 5 August 2019 AD; Dedication of Basilica of St Mary Major in Rome (Our Lady of the Snows); St Emygdius

The sum of the pairs of elements will form either a triangle (sum of the internal angles equal to 180 deg) - this is a pair of Quark and Graviton.

Or another sum of the angles of the pair will form a quadrangle - probably a rectangle, with the sum of internal angles 360 deg this is the sum of Neutrino and Boson. If you add all four elements the sum of internal angles will be 540 deg forming a pentagon (the sum of Quark angles, Graviton angles, Neutrino angles, and Boson angles). Other sums will have values of the angles between 180 deg and 360 deg.

To get the spherical angles correctly, I have to introduce an idea of a unit angle, which is about 90 deg.

For quark and graviton pair the calculation is as follows:

The sum of the quark angles equals to 90.4706 deg + sum of the graviton angles equals to 90.5494 deg = 181.0200 deg.

The arithmetic mean of this sum equals to 181.0200 deg / 2 = 90.5100 deg.

This is the unit angle for quark and graviton pair.

In a similar way unit angle of neutrino and boson, the sum equals to 257.8801 deg + 102.8371 deg = 360.7172 deg

Dividing this value by 4 we get

360.7172 deg / 4 = 90.1793 deg

This is the unit angle for neutrino and boson pair.

Unit angles are necessary to calculate exactly value s of pairs, triplets, and singles as well as quadruplet of the sum of the angles.

The first pair of the angles - quark and neutrino:

Both neutrino/quark and neutrino/graviton pairs follow this rule:

CP VilationPhase Angle δ = ( θ - 90 deg ) * ( 4 / 15 )

where θ is the sum of the angles of a given pair of angles.

90.4706 deg (quark) + 257.8801 deg (neutrino) = 348.3507 deg

δ CP quark/neutrino = 348.3507 deg - [( 90.5100 deg (quark unit angle) + 90. 1793 deg (neutrino unit angle) / 2 ] 348.3507 deg - 90. 3446 deg = 258.0060 deg 8 ( 4 / 15 ) = 68.8016 deg (spherical angle)

The second pair of angles from the same family is neutrino/graviton pair:

257.8801 deg + 90.5494 deg = 348. 4295 deg

348.4295 deg - [ (90.5100 deg + 90.1793 deg ) / 2 ] = 258.0849 deg 8 ( 4 / 15 ) = 68.8226 deg (spherical CP angle)

The third pair is quark/boson pair of angles.

CP VilationPhase Angle δ = ( θ - 90 deg ) * ( 2 / 3 )

where θ is the sum of the angles of a given pair of angles.

90.4706 deg (quark) + 102.8371 deg (boson) = 193.3077 deg - 90.3446 deg = 102.9630 deg * ( 2 / 3 ) = 68.6420 deg (spherical CP Angle)

The fourth pair is boson/graviton pair:

δ CP Angle = 193.3865 deg - 90. 3446 deg = 103.0419 deg * ( 2 / 3 ) = 68.6946 deg (spherical CP Angle)

The fifth pair is quark/graviton pair:

The rule is:

δ CP Angle = ( θ + 180 deg ) * ( 4 / 21 )

where θ is the sum of the angles of a given pair of angles.

δ CP Angle = 90.4706 deg + 90.5494 deg + ( 181.0200 deg ) = 363.0399 deg * ( 4 / 21 ) = 68.9600 deg (spherical CP Angle)

The sixth pair is boson/neutrino pair.

It follows the rule:

δ CP Angle = ( θ ) * ( 4 / 21 )

where θ is the sum of the angles of a given pair of angles.

δ CP Angle = 102.8371 deg + 257.8801 deg = 360.7172 deg * ( 4 / 21 ) = 68.7080 deg (spherical CP Angle)

The arithmetic mean of all the six pairs is:

δ CP Mean Angle (pairs) = 412.6288 deg / 6 = 68.77147 deg

To the same family belongs the sum of all four elements (quark/neutrino/boson/graviton), i.e. pentagon.

It follows the rule:

δ CP Angle = ( θ - 180 deg ) * ( 4 / 21 )

δ CP Angle = 541.7372 deg (sum of all) - 2 * 90.3446 deg = 361.0479 deg * ( 4 / 21 ) = 68.7710 deg (spherical CP Angle for pentagon).

The arithmetic mean of 6 pairs and a pentagon is equal to

68.77147 deg * 6 + 68.7710 deg = 481.39987 deg / 7 = 68.77141 deg

Next article - sums of the singles and triplets.