The eight certain digits of the fine structure constant^{1}are

0.00729735. . . .

In the infinite pentagonal arrangement of 'golden bowls' at thef-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}

Wherefis the 'golden section' and 8.2944 is the first factor of the number uniquely comparable to unityfquaintegral product,^{3}and itself a function of-geometry,

[137.082039324994 - (8.2944φ^{2})^{-}^{1}]^{-}^{1}=

0.00729735 314. . . ,

that is,

[20_{}^{4}- (8.2944φ^{2})^{-}^{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,

(_{})^{-1},

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

(E^{2}- p^{2}c^{2}-m^{2}c^{4})^{-1},

such that

(E^{2}- p^{2}c^{2}-m^{2}c^{4})^{ }_{ }(p,E) = 1

parallels

[20φ^{4}- (8.2944φ^{2})^{-}^{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,f_{ }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-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 ordinarymathematics and nature, φ + 1 = φ^{2}.

The exchangeability of logical 0 and with the governing ratios of theunum-founded geometric series, wherex= 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.

**Notes
**

^{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.
http://physics.nist.gov/constants (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 **F*_{1} 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**.

--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
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 = 1^{I} = 1(111 or
111) = 1, so the "square" or "square root" of 0 = 0^{I} = 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, 0^{I} = 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. . .** ).