Anomalies within the dimensioned physical constants (G, h, c, e, me, kB) suggest a mathematical relationship linking the units (kg, m, s, A, K).
A dimensioned physical constant, sometimes denoted a fundamental physical constant, is a physical quantity that is generally believed to be both universal in nature and have constant value in time. Common examples being the speed of lightc, the gravitational constantG, the Planck constanth and the elementary chargee. These constants are usually measured in terms of SI units mass (kilogram), length (meter), time (second), charge (ampere), temperature (Kelvin) ... (kg, m, s, A, K ...).
These constants form the scaffolding around which the theories of physics are erected, and they define the fabric of our universe, but science has no idea why they take the special numerical values that they do, for these constants follow no discernible pattern. The desire to explain the constants has been one of the driving forces behind efforts to develop a complete unified description of nature, or "theory of everything". Physicists have hoped that such a theory would show that each of the constants of nature could have only one logically possible value. It would reveal an underlying order to the seeming arbitrariness of nature [1].
Notably a physical universe, as opposed to a mathematical universe (a computer simulation), has as a fundamental premise the concept that the universe scaffolding (of mass, space and time) exists, that somehow mass is, space is, time is ... these dimensions are real, and independent of each other ... we cannot measure distance in kilograms and amperes, or mass using length and temperature. The 2019 redefinition of SI base units resulted in 4 physical constants (h, c, e, kB) having independently assigned exact values (they cannot be derived in terms of each other), and this confirmed the independence of their associated SI units as shown in this table.
However there are anomalies which occur in certain combinations of the physical constants (G, h, c, e, me, kB) which suggest a mathematical relationship between the units (kg, m, s, A, K) [2]. In order for the dimensioned physical constants to be fundamental, the units must be independent of each other, there cannot be a unit number relationship ... however these anomalies question this fundamental assumption. Every combination predicted by the model returns an answer consistent with CODATA precision. Statistically therefore, can these anomalies be dismissed as coincidence?
The Planck units are direct measures of the SI units; Planck mass in kg, Planck length in m, Planck time in s ... and so they are analogues to the attributes in the above table. The SI Planck units have numerical values, however to derive a mathematical relation between these SI units we cannot use numerical values, this is because numerical values are simply dimensionless frequencies of the SI unit, 299792458 could refer to the speed of light 299792458m/s or equally the number of apples in a container, numbers such as 299792458 carry no unit-specific information and so the units are treated as independent by default.
This can be resolved by assigning to each Planck unit a geometrical object (denoted MLTVA), and for which the geometry embeds the attribute (for example, the geometry of the time object T embeds the function time and so a descriptive s is not required). We may then combine these objects to form more complex objects; from electrons to galaxies, while still retaining the underlying attributes (of mass, wavelength, frequency …). As this particular geometrical approach requires that the objects be interrelated (they are not independent of each other), a unit number relationship hypothesis can be tested. This is because, if there are natural Planck objects, then these MLTVA objects will be embedded within our dimensioned SI physical constants.
Table 2. is an example of object orientated units; it assigns MLTA objects as the geometry of 2 dimensionless physical constants; the Sommerfeld fine structure constant alpha and Omega. As alpha and Omega are dimensionless (alpha = 137.035999084, Omega = 2.0071349496), so too are these objects.
As Alpha and Omega can have (dimensionless) numerical values, we can use dimensioned numerical scalars to convert from the MLTVA objects to their SI equivalents.
As scalar v carries the unit designation m/s (v = 11843707.905 m/s), scalar v is dimensioned, and so we can assign a unit number to v (θ = 17).
Table 4. Scalars
attribute
geometrical object
scalar
mass
k (θ = 15)
time
t (θ = -30)
velocity
v (θ = 17)
length
l (θ = -13)
ampere
a (θ = 3)
As this unit number relationship is expressed within the scalars (included in formulas as unit = uθ), we need only 2 scalars to define the others, for example if we know the numerical values for a and l then we know the numerical value for t (a3l3/t = 1 and so a3l3 = t), and from l and t we know k … and so we can solve the SI Planck units (α and Ω have fixed values), and from these, we can solve (G, h, c, e, me, kB).
Once any 2 scalars have been assigned values, the other scalars are then defined by default, consequently the CODATA 2014 values are used here as only 2 constants (c, μ0) are assigned exact values, following the 2019 redefinition of SI base units 4 constants have been independently assigned exact values which is problematic in terms of this model. Here the attributes are defined in terms of 2 scalars; from c is v (θ = 17), and from μ0 is r (θ = 8).
If we can reduce the 5 SI units to 2 scalars (example; r, v in tables 5, 6), then we can find combinations of the physical constants (G, h, c, e, me, kB) where the unit numbers θ and the scalars will cancel, these combinations, which are unit-less (units = 1), will then return the same numerical value as the MLTVA object equivalents. This is because if the scalars have cancelled, and as the scalars embed the SI conversion values as well as the SI units, then these combinations are defaulting to the underlying MLTVA objects (the SI component has cancelled).
This should therefore apply to any set of units, even extraterrestrial and non-human ones, suggesting that these MLTVA objects could be 'natural' units.
...ihre Bedeutung für alle Zeiten und für alle, auch außerirdische und außermenschliche Kulturen notwendig behalten und welche daher als »natürliche Maßeinheiten« bezeichnet werden können...
...These necessarily retain their meaning for all times and for all civilizations, even extraterrestrial and non-human ones, and can therefore be designated as "natural units"... -Max Planck [4][5]
For example;
0.228 473 759... 10-58
0.228 473 639... 10-58
Here we solve physical constant combinations using only α, Ω (and the mathematical constants 2, 3, π). As the scalars (v, r) have cancelled, we do not need to know their values or the units. The precision of the results depends on the precision of the SI constants; combinations with G and kB return the least precise values.
Note: the geometry (integer n ≥ 0) is common to all ratios where units and scalars cancel
The electron is an example of a combination that is dimensionless (units = scalars = 1). The electron function (the mathematical formula for the electron) fe can be defined, for example, in terms of ALT where AL as an ampere-meter (ampere-length = e*c) are the units for a magnetic monopole.
Associated with the electron are dimensioned parameters, these parameters however are a function of the base MLTVA units, the formula fe dictating the frequency of these units. By setting MLTVA to their SI Planck unit equivalents;
↑J. Barrow, J. Webb "Inconsistent constants". Scientific American292: 56. 2005.
↑Macleod, M.J. "Programming Planck units from a mathematical electron; a Simulation Hypothesis". Eur. Phys. J. Plus113: 278. 22 March 2018. doi:10.1140/epjp/i2018-12094-x.
↑Are these physical constant anomalies evidence of a mathematical relation between the SI units?. doi:10.13140/RG.2.2.15874.15041/6."