{% include list.liquid all=true %}
Every second, trillions upon trillions of the tiny particles shoot down to Earth from space almost completely unaffected by any matter they come across.
Feynman diagram of neutrinoless ***double beta decay***, with two neutrons decaying to two protons.
- The only emitted products in this process are two electrons, which can occur if the neutrino and antineutrino are the same particle (i.e. Majorana neutrinos) so the same neutrino can be emitted and absorbed within the nucleus.
- In conventional double beta decay, two antineutrinos — one arising from each W vertex — are emitted from the nucleus, in addition to the two electrons.
The detection of neutrinoless double beta decay is thus a sensitive test of whether neutrinos are Majorana particles. _([Wikipedia](https://en.wikipedia.org/wiki/Double_beta_decay#Neutrinoless_double_beta_decay))_
We analyze a simple extension of the Standard Model (SM) with a dark sector composed of a scalar and a fermion, both singlets under the SM gauge group but charged under a dark sector symmetry group.
- Sterile neutrinos, which are singlets under both groups, mediate the interactions between the dark sector
and the SM particles, and generate masses for the active neutrinos via the seesaw
mechanism.
- We explore the parameter space region where the observed Dark Matter relic abundance is determined by the annihilation into sterile neutrinos, both for fermion and scalar Dark Matter particles. The scalar Dark Matter case provides an interesting alternative to the usual Higgs portal scenario.
We also study the constraints from direct Dark Matter searches and the prospects for indirect detection
via sterile neutrino decays to leptons, which may be able to rule out Dark Matter masses below and around 100 GeV. _([Sterile Neutrino portal to Dark Matter II - pdf](https://github.com/eq19/maps/files/13822870/1607.02373.pdf))_
The current status of the nucleon decay experiments is as follows: the partial lifetime
limit on p → π0e+ is τ (p → π0e+) > 1.67 × 1034 years, and the bound on the partial lifetime for p → K+ν is τ (p → K+ν) > 6.6 × 1033 years [42, 43]. It is expected that a future experiment, the Hyper-Kamiokande, may achieve a sensitivity of 5-10 times the present bound. _([Proton Decay - pdf](https://github.com/eq19/maps/files/13884072/1603.03568.pdf))_
p r i m e s
1 0 0 0 0 0
2 1 0 0 0 1 ◄--- #29 ◄--- #61 👈 1st spin
3 2 0 1 0 2 👉 2
4 3 1 1 0 3 👉 89 - 29 = 61 - 1 = 60
5 5 2 1 0 5 👉 11 + 29 = 37 + 3 = 40
6 👉 11s Composite Partition ◄--- 102 👈 4th spin
6 7 3 1 0 7 ◄--- #23 👈 f(#30) ◄--- break MEC30 symmetry
7 11 4 1 0 11 ◄--- #19 👈 30
8 13 5 1 0 13 ◄--- #17 ◄--∆32-- #49 👈 30 ✔️
9 17 0 1 1 17 ◄--- 7th prime 👈 f(#36) ◄--- antisymmetric state ✔️
18 👉 7s Composite Partition ◄--- 168 👈 7th spin
10 19 1 1 1 ∆1 ◄--- 0th ∆prime ◄--- Fibonacci Index #18
-----
11 23 2 1 1 ∆2 ◄--- 1st ∆prime ◄--- Fibonacci Index #19 ◄--- #43
..
..
40 163 5 1 0 ∆31 ◄- 11th ∆prime ◄-- Fibonacci Index #29 👉 11
-----
41 167 0 1 1 ∆0
42 173 0 -1 1 ∆1
43 179 0 1 1 ∆2 ◄--- ∆∆1
44 181 1 1 1 ∆3 ◄--- ∆∆2 ◄--- 1st ∆∆prime ◄--- Fibonacci Index #30
..
..
100 521 0 -1 2 ∆59 ◄--- ∆∆17 ◄--- 7th ∆∆prime ◄--- Fibonacci Index #36 👉 7s
-----
lightning speed ÷ shockwave speed = 300000km/s ÷ 3km/s = 100000 ÷ 1
Sub | i | β | f
=======+====+=======+======= === === === === === ===
1:1:0 | 1 | 1 | 2 {71} 1 1 | | | |
-------+----+-------+------- --- --- | | | |
1:2:1 | 2 | 2 | 3 {71} | | | | |
-------+----+-------+---- | | | | |
*1:2:2 | 3 | 3 | 7 = 1 + 2x3 | | | | |
-------+----+-------+---- | | | | |
*1:3:3 | 4 | 4 | 10 = 9 + 1 | | | | |
-------+----+-------+---- | | | | |
1:3:4 | 5 | 5 | 11 = 9 + 2 | | | | |
-------+----+-------+---- 9 1‘ | Δ100 |
*1:3:5 | 6 | 6 | 12 = 9 + 3 | | | | |
-------+----+-------+---- | | | | |
*1:4:6 | 7 | 7 | 13 = 9 + 4 | | | | |
-------+----+-------+---- | | | | |
1:4:7 | 8 | 8 | 14 = 9 + 5 | | | | |
-------+----+-------+---- | | | | |
*1:4:8 |{9} | 9 | 15 = 9 + 6 | | | | |
-------+----+-------+---- | | | | |
*1:4:9 |{10}| 10 | 19 = 9 + 10 | | | | |
=======+====+=======+==== === --- 1" --- |
2:1:0 | 11 | 20 | 20 = 19 + log 10¹ | | | |
-------+----+-------+---- | | | |
2:2:1 | 12 | 30 | 26 = 20 + 2x3 | | | |
-------+----+-------+---- | | | |
*2:2:2 | 13 | 40 | 27 = 26 + 1 | | | |
-------+----+-------+---- | | | |
*2:3:3 | 14 | 50 | 28 = 26 + 2 | | | |
-------+----+-------+---- | | | |
2:3:4 | 15 | 60 | 29 = 26 + 3 9‘ | Δ200 |
-------+----+-------+---- | | | |
*2:3:5 | 16 | 70 | 30 = 26 + 4 | | | |
-------+----+-------+---- | | | |
*2:4:6 | 17 | 80 | 31 = 26 + 5 | | | |
-------+----+-------+---- | | | |
2:4:7 |{18}| 90 | 32 = 26 + 6 | | | |
-------+----+-------+---- | | | |
*2:4:8 |{19}| 100 | 36 = 26 + 10 | | | |
=======+====+=======+==== === --- --- ∆1000
*2:4:9 | 20 | 200 | 38 = 36 + log 10² | | |
-------+----+-------+---- | | |
3:1:0 | 21 | 300 | 40 = 36 + 2 x log 10² | | |
-------+----+-------+---- | | |
3:2:1 | 22 | 400 | 41 = 40 + 1 | | |
-------+----+-------+---- | | |
*3:2:2 | 23 | 500 | 42 = 40 + 2 | | |
-------+----+-------+---- | | |
*3:3:3 | 24 | 600 | 43 = 40 + 3 9" Δ300 |
-------+----+-------+---- | | |
3:3:4 | 25 | 700 | 44 = 40 + 4 | | |
-------+----+-------+---- | | |
*3:3:5 | 26 | 800 | 45 = 40 + 5 | | |
-------+----+-------+---- | | |
*3:4:6 | 27 | 900 | 46 = 40 + 6 | | |
-------+----+-------+---- | | |
3:4:7 |{28}| 1000 | 50 = 40 + 10 | | |
=======+====+=======+==== === ==== |
*3:4:8 |{29}| 2000 | 68 = 50 + 3 x (2x3) {10³} | |
-------+----+-------+---- Δ | |
3:4:9 |{30}| 3000 |{71}= 68 + log 10³ | |
-------+----+-------+---- | |
3:2:1 | 31 | 4000 | 72 = 71 + 1 | |
-------+----+-------+---- | |
*3:2:2 | 32 | 5000 | 73 = 71 + 2 | |
-------+----+-------+---- | |
*3:3:3 | 33 | 6000 | 74 = 71 + 3 Δ400 |
-------+----+-------+---- | |
3:3:4 | 34 | 7000 | 75 = 71 + 4 | |
-------+----+-------+---- | |
*3:3:5 | 35 | 8000 | 76 = 71 + 5 | |
-------+----+-------+---- | |
*3:4:6 | 36 | 9000 |{77}= 71 + 6 | |
-------+----+-------+---- | |
3:4:7 |{37}| 10000 | 81 = 71 + 10 = 100 - 19 | |
=======+====+=======+==== ==== ----
32-5 = 27 = 9x3
The four faces of our pyramid additively cascade ***32 four-times triangular numbers*** (Note that 4 x 32 = 128 = the perimeter of the square base which has an area of ***32^2 = 1024 = 2^10***). These include Fibo1-3 equivalent 112 (rooted in T7 = 28; 28 x 4 = 112), which creates a pyramidion or capstone in our model, and 2112 (rooted in T32 = 528; 528 x 4 = 2112), which is the index number of ***the 1000th prime*** within our domain, and equals the total number of 'elements' used to construct the pyramid. _([PrimesDemystified](https://www.primesdemystified.com/First1000Primes.html))_
109 = 29th prime = ((10th)th prime)
|-------------------------------- 2x96 ---------------------|
|--------------- 7¤ ---------------|---------- 5¤ ----------|
✔️👉|- π(32(109²-89²))=109² -|-- {36} -|-------- {103} ---------|
+----+----+----+----+----+----+----+----+----+----+----+----+
| 5 | 7 | 11 |{13}| 17 | 19 | 17 |{12}| 11 | 19 | 18 |{43}|
+----+----+----+----+----+----+----+----+----+----+----+----+
|--------- {53} ---------|---- {48} ----|---- {48} ----|---👉109²-89² ✔️
|---------- 5¤ ----------|------------ {96} -----------|-1¤-|
|-------- Bosons --------|---------- Fermions ---------|-- Graviton
13 variations 48 variations 11 variations
The origin of multiple generations of fermions, and the particular count of 3, is an ***[unsolved problem of physics](https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_physics).***
- [String theory](https://en.wikipedia.org/wiki/String_theory) provides a cause for multiple generations, but the particular number depends on the details of the [compactification](https://en.wikipedia.org/wiki/Compactification_(physics)) of the [D-brane](https://en.wikipedia.org/wiki/D-brane) intersections.
- Additionally, [E8](https://en.wikipedia.org/wiki/E8_(mathematics)) [grand unified theories](https://en.wikipedia.org/wiki/Grand_Unified_Theory) in 10 dimensions [compactified](https://en.wikipedia.org/wiki/Compactification_(physics)) on certain [orbifolds](https://en.wikipedia.org/wiki/Orbifold) down to 4‑D naturally contain 3 generations of matter.[[11]](https://en.wikipedia.org/wiki/Generation_(particle_physics)#cite_note-11) This includes many [heterotic string theory](https://en.wikipedia.org/wiki/Heterotic_string_theory) models.
In standard quantum field theory, under certain assumptions, a single fermion field can give rise to multiple fermion poles with mass ratios of around eπ≈23 and e2π≈535 potentially explaining the large ratios of fermion masses between successive generations and their origin. _([Wikipedia](https://en.wikipedia.org/wiki/Generation_(particle_physics)))_
$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 ✔️ |
| +-----+-----+ 👆 | Double
| | | 9 | ∆9+∆(89-71)=∆27= { ∆9 ✔️ |‹--109² { Beta
2 +------| 5* +-----+----- 👇 | Decay
| | | 10 | ∆9 ✔️ |
| |-----+-----+ |
| 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}
Tabulate Prime by Power of 10
loop(10) = π(10)-π(1) = 4-0 = 4
loop(100) = π(100)-π(10)-1th = 25-4-2 = 19
loop(1000) = π(1000) - π(100) - 10th = 168-25-29 = 114
-----------------------+----+----+----+----+----+----+----+----+----+-----
True Prime Pairs Δ | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | Sum
=======================+====+====+====+====+====+====+====+====+====+=====
19 → π(10) | 2 | 3 | 5 | 7 | - | - | - | - | - | 4th 4 x Root
-----------------------+----+----+----+----+----+----+----+----+----+-----
17 → π(20) | 11 | 13 | 17 | 19 | - | - | - | - | - | 8th 4 x Twin
-----------------------+----+----+----+----+----+----+----+----+----+-----
13 → π(30) → 12 (Δ1) | 23 | 29 | - | - | - | - | - | - | - |10th
=======================+====+====+====+====+====+====+====+====+====+===== 1st Twin
11 → π(42) | 31 | 37 | 41 | - | - | - | - | - | - |13th
-----------------------+----+----+----+----+----+----+----+----+----+----- 2nd Twin
7 → π(60) → 19 (Δ12) | 43 | 47 | 53 | 59 | - | - | - | - | - |17th
-----------------------+----+----+----+----+----+----+----+----+----+----- 3rd Twin
5 → π(72) → 18 (Δ13) | 61 | 67 | 71 | - | - | - | - | - | - |20th
=======================+====+====+====+====+====+====+====+====+====+===== 4th Twin
3,2 → 18+13+12 → 43 | 73 | 79 | 83 | 89 | 97 | 101| 103| 107| 109|29th
=======================+====+====+====+====+====+====+====+====+====+=====
Δ Δ
12+13+(18+18)+13+12 ← 36th-Δ1=151-1=150=100+2x(13+12) ← 30th = 113 = 114-1
Sequence:
By the next layer the 89² will become 89 and 5 become 5² or 25.
This 89 and 25 are in the same layer with total of 114 or prime 619
So sequence from the first prime is 1,4,7,10,29,68,89,114,139,168,329,618.Using Euler's method to find p(40): A ruler with plus and minus signs (grey box) is slid downwards, the relevant terms added or subtracted. The positions of the signs are given by differences of alternating natural (blue) and odd (orange) numbers. In the SVG file, hover over the image to move the ruler _([Wikipedia](https://en.wikipedia.org/wiki/Partition_function_(number_theory)#Generating_function))_.
π(π(π(1000th prime))) + 1 = 40
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 
Plottng 40th prime scheme of the three (3) layers with all the features of 3rd prime identity as explained above then they would form their recycling .
89^2 - 1 = 7920 = 22 x 360 = 66 x 120 = (168 - 102) x 120
Fermion | spinors | charged | neutrinos | quark | components | parameter
Field | (s) | (c) | (n) | (q=s.c.n) | Σ(c+n+q | (complex)
===========+=========+=========+===========+===========+============+===========
bispinor-1 | 2 | 3 | 3 | 18 | 24 | 19
-----------+---------+---------+-----------+-----------+------------+-- 17
bispinor-2 | 2 | 3 | 3 | 18 | 24 | i12 ✔️
===========+=========+=========+===========+===========+============+===========
bispinor-3 | 2 | 3 | 3 | 18 | 24 | 11
-----------+---------+---------+-----------+-----------+------------+-- 19
bispinor-4 | 2 | 3 | 3 | 18 | 24 | i18 ✔️
===========+=========+=========+===========+===========+============+===========
SubTotal | 8 | 12 | 12 | 72 | 96 | 66+i30
===========+=========+=========+===========+===========+============+===========
majorana-1 | 2x2 | - | 18 | - | 18 | 18 ✔️
-----------+---------+---------+-----------+-----------+------------+-----------
majorana-2 | 2x2 | - | 12 | - | 12 | 12 ✔️
-----------+---------+---------+-----------+-----------+------------+-----------
majorana-3 | 2x2 | - | 13 | - | 13 | i13
===========+=========+=========+===========+===========+============+===========
SubTotal | 12 | - | 43 | - | 43 | 30+i13
===========+=========+=========+===========+===========+============+===========
Total | 20 | 12 | 55 | 72 | 139 | 96+i43 ✔️
Scheme 13:9
===========
(1){1}-7: 7'
(1){8}-13: 6‘
(1)14-{19}: 6‘
------------- 6+6 -------
(2)20-24: 5' |
(2)25-{29}: 5' |
------------ 5+5 -------
(3)30-36: 7:{70,30,10²}|
------------ |
(4)37-48: 12• --- |
(5)49-59: 11° | |
--}30° 30• |
(6)60-78: 19° | |
(7)79-96: 18• --- |
-------------- |
(8)97-109: 13 |
(9)110-139:{30}=5x6 <--x-
--
{43}
The True Prime Pairs:
(5,7), (11,13), (17,19)
|-------------------------------- 2x96 -------------------------------|
|--------------- 7¤ ---------------|------------ 7¤ ------------------|
|-------------- {89} --------------|{12}|-- {30} -|-- {36} -|-- {25} -|
+----+----+----+----+----+----+----+----+----+----+----+----+----+----+
| 5 | 7 | 11 |{13}| 17 | 19 | 17 |{12}| 11 | 19 | 18 | 18 | 12 | 13 |
+----+----+----+----+----+----+----+----+----+----+----+----+----+----+
|--------- {53} ---------|---- {48} ----|---- {48} ----|---- {43} ----|
|---------- 5¤ ----------|------------ {96} -----------|----- 3¤ -----|
|-------- Bosons --------|---------- Fermions ---------|-- Gravitons--|
13 variations 48 variations 11 variations 
The Prime Recycling ζ(s):
(2,3), (29,89), (36,68), (72,42), (100,50), (2,3), (29,89), ...**infinity**
----------------------+-----+-----+-----+ ---
7 --------- 1,2:1| 1 | 30 | 40 | 71 (2,3) ‹-------------@---- |
| +-----+-----+-----+-----+ | |
| 8 ‹------ 3:2| 1 | 30 | 40 | 90 | 161 (7) ‹--- | 5¨ encapsulation
| | +-----+-----+-----+-----+ | | |
| | 6 ‹-- 4,6:3| 1 | 30 | 200 | 231 (10,11,12) ‹--|--- | |
| | | +-----+-----+-----+-----+ | | | ---
--|--|-----» 7:4| 1 | 30 | 40 | 200 | 271 (13) --› | {5®} | |
| | +-----+-----+-----+-----+ | | |
--|---› 8,9:5| 1 | 30 | 200 | 231 (14,15) ---------› | 7¨ abstraction
289 | +-----+-----+-----+-----+-----+ | |
| ----› 10:6| 20 | 5 | 10 | 70 | 90 | 195 (19) --› Φ | {6®} |
--------------------+-----+-----+-----+-----+-----+ | ---
67 --------› 11:7| 5 | 9 | 14 (20) --------› ¤ | |
| +-----+-----+-----+ | |
| 78 ‹----- 12:8| 9 | 60 | 40 | 109 (26) «------------ | 11¨ polymorphism
| | +-----+-----+-----+ | | |
| | 86‹--- 13:9| 9 | 60 | 69 (27) «-- 3xΔ9 (2xMEC30) | {2®} | |
| | | +-----+-----+-----+ | | ---
| | ---› 14:10| 9 | 60 | 40 | 109 (28) ------------- | |
| | +-----+-----+-----+ | |
| ---› 15,18:11| 1 | 30 | 40 | 71 (29,30,31,32) ---------- 13¨ inheritance
329 | +-----+-----+-----+ |
| ‹--------- 19:12| 10 | 60 | {70} (36) ‹--------------------- Φ |
-------------------+-----+-----+ ---
786 ‹------- 20:13| 90 | 90 (38) ‹-------------- ¤ |
| +-----+-----+ |
| 618 ‹- 21,22:14| 8 | 40 | 48 (40,41) ‹---------------------- 17¨ class
| | +-----+-----+-----+-----+-----+ | |
| | 594 ‹- 23:15| 8 | 40 | 70 | 60 | 100 | 278 (42) «-- |{6'®} |
| | | +-----+-----+-----+-----+-----+ | | ---
--|--|-»24,27:16| 8 | 40 | 48 (43,44,45,46) ------------|---- |
| | +-----+-----+ | |
--|---› 28:17| 100 | {100} (50) ------------------------» 19¨ object
168 | +-----+ |
| 102 -› 29:18| 50 | 50(68) ---------> ∆27-∆9=Δ18 |
----------------------+-----+ ---