**A. Appendix**

Essential differences are present in each panel of **Figure 10**, there are both plots, which are simply related to the elaboration pursued in their generation. A panel of the left for the protons gives just the proton flux observed in a steady (undisturbed) strong **B-**field interval as a function of a very well-collimated radial velocity from the Sun of the proton population selected through a process of triple coincidence that includes detectors independent evaluation of the start time each proton enters, followed of the time of impact in the stopping location, and charge, mass, and energy by a strong magnetic field sweep that by deflection of the particle identifies the ion's unique properties of interest, see, e.g. [56]. Further, the plot evaluates the intensity of the proton's flux from the Sun as a physical invariant property of the interval of observation which is extremely small and consistent with the well-known high vacuum of the interplanetary medium near the distance Ulysses is found at the time of observation in its initial trajectory from the Earth to Jupiter, as indicated in the work of these observations, i.e. [37]. While in the case of the very detailed study of the electron distribution in Nieves-Chinchilla and Figueroa Viñas [38] with the SWE instrument in Wind SC

$$\begin{aligned} \text{(\$\text{Energy}\$ (electron \sim \text{\textquotedblleft } \text{m}\_{\text{e}} \text{ (0.8 kms}^{-1})\$^{2}\$) / Energy} &(\text{proton} \sim \text{\textquotedblleft } \text{m}\_{\text{p}} \text{ (21 kms}^{-1})^{2}\$) ^{\*} \\ \text{mass} (\text{p}) / \text{mass} (\text{electron}) &\text{ is in the order of 1836.6.} \end{aligned} \tag{24}$$

But still, we can do one more normalization related to the general radiative cooling of the strong magnetized matter, which increases with SC distance from the Sun, which for Ulysses is close to 3 AU from the Sun versus 1 AU for the location of Wind SC, and the time of the respective measurements. This is valid when we consider the radiative cooling of matter with time in the interplanetary space and relate the observations with Ulysses of the proton. This behavior is well understood for a frozen matter estate of nature which depending on its mass produces tighter oscillation for smaller mass; versus larger oscillation amplitude for larger masses. Henceforth, strong support for the constitutive nature of the ideal magnetized matter is identified through very delicate analysis using very clean SWE in Wind SC, and SWICS in Ulysses SC plasma instruments. Here we deal with far smaller uncertainties resolution than what is available with most instruments from past and planned missions.

Hence, the obtained ratio close to one is a quite strong argument favoring the 3-D Langmuir amorphous crystalline nature of the low beta ideal MHD matter structure.
