_{∞}crucial in explaining spectrums of EM radiation emitted from material such as hydrogen. On the other hand, if α appears in the proposed length conversion functions, then α is fundamental to all physical phenomena, not just EM effects, because experimental length measurements in study of any physical phenomenon could be mapped from corresponding lengths in BM space containing the underlying mechanisms for the studied phenomenon.

The inverse length conversion functions (eqs. 6 and 8 in [2]) were

d = a * ln(d') + b [Eq. 1]

d' = e

^{(d - b)/a}[Eq. 2]

where d and d' are length in meters in BM space and experimental measurement space respectively. The numeric constants a = 0.129838012936585 and b = 7.44495436913956 in Eqs. 1 and 2 were obtained by linear regression fitting test data points for electron, nucleon and Bohr radii in each of the two spaces. The number of significant figures in constants a and b reflect values obtained from the test data set and used in calculations presented below and do not indicate degree of accuracy.

ln(4π/α) = 7.451267835 [Eq. 3]

agrees with the empirical value b within 0.0085 percent.

Substituting in Eqs. 1 and 2,

d = a * ln(d') + ln(4π/α) [Eq. 4]

d' = e

^{(d - ln(4π/α))/a}[Eq. 5]

Numeric constant a in Eqs. 4 and 5 might be a new value pertaining to these length conversion functions, or simply be not understood. On the other hand,

1/ln(4π/α) = 0.134205349 [Eq. 6]

agrees with the empirical value a within 3.4 percent. The original test data may not in aggregate have been even this accurate. In an admittedly more speculative step, substituting Eq. 6 in Eqs. 4 and 5 with minor rearranging terms,

d = ln(d')/ln(4π/α) + ln(4π/α) [Eq. 7]

d' = e

^{ln(4π/α)(d - ln(4π/α))}[Eq. 8]

**Discussion**

The length conversion functions map between quantized BM space and continuous space assumed in classical physics up to and including the Standard Model and its associated experimental length measurements. If usage of the fine-structure constant α in expressions for numeric constants a or b or both (Eqs. 1 to 8) is correct, no less than four fundamental physics constants appear to be involved in these inter-space mapping functions -- elementary charge e, Planck's constant h, light speed in vacuum c

_{0}and the electric constant ε

_{0}or permittivity of free space.

In other words, these four constants express how fundamental events in BM space result in specific experimentally measured values quantified in familiar units of mass, length, time and charge. Each of these four constants involves a length unit. Hence, it may be no surprise that these heavy-hitters among physics constants would be involved in the mapping of the length parameter between phenomenological and BM spaces.

**Elementary Charge e.**Classically and experimentally, elementary electric charge e is equal to the absolute value of proton charge or electron charge.

According to the CODATA method [3], elementary charge e is defined in terms of constants listed above or their equivalents, namely the fine-structure constant α, Planck's constant h, vacuum light speed c

_{0}and the magnetic constant µ

_{0}. BM breaks this apparent circularity in constant definitions [4] by independently defining each α component. For example, elementary charge is nominally defined as 3 times the sign of mite bits, which is based on a value called I parity of mite position in BM space [1]. That is, in the abstract world of BM, the sign of electric charge is independently defined by mite position I parity. Hence, one might argue that elementary charge e might necessarily be required to properly project events in BM space such as length to observations in phenomenological space caused by these events.

**Planck's Constant h.**Defined with the mass, length and time units fundamental to constants and experimental measurements in physics, Planck's constant h is typically expressed as a quantum unit of energy such as, for example, mc

^{2}, times a time unit such as seconds. The energy component of this quantum unit appears to correspond to the 1-state of mite and lite bits in BM, where each 1-state bit may be said to be one unit of energy, as reckoned in phenomenological space where experimental observations are conducted. Thus, BM provides an independent definition of a single quanta of energy.

As with elementary charge, it is reasonable to suppose that Planck's constant might be a necessary value to correctly project the abstract value of one for the energy quanta of 1-state bits in BM space into the observable world of experimental measurements.

**Light Speed in Vacuum c**As described previously (e.g., [5]), all velocities are quantized in BM as a consequence of its quantization of both length (space) and time. Hence, for the smallest possible time interval, denoted as a single

_{0}.**tick**, the time unit

*t*in BM, velocity v of 1-state bits can be only one or zero. If v = 1 in a single tick interval, a 1-state bit moves exactly one unit of BM length

*d*(Eqs. 1 to 8 above). In short, BM provides an independent definition of velocity.

Given the lattice-like structure of BM space, this velocity v must necessarily be greater than so-called vacuum light speed, which is an "as the crow flies" measure expressed in large multiples of

*d*and

*t*, where in each tick, a 1-state bit moves at v = 1 or 0. Hence, by the same logic applied to elementary charge and Planck's constant, if light speed were not a defining part of the fine-structure constant α, it probably would be explicitly required in the length conversion functions presented above.

**Electric Constant ε**Permittivity, known as the electric constant, used to define the fine-structure constant α, may be expressed in terms of the permeability µ

_{0}._{0}and light speed c

_{0}constants. Given that vacuum light speed is a phenomenological measurement, the permittivity ε

_{0}and permeability µ

_{0}constants required to correctly quantify experimental observations of EM interactions and radiation might also be required for length mapping between BM and observational spaces. This supposition is supported by the fact that both of these constants are independently defined as one in BM space or in other words, are not explicitly needed.

**Length Conversions Functions.**If one or both of the numeric constants found from a small test data set in the length conversion functions (Eqs. 1 and 2, [2]) are indeed expressible in terms of the fine-structure constant α, it is noteworthy that each α component briefly described above is independently defined with exact precision in BM space.

The prototype length conversion functions presented previously [2] and expressed in terms of the fine-structure constant α in Eqs. 4 and 5, or alternatively in Eqs. 7 and 8, in the present report, may now be used in BM space to examine EM spectrums such as the fine-structure lines of radiation from hydrogen atoms and blackbody radiation [Keene, unpublished data]. Both of these spectrums appear to be reproducible using present BM simulation software [6]. Such further studies could provide much larger, new data sets which can further polish the exact values of the a and b constants in Eqs. 1 and 2 and perhaps confirm their presently proposed relation to the fine-structure constant α.

On the other hand, using measured results for length d' such as one meter or one kilometer produce resulting lengths d in BM space that are much too small to be credible. This apparent flaw would support a much simpler linear form for the length conversion functions, such as

d = aαd' + b [Eq. 9]

d' = (d - b)/(aα) [Eq. 10]

Hence, much larger data sets such as the EM spectrums mentioned above are mandatory to extend the exercise presented previously [2] to address its possible weaknesses. Only three pairs of data points were used so that one or more errors in the test data set could easily distort an effort to model correct length conversion functions.

Further work confirming or updating the length conversion functions might place BM as perhaps

**the only theory of everything to obtain the numeric value of the fine-structure constant α from first postulates and principles**.

**References**

[1] Keene, J. J. "Binary mechanics" July, 2010.

[2] Keene, J. J. "Fundamental physics constants" June, 2011.

[3] CODATA. "CODATA Resources" June, 2011.

[4] Keene, J. J. "Captives in a binary mechanical universe" March, 2011.

[5] Keene, J. J. "Electron acceleration and quantized velocity" April, 2011.

[6] Keene, J. J. "Binary mechanics simulator updated" March, 2011.

© 2011 James J Keene