**For the first time, the empirically measured value of Planck's constant h is calculated from first principles of a physical theory to the full precision allowed by CODATA values.**Using the postulates of binary mechanics (BM) where both space and time are quantized [1], this report describes the key steps in this calculation and proposes values for the fundamental length d and time t constants.

(1) Bit velocity v was defined as greater than the speed of light in a vacuum c consistent with the BM constraint that v > c [2].

(2) A physical interpretation of BM space [3] suggested a proposed value for the fundamental BM length constant d as approximately 0.6 fm.

(3) d/v = t' = approximately 6.35E-25 s, the fundamental time constant in BM space-time.

(4) The fine structure constant α maps this quantized time unit t' from BM space to observational space with t = αt' = approximately 4.63325E-27 s.

(5) Intrinsic electron spin and hence the Planck constant h was calculated using only electron rest mass m

_{e}and the proposed length d and time t constants.

(6) In addition to steps (3) and (4) above, another method was used to calculate quantized time t based only on m

_{e}, h and quantized length d.

(7) Finally, eq. 9 calculates Planck constant h directly from the independently determined length constant d (step 2 above) and familiar physical constants.

**1. Bit Velocity.**

An analysis of the relativistic Dirac spinor equation led to an upgrade in quantum mechanical (QM) formalism to include quantized space and time units d and t respectively. The QM complex wave function evolved into the binary mechanical (BM)

**bit function**(eq. 2 in [1]) as an irreducible representation of any physical system or object as a spatial pattern of one and zero binary bits. Likewise, with quantized space-time, QM infinitesimal operators were clearly inadequate to describe system time evolution, especially for small spatial or time intervals, since only integer increments in position and time parameters are allowed with full quantization. Thus, infinitesimal QM operators have evolved into four BM

**bit operations**which exactly describe bit function time-development. In each bit operation -- unconditional, scalar, vector and strong, a 1-state bit may move exactly one unit of BM distance d in one unit of BM time t', defining bit velocity v (eq. 1).

v = d/t' (eq. 1)

Given a physical interpretation of BM space [3], a constraint on velocity v is that it must be substantially greater than the speed of light in vacuum c [2]. In this report we try

v = πc (eq. 2)

estimating v as 941825783.7 m/s.

**2. BM Fundamental Length Constant d.**

**Fig. 1: Electron Spot XYZ Parity 111 in Spot Cube**

>

Legend: Electron spot (yellow). Matter d quark spots (dark red, green, blue). Antimatter d quark spots (light red, green, blue). 001 to 111: integer XYZ spot position coordinates. From Fig. 3 in [4].

Since velocity v is a ratio of length and time parameters, some leverage is required to begin with a credible estimation of one of them. The BM fundamental length constant d may be the best starting point. Empirical scattering data estimates a nucleon radius of approximately 1.2 fm [5]. With the association between the nucleon and the spot cube (Fig. 1) derived from BM postulates [1] and a physical interpretation of BM space [3], this radius roughly corresponds to 2d, setting length constant d at about 0.6 fm. This report will show that this value can be used to set the value of the BM fundamental time constant t and calculate from BM first principles the electron intrinsic spin and hence, Planck's constant h. With a step-wise interpolation of the these results to a similar level of precision as the CODATA physical constants used (e.g., c, h, electron mass m

_{e}),

**the proposed BM fundamental length constant d is approximately 5.9798551E-16 m.**

**3. BM Fundamental Time Unit.**

From eq. 2 and an independent estimate of constant d, the quantized time t' is

t' = d/v = 5.9798551E-16 / 941825783.7 = approx. 6.34922E-25 s (eq. 3)

where t' is the fundamental time unit, called a

**tick**, in the frame of BM space-time. To complete our calculations, the tick value t in observational space is required.

**4. Fine Structure Constant α Maps BM Space to Observational Space?**

Using eq. 3, the fundamental time constant t in observational space may be

t = αt' = 7.2973525698E-3 x 6.34922E-25 s = 4.63325E-27 s (eq. 4)

where α is the fine structure constant. This relation suggests a physical significance of the fine structure constant and defines how results obtained in BM space (e.g., from simulations) correspond to result values in observational space, and visa versa [6].

Notice the value of light speed in vacuum c in observational space was used in eq. 2. The use of fine-structure constant α in eq. 4 would not have been required if the value of c in eq. 2 had been adjusted with α to its value in BM space in eq. 2. This amounts only to a question of which arrangement of terms might be most convenient in eqs. 1 to 4 and their physical interpretation, but does not affect the results from eqs. 5 to 9 below.

**5. Intrinsic Electron Spin.**

**Fig. 2: Electron Spot Geometry for Spin Calculation**

This spin radius r, electron rest mass m

_{e}and t from eq. 4 may be used to calculate the intrinsic electron spin or angular momentum magnitude

m

_{e}r

^{2}/t = (1/2)(h/2π) (eq. 5)

where m

_{e}is electron rest mass, h is Planck's constant and t is observational tick time in seconds. Eq. 5 sets a scalar expression of the classical angular momentum equal to the observed value of intrinsic electron spin of 1/2 of the reduced Planck constant (h/2π). Expressing radius r in fundamental length d units (sqrt(3)d/2) and solving for the Planck constant, we have

h = 3πm

_{e}d

^{2}/t = 6.62606957E-34 Js (eq. 6)

where d is the interpolated value 5.9798551E-16 m cited in section 2 above. In summary, Planck's constant h in Joule seconds may be derived from observed electron rest mass m

_{e}and first principles of BM including proposed fundamental quantized length and time constants d and t respectively.

**6. Alternate Calculation of BM Fundamental Time Unit t.**

From eq. 6, an alternate calculation to the BM fundamental time unit t is simply

t = 3πm

_{e}d

^{2}/h = approx. 4.63325E-27 s (eq. 7)

where the precision depends only on the CODATA constants m

_{e}and h, noting that the proposed value for the fundamental length constant d was "fit" from the original estimated 0.6 fm using the electron mass and Planck constants. This tick t value in observational space is identical to that obtained with eq. 4. Eqs. 4 and 7 depend on the "input" value of the estimated length d from which fundamental time constant t was obtained. Eq. 7 may be little more than a trivial rearrangment of eq. 6. However, from eqs. 1 to 6, it may be evident that there is no circularity in the logic used to determine time tick t (eq. 4) and Planck's constant (eq. 6). In any case, whatever their correct values may be, length d and time t have the status of postulated fundamental constants based on first principles of BM.

**7. From BM Fundamental Length d to Planck's Constant h.**

Eqs. 1 to 4 explicitly express a constraint of BM (eq. 2) and an issue regarding possible difference between the BM space-time frame and the more familiar observational frame (eq. 4). However, central to the present report is demonstrable lack of circularity in computation of h from d. Hence, it might be convenient to express tick time t in eqs. 4 and 6 in terms of d. From eqs. 1 to 4,

t = αd/πc (eq. 8)

Substituting eq. 8 in eq. 6 and simplifying, we have

h = 3π

^{2}m

_{e}cd/α (eq. 9)

In summary, given an independent estimate of the fundamental length d derived from BM postulates (section 2 above), Planck's constant h may be directly computed with d and familiar physical constants.

**Discussion**

*"She's got legs" (ZZ Top).*BM is the only physical theory known to the author where Planck's constant can be calculated from first principles or postulates. If this is true, the inescapable conclusion is that this milestone success in physics resulted from none other than the postulate of quantized space. Clearly, the equations presented depend on a realistic proposal for the fundamental length constant d. The approximate value of d (0.6 fm) was based on (1) the BM-derived association of the nucleon particle with the spot cube (Fig. 1) and (2) experimental data for nucleon radius of about 1.2 fm. In contrast, various formula for Planck's constant h in physics literature are without exception based on experimental measurements (e.g. energy E, frequency

*v*, electron charge e, fine structure constant α, etc). In summary, the present report appears to be the first to derive h from the postulates of a comprehensive coherent physical theory, namely BM. As a further confirmation that BM has "legs", an upcoming report calculates the electron magnetic dipole moment from BM fundamentals without invoking the so-called g-factor (Keene, in preparation). Hence, the gyromagnetic- or g-factor is revealed to be an additional explicit admission of the failure of conventional formalism assuming continuous space-time with no known justification.

*Size matters.*Thus far, analysis of the electron spot alone has not led to an unambiguous best guess of the BM fundamental distance d, presumably because observations of electron presence and behavior appear to be based on 1-state bits absorbed by (entering) or emitted from (exiting) the three spot unit components of the electron spot (Figs. 1 and 2). Indeed, the physical size of a 1-state bit is not addressed by BM postulates beyond the stipulation that it represents a physical quanta of energy and fits in a size d spatial cube. Likewise, the question of possible internal structure of these energy quanta (1-state bits) remains unanswered. Meanwhile, experimentalists reportedly continue pushing size estimates for these readily observable 1-state energy bits to ever smaller, seemingly point-like values. While this work is important, it apparently addresses other issues and has yet to reveal anything useful to estimate bit locus cube size d.

However, the discovery of the central baryon bit cycle [9], the basis for quark (color) confinement in nucleons, provided an opportunity to estimate the value of length d. Consider that 1-state bits circulate through d quark spots spanning all sides of the spot cube. That is, these baryons are multi-spot particles (Table 3 in [1]). This larger spatial volume of spots had yielded size measurements of nucleons [5] due to at least two phenomena. First, the spot units involved in baryon bit cycles may absorb and emit 1-state bits by the same mechanisms seen in single-spot electron and positron particles. Second, the cycling 1-state bits are the electrostatic potential field when in mite cube loci and the magnetic potential field when in lite cube loci (see, e.g., [4]). Processes such as these are the basis for scattering of incident beams by targets. In short, it is quite plausible that nucleon scattering data [5] may provide an excellent window to estimate the quantized BM fundamental length constant d as presented in this report.

*Captives in a binary mechanical universe.*In eq. 4, the fine structure constant α is used to define two values for the BM fundamental time constant: t' in the abstract BM space-time and t in physical observational space. The physical senses or event detection instruments of investigators are, of course, embedded in observational space-time. At present, BM postulates do not require that this observational frame is identical to that of postulated, quantized BM space-time. Eq. 4 tries the hypothesis that these two frames are in fact different and that the fine structure constant α may represent a scale factor relating the two space-time frames. If this scale factor role of α is correct, the present report may help clarify the physical significance of the α constant.

*New gold rush?*Whether the observational and BM space-time frames are identical or different, as long as measurements made in one frame can be projected to corresponding values in the other frame, results from simulations in the BM frame can be expressed in length and time units in the observational frame, facilitating hypothesis testing and engineering applications developed from the simulations. Conversely, physical problems defined in the observational frame can be set up with proper length and time units for exploration and testing in BM simulations. In other words, a few nerds with lap-tops could conceivably accomplish more advances in basic science in months than the entire staff and budget of a facility like CERN might accomplish in decades, if ever. Consider that CERN has yet to provide a downloadable bit function 3D matrix for the "Higgs boson". Do they have any clue about what this object really is? If so, show us its bit function so anybody can study its properties and behavior on any decent lap-top. Likewise, apply a similar efficiency ratio to commercial research and development for new technology products, given the current thirst for nanotechnology, which is only the beginning of more microscopic levels of analysis with the capabilities of the software "bit operations engine" in the BM simulator.

*What about the famous Planck length and time?*What about them? They are science fiction creations of the unwarranted assumption of continuous space-time. The famous Planck length and time constants in physics literature are proof that imagined creations can be very famous indeed. For example, consider the Terminator movies. At present, the major usefulness of the Planck length and time is heuristic, namely to exclude any physical theory which depends on those values as probably seriously flawed. That is, the BM length d and time t constants are by definition the minimum possible values for these parameters with reference to time-development (bit operations) of a system state (bit function). Length and time measurements are typically integer multiples of d or t. As a consequence, for example, velocity is quantized [10]. The advisory for physicists is that reliance on fictional creations in the work place outside the movie theater can severely limit career development.

*Two-for-one sale.*Eq. 5 used the positive value for intrinsic electron spin instead of the usual ±(1/2)(h/2π), which is apparently a misleading combination of two completely independent quantum numbers. (1/2)(h/2π) is a particle property as described above regarding an apparently real and easily visualized intrinsic spin which occurs in the electron particle cycle [3]. As reported previously [4], the up-down component is an independent quantum number with allowed values of ±1 due to

**elemental electromagnetic resonance**in potential fields affecting electron motion and to electron

**particle time phase**, both with reference to odd and even quantized time tick parity. In short, with this previous and the present reports, one may trade in a confused ±(1/2)(h/2π) quantum number and get two in return: (1/2)(h/2π) representing a real intrinsic spin and ±1 (for the electron intrinsic spin component) representing up-down splitting of charged particle beams in inhomogeneous magnetic fields.

*Homework assignment.*This report choose to assume that length measurements, such as the fundamental constant d, are identical in BM and observational space-time frames. Why? Perhaps this choice might make the calculation steps described easier for everybody to visualize. With apologies if the reader might have thought this was summer camp with only fun and mirth, there is a homework assignment: What if time measurements were constant between the frames, and length values were scaled by α instead? Would this be a different, but physically equivalent representation or not?

*Take-home message.*Whether or not the considerations in eqs. 1 to 4 are justified and correct, the major results of the present report are that

**a plausible value in meters for the BM fundamental length constant d was proposed (section 2), this distance d was used to set a value in seconds of BM fundamental time tick constant t (eqs. 1 to 4), and Planck's constant h was calculated directly from d, t and the electron rest mass (eq. 6) or with t expressed in d units (eq. 9).**. Hence, in effect, experimental measurements of Planck's constant h might be viewed as confirmations of the hypothetical value of h derived from BM postulates and the classical definition of angular momentum. Notice that BM predicts that h, based on quantized space and time values (eq. 6), must also be quantized as Planck and colleagues stated better than a century ago.

**Editor's note: The reader is invited to post comments in agreement or disagreement with this or other**

*Journal of Binary Mechanics*articles at the Binary Mechanics Forum. The Journal also welcomes on-topic articles from other investigators and persons considering serving on the Journal's editorial board.**References**

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

[2] Keene, J. J. "Captives in a binary mechanical universe" J. Bin. Mech. March, 2011.

[3] Keene, J. J. "Physical interpretation of binary mechanical space" J. Bin. Mech. February, 2011.

[4] Keene, J. J. "Particle up-down spin and quantized time parity", J. Bin. Mech. January, 2015.

[5] Krane, K. S., Introductory Nuclear Physics, Wiley, 1987.

[6] Keene, J. J. "Fundamental physics constants" J. Bin. Mech. June, 2011.

[7] Keene, J. J. "Zero electron electric dipole moment" J. Bin. Mech. January, 2015.

[8] Keene, J. J. "Fundamental forces in physics" J. Bin. Mech. Oct, 2014.

[9] Keene, J. J. "The central baryon bit cycle" J. Bin. Mech. March, 2011.

[10] Keene, J. J. "Electron acceleration and quantized velocity" J. Bin. Mech. April, 2011.

© 2015 James J Keene