# The Pairwise Disjoint

This section is referring to *wiki page-6* of *gist section-2* that is *inherited * from *the gist section-29* by *prime spin-11* and *span-* with *the partitions* as below.

## Mobius Strip

There are some mathematical shape of this residual objects. Torus is basically a donut shape, which has the property of of having variable Gaussian curvature.

The blue parts of the torus above have positive curvature, the red parts negative and the top grey band has zero curvature. If our 3 dimensional space was like the surface areas of a 4 dimensional torus, the parts would have different angle sums.

Some parts of the surface has positive curvature, others zero, others negative.

If you start anywhere on its surface and follow the curvature round you will eventually return to the same place having travelled on every part of the surface.

Mobius strip only has one side, there are two more bizarre shapes with strange properties.

## The Klein bottle

The Klein bottleis in someways a 3D version of the Mobius strip and even though it exists in 3 dimensions, to make a true one you need to "fold through" the 4th dimension.

In mathematics, the Klein bottle (/ˈklaɪn/) is an example of a non-orientable surface; that is, informally, a one-sided surface which, if traveled upon, could be followed back to the point of origin while flipping the traveler upside down.

- More formally, the Klein bottle is a two-dimensional manifold on which one cannot define a normal vector at each point that varies continuously over the whole manifold.
- Other related non-orientable surfaces include the Möbius strip and the real projective plane.

While a Möbius strip is a surface with a boundary, a Klein bottle has no boundary. For comparison, a sphere is an orientable surface with no boundary.

A sign inversion visualized as a vector pointing along the Möbius band when the circle is continuously rotated through a full turn of 360°.

## The Spinors

A spinor associated to the conformal group of the circle, exhibiting a sign inversion on a full rotation of the circle through an angle of 2π.

*(17+13) + (11+19) = (7+11) + (19+23) = 60*

Eigennvalue curves (right) showing a triple eigenvalue at zero for τ = 1 and double eigenvalues at 1 ± √2i for τ = √43. On the left the graph of 1/|Q(λ)| with the same eigenvalue curves plotted in the ground plane. Green stars indicate the eigenvalues of A, blue stars the roots of puv(λ) and triangles the zeroes of Q0(λ)

## Global Properties

*7 + 11 + 13 = 31**1 + (26+6) + (27+6) = 66*

```
0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17
---+---+---+---+---+---+---+---+---+---+----+----+----+----+----+----+----+----
- | - | 20| 21| 22| 23| 24| 25|
---+---+---+---+---+---+---+---+
- | - | - | - | 28| 29| ◄--- missing 26 & 27 ✔️
---+---+---+---+---+---+
30| 31| - | - | ◄--- missing 32 & 33 ✔️
---+---+---+---+
36|
```

This behaviour finaly brings us to a suggestion that the dimension in string theory are linked with ** the prime distribution level** as indicated by the

*self repetition*on MEC30.

*7th spin - 4th spin = (168 - 102)s = 66s = 6 x 11s = 30s + 36s*

```
$True Prime Pairs:
(5,7), (11,13), (17,19)
layer | node | sub | i | f. MEC 30 / 2
------+------+-----+-----+------ ‹------------------------------ 0 {-1/2}
| | | 1 | --------------------------
| | 1 +-----+ |
| 1 | | 2 | (5) |
| |-----+-----+ |
| | | 3 | |
1 +------+ 2 +-----+---- |
| | | 4 | |
| +-----+-----+ |
| 2 | | 5 | (7) |
| | 3 +-----+ |
| | | 6 | 11s ‹-- ∆28 = (71-43) √
------+------+-----+-----+------ } (36) |
| | | 7 | |
| | 4 +-----+ |
| 3 | | 8 | (11) |
| +-----+-----+ |
| | | 9 |‹-- ∆9 = (89-71) / 2 √ |
2 +------| 5* +-----+----- |
| | | 10 | |
| |-----+-----+ |
| 4 | | 11 | (13) ---------------------
| | 6 +-----+ ‹------------------------------ 15 {0}
| | | 12 |---------------------------
------+------+-----+-----+------------ |
| | | 13 | |
| | 7 +-----+ |
| 5 | | 14 | (17) |
| |-----+-----+ |
| | | 15 | 7 x 24 = 168 √
3* +------+ 8 +-----+----- } (36) |
| | | 16 | |
| |-----+-----+ |
| 6 | | 17 | (19) |
| | 9 +-----+ |
| | | 18 | --------------------------
------|------|-----+-----+----- ‹----------------------------------- 30 {+1/2}
```

This model may explains the newly discovered prime number theorem in relatively simple layman's terms for anyone with a slight background in theoretical physics.

The property gives an in depth analysis of the not so random distribution of primes by showing how it has solved Goldbach’s conjecture and the Ulam spiral.

The model suggests a possible origin for both charge and half-integer spin and also reconciles the apparently contradictory criteria discussed above.

** Arbitrary sequence of three (3) consecutive nucleotides** along a helical path whose metric distances satisfy the relationship dn,n+3dn,n+2dn,n+1.

- Sketch showing a characteristic duplex DNA helical standing-wave pattern.
- The vertical lines depict the cross-section projections of each bp along the helix axis, their length providing a measure of their twist magnitude.
- Thick lines represent the sugar-phosphate profile.

Optimally overlapping bps are indicated by the presence of the ovals (m) measures the overlapping resonance correlation length. *(π − π orbital resonance in twisting duplex DNA)*

Under certain conditions, energy could not take on any indiscriminate value, the energy must be some multiple of a very small quantity (later to be known as a ** quantum**).

Twisted strip model for one wavelength of a photon with circular polarisation in at space. A similar photon in a closed path in curved space with periodic boundary conditions of length C.

- The B-field is in the plane of the strip and the E-field is perpendicular to it (a).
- The E-field vector is radial and directed inwards, and the B-field is vertical (b).

The magnetic moment ~, angular momentum L~, and direction of propagation with velocity c are also indicated. *(Is the electron a photon with toroidal topology? - pdf)*

A deeper understanding requires a uni cation of the aspects discussed above in terms of an underlying principle.