The Difference


The eight certain digits of the fine structure constant1 are

0.00729735. . . .

In the infinite pentagonal arrangement of 'golden bowls' at the f-level of the structure of existence itself, the square of the diameter of the rim of the 'defining bowl' -- at once the square of tenfold the constant ratio of the base diameter of a bowl to the rim diameter of the next smaller in the infinite pentagonal series of bowls -- at once the square of tenfold the ratio of origin in the logarithmic spiral -- is

137.082039324994. . . .2

Where f is the 'golden section' and 8.2944 is the first factor of the number uniquely comparable to unity qua integral product,3 and itself a function of f-geometry,

[137.082039324994 - (8.2944φ2)-1]-1 =

0.00729735 314. . . ,

that is,

[204 - (8.2944φ2)-1]-1 = 0.00729735 314. . . .4

This equation is fundamentally in the form of the formula for the internal element in the calculation of the probability amplitude in quantum electrodynamics. Compare the form of the "Feynman propagator,"

[(E - E')2 - (')2]-1,

the similar propagator in Feynman diagrams,


and, beginning with the Fourier transformation of the relativistic version of the Schrödinger equation for a free particle, the propagator in momentum space,

(E2 - p2c2 - m2c4)-1,

such that

(E2 - p2c2 - m2c4) (p,E) = 1


[20φ4 - (8.2944φ2)-1] 0.00729735 3146047. . . = 1.

This coincidence is to be distinguished from all which are purely numerical and without physical foundation. This coincidence is without physical derivation. But, unlike purely numerical coincidences, it is derived ultimately from the prenumeral trinary logic which is the real ground of both ordinary mathematics and physical action.5

Initial insight into how the trinary logic prefigures the physics can be gotten by letting the moving electron be logical 1 and the moving photon logical 0. Then the emission or absorption of a photon by an electron is conveyed by the trinary logic expression, 1 0 = .6 The "stimulated emission" of a photon by an electron, discovered by Einstein, is also expressed by this logical equation. And, bearing in mind that there is no 'nothing' of any kind conceivable in the terms of this logic, the Pauli "exclusion principle" accords with the trinary logic expression, 1 1 = 1.

In the trinary logic, 0 = 1.7 Thus, reducing to the appropriate mathematical operator, (1 - 0)0 = 1, or, (1 - 0)-1 = 0, the propagator in the Feynman diagrams is the photon.8 The material and formal coincidence of f-geometry, the Feynman diagrams, and the fine structure constant arises directly from the logical ground. That the trinary logic is the ground of extreme and mean proportionality is evidenced by the prenumeral logical equation, 0 = 1 = .9 This equation is the founding template for that unique equation of ordinary mathematics and nature, φ + 1 = φ2.

The exchangeability of logical 0 and with the governing ratios of the unum-founded geometric series, where x = φ, is explicit for the integrated form of the fundamental QED process when the following numerical substitutions are made: let 0 = φ-2, = φ-1, and 1 = 1.10 Then (1 - 0)-1 becomes (1 - φ-2)-1, and (1 - φ-2)-1 = φ-1 + 1. With these substitutions, numerical (1 - 0)-1, logical 0 (= 1), = numerical + 1, while numerical (1 - 0)( + 1), logical 0 (= 1), = numerical 1.



1 For 1996 scientific attempts to more precisely calculate the fine structure constant, see Kinoshita.    For the most recent recommended values of the fundamental physical constants, cf. (National Institute of Standards and Technology, Gaithersburg, MD 20899).  See also, on this web, 82944 & the Four Fundamental Forces & the God Particle.
2 This number is also the square of tenfold the 'retrospective square' value of *F1 in the unum-founded Fibonacci sequence. Cf. D.G. Leahy, Foundation: Matter the Body Itself (Albany, 1996), Section III.2, and p. 444, n. 2.
3 Cf. Leahy, Foundation, p. 357, n. 13, et passim. For 'integral product', see, on this web, Transdecimal Calculation of Number Identity: A Note on Integral Product and Related Terms.
4 (20 - 8.2944)2 137.082039324994. . . .
5 The fact that the form of this derivation is proximately geometrical is nicely consistent with what R. P. Feynman says about the quantum world in QED: The Strange Theory of Light and Matter (Princeton, 1988), p. 78: "In the wild and wonderful world of quantum physics, probabilities are calculated as the square of the length of an arrow: where we would have expected to add the probabilities, we 'multiply' arrows. The peculiar answers that we get from calculating probabilities in this manner match perfectly the results of experiment. I'm rather delighted that we must resort to such peculiar rules and strange reasoning in order to understand Nature, and I enjoy telling people about it. There are no 'wheels and gears' beneath this analysis of Nature; if you want to understand Her, this is what you have to take." Yes, there are no "wheels and gears," but there is a logic. For the trinary logic as the foundation of ordinary mathematics and geometry, cf. Leahy, Foundation, Section III, passim; cf. also, on this web, Real Trinary Logic Geometric Series Infinite Iterative Sequential Summing. For the trinary logic as foundation of physical action, see what follows on this page. For the triangulation of the logic, φ-geometry, and physics, cf. also the links above on this page, and, on this web, Measure of Superconductive YBa2Cu3O6+x.
6 Cf. Feynman, QED, p. 85: "I present to you the three basic actions, from which all the phenomena of light and electrons arise.

--Action #1: A photon goes from place to place.
--Action #2: An electron goes from place to place.
--Action #3: An electron emits or absorbs a photon.

Each of these actions has an amplitude--an arrow--that can be calculated according to certain rules."
7 Cf. Leahy, Foundation, Section III.1.
8 Since logical 0 and u12wh.bmp (1190 bytes) are sorts of 1, the logical form, (1 - 0)-1, indifferently expresses the numerical elements as squares or square roots. The logical equivalent of any natural numerical power or root of a logical digit is identity, I (= 000, , 111, for , 0, and 1 respectively). Just as the "square" or "square root" of 1 = 1I = 1(111 or 111) = 1, so the "square" or "square root" of 0 = 0I = 0( or 000) = 0, and the "square" or "square root" of = I = (000 or ) = . (The multiplication in the latter two cases is reduced by canceling the identity form of the exponential factor, as the rule requires when the exponent next follows the base digit in the rational order of digits. [Cf. Leahy, Foundation, Section III.1, Rule 6, pp. 263f.] Here the difference from 1 of 0 and qua sorts, combined with the indifferent reference of I to "square" and "square root," overrides the otherwise equally true reduction by pairing, 0I = 1 and I = 1.)
9 Cf. Leahy, Foundation, Section III.1.
10 Ibid., Section III.2. The geometry from which, above, the fine structure constant is derived is built precisely on the identical geometric proportion in which the extreme and mean ratio is the mean proportional (cf., on this web, The Golden Bowl Structure. . . ).


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