^{1}

^{*}

^{2}

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The Harmonic Neutron Hypothesis, HNH, has demonstrated that many of the fundamental physical constants including particles and bosons are associated with specific quantum integers,
n
. These integers define partial harmonic fractional exponents, 1 ± (1/
n
), of a fundamental frequency,
V_{f}
. The goal is to evaluate the prime and composite factors associated with the neutron n
^{0}
, the quarks, the kinetic energy of neutron beta decay, the Rydberg constant,
R
,
e
,
a
_{0}
,
H
^{0}
,
h
,
α
,
W
,
Z
, the muon, and the neutron gluon. Their pure number characteristics correspond and explain the hierarchy of the particles and bosons. The elements and black body radiation represent consecutive integer series. The relative scale of the constants cluster in a partial harmonic fraction pattern around the neutron. The global numerical organization is related to the only possible prime factor partial fractions of 2/3, or 3/2, as pairs of 3 physical entities with a total of 6 in each group. Many other progressively resonant prime number factor patterns are identified with increasing numbers of smaller factors, higher primes, or larger partial fractions associated with higher order particles or bosons.

The primary method organizing and conceptualizing the fundamental physical constants is the Standard Model, SM, (

Quantum physics demonstrates many pure number properties that are expressed simultaneously as physical characteristics, but this is the minority of all physical phenomena. There are many unanswered questions that go to the very heart of a system of physical constants for which there is no logical pure number structure. These questions include: Why are there are six leptons and six quarks? Why there are 3 up type quarks and 3 down type quarks? Why are there three charged leptons, and three with zero or minimal mass neutral leptons? Why do the quarks have fractional charges of ±1/3 and ±2/3? Why are the fundamental constants clustered around the relative mass of the neutron and become increasingly sparse as one moves away? Physicists know that black body radiation is integer-based harmonic system, but how are other particles and bosons scaled by an integer harmonic system? If the physical constants possess a numerically and computationally single pure number system what is its common organization? Why are some physical constants quantum in character and others apparently continuous? What has prevented many more physical systems from pure number representation?

The goal is to evaluate the prime numbers, prime factors, and their composite integers of the partial harmonic fractions derived from standard experimental data utilizing methods of the Harmonic Neutron Hypothesis, HNH. The HNH is a unified model associated with the some of the most important physical constants including particles and bosons,

u, −10 −1/10, 9/10 −1/(2 × 5), (3 × 3)/(2 × 5) | c, +109 1/109, 110/109 1/109, (2 × 5 × 11)/109 | t, +10 +1/10, 11/10 1/(2 × 5), 11/(2 × 5) | photon, −3 −1/3, 2/3 hydrogen ionization energy, R |
---|---|---|---|

d, −11 −1/11, 10/11 −1/11, (2 × 5)/11 | s, −28 −1/28, 27/28 −1/(2 × 2 × 7), (3 × 3 × 3)/(2 × 2 × 7) | b, +32 +1/32, 33/32 1/(2 × 2 × 2 × 2 × 2), (3 × 11)/(2 × 2 × 2 × 2 × 2) | gluon, n^{0}, −48 −1/48, 47/48 1/(2 × 2 × 2 × 2 × 3), 47/(2 × 2 × 2 × 2 × 3) |

_{ } (3/4 + 1/10), (3/(2 × 2)) + (1/(2 × 5)) | Z, +12 and W^{±}, +12 +1/12, 13/12 +1/(2 × 2 × 3), 13/(2 × 2 × 3) | ||

e, −7 −1/7, 6/7 −1/7, (2 × 3)/7 | H^{0}, +11 +1/11, 12/11 1/11, (2 × 2 × 3)/11 | ||

KE, −8 −1/8, 7/8 beta decay kinetic energy −1/(2 × 2 × 2), 7/(2 × 2 × 2) | n^{0}, 1 | p^{0}, 1 |

^{−1} is 11, as well as, the down quark, and H^{0}.

Constant unit | n_{ie} or n_{ife} | n_{ie}, or 1 ± 1/n_{ife} |
---|---|---|

2 × elemental graviton, binding kinetic energy of the electron in hydrogen, boson 2 × 2.90024 (22) × 10^{−24} Hz = 5.80048 (44) × 10^{−24} Hz | −1 n_{ie} | −1 |

Electromagnetic energy, h, boson 6.62606957 (29) × 10^{−34} Js, 1 Hz | 0 n_{ie} | 0 |

Neutron, elemental mass, strong force, fermion 939.565378 (21) × 10^{6} MeV/c^{2}, 2.2718590 (01) × 10^{23} Hz | 1 n_{ie} | 1 |

Beta decay kinetic energy, fermion, 2.19698(13) × 10^{−3} eV/c^{2}, 5.3122910 (03) × 10^{11} Hz | −2 n_{ife} | 1/2, 1 − (1/2) |

Rydberg constant, R, EM energy, boson 13.60569 (07) × eV/c^{2}, 3.28984196 (17) × 10^{15} Hz | −3 n_{ife} | 2/3, 1 − (1/3) |

Beta decay, kinetic energy, related to leptons 1.579218(08) × 10^{3} eV/c^{2}, 3.818533(19) × 10^{17} Hz | −4 n_{ife} | 3/4, 1 − (1/4), 3/(2 × 2) |

Bohr radius, a_{0}, distance, beta decay kinetic energy 5.2917721092 (17) × 10^{−11} m, 5.66525639 (28) × 10^{18} Hz | −5 n_{ife} | 4/5, 1 − (1/5), (2 × 2)/5 |

Beta decay kinetic energy, related to leptons 1.4146439 (71) × 10^{5} eV/c^{2}, 3.42059 (02) × 10^{19} Hz | −6 n_{ife} | 5/6, 1 − (1/6), 5/(2 × 3) |

Electron, e, mass, matter, fermion 5.10998925 (20) × 10^{5} eV/c^{2}, 1.23558996 (05) × 10^{20} Hz | −7 n_{ife} | 6/7, 1 − (1/7), (2 × 3)/7 |

Beta decay kinetic energy, related to leptons 1.338904 (07) × 10^{6} eV/c^{2}, 3.237456 (16) × 10^{20} Hz | −8 n_{ife} | 7/8, 1 − (1/8), 7/(2 × 2 × 2 × 2) |

Up quark, u, matter, fermion 3.0 × 10^{6} eV/c^{2}, 7.253 × 10^{20} Hz to 1.8 × 10^{6} eV/c^{2}, 4.352 × 10^{20} Hz | −10 n_{ife} | 9/10, 1 − (1/10), (3 × 3)/(2 × 5) |

Top quark, u, matter, fermion 174.1 × 10^{9} eV/c^{2}, 4.2097 × 10^{25} Hz to 172.7 × 10^{9} eV/c^{2}, 4.1759 × 10^{25} Hz | +10 n_{ife} | 11/10, 1 + (1/10), 11/(2 × 5) |

Down quark, d, matter, fermion 5.5 × 10^{6} eV/c^{2}, 1.328 × 10^{21} Hz to 4.5 × 10^{6} eV/c^{2}, 1.088 × 10^{21} Hz | −11 n_{ife} | 10/11, 1 − (1/11), (2 × 5)/11 |

Reciprocal fine structure constant, coupling constant, ^{ } ^{2} | −11 n_{ife} | 1/11, 1 − (10/11), |

Higgs boson, H^{0}, boson 125.91 × 10^{9} eV/c^{2}, 3.0445 × 10^{25} Hz to 124.61× 10^{9} eV/c^{2}, 3.0131 × 10^{25} Hz | +11 n_{ife} | 12/11, 1 + (1/11), (2 × 2 × 3)/(2 × 5) |

W, weak force boson 80.385 (15) × 10^{9} eV/c^{2}, 1.9437 (08) × 10^{25} Hz | +12 n_{ife} | 13/12, 1 + (1/12), 13/(2 × 2 × 3) |

Z, weak force boson 91.1876 (21)× 10^{9} eV/c^{2}, 2.204 (05) × 10^{25} Hz | +12 n_{ife} | 13/12, 1 + (1/12), 13/(2 × 2 × 3) |
---|---|---|

Muon, lepton, matter, fermion 105.6583715 (35) × 10^{6} eV/c^{2}, 2.554808 (08) × 10^{22} Hz | −24 n_{ife} | 23/24, 1 − (1/24), 23/(2 × 2 × 2 × 3) |

Strange quark, matter, fermion 100 × 10^{6} eV/c^{2}, 2.418 × 10^{22} Hz to 90 × 10^{6} eV/c^{2}, 2.176 × 10^{22} Hz | −28 n_{ife} | 27/28, 1 − (1/28), (3 × 3 × 3)/(2 × 2 × 7) |

Bottom quark, matter, fermion 4.21 × 10^{9} eV/c^{2}, 1.018 × 10^{24} Hz to 4.15 × 10^{9} eV/c^{2}, 1.003 × 10^{24} Hz | +32 n_{ife} | 33/32, 1 + (1/32), (3 × 11)/(2 × 2 × 2 × 2 × 2) |

Gluon, neutron, strong force, boson 3.092687 (15) × 10^{8} eV/c^{2}, 7.478086 (33) × 10^{22} Hz | −48 n_{ife} | 47/48, 1 − (1/48), 47/(2 × 2 × 2 × 2 × 3) |

Charm quark, matter, fermion ^{2}, ^{2}, | +109 n_{ife} | 110/109, 1 + (1/109), (2 × 5 × 11)/109 |

_{ie}, n_{ife}, exponents known, δ, and the partial harmonic fractions.

factor patterns, using pure number properties elucidated by the HNH, are interrogated to see if they correlate and explain the known hierarchy and organization of the physical constants. The HNH is an innovative novel method to evaluate the relationships between the physical constants. Defining the origin of the SM organization would be a significant theoretical and computational advancement.

The following is a limited review and explanation of the HNH. The details have been described in multiple previous publications, and will not be repeated [^{0}, electron, e, Bohr radius,

The model utilizes the dimensional analysis methods similar to that of Rayleigh and Buckingham’s Pi Theorem, where the exponential base is the dimensionless neutron annihilation frequency, ^{23} Hz-s,

of the form of the physical units. Therefore, acceptable laws of physics are homogeneous in all dimensions. All of the physical phenomena are evaluated as frequency equivalents, and secondarily as dimensionless coupling constant ratios. The system is physical unit-independent. The fundamental frequency,

Our model has derived Planck’s time, t_{P}, the Higgs boson,

The unit system of the HNH is simplified with the units for Planck’s constant, unit electric charge, time, the distance light travels in one unit of time, and the speed of light all equaling 1. All entities are defined as in terms of ratios and exponents of

1 divided by frequency. The speed of light equals

neutron. In this type of single-variable physical system the units are all dimensionless coupling constants and completely defined by exponents or integer values of

The Equality Pair Transformations (EPTs), which we describe, inter-relate matter, electromagnetic energy, and kinetic energy transformations. These EPT are common physical phenomena, but necessitate a definitional approach when utilized in this model. Each EPT is associated with a point transformation from one state to another, or from one force to another, such as, kinetic energy to electromagnetic energy, electromagnetic energy to matter, or vice-versa. This occurs whenever there is a relative scale equality of two different states or forces. The pair is identically scaled phenomena, but can represent different (paired) physical manifestations of different forces or states. This is the essence of the particle-wave duality paradox. Examples are matter-antimatter pair production or annihilation; or the transformation of electromagnetic energy to kinetic energy as in the photoelectric effect. Not only is there a conservation-equality of total energy-matter, but also a transformation of state or force.

The primary fundamental EPT scaling in the HNH model is neutron anti-neutron pair production, and is the scaling factor used to derive further observable phenomena. The fundamental EPT ratio set is composed of a natural physical unit within a consecutive integer series,

There are multiple possible integer “n’s” described in this model and they are differentiated. Integer exponents of _{ie}”. Partial or harmonic integer fractional “n” exponent values are described as “n_{ife}”, whereas the “n” of the consecutive integer series are described as “

All of the data for the fundamental constants were obtained from the websites: http://physics.nist.gov/cuu/Constants/ and www.wikipedia.org. The NIST site http://physics.nist.gov/cuu/Constants/energy.html has an online physical unit converter that can be used for these types of calculations so the values used in the model are all standard unit conversions.

The floating point (the number of accurate digits) is based upon known experimental atomic data, of approximately 5 × 10^{−8}. All of the known fundamental constants are converted to frequency equivalents, v_{k}. Equation (1) demonstrates the frequency equivalent conversion of the neutron as an example. _{ie}, n_{ife}, partial fraction, δ, and known exponents, exp_{k} of the constants evaluated in this paper. Conversation of the other constants have been previously published. Masses are converted by multiplying by c^{2} (speed of light squared) then dividing by h (Planck’s constant). Distances are converted by dividing the wavelength into c. Energies in Joules are converted to Hz by dividing by h. The eV value for the neutron is 939.565378 (21) × 10^{6}. Its frequency in Hz is converted to eV by multiplying by the constant, 4.13566750 (21) × 10^{?15} eV/Hz. The eV was converted to frequency by multiplying by the constant 2.41798930 (13) × 10^{14} Hz/eV.

Though not intuitively obvious in this type of a defined single physical unit system, _{A}. Any change in the arbitrary time or arbitrary distance units are canceled as well.

This model has two parallel domains both describing identical physical values. One domain is the frequency equivalent of any physical value. This is the linear domain of possible physical states. The other domain is the

exponent of the base

cific value, Equation (2). The known exponent, exp_{k}, of a fundamental constant is the ratio of the log_{e} of the fre-

quency equivalent, v_{k}s, divided by the

(22). Subscript k denotes a known experimental value and subscript d represents a derived value.

The known exp_{k} minus the quantum fraction, qf, or partial harmonic fraction equals the known

The known frequency equivalent of a constant, v_{k}, is calculated by raising _{k}.

Quark | Prime factors | Partial harmonic fraction | Prime factor composites |
---|---|---|---|

u | 2, 3, 5 | 9/10 | (3 × 3)/(2 × 5) |

d | 2, 5, 11 | 10/11 | (2 × 5)/(11) |

s | 2, 3, 7 | 27/28 | (3 × 3 × 3)/(2 × 2 × 7) |

c | 2, 5, 11 | 110/109 | (2 × 5 × 11)/(109) |

b | 2, 3, 11 | 33/32 | (3 × 11)/(2 × 2 × 2 × 2) |

t | 2, 5, 11 | 11/10 | (11)/(2 × 5) |

Quark versus prime factors | 2 | 3 | 5 | 7 | 11 | 109 |
---|---|---|---|---|---|---|

u, even | ^{* } | ^{* } | ^{* } | |||

d, odd/prime | ^{* } | ^{* } | ^{* } | |||

s, even | ^{* } | ^{* } | ^{* } | |||

c, odd/prime | ^{* } | ^{* } | ^{* } | ^{* } | ||

b, even | ^{* } | ^{* } | ^{* } | |||

t, even | ^{* } | ^{* } | ^{* } |

^{*}Associated with the quarks’ composite harmonic fractions. They demonstrate specific progressive patterns.

Lepton or boson | 2 | 2 | 2 | 2 | 3 | Partial fraction | Exponent of factor 2 | Even or odd |
---|---|---|---|---|---|---|---|---|

R, boson | 3 | 2/3 | 0 | |||||

KE beta decay, lepton | 2 | 3 | 5/6 | 1 | odd | |||

Z, W, boson | 2 | 2 | 3 | 13/12 | 2 | even | ||

Muon, lepton | 2 | 2 | 2 | 3 | 23/24 | 3 | odd | |

Gluon of the neutron, boson | 2 | 2 | 2 | 2 | 3 | 47/48 | 4 | even |

evaluated. Their standard unit values, frequency equivalents, v_{k}, known exponents, exp_{k}, known deltas, δ_{k}, n_{ie}, n_{ife}, partial fractions, prime factors, composite factors, and inter-relationships are listed.

Each of the pure number properties of integers that define the HNH are manifest as the organization of physical systems. There are a number of consecutive integer series quantum systems. Black body radiation is one. The other is the chemical periodic chart. There are many others including Moseley’s law and the Rydberg series.

The observable physical constants are spaced in a classic partial fraction pattern around the neutron. The relative scale of the physical constants in a single physical unit are clustered near the neutron by unstable higher order quark baryons. The mesons and more massive leptons are next. The recent discovery of the Higgs boson falls precisely near a partial fraction, 12/11. The products of neutron beta decay including hydrogen are farther out. Finally cosmic background microwave radiation, 1/2, H_{0}, −3/4, and t_{P} squared, −128/35. The fundamental constants are only found near partial fraction values.

The origin of the general numerical organization of the constants is defined by the pure number imperative of partial fractions and prime numbers. The only partial harmonic fractions composed of the consecutive prime numbers 2 and 3 are 2/3 and 3/2. This is the unique intersection of a system based on a consecutive integer series, partial fractions, and primes. These prime factors and their composite, 6, are logically assumed to represent the most important relationship defining the global hierarchy of the physical constants. There must be paired groups of 3 physical entities. The other option is 2^{3} or 8. This assumption is supported since there are many examples of 2, 3, 6, 8, 12 physical entities defining an associated group. This pattern does define the global organization of the Standard Model, _{ife} of the quarks are even, and one third are primes. There are 3 matter leptons and 3 anti-matter leptons, totaling 6. There are 3 charged leptons and 3 neutral leptons. Beta decay is associated with three matter components, and three kinetic energy components. Baryons are composed of 3 quarks. Mesons are composed of 2 quarks. Bosons are frequently paired, as in pair production. There are no single isolated quarks. The charges of the quarks are ±1/3, and ±2/3. The six (6) quarks have subatomic properties of up, down, strangeness, charmness, bottomness, or topness. There are three physical states: matter, distance and time. The Lorentz factors are associated with three different ratios of the three states, matter/distance, distance/time, and time/matter. The product of these three fractions is dimensionless. There are 8 gluons. When using the computational approach developed in the HNH model, the consecutive primes 2 and 3 and the perfect integer 6, are fundamental, and not coincidental to physical reality.

The quarks demonstrate a very specific prime factor progression

There is a progression of a consecutive integer series of the powers of 2 in a composite product with 3 representing an alternating pattern (e.g. 2^{n} × 3) of the most fundamental bosons and leptons,

A robust model that accurately scales from classic, to quantum, to cosmic physical constants does not exist [_{f}.

There are three primary number patterns inherent in the definition of the HNH that explain the most important patterns of the physical constants. First, is that all physical constants and possibly physical variables, which includes examples such as time, distance, or astronomically observable constants, are associated with a consecutive integer series. This accounts for many physical systems. Second, that many other physical constants are associated with partial harmonic fractions. This accounts for the clustering of the physical constants near the neutron scale, and increasingly sparse farther away, but in a very specific spacing pattern, _{ife}, represent observable physical constants, but only those that demonstrate harmonic resonant characteristics of the lower order prime factors. This pure number property is related to classic resonant properties of prime factor composites.

Pure number properties manifest physically in many quantum phenomena. These pure number properties are well established within a limited number of physics systems. The following are a few examples. Black body radiation, E equals n_{cis}h, and is associated with integer-valued frequencies. All elements are based on an integral number of nucleons. Both of these can appear continuous experimentally, but conceptually and mathematically they must be integer-based. Even numbers of protons and neutrons, within nucleons, are not associated with well-known nuclear magnetic resonance properties whereas those associated with odd numbers of nucleons are. The property of strangeness is integer-related as well. Charge is either integrally quantized, or fractional in units of 1/3 or 2/3. In this HNH model every possible degenerate mass, frequency, energy, temperature, distance, time, velocity, density, momentum is quantized, and defined by an integer in the collection of V_{f}.

There has been a long search for a clear example of the unequivocal dominance of primes manifest in a physical system [^{0}, and the down quark, respectively [

The entire hierarchy of the organization and scaling of the physical constants is dominated by the smallest prime factors. The primes 2, 3, 5, 7, 11, 13, and 47 are associated with the first generation of particles and bosons,

There are also classic relationships of harmonic resonances defined by the product of common prime factors. These types of resonance patterns are ubiquitous in many physical systems, but they are not limited to prime fac- tors. This is not the case as demonstrated by the HNH for the hierarchy of the constants, which is inherently dependent upon the primes. The HNH demonstrates that there is a logical pure number explanation of the hierarchy of the fundamental particles and bosons, Tables 1-5.

The primary pure number property elucidated on previous published work using the HNH is that there is only one possible partial harmonic faction defined by the only possible consecutive primes, 2 and 3, namely 2/3. The factors, 2 and 3 as individual primes, products, and fractions define the global numerical hierarchy of the whole physical constant system, Tables 1-5. The higher order possibilities are defined by a series of logical progressions of products of lower prime factors.

There is no known physical entity associated with the partial fraction

If one carefully looks at the other three primes associated with the quarks there are no other possible combinations that can represent the known physical partial fractions starting with three primes. For example, the primes 2, 3, 7 are associated with the strange quark, 27/28. The other possible composites are 12 = (2 × 2 × 3), 14 = (2 × 7), 18 = (2 × 3 × 3), 21 = (3 × 7), 24 = (2 × 2 × 2 × 3), 36 = (2 × 2 × 3 × 3), 42 = (2 × 3 × 7), 49 = (7 × 7), 54 = (2 × 3 × 3 × 3), 63 = (3 × 3 × 7), 81 = (3 × 3 × 3 × 3). The associated n values of a partial fraction would have to be 11, 13, 15, 17, 19, 20, 22, 23, 25, 35, 37, 41, 43, 48, 50, 53, 55, 62, 64, 80, 82. It would be possible to generate the denominators for the partial fractions of 47/48 and 63/64 from 2 and/or 3, but there are many more prime factors. These should be associated with higher order entities. 27/28 is the lowest possible number of prime factors fulfilling the partial fraction imperative. The numerically restricted values defined by the prime factors and simultaneously defining a partial fraction represent actual physical values.

Many of the known exponents are not associated with their closest possible partial fraction. This manifestation is a demonstration that the actual partial fraction values must fulfill a prime product imperative [_{k}. Good examples of this pattern are seen with the up, down, and strange quarks. Their _{s} are associated with multiple integer factors of the natural unit slope bem. The actual partial fractions and δ values all fulfill power law relationships with the natural unit values so they can be logically computed.

The properties of a consecutive integer series and a partial fraction series are both the primary organization and scaling of the fundamental physical constants. Pure prime number factor properties starting with 2 and 3, as the only possible prime partial harmonic fraction, define the dominant numerical hierarchy of groups of constants.

I would like to thank Richard White MD, Yu Ding PHD and Tom Budinger PHD for their support, and help.