The Binary Mechanics Lab (BML) Simulator v3.13 was used in Random Mode to demonstrate the Hall effect in a cubic volume. Analysis of charge displacements of 1-states revealed the role of the proton cycle as the primary mechanism producing the Hall effect. This Hall effect demonstration and description of its mechanism further establish the heuristic value of binary mechanics postulates and equations.
Introduction
In 1879, Edwin Hall discovered that an electric current produced a transverse voltage in a conductor [1]. In Random Mode, the BML Simulator records displacements of positive and negative fractional charges, associated with the proton and electron cycles respectively [2], for each application of the binary mechanics (BM) update to the Schrödinger equation (eq. 17 in [3]).
Fig. 1 shows displacements of 1-states in the proton cycle for 1000 applications of the Schrödinger equation update, with Simulator cube size Dim = 72 and 1-state (energy) density d = 0.25. The displacements in two directions (up-down and left-right) are changes in the mean position (or "center of gravity") of net positive 1/3 fractional charges in the proton cycle.
Charge displacement (or current) in the upward direction was associated with displacement to the right (Fig. 1, top). Similarly, charge displacement in the downward direction was associated with displacement to the right with respect to that current direction (Fig. 1, bottom). In sum, the Pearson product-moment correlation r = 0.7844 between these currents accounted for 61.5% (r2 = 0.615) of the variance of each set of displacements (up-down and left-right).
Each data point in Fig. 1 is the result of one application of a sequence of four BM bit operations [3] which defines a single BML Simulator Tick. For the three Simulator runs summarized in Fig. 2, the simulated volume was seeded randomly with 1-states to approximate the energy density input parameter d. As reported previously [4], some time is required for the bit operations to "organize" the configuration of 1-states. Hence, initial data was discarded, typically about 350 Ticks.
A single 1-state in the proton cycle returns to its initial locus in 21 Ticks. If the "A" key is pressed during a Simulator run, a record is written to the output *.csv file only once per proton cycle, producing an output file size about 5% of the size of a file including every Tick (Fig. 1 and Fig. 2, top). In effect, this "A OFF" option captures data only for a single phase of the 21 Tick proton cycle and was used for the second and third runs in Fig. 2. That is, the 79 Tick sample represents one phase of a 1659 (21 x 79) Tick run (Fig. 2, middle) and the 83 Ticks analyzed represent 1743 (21 x 83) Ticks (Fig. 2, bottom).
In this pilot study, similar results were obtained in three BML Simulator runs with different cube size (Dim) and initial energy density (d) settings. Further, the high correlations were observed based on all Simulator Ticks or on an arbitrary selection of just one phase of the proton cycle.
The correlations among the proton cycle state displacements generally accounted for more than 50% of the current variance (Fig. 2, left). On the other hand, the correlations among the electron displacements were much lower and displayed a greater range (Fig. 2, right).
Discussion
Methodology. Hall effect has often been observed as a "Hall voltage" transverse to current in a "flat 2D" conductor material. The present study looked at currents in cubic volumes, adding a third dimension to the experimental setup. Thus, Hall voltages in two dimensions transverse to current flow might be expected. Using Ohm's law, charged particle current may be used to operationally define a voltage potential. Hence, one might conclude that the transverse fractional charge displacements reported are consistent with the presence of transverse Hall voltages. Ongoing studies specifically address this issue.
The present method analyzed naturally occuring charge displacements instead of inducing a current in the simulated volume by applying a voltage from an external source. With sufficient energy density d in Random Mode operation of the BML Simulator, kinetic energy and electromagnetic radiation in the simulated volume result in current flows. In addition, radiation and particles may exit the volume. With these events, the volume is "broadcasting" and further studies may identify known spectral lines. In Random Mode, the Simulator maintains a relatively constant energy density in the volume by injecting 1-states at randomly selected locations in the six sides of the volume. It is as if the simulated volume was located in a "larger volume" with similar energy density. These random injections may also produce transient currents in the simulated volumes analyzed in this report.
Note that if r = 0.22, less than 5% of the variance is shared by the two variables. In contrast, the present data suggest that the proton cycle correlations represent more than 50% shared variance. Hence, one might safely conclude that the proton cycle has a 10x greater link to Hall effect phenomena than the electron cycle. However, effects of the electron cycle may be required to achieve more precise quantitative analysis related to the Hall effect.
Finally, the BML Simulator implements BM eqs. 1-17 without any input from, or reference to, so-called "fundamental constants" such as elementary charge e. Along with the full quantization postulates, the data in this report depend only on the first principles of BM [3].
Causality. In an experimental setup with a "flat 2D" conductor, results may be described as applied voltage causes a current flow which then causes the transverse Hall voltage. In the present experimental design, switching an "applied voltage" on and off might be represented by data points far from, or near to, the center of Fig. 1 for On and Off respectively.
In any case, the correlations between orthogonal current events in Figs. 1 and 2 do not establish causality among the currents observed, nor is that required. Instead, the data suggest that the correlated currents are produced by the proton (hadron) cycle [2].
Mechanism. As documented in the milestone derivation of the measured value of Planck's constant h from first principles [5], the axis of 1-state rotation in the proton cycle is at a 45 degree angle to each side of the simulated volumes. In other words, motion of positive fractional charges in segments of the proton cycle projects directly to planes parallel with the six sides of the simulated volumes.
This key feature of the geometry of the present experimental design supports the conclusion that the proton cycle is both linked with the Hall effect and is the underlying mechanism producing the Hall effect.
References
[1] Edwin Hall (1879). "On a New Action of the Magnet on Electric Currents". American Journal of Mathematics. 2 (3): 287–92.
[2] Keene, J. J. "Proton and electron bit cycles" JBinMech April, 2015.
[3] Keene, J. J. "Binary mechanics postulates" JBinMech November, 2020.
[4] Keene, J. J. "Binary mechanics cosmology" JBinMech December, 2025.
[5] Keene, J. J. "Intrinsic proton spin derivation" JBinMech December, 2018.
© 2026 James J Keene
