Hubble’s law refers to the observation that objects at greater than 10 megaparsecs have a Doppler shift interpretable as a relative velocity. The Doppler shift is most commonly quoted as a velocity in (km/s)/Mpc. Galaxies appear to be moving at a rate proportional to their distance from the Earth. This is typically interpreted as evidence of the expansion of the Universe. A high precision Hubble constant H₀, is an important physical constant, [1] -[6] . Hubble’s Law relates a velocity to H₀ as a proportionality constant with units of s⁻¹ times the proper distance, D. The reciprocal of H₀ is the Hubble time. The reported velocities at one Mpc vary with the model and published values include: 76.9^+3.9_−3.4^+10.0_−8.0 km・s⁻¹・Mpc⁻¹, 69.32 ± 0.80 km・s⁻¹・Mpc⁻¹, 74.3 ± 2.1 km・s⁻¹・Mpc⁻¹, 67.3 ± 1.2 km・s⁻¹・Mpc⁻¹, Table 1, [1] -[6] . The methods, probes, of measurement and derivation of the H₀ are different. This leads to divergent estimated values based on the methods and model. The methods include the Hubble telescope, Chandra and Sunyaev-Zeldovich Effect data from the Owens Valley Radio Observatory and the Berkeley-Illinois-Maryland Association interferometric arrays, Wilkinson Microwave Anisotropy Probe, and cosmic microwave background, CMB, temperature and lensing-potential power spectra The experimental Hubble rates range from 2.18(4) × 10⁻¹⁸ s⁻¹ to 2.49(12) × 10⁻¹⁸ s⁻¹. The reported Hubble time, , equals approximately 4.35 × 10¹⁷ s or 13.8 byr. The approximate Hubble length equals or 13.8 blyr. In this model the predicted H₀ is assumed to be present experimental value with an of 0.

The goal of this work is to derive a high precision H₀, and subsequently a Hubble length, and Hubble time from natural unit frequency equivalents as integer sets of the neutron, , ,; the hydrogen ionization energy; the Rydberg constant R,; the Bohr radius,; and the electron, e,; and the finite integer and harmonic integer set of N₁₂ and the harmonic fractions [7] -[14] . N₁₂ are referred to as integer fraction exponents,. Another goal is to demonstrate that experimental Planck time squared, and H₀ follow a previously predicted power law relationship used to derive. [9] The proportionality constants of, and H₀ ratios with are derived, and compared to the known experimental values.

The following is a review and explanation of the harmonic neutron hypothesis. It was initially copyrighted in 2006 and published in 2009, [7] -[14] . All of the physical constants are evaluated as frequency equivalents, and secondarily as dimensionless coupling constant ratios in an exponential integer or integer fraction system. Any single physical unit could be utilized, but Hertzian frequency (Hz) was arbitrarily chosen since the whole physical

system can logically be evaluated as a unified quantum spectrum. The primary natural unit is. All possible physical values are defined by and integer fractions. The unit system of the HNH is maximally simplified with the units for Planck’s constant, charge, and speed of light all equaling 1. Energy, mass, frequency, temperature are all equal. The distance is 1 divided by frequency. The speed of light equals times the dis-

tance, Compton radius of the neutron. The maximum Lorentz factor equals. Velocity squared is.

The primary hypothesis is that the fundamental constants are inter-related by simple, ubiquitous mathematical and geometric integer patterns. The first twelve integers, N₁₂, and their harmonic fractions, , provide the basis for a coherent theory defining the first generation of bosons and particles, used in a power law computation for defining physical phenomena, Table 2. The model utilizes the dimensional analysis methods of Rayleigh’s method and the Buckingham Pi Theorem, where the exponential base is the dimensionless neutron annihilation frequency, Hz-s, , , and the exponents are combinations of 1 plus or minus. Buckingham’s pi theorem states that physical laws are independent of the form of the physical units. Therefore, acceptable laws of physics are homogeneous in all dimensions.

The Equality Pair Transformations (EPTs) inter―relate matter, electromagnetic energy, and kinetic energy transformations. EPT are common physical phenomena that can be described by Feynman diagrams, 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 when there is a scale equality of two different states or forces. The pair is identically scaled phenomena, but can represent two different dual (paired) physical manifestations of different forces or states. This is the essence of 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. These transformations are always associated with symmetric pairs.

The primary fundamental EPT scaling HNH 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 as a consecutive integer series representing the transformation of electromagnetic energy into frequency multiples matter associated with neutron/anti-neutron pair production. The integrally spaced dimensionless elements, v_f in V_f, are based on the ratio of the respective annihilation frequencies of that physical constant to that of the neutron. At the point where the photon integer frequency series has enough energy to be scaled identically with elemental neutral matter equivalent represents the fundamental EPT. The series restarts again at 1 with each integer representing the number of nucleons in elemental matter or groups of nucleons at the EPT of pair production point. Elements in the set V_f scale all of the possible physical phenomena under consideration.

The neutron annihilation frequency, , is the fundamental coupling constant, and the basis for EPTs on the set V_f. We assume that there exists a gravitational binding energy of the electron to the proton in hydrogen, equally as important to the units of gravitational energy, as the ionization energy of hydrogen is to the electromagnetic force. The genesis of this hypothesis was based on the empirical observation that there are integer exponent relationships of between the twice the frequency equivalent of the gravitational binding energy of the electron in hydrogen, the energy of Planck’s constant, h-Hz, and the frequency equivalent of the neutron.

Twice the frequency equivalent of the gravitational binding energy of the electron in hydrogen, 2 × 2.90024(22) × 10⁻²⁴ Hz equals 5.80048(44) × 10⁻²⁴ Hz. We label here the gravitational binding energy of the electron in hydrogen as the elemental graviton. The frequency equivalent of Planck’s constant, h, is 1 Hz. Here,

Hz is 2.271859078(50) × 10²³ Hz. The reciprocal of is 4.4016815(1) × 10⁻²⁴, and is almost iden-

tical to twice the binding gravitational frequency of the electron in hydrogen.

The factor two in the gravitational binding energy of the electron to the proton arises from the fact that it is a kinetic energy. This “2” has the same origin as the “2” in the Schwarzschild radius equation, , and both transform kinetic energy into an electromagnetic equivalent. This is also the “2” associated with an equality “pair”. All three force-state unit values are separated by a ratio of Each represents unit step values for elemental gravitational kinetic, electromagnetic, and the elemental strong nuclear forces. The frequency

equivalent of the gravitational electron binding energy when multiplied by equals 1.3178, and the nor-

malized value of 1 Hz for Planck’s constant h, times, equals in Hz. The coupling constant ratios are

represented by an injective mapping of the sequence: {−1, 0, 1, 2} to exponents of Hz. These integer

exponents, are referred to as for integer exponents. Each is associated with a different force and state. Thus there is a discrete quantum relationship similar to blackbody spectra, and a power law. The distribution of energy over time is exponential for many physical systems including magnetic resonance relaxation.

Energy multiples of Planck’s constant, h, is also an integer based wavelength or frequency system. This is identical to the resonant modes of a vibrating string. Planck’s constant represents integral units of electromagnetic energy. Though h is quantum by definition, its actual physical manifestation in black body radiation appears to be continuous. When the divisions between the physical values associated with each n unit are smaller than experimental accuracy then the physical system appears to be Almost Everywhere (A.E.) continuous, but is none the less conceptually and mathematically integer-based Figure 1.

If this initial EPT observation is valid, then there logically should be a similarly scaled transformation between the unit values of the gravitational and electromagnetic forces. At integral steps of gravitational units, i.e. twice the elemental graviton, this energy could represent an electromagnetic wave with a frequency of

1 Hz. At integral steps of electromagnetic units, at 1 Hz, this energy could represent a neutron, at a fre-

quency of Hz. At integral steps of nucleon units, Hz, this mass could represent the unit mass

of a black hole, at a frequency of Hz, 5.16134367(12) × 10⁴⁶ Hz. This is an extremely dense from of

matter, and must be the mass of individual unit forming a black hole analogous of neutrons forming elemental matter. This latter mass’s Compton radius is smaller than its Schwarzschild radius, and must represent the matter of a black hole.

In this method all of the physical phenomena are evaluated as exponents of. Each physical constant is associated with an integer identical to all quantum systems. The known exponent, exp_k, of the base

for any physical constant is the natural log of the frequency equivalent of that constant divided by

. This is a classic exponent base transformation. The difference between the known exponent and

its associated partial fraction represents the δ factor.

Inspection of an integer-based exponent system of the forces/states, Hz is a linear plot in the ex-

ponential domain Figures 1-4. The only other possible point values are the harmonic fractions, ±1/n_ife and harmonic mixed partial fractions, 1 ± 1/n_ife. These integer based exponents are associated with the observable fundamental constants. For this paper the only n_ife utilized are points in N₁₂, since they are associated with the first generation of kinetic energies, particles, and bosons of neutron beta decay, Table 2, Table 3.

Assignment of the n_ife is dependent on that value fulfilling a power law relationship with the natural unit values of the first two natural integer sets. For the simplest situation the fundamental constant is related to the n_ife closest to 1 divided by 1 minus the exponent of the constant. If that value is positive then the partial harmonic fraction is 1 − 1/n_ife. If the value is negative then the partial harmonic fraction is 1 + 1/n_ife. This is not the greatest integer function. However, the n_ife value can be driven far from the closest n_ife value by the power law imperative or the fact that some of the constants are divided or multiple by 2. Utilizing these relationships it is possible to logically assign the physical phenomena of the neutron beta decay process to a specific partial harmonic fraction, Table 2 [7] -[14] .

The simplest sinusoidal system that is related to a consecutive integer series is the possible wavelengths and frequencies of a vibrating string. This is associated with the harmonic sequence, , and the origin of “harmonic” in the model’s name. Figure 1. Many of the properties of the model are analogous to or identical with the mathematics and physics of music. Pure number properties are important in music, and the HNH physical model such as positive or negative, even or odd, and prime, and composite. These properties also represent imperatives that define the n_ife of associated entities and the hierarchy of the physical constants.

Thus, the HNH model hypothesizes that physical constants represent a multi-layered simultaneously linear and exponential inter-locking, transformational integer series, with classic harmonic properties, in both the linear and exponential domains. These types of harmonic/repeating pattern systems are remarkably unified, where if any frequency and its associated integer value are known, then an infinite number of associated possible harmonic fractions and frequencies of the system are defined in Tables 2-4, Figure 1 of reference [14] .

The Harmonic Neutron Hypothesis was previously used to derive the energy/matter lost in the transformation of a neutron to hydrogen, the masses of the quarks, the Higgs boson, and the Planck time [9] [10] [14] . Purely kinetic factors of neutron beta are related to the even-numbered denominators in the harmonic fractions, 1/2, 3/4, 5/6, and 7/8. It is logically hypothesized that the additive inverse partial harmonic fractions, which includes −1/2, −3/4, −5/6, and −7/8 should be related to the cosmic kinetic fundamental constants including, the, cosmic microwave background radiation, CMB, and dark matter/dark energy.

The HNH has accurately derived Planck time, , from this same sets of integers. [9] . It is also logical to hypothesize that other cosmic fundamental constants should be related to. Our derivation shows that is identical to the Newtonian gravitational constant, G, in the frequency domain. The product of and the frequency equivalents of two masses and distance separating them equals the gravitational binding energy as a frequency equivalent of that system. The derived values associated with the -line at the partial harmonic fractions −1/2, −3/4, −5/6, and −7/8 should be related to the cosmic kinetic fundamental constants including, cosmic microwave background radiation, CMB, dark matter and energy.

None of the individual elements of the HNH are new or radically depart from standard physics or mathematical methods. The perspective taken, the nomenclature used, and methods are not standard, but are nonetheless logically and mathematically valid. To understand this method, a significant intellectual investment is essential since it is not intuitively obvious. The model is from a global perspective very similar to magnetic resonance imaging. MRI, which was assumed to be impossible at the time of its introduction based on classical physics interpretations of optical imaging criteria, yet disproven [15] [16] . Both MRI and the HNH are spectral analysis methods. Both have two different domains that simultaneously define the identical object/system. One domain is in standard linear physical 3d space, and the other is in a mathematically transformed domain related to phase and frequency, the k-space for MRI. In this model it relates to an exponential universal harmonic plane, Figures 2-4. Both represent the identical physical system, but are defined in different mathematical terms.

The standard components of the HNH will be highlighted in italics. The HNH model is not in conflict with the Standard Model or overturns its methods or tenants. The actual physical values used are equivalent to standard unit values, but they are all transformed into frequency; harmonic fraction plus; or exponential equivalents. They are evaluated as dimensionless coupling constant ratios, and proportionality constants. For example the ratio of the mass of the electron divided by the mass of the neutron equals the frequency equivalent of the electron divided by the frequency equivalent of the neutron independent of unit.

This model is independent of any specific physical unit system. Converting the standard units to this unit 1 format does not change the absolute integer value, The reason for this non-intuitive mathematical imperative is that any arbitrary mass unit changes the arbitrary Avogadro’s number, , canceling out any effect of the initial mass unit. Transforming the arbitrary distance, time, and speed of light units to 1 are canceled out by changing the unit distance ratio with the fixed Compton radius, and changing the time unit to 1. Time and distance in the speed of light are proportional so the changes parallel each other, and cancel out any choice of those initial arbitrary values.

The combined components of the hypothesis are controversial because they are not well understood, are different from standard nomenclature, and novel. The concepts and mathematics are actually not complicated, but require a significantly different conceptual approach. All of the physical relationships are viewed solely as ex-

ponents (integer fractions plus) or ratios of.In the linear domain these are ratios, coupling con-

stants/proportionality constants. One example would be the ratio of the frequency equivalent of the ionization energy of hydrogen divided by the frequency equivalent of the Bohr Radius, [11] . This is equivalent to Coulomb’s constant in the frequency domain.

All of the ratios are scaled by , and therefore proportionality constants. Therefore, in this paper there should be a derivable proportionality constant that inter-relates, , and H₀, Table 4. Presently there is no known proportionality constant relating these three physical constants.

In this model the identical consecutive integer and harmonic fraction sequences are seen, as exponents of the fundamental frequency, , of the system, Figures 1-3. The origin of this exponential pattern is not from a standing wave sinusoidal pattern, but from a consecutive integer exponential sequence defining the forces. In this type of ratio system, when is raised to an exponent in the sequence, the results represent the only possible observable frequencies of the derived or known physical constants. When this sequence of points is plotted on a 2d-plane, the only possible exponents are the integer, the harmonic fraction, and the partial fractional sequence of 1/n and 1 ± 1/n, where n equals N₁₂, Figure 3, Figure 4. The resulting 2d-plane represents a universal 2d-exponential “slide rule”, Figure 3 [17] .

The hypothesis is based on classic harmonic fractions, and partial harmonic fractions, 1 ± 1/n for. This offers a tremendously powerful, predictive attribute of the model since associating a physical constant with a specific harmonic fraction creates a series of other discrete harmonics and characterizes the whole system from an initial two-element data set. This is how H₀ can be derived with no direct physical measurement. An analogy is Moseley’s law on ion emission spectra.

Another important concept is resonance and products of harmonic numbers. Resonance is the tendency of a system to oscillate with greater amplitude at some frequencies rather than at others. If two systems have common frequency components there will be greater coupling and potential transfer of energy between them. For a musical example, two prime number frequencies can only resonant at the product of the two prime frequencies. This pure number property defines a higher order hierarchy of physically associated entities.

The HNH presents itself as a natural unit system. A natural unit system incorporates known physical units rather than arbitrary units. A natural unit model with all of the other constants driven to 1 greatly simplify the mathematics. This model is based on the annihilation frequency of the neutron, as the fundamental frequency similar to Planck units. In Planck units all of the different fundamental constants are converted into a single common standard unit such as Hz, seconds, kilograms, or meters. The neutron is a logical unified fundamental physical entity that is centered between atomic, subatomic, and cosmic entities.

In the simplest exponential harmonic series, all of the possible frequency values could be defined and related solely to a single fundamental frequency and the harmonic fraction series, Figure 2, Figure 3 [14] . The HNH and physical reality are far more complicated. This arises from a mathematical imperative. Known fixed number values of the products of 2 and pi are associated with specific exponent integer-fraction values. These arise from the product ratio relationships of R, , e, and, [11] .

There are four product ratio relationships of these entities associated with the hydrogen atom. For example the

integer fraction associated with 2 must be related to raised to (10/1155). Also 2pi must be related to raised to (39/1155). Additionally, 2 raised to (1/(10/1155)) is 5.8744 × 10³⁴, and (2pi) raised to

(1/(39/1155)) is 4.34916 × 10²³. There is no common fundamental frequency that can fulfill these conflicting mathematical imperatives. Nature’s solution is to have small δ-values added to the quantum harmonic fractions that “shim” these various values to a common fundamental frequency, in this case Hz. These δ-values represent the known or derived exponent minus the harmonic fraction. This makes the system more complicated, but fulfills the imperative of a resonant system.

Each physical constant is plotted as a harmonic fraction minus one on the X-axis and the Y-axis is the plot of δ (the exponent minus the harmonic fraction), Figure 3, Figure 4. This is described as the universal harmonic plane. It is universal since it can perform any ratio, product, or power calculation, and all of the physical constants are plotted on this common 2d space. [7] [14] [17] . It is harmonic since the X-axis is defined by sums and differences of integers and harmonic fractions only.

The physical datum of each point has the identical value as its standard exponent, and can be translated to its standard routine physical value. The difference between two points on the 2d universal plane represents a proportionally constant, a ratio in the linear domain, a power law. This harmonic plane also has all of the classic mathematical properties of lines. A line connecting any two points can define a proportionality relationship of two or more physical constants. It is possible to derive any harmonic value from the slope and y-intercept of a

δ-line and, if the harmonic fraction is known or derived. Different forces can be associated with different

or common δ-lines.

In Physics, under most circumstances if one knows a natural unit within a ratio or product relationship that value can be used as a constant, but it does not have inherent predictive value to other multiple other related physical constants. In a harmonic system if the natural unit is known and its associated quantum integer value then an infinite series of other values/constants, can potentially be derived. [14] This is a classic property or quantum spectrum.

Many of these constants can be derived since the actual δ-values can be derived from the three finite point sets described above sets, provided the harmonic fraction is logically derived. [7] . The data define two lines on

the universal harmonic plane. Their slopes and y-intercepts along with are all that is necessary to derive

many of the other physical constants in this model. One line is related to the weak kinetic entities, and one related to the electromagnetic entities. The only other possible valid force δ lines are related to sums and differences of the slopes and y-intercepts of those two lines, Figure 4. Harmonic fractions, other than the very limited set used here, opens the possibility to derive the actual δ-values since they would represent just another point on a valid line following the power law and quantum spectrum properties.

The value of the HNH method is that it can derive and predict physical constants beyond what can be experimentally measured [9] [10] [14] . The Standard Model today cannot unify or scale quantum and cosmic phenomena simultaneously. The HNH also demonstrates the inter-relationship of the constants. The derived values have high precision since the calculations are based on high precision atomic data to begin with, and not on experimental data related to the physical constant in question [14] . In the present paper no astronomical data is utilized in the derivation of H₀.

The methodology of HNH derivation generates new insights into connections between the subatomic entities of neutron beta decay, kinetic energy, and the neutrinos, leading to frequency expressions of the neutron, t_P, gravity, H₀; and the apparent expansion of the universe.

Floating point accuracy is based upon known experimental atomic data, of approximately 5 × 10⁻⁸. All of the known fundamental constants are converted to frequency equivalents, , , for the four (4) physical constants used in the HNH derivation of H₀. Equations (1)-(4), Table 3. The masses are converted by multiplying by c² (speed of light squared) then dividing by h (Planck’s constant). The distances are converted by dividing the wavelength into c. Energies in Joules are converted by dividing by h. The eV value for the neutron is 939.565378(21) × 10⁶. 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) × 10¹⁴ Hz/eV. N_A is Avogadro’s number, 6.02214129(27) × 10²³ mol⁻¹. Converting the standard units to where they are all 1’s does not change.

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 conversions.

This model has two parallel domains both describing the identical physical values. One domain is the frequency

equivalent of any physical value. The other domain is the exponent of the base,which when raised to

that exponent equals the frequency equivalent of that specific value. Equation (5). The known exponent, exp_k, of

a fundamental constant is the ratio of the log_e of the frequency equivalent, n_ks, divided by the, Equation (5). Here,. Subscript k denotes a known experimental value and

subscript d represents a derived value.

Every value in the physical domain is defined completely in terms of its ratio and exponent relationship with

. All other constants are 1. Each domain describes the identical physical value, but one does so in the

standard physics linear/frequency domain, and the other in the exponential domain which is unique to the HNH model. The value of viewing the fundamental constants in the exponential domain is that their harmonic integer inter-relationships are clearly defined. The other is that any physical relationship can be displayed and calculated across all of the forces, and at any scale in this virtual 2d space. Despite the fact the HNH model utilizes virtual space, that space accurately defines true physical phenomena.

In the Standard Model, only a subset of physical values are quantum by definition or computational in character. In the HNH model every aspect of physical systems is quantized by integral steps. There are regions where the system appears to be experimentally continuous in the Standard Model, but this is not true in the HNH model, Figure 1. In the Standard Model there are physical values that equal 0, such as a velocity, but in the HNH this is not the case. In regions that are A.E. continuous the true quantum values can be evaluated in the HNH model.

The known exp_k minus the harmonic/quantum fraction, qf, equals the known δ_k, Equation (6). The known

frequency equivalent of a constant, v_k, is calculated by raising to the exp_k in Equations (6). Equation (7) shows that many of the fundamental constants do not have n_f equal to, but.

The n_ife (integer-fractional exponents, “ife”) and the associated quantum fractions, qf, the harmonic fractions, and the partial harmonic fractions must fulfill a power law relationship with a natural unit constant. The closest n_ife to the experimental value is derived from the exp_k, Equations (8), but that value may not be the actual n_ife since it may not fulfill a power law relationship [10] . The first n_ife assignments from 1 to 12 are listed in Table 2. The higher principal quantum number values are harder to assign unless the constant is known with a high precision. There have been errors in assignment in the past, for example, the strange quark was initially assigned to the qf 29/30, but later corrected to 27/28, [10] [14] .

As is raised to exponents of a consecutive harmonic quantum fraction for

it represents many of the degenerate exponent values of the fundamental constants, Equations (9)-(11). The degenerate ratios of the constant’s frequency when divided by represent raised to the quantum fractions qf, where the harmonic quantum fraction exponents equal 1/±n where the natural number,. These represent the degenerate proportionality constants relating to the degenerate n value.

for principal quantum number to ∞ (9)

for principal quantum number (10)

for principal quantum number (11)

Equation (12) calculates the X-axis value for a specific quantum fraction, partial fraction, and n_ife.

By definition the exp_k of the n⁰ is 1, and the exp_k of h is 0. Both have a δ_k of 0. All of the electromagnetic spectrum, quantized n-Hz, have an effective (i.e. degenerate). All photons are actually not degenerate, but split with a Lorentz factor of for energy and Lorentz factor of for time dilation. The other quantized numbers used for e, , R, are related to the first four odd prime numbers. The frequency equivalent of the, , is 5.66525639(28) × 10¹⁸ Hz; where is 0.80291631(05). The qf is 4/5, on the X-axis location of −1/5, with principal quantum number 5, and its equals 2.91631043(14) × 10⁻³. The exponent of is 1 + 3.6453880(02) × 10⁻³, 3.6453880(02) × 10⁻³, and of 2.76391359(14) × 10²³ Hz. The y-intercept of line is referred to as the b entity value, in this case.

The frequency equivalent of the electron n_e, is 1.23558996(05) × 10²⁰ Hz; exp_e is 8.6023061(06). The qf is 6/7, X-axis location of −1/7, principal quantum number, 7, and its is 3.08775982(21) × 10^?3. The exponent of e is 1 + 3.60238646(18) × 10^?3, be 3.60238646(18) × 10^?3_, of 2.7575290(01) × 10²³ Hz.

The frequency equivalent of hydrogen ionization energy, is 3.28984196(17) × 10¹⁵ Hz; is 6.64365544(33). The qf is 2/3, X-axis location of −1/3, principal quantum number 3, and its is −2.30112231(11) × 10⁻³. The exponent of R is 1 −3.45168347(17) × 10⁻³, −3.45168347(17) × 10⁻³, of 1.88695938(09) × 10²³ Hz.

The frequency equivalent of is 1.370359991(69) × 10²; is 9.14882590(46) × 10⁻². The qf is 1/11, X-axis location of −10/11, principal quantum number 11, and its is 5.7916811(03) × 10⁻⁴.

The known and derived exponents of physical constants are plotted/ transformed to the 2d universal harmonic plane. The X-axis is a multi-dimensional physical descriptor, Figures 2-4. The point (0, 0) represents the neutron since that is the exponent of Hz divided by Hz. The Y-axis is the related to the difference of the known exponent minus its associated qf, Equation (6). The Y-axis is not continuous in terms of the inter-relationships of the fundamental constants either. The slopes and y intercepts of the wk and EM lines also represent three “quantum” unit values though not integers. The only possible δ-values are a function of qf-1, and are discrete.

The points for the e, and fall above the X-axis since their are positive. Their respective X-axis values are −1/7 = (6/7 − 1), −1/5 = (4/5 − 1), −10/11 = (1/11 − 1). The point for R falls below that X-axis at x equals −1/3 = (2/3 − 1) since the is negative. The differences between these points on the universal harmonic exponent plane represent the classic ratios of these same values in the frequency domain [12] .

The only possible derived exponents are discrete since the only possible qf and values are discrete. The lines described below transpose the natural unit sets to the harmonic 2d plane. The line connecting the e, , and, , points is logically related to the weak kinetic force, -line. Here, is a distance and related to a length dimension and velocity, therefore kinetic properties. The electron is a mass related to the weak force. The other 1/±n points on the wk line represent the other possible derived s that are logically related to the weak force. These possibilities graphically represent the intersection points of the line with x values of ±1/n. The slope is awk, 3.000364286(15) × 10⁻³ and the y-intercept is bwk, 3.51638329(18) × 10⁻³, Equations (13), (14).

These derived values when proposed had no known physical significance. [7] It was assumed that these other “possible valid” harmonic fraction points along a line are identical in concept and physical reality to other possible “spectral lines” in a standard quantum spectrum where each spectral line is associated with a different quantum integer. In the interval these hypothesized points have been shown to be associated with many known physical entities. [9] [10] [14]

The EM, line is defined by the points for the Planck constant, (−1, 0) and Rydberg R, Figure 4, Equation (15). This is the second line that was previously published, [7] . This is logical since h is the unit for electromagnetic energy, and R is the unit for the atomic ionization energy. It is referred to as the electromagnetic, EM -line. Its slope and y-intercept, bem, are identical and equal to −3.45168347(17) × 10⁻³, Equation (15). This line is related to the principal quantum number 3, qf, 2/3. The other potential qf values have also been shown to be related to the quarks and mesons, [10] .

The derived values are a linear function of the qf-1, and the slope and y intercepts of the -lines. An alternate definitions of slope and y intercept are the y values at x equals 0, and −1, Equations (16). These are interchangeably equivalent, and both are used in this method, Figure 4. It is possible to predict the exact exponents of unmeasured fundamental constants from these lines just like it is possible to derive lines of quantum spectrum without a physical measure if its natural unit and principal quantum number are known, Equation (17), (18).

Many of the fundamental constants are compound product/ratios of other fundamental constants. Therefore is a compound of four other entities. All of the possible valid force lines of these composite forces can be derived since they must represent sums and differences of the slopes and y intercepts of the wk and EM lines, bwk, awk, and bem, as in the -line, Equations (19), (20). [14] These are the only possible valid values in a unified harmonic system, Figure 3, Figure 4. All of the compound values are derived.

for to 12 and products (19)

The compound values are plotted at their qf_d − 1 X-axis values and their Y-axis values identical to the other single physical value constants, Figure 4.

Equation (21) is the proportionality constant, known or derived or, multiplied with that is associated with another fundamental constant. Equation (22) is the basic form of this relationship used in the proportionality constant derivations.

The known experimental h-bar is 5.39106(32) × 10⁻⁴⁴ s, relative error 6 × 10⁻⁵. The known experimental non h-bar is 1.35134(81) × 10⁻⁴³ s, Table 1. The s² non h-bar is 1.82611(11) × 10⁻⁸⁶ s². The known mean experimental non h-bar exp_k is −3.670879(12). The known experimental non h-bar is −1.37371(1) × 10⁻². The qf of is the sum of −1, −1, −6/7, −4/5 or −128/35 or −3.657142857142. The point is plotted at x = −128/35 −1, which equals −163/35.

The harmonic neutron hypothesis has derived a high accuracy Planck time. [9] The qf for these four entities defining are −1, the gravitation binding energy of the electron −1, negative qf proton, −6/7, negative qf electron; −4/5; negative qf Bohr radius. The sum equals −128/35. This is plotted at the x-value of −163/35, −128/35-1.

The derived compound line has a slope of awk ? bwk ? bem, and a y intercept of ?bwk ? bem at x equals 0, and y intercept of ?awk at x equals −1, Equations (23)-(25), Figure 4. [9] The derived is −3.670879366(1), the is −1.3736509(1) × 10⁻² and the derive is 1.82617126(91) × 10⁻⁸⁶ s² relative errors of 5 × 10⁻⁸. The derived no h-bar equals 1.35135904(68) × 10⁴³. The equivalent h-bar value is 5.39114257(27) × 10⁻⁴⁴ s. All these values are within the known experimental values.

The known experimental velocities at Mpc⁻¹ are 76.9^+3.9_−3.4^+10.0_−8.0 km・s⁻¹・Mpc⁻¹, 74.3 ± 2.1 km・s⁻¹・Mpc⁻¹, 69.32 ± 0.80 km・s⁻¹・Mpc⁻¹, 67.3 ± 1.2 km・s⁻¹・Mpc⁻¹, Table 1. These are converted to standard s⁻¹ values by dividing the velocity by one Mpc, 3.085677580 × 10²² meters. These value are 2.49(12) × 10⁻¹⁸ s⁻¹, 2.408(67) × 10⁻¹⁸ s⁻¹, 2.24(2) × 10⁻¹⁸ s⁻¹, and 2.18(4) × 10⁻¹⁸ s⁻¹. Their respective exp_k are −7.53(12) × 10⁻¹, −7.543(8) × 10⁻¹, −7.556(16) × 10⁻¹, and −7.562(3) × 10⁻¹. The respective are: −3.70(90) × 10⁻³, −4.33(50) × 10⁻³, −5.62(20) × 10⁻³, and lastly, −6.17(32) × 10⁻³.

A line fit of the known and data points was completed. The line fit and values are respectively 1.82611(11) × 10⁻⁸⁶ s² and 2.33(11) × 10⁻¹⁸ s⁻¹. The slope is 3.021 × 10⁻³, and the y-intercept at x = 0 is 3.315 × 10⁻⁴. The line fit and for are −3.670879(12), and −1.37371(1) × 10⁻². This is compared to the derived values in Table 1, Figure 4. The derived and known values correlate well.

The generalized line for any qf value is Equations (26)-(28). This equation is used to derive the possible hypothesized negative value harmonic fractions of beta neutron decay, −1/2, −3/4, −5/6, −7/8. One of these is hypothesized be related to.

The derived, exponents, and s⁻¹ calculated values from the line for the possible qfs of −1/2, −3/4, −5/6, −7/8 are respectively for −1/2: −4.4681966(3) × 10⁻³, −5.0446820(13) × 10⁻¹, 1.6498650(1) × 10⁻¹² s⁻¹, for −3/4: −5.20211263(26) × 10⁻³, −7.5520211(04) × 10⁻¹, 2.2972668(2) × 10⁻¹⁸ s⁻¹, for −5/6: −5.4467515(3) × 10⁻³, −8.3878008(4) × 10⁻¹, 2.5652661(7) × 10⁻²⁰ s⁻¹, for −7/8: −5.5690708(3) × 10⁻³, −8.8056907(5) × 10⁻¹, 2.7107717(3) × 10⁻²¹ s⁻¹, Figure 4. The qf of −3/4 is most the associated qf of. The derived from this method is the intercept of the line at an x value of −3/4-1. This is the harmonic fraction x location of −3/4-1, −7/4. The specific derivation of these factors for are shown in Equations (29)-(31).

The derived Hubble time is the inverse of, 4.3529990(17) × 10¹⁷ seconds. There are 3.1556926 × 10⁷ seconds per year. The derived Hubble time equals 13.7941161(13) × 10⁹ years. The derived Hubble length is 13.7941161(13) × 10⁹ light years. The reported experimental value is approximately 13.8 × 10⁹ light years.

The known proportionally constant of the ratio of line fit divided by is 1.02(15) × 10⁻⁴¹. The derived value is 1.01118370(16) × 10⁻⁴¹, Equation (32), Table 4.

The known proportionally constant of the ratio from the line fit divided by is 8.03(40) × 10⁻¹¹⁰. The derived value is 8.0382241(5) × 10⁻¹¹⁰, Equation (33).

The known proportionally constant of the ratio from the line fit divided by is 1.272(64) × 10⁶⁸. The derived value is 1.25796898(7) × 10⁶⁸, Equation (34).