## Book 6 - Derivation of General Formula for Constants of the Cosmology and Quantum Mechanics - part I.

22 September 2017 AD, feast of St Maurice

Finding General Formula for Equations from the previous book, “Book 5 Integer Formula for Dimensionless Coupling Constants of Fundamental Forces”.

All Coupling Constants use the same equations, from which constants can be calculated.

**✠Part A of the equation (A/B) ^{C} **

Integer Formula for fine structure constant alpha, *α*_{E}, ruling electromagnetic force is

A_{16} = ( C_{0})^{(24 ⁄ 24)}

Integer Formula for weak force, *α*_{W}, ruling force of decays is

A_{17} = ( C_{0})^{(27 ⁄ 24)}

Integer Formula for strong nuclear force, *α*_{S}, ruling quarks, and nucleons is

A_{1} = ( C_{0})^{( − 21 ⁄ 24)}

From this partial sequence can be obtained a complete sequence.

**At n=8 there are axis of symmetry. Axis at n=8 divide (as it will be shown on the Graphs of the Function) values between Real and Complex. This can be achieved only when the denominator equals 24.**

**For n=8 the nominator equals 0.**

For explanation how to find formulae for some sequences see webpage of Brian Kell^{(1)}

So we have in the first row number of a constant, in the second row value of the exponent of “A”,

and in the third row the difference between the two adjoining values of the exponent “A”.

**n = 11 12 13 14 15 16 17**

**E(n) = (9)/(24) (12)/(24) (15)/(24) (18)/(24) (21)/(24) (24)/(24) (27)/(24)**

**Δ(E(n)) = (3)/(24) (3)/(24) (3)/(24) (3)/(24) (3)/(24) (3)/(24)**

**Constant difference between the values of the Exponents Δ(E(n)) = (3)/(24) and this is the slope ( ((Δ y)/(Δx))= a ) of the function (y = a*x + b) we are trying to find. Y-intercept equals to b. Y-intercept is at C_{0} and equals b = -(24)/(24). **

**So, the exponent can be written as a linear function: **

**y = ((3)/(24))*x + (-(24)/(24)) = (3* x − 24)/(24) = (x − 8)/(8) **

**Now, the “A” part of the equation can be written as: **

**✠A _{x} = ( C_{0})^{((}^{x}^{ − 8)/(8)) }(Eqn. A)**

**✠Part B **

**Next sequence is “B” part of the equation (A/B) ^{C} **

B_{16} = ( C_{16} / ( 8 + 2 * (24/24) ) )^{(88 ⁄ 24)}

B_{17} = ( C_{17} / ( 9 + 2 * ( 24/27 ) ) )^{(99 ⁄ 24)}

B_{1} = ( C_{1} / ( -7 + 2 * ( -24/21 ) ) )^{( − 77 ⁄ 24)}

**Taking under consideration the Exponent part first - it can be seen that this sequence is the one in “A” multiplied by factor of 11, giving:**

**n = 6 7 8 9 10 11 **

**E (n ) = -(22)/(24) -(11)/(24) (0)/(24) (11)/(24) (22)/(24) (33)/(24) **

**Δ(E(n)) = (11)/(24) (11)/(24) (11)/(24) (11)/(24) (11)/(24) **

**Slope, a = (Δ y)/(Δx) = (11)/(24) **

**Y-intercept, at C _{0}, equals b = -(88)/(24) **

**And formula for Exponent E _{B} sequence is: **

**E _{B}= ((11*x − 88)/(24)) **

**Sequence of the base of the power is:**

Constant C_{16} is ( 8 + 2 * (24/24)

Constant C_{17} is ( 9 + 2 * ( 24/27 )

Constant C_{1} is ( -7 + 2 * ( -24/21 )

**It can be seen that in the 2 * d part, the “d” equals to reciprocal of the Value of exponent from part A, which we calculated right at the beginning so this sequence will be equal to: **

**E _{part}B = 2 * ((8)/(x − 8)) = ((16)/(x − 8)) **

**Remaining factor from the sum: **

**8, 9, -7 gives sequence = (x-8)**

**So, the base of the power equals to: **

**Base of the power = (x-8) + ((16)/( x − 8)) = (x^{2} − 16x + 80)/(x − 8) **

**If we take reciprocal of this function we will have product of C _{x}and this reciprocal raised to the Exponent E_{B}. Constants in the nominator of the base of the power C_{16}, C_{17}, C_{1}, etc., are just the number of the constant, “n”, which we replaced by “x”, and each constant can be calculated from formula from “Book 1”**

**C _{x} = ( C_{0}) ( π/ e )^{x} (Eqn. I)**

**Formula for sequence of the part “B” is then:**

**✠B _{x} = {[(C_{0})(π ⁄ e)^{x}][(x − 8)/(x^{2} − 16x + 80)]}^{[(11}^{x}^{ − 88)/(24)]} (Eqn. B)**

**✠Part C of the equation (A/B) ^{C} **

**The Exponents of the equation (A/B) ^{C} are:**

Exponent of the constant C_{16} is ( C_{16} ) * (8/24)

Exponent of the constant C_{17} is ( C_{17} ) * (9/24)

Exponent of the constant C_{1} is ( C_{1} ) * (-7/24)

**The sequence : -(7)/(24),..., (8)/(24), (9)/(24),... has an equation equal to ((1)/(3)) of the first one in this article, i.e., **

**y _{c} = ((1)/(3)) * ( (x − 8)/(8) ) = (x − 8)/(24) **

**And C _{Ex} = ( C_{0}) ( π/ e )^{x} is the same as in the Eqn. I**

(It should be C_{x} instead of C_{Ex}But that would mixed up with final result.)

**Product of ( C _{Ex}) and ( y_{c} = (x − 8)/(24) ) is the Exponent C_{x}we are looking for: **

**✠C _{x} = ( ( C_{0}) ( π/ e )^{x} ) * ((x − 8)/(24)) (Eqn. C)**

**And the final result of Exponent Main, ExpM is ExpM = ( A / B ) ^{C} **

**Substituting the Equations, A, B, C, and I we get long formula for ExpM, which is way too long to write, therefore I will use parts A, B, C of the Equation**

**ExpM = ( A / B ) ^{C} (Eqn. EM)**

In part II I will derive the General Formula for any value of the constant alpha, and similar to alpha.

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