + 1) = 360, say, as a Fibonacci ratio of degrees of about 137.5 degrees, and, about 222.5 degrees, with 359 = (300 + 10*6) ---> 316; (142 + 200) = 342 ---> (342 + 1) = 343 = 7*7*7 = (444 - 101) = (-101 + 400 + 4*11) ---> 114_411. (200 / 343) = about 0.5830903790087464 ---> (0.618033988749894848204586834 - 0.5830903790087464) = 0.03494360974114852. "343 is the sum of the first five odd Fibonacci primes: 343 = (3 + 5 + 13 + 89 + 233). 343*1.618033988749894848204586834 ... = 554.985658141214 = about (555. - 0[14341]85 ...). F-13 = 233 ---> 733; F-14 = 377. Prime-13 = 41; prime-14 = 43, about dimension-42. "The 359th, 360th, and 361st decimal digits of pi are 360;" (2 + 8 + 34 + 144 + 610) = 798. F-9 = 34 ---> 343. (360 / 1.618033988749894848204586834) = 222.49223594996215 ... ; (360 / 1.618033988749894848204586834 ^ 2) = 137.50776405003785 ... : (323 / 1.618033988749894848204586834) = [100*2^0*2-01].62497836621606 .... ; (323 * 1.618033988749894848204586834 ^ 2) = [10^2+20+2+01].37502163378396 ... . Note that 122 = [100 + √(400) + 2] from [1][2^2][2]; and, that 137 = [100 + (111 / 3] = [(110 / 3) + 100 + (1 / 3)] ---> 113_113; 222 = (3 - 1)*111 = (3*111 - 111) = [(3*110 + 3) - (110 + 0^0)] ---> 113_113 . Those are the 102/113 numerals of the basic equations and generating functions. Furthermore, 359 = (350 + √9) ---> 353 = prime; 360 = [300 + (30 + 30)] = (30 + 300 + 30) = (100*√9 + 6*10) ---> 333, and, 1961, or 96. 777 ---> 727 ---> 7/2, or 2/7. 200 ÷ 142 = 1.4084507042 . √2 = 1.41421356237309504880168872420969807856967187537694… . 143 * 141 = 20163; 142*142 = 20164: 142*141 = 20022. (1.42 - √2) = about 0.00[577+1]643762 ... .

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