Professor, The Ohio State University
The mathematical origin of δ values
First let us focus on the mathematical origin of δ values. These δs are very hard to understand. There is a mathematical imperative which arises from the quantum fractions of the electron, Bohr radius, and ionization energy of hydrogen that specific numbers be associated with specific integer fraction exponents. For example the number 2 must be associated with the quantum fraction 10/1155. Therefore its fundamental frequency, v?, is related to that number raised to the reciprocal exponent, see below. Its v? is in the range of 1034.
Another example is 2π and 39/1155, and its v? is in the range of 1023.
These two v? are massively different. There are a total of 4 of these similar imperatives. The mathematical solution to bring all of the possible fundamental frequencies to a common one is to add small δs to the integer fractions. These are small “shims” like tuning a magnet in magnetic resonance systems.
There is a common misunderstanding related to δ. In the harmonic neutron hypothesis it is true that the δ of any physical constant can be calculated if its quantum fraction and the actual frequency equivalent of a physical constant are known. These have been listed in the publications. In the model there are only three δ values used in all of the other derivations. These are natural unit values. These are the δs related to the electron, Bohr radius, and the ionization energy of hydrogen. All of the other δs in the predictions are derived from only those three so there is no circular logic. This is a good observation by the reviewer and is confusing for a reader. In the future the known δs will be notated as δk, and δd for the derived ones.
How is it possible to derive a δ for a physical constant when its exact exponent is not known? This was first an empiric observation that physical constants that are logically related all fell on a common line. For example the kinetic energy lost in the neutron beta decay process fall on a line. The charm, bottom and top quark all fall on another common line.
First one needs to understand the graphical display of a classic proportionality constant equation as plotted in the model. The standard equation is shown below where y is the output, x is the input, and k is the proportionality constant. k is the ratio of y/x. k is a slope.
In the exponent domain this equation is shown below. The sum of the log of the output is the sum of the log of the input and log of k. The log(k) is the difference of log(y) and log(x). All of these values are related to specific qfs and δs.
The ratio of two fundamental constants represents the difference of the quantum fractions and the difference δ, or another qfdiff +δdiff in the exponent domain. On the model’s exponent plane this represents the difference between two points. The qf difference is the x axis spread and the δ difference represents the y axis spread. This defines a line and a slope. Each k has its own slope.
If x equals 1 then y equals k. On the model’s exponent plane a ratio of 1 is defined by the point at (-1, 0), a quantum fraction of 0. That point plus the qfdiff +δdiff defines the proportionality constant equation relationships on the model exponent plane. If the x equals qfdiff +δdiff that point falls on the same line, but at an x axis value qfdiff to the right or left. If the x value equals 2 times qfdiff +δdiff , the square, that point falls on the same line, but now 2 qfdiff to the right or left. Physical constants that are related in a proportionality constant relationship in classic physics, which are the vast majority of relationships, all fall on a common line on the model’s exponent plane. For example, differences between the points for the ionization energy of hydrogen and the Bohr radius is identical to Coulomb’s law.
Therefore if the δ line associated with a proportionality constant relationship is known or predicted then the δd for any qf can be derived. Since most physics equations are proportionality constant equations the known fundamental constants related to that relationship all naturally fall on a line. It is possible to predict the δ lines since logically related entities should all fall on the same line. In this case it is assumed that some of the quarks fall on a common δ line. It is assumed that all of the other δs are defined by the δs of the electron, Bohr radius, and the ionization energy of hydrogen values only as slopes and y intercepts. This is how it is possible to derive the δd ,and calculate exact values for physical constants with no other physical data.