# The Angular Momentum (spin 9)

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**).

This section is referring to *wiki page-11* of *gist section-7* that is *inherited * from *the gist section-67* by *prime spin-20* and *span-* with *the partitions* as below.

This is also consistent with the fact that the quadratic residues for modulo 30 (making them congruent with perfect squares) are ** 1 and 19**.

## Perfect Squares

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

## Reversal behaviour

*329 + 109 + 109 + 71 = 329 + 289 = 618 = 1000/1.618 = 1000/φ*

*2 + 60 + 40 = 102*

```
1st layer:
It has a total of 1000 numbers
Total primes = π(1000) = 168 primes
2nd layer:
It will start by π(168)+1 as the 40th prime
It has 100x100 numbers or π(π(10000)) = 201 primes
Total cum primes = 168 + (201-40) = 168+161 = 329 primes
3rd layer:
Behave reversal to 2nd layer which has a total of 329 primes
The primes will start by π(π(π(1000th prime)))+1 as the 40th prime
This 1000 primes will become 1000 numbers by 1st layer of the next level
Total of all primes = 329 + (329-40) = 329+289 = 618 = 619-1 = 619 primes - Δ1
```

The Ulam spiral or prime spiral is a graphical depiction of the set of prime numbers, devised by mathematician Stanisław Ulam in 1963 and popularized in Martin Gardner's Mathematical Games column in Scientific American a short time later.

Both Ulam and Gardner noted that the existence of such prominent lines ** is not unexpected**, as lines in the spiral correspond to quadratic polynomials, and certain such polynomials, such as Euler’s prime-generating polynomial x²-x+41, are believed to produce a high density of prime numbers. Nevertheless, the Ulam spiral is connected with

**major unsolved problems**in number theory such as Landau’s problems

*(Wikipedia)*.

## Fibonacci Retracement

The weak mixing angle or Weinberg angle[2] is a parameter in the Weinberg–Salam theory of the electroweak interaction, part of the Standard Model of particle physics, and is usually denoted as θW. ** It is the angle by which spontaneous symmetry breaking rotates the original W0 and B0 vector boson plane, producing as a result the Z0 boson, and the photon.[3]**. Its measured value is slightly below 30°, but also varies, very slightly increasing, depending on how high the relative momentum of the particles involved in the interaction is that the angle is used for

*(Wikipedia)*

More interesting is that, like the Prime Hexagon it self, they are newly discovered. See how these layers will *behave* there:

This progression 41,43,47,53,61,71,83,97,113,131 whose general term is ** 41+x+xx**, is as much remarkable since the

**are all prime numbers**

*40 first terms**(Euler’s letter to Bernoulli)*.

So here we are going to discuss about this number particularly with the said recombination which resulting the above **Δ1 with 619**.

There are many other prime curiousity has been stated for this number 619 but almost none about 619-1 which is 618.

*(786/1000)² = 618/1000*

There are set of sequence known as *Fibonacci retracement*. For unknown reasons, these Fibonacci ratios seem to play a role in the *stock market*, just as they do in nature.

The mathematically significant Fibonacci sequence defines a set of ratios known as *Fibonacci retracements* which can be used to determine probable ** entry and exit points** for the equities when paired with additional momentum. The Fibonacci retracement levels are 0.236, 0.382,

**.**

*0.618, and 0.786*- The key Fibonacci ratio of 61.8% is found by dividing one number in the series by the number that follows it. For example, 21 divided by 34 equals 0.6176, and 55 divided by 89 equals about 0.61798.
- The 38.2% ratio is discovered by dividing a number in the series by the number located two spots to the right. For instance, 55 divided by 144 equals approximately 0.38194.
- The 23.6% ratio is found by dividing one number in the series by the number that is three places to the right. For example, 8 divided by 34 equals about 0.23529.
- The 78.6% level is given by the
*square root*of 61.8%, while not officially a Fibonacci ratio, 0.5 is also commonly referenced (50% is derived not from the Fibonacci sequence but rather from the idea that on average stocks retrace half their earlier movements).*(Golden Ratio - Articles)*

*(√0.618 - 0.618) x 1000 = (0.786 - 0.618) x 1000 = 0.168 x 1000 = 168 = π(1000)*

They are used to determine critical points where an *asset's momentum is likely to reverse*. This study cascade culminating in the Fibonacci digital root sequence (also period-24).

## Truncated Perturbation

I wondered if that property might hold for the incremental powers of phi as well. For this reason I chose to see numbers in the hexagon as quantum, and truncate off the decimal values to determine which integer cell they land in.

```
That is what I found. Phi and its members have a pisano period if the resulting fractional numbers are truncated.
```

```
Direction:
- The initial of 168 & 329 brings the 102 as 100+2 to π(π(10000))-1=200 or 100 x 2.
- Then the 289 lets this 100x2 to 100² so it brings 100 to 10000 by the power of 2.
- At the last it will be separated by the scheme of 168 to 102 goes back 100 and 2.
Conclusion:
- All of the other primes than 2 is 1 less than the number n times the number of 2.
- Those Mersenne primes is generated as 1 less than the power n of the number of 2.
- Thus they will conseqently be carried out by the same scheme of this number of 2.
```

Speaking of the Fibonacci number sequence, there is symmetry mirroring the above in the relationship between the terminating digits of Fibonacci numbers and their index numbers equating to members of the array populating the Prime Spiral Sieve.

*103 - 43 = 60*

```
|-------------------------------- 2x96 -------------------------------|
❓ |--------------- 7¤ ---------------|------------ 7¤ ------------------|
〰️43👉------------- {89} --------------|-------------- {103} -------------|
+----+----+----+----+----+----+----+----+----+----+----+----+----+----+
| 5 | 7 | 11 |{13}| 17 | 19 | 17 |{12}| 11 | 19 | 18 | 18 | 12 | 13 |
+----+----+----+----+----+----+----+----+----+----+----+----+----+----+
|--------- {53} ---------|---- {48} ----|---- {48} ----|---- {43} ----👉89〰️
|---------- 5¤ ----------|------------ {96} -----------|----- 3¤ -----| ❓
|-------- Bosons --------|---------- Fermions ---------|-- Gravitons--|
13 variations 48 variations 11 variations
```

To date, I have found only one number sequence that visibly produces non-random results: pi and its powers, shown as truncated for display purposes. I believe these data suggest prime numbers are linked in some way to pi. *(Hexspin)*