The universe began with the Big Bang, a prevailing scientific opinion. However, this Big Bang was not an explosion of 4-dimensional spacetime, which is also a current prevailing scientific opinion, but an explosion of dimensions that started in $(-1)$-dimensional void. The only exception to the general formula for the integral of a volume unit $a^n$, where a is the length of the edge in any dimension n is $n=-1$, where a jump discontinuity occurs and $\int a^{-1}da=ln\left(a\right)$. A first 0-dimensional point appeared in the $(-1)$-dimensional void, inducing the appearance of other points that were indistinguishable from the first. The breach made by the first operation of the dimensional successor function of the Peano axioms inevitably continued leading to the formation of 1-dimensional, real and imaginary lines, allowing for an ordering of points using multipliers of real units (ones) or imaginary units ($a\in \mathbb{R}\iff a=1b$, and $a\in \mathbb{I}\iff a=ib$, where $b\in \mathbb{R}$). Then, out of the two lines of each kind, crossing each other only at one initial point $(0,0)$, the dimensional successor function formed 2-dimensional ${\mathbb{R}}^2$, ${\mathbb{I}}^2$, and $\mathbb{R}\times \mathbb{I}$ Euclidean planes, with ${\mathbb{I}}^2$ being a mirror reflection of ${\mathbb{R}}^2$. Thus, forming n-dimensional Euclidean spaces ${\mathbb{R}}^a\times {\mathbb{I}}^b$ with $a\in \mathbb{N}$ real and $b\in \mathbb{N}$ imaginary lines, $n=a+ib$, and the scalar product defined by where $\mathbf{x},\mathbf{y}\in {\mathbb{R}}^a\times {\mathbb{I}}^b$. With the appearance of the first 0-dimensional point, information has begun to evolve [1,2,3,4,5,6,7,8,9], initially using undirected exploration in a selectionless [8] and a timeless [9] assembly process.

The mathematical properties of particular dimensions are not the same. For regular convex n-polytopes, for example, there are countably infinitely many regular convex polygons, five regular convex polyhedra (Platonic solids), six regular convex 4-polytopes and only three regular convex n-polytopes if $\left|n\right|>4$ and $\left|n\right|<0$ [10]. In particular, a 4-dimensional Euclidean space is endowed with a peculiar property known as exotic ${\mathbb{R}}^4$ [11], absent in other dimensionalities. Owing to this property, ${\mathbb{R}}^3\times \mathbb{I}$ Euclidean space provides a continuum of homeomorphic but non-diffeomorphic differentiable structures. Each piece of individually memorized information is homeomorphic to the corresponding piece of individually perceived information but remains non-diffeomorphic (non-smooth). This allows the variation of phenotypic traits within populations of individuals [12]. Hence, selection [8] and time [9] emerged and the evolution of information had to inevitably exploit directed exploration provided by biological evolution. Exotic ${\mathbb{R}}^4$ solves the problem of extra dimensions of nature, and perception requires a natural number of (thus independent) dimensions to form perceived space [13]. Each biological cell and each biological agent perceives an emergent space of three real dimensions and one imaginary (time) observer-dependently [14] and at present, when $i0=0$ is real, through a spherical Planck triangle corresponding to one bit of information in units of $-c^2$, where c is the speed of light in vacuum. This is the principle of emergent dimensionality (ED) [5,6,9,12,15,16].

Human perception involves measuring and measuring requires measurement units. In 1899 Max Planck derived the natural units of measure as "independent of special bodies or substances, thereby necessarily retaining their meaning for all times and for all civilizations, including extraterrestrial and nonhuman ones" [17]. Planck units utilize the Planck constant h introduced in his black-body radiation formula. Earlier, in 1881, George Stoney derived a system of natural units [18] based on the elementary charge e (Planck’s constant was unknown at this time). The ratio of Stoney units to Planck units is $\sqrt{\alpha }$, where $\alpha$ is the fine-structure constant. This study derives three complementary sets of Planck units based on the three complementary fine-structure constants established by the Fresnel coefficients for the normal incidence of electromagnetic radiation on monolayer graphene (MLG) and hints at certain areas of their applicability.

The paper is organized as follows. Section 2 shows that the Fresnel coefficients for the normal incidence of electromagnetic radiation on MLG include three complementary fine-structure constants, including two negative ones and hinting that the negative complementary fine-structure constant ${\alpha }_2$ established by the graphene reflectance is dual to the fine-structure constant $\alpha$. Section 3 presents complementary sets of ${\alpha }_{*}$-Planck units established by these three fine-structure constants. Section 4 introduces the concept of a black-body object in thermodynamic equilibrium that emits perfect black-body radiation and reviews its necessary properties. Section 5 introduces complex energies expressed in terms of real and imaginary ${\alpha }_{-}$-Planck units and applies them to black-body objects to show that in this model $\alpha$ and ${\alpha }_2$ are indeed dual to each other. Section 6 considers the observed mergers of black-body objects to show that the observed data can be explained without the need to introduce hypothetical exotic stellar objects. Section 7 discusses fluctuations of black-body objects. Section 8 defines complex forces based on the products of the complex energies. The complex force between real masses and imaginary charges is applied in Section 9 to the Bohr model of the hydrogen atom and in Section 10 to derive a black-body object surface gravity and the generalized Hawking radiation temperature. Section 11 summarizes the findings of this study. Certain prospects for further research are provided in the Appendices.

Numerous publications provide Fresnel coefficients for the normal incidence of electromagnetic radiation (EMR) on monolayer graphene (MLG), which are remarkably defined only by $\pi$ and the fine-structure constant $\alpha$ having the inverse where $q_{\mathrm{P}}$ is the Planck charge, ℏ is the reduced Planck constant, ${ϵ}_0\approx 8.8542e-12\left[{\mathrm{kg}}^{-1}·{\mathrm{m}}^{-3}·{\mathrm{s}}^2·{\mathrm{C}}^2\right]$ is the vacuum permittivity (electric constant), and e is the elementary charge. We choose this set of units over [F$·{\mathrm{m}}^{-1}$] for ${ϵ}_0$, since the mass, length, time, and charge units can express all the electrical units and together with temperature, amount of substance, and luminous intensity, these are the base units of the International System of Quantities (ISQ). Furthermore, in this notation we see that ${ϵ}_0$ is dependent on the unit of time. Transmittance (T) of MLG for normal EMR incidence was derived from the Fresnel equation in the thin-film limit [19] (Eq. 3), whereas a spectrally flat absorptance (A) $\mathrm{A}\approx \pi \alpha \approx 2.3\%$ has been reported [20,21] for photon energies between approximately $0.8011e-19$ and $4.0054e-19$ [J]. T is related to reflectance (R) [22] (Eq. 53) as $\mathrm{R}={\pi }^2{\alpha }^2\mathrm{T}/4$, i.e, The above equations for T and R, as well as the equation for the absorptance were also derived [23] (Eqs. 29-31) based on the thin-film model (setting $n_s=1$ for the substrate). The sum of the transmittance (3) and reflectance (4) at normal EMR incidence on the MLG was derived [24] (Eq. 4a) as where $\eta \approx 376.73$ [$\mathrm{kg}·{\mathrm{m}}^2·\mathrm{s}·{\mathrm{C}}^{-2}$] is the vacuum impedance, $\sigma =e^2/(4\hslash )=\pi \alpha /\eta \approx 6.0853e-5$ [${\mathrm{kg}}^{-1}·{\mathrm{m}}^{-2}·{\mathrm{s}}^{-1}·{\mathrm{C}}^2$] is the MLG conductivity [25], k is the wave vector of light in vacuum, and $\chi =0$ is the electric susceptibility of vacuum. Therefore, these coefficients are well established both theoretically and experimentally [19,20,21,24,26,27].

As a consequence of the conservation of energy In other words, the transmittance in the Fresnel equation describing the reflection and transmission of EMR at normal incidence on a boundary between different optical media is, in the case of the 2-dimensional (boundary) MLG, modified to include its absorption (cf. Appendix C). We note that each conservation law is associated with a certain symmetry, as asserted by Noether’s theorem. In this case, the symmetry involves the fine-structure constant $\alpha$ and $\pi$.

Natural units can be derived from numerous starting points [6,29] (cf. Appendix E). The central assumption in all systems of natural units is that the quotient of the unit of length ${\ell }_{*}$ and time $t_{*}$ is a unit of speed; we call it $c={\ell }_{*}/t_{*}$. It is the speed of light in vacuum c in all systems of natural units, except for Hartree and Schrödinger units, where it is $c\alpha$, and Rydberg units, where it is $c\alpha /2$. Hence, $\alpha$ is coupled with c in Hartree, Schrödinger, and Rydberg measurement units.

We chose Planck units over other natural unit systems not only because they incorporate the fine-structure constant $\alpha$ and the Planck constant h. Other systems of natural units (except for Stoney units) also incorporate them. This is because only the Planck area defines one bit of information on a patternless black hole surface given by the Bekenstein bound (51) and the binary entropy variation [5,6].

The fine-structure constant can be defined as the quotient (2) of the squared (and thus positive) elementary charge e and the squared Planck charge $\alpha =e^2/q_{\mathrm{P}}^2$. To accommodate the negativity of the fine-structure constants ${\alpha }_{-}$ discovered in the preceding section, we must introduce the imaginary Planck charge such that its square would yield a negative value.

For example, in the case of ${\alpha }_2$ Therefore We note that an imaginary $q_{\mathrm{P}-}$, which must have a physical definition analogous to $q_{\mathrm{P}}$, requires either a real, negative speed of light or some complementary real, negative electric constant (h is positive). Let us call them $c_{-}$ and $\widetilde{{ϵ}_0}$ From this equation, we find that $\widetilde{{ϵ}_0}{c}_{-}<0$, because the values of the other constants are known.

The fine-structure constant can also be defined as the ratio of Coulomb’s law for two elementary charges to Newton’s law of gravity for two Planck masses separated by the same distance. Constructing the same ratio for the remaining ${\alpha }_{*}$ leads to where the area of a 3-ball ($4\pi R^2$) in the denominator of the Coulomb force requires further investigation. Hence, the quotient of the squared Planck charge and mass must be the same for all sets of Planck units. Therefore is independent of the unit of time (even though ${ϵ}_0$ and G are) and introduces the imaginary Planck masses $m_{\mathrm{P}-}$. The masses $m_{\mathrm{P}*}$ can be calculated from the equation (29) by determining the value of the Planck charge $q_{\mathrm{P}*}$ from the equation (26). From (29) we also conclude that $\widetilde{{ϵ}_0}={ϵ}_0>0$ and then by (30) that ${\mu }_2>0$ and $c_2<0$. Next, we assume that the solution of Maxwell’s equations in vacuum is valid for other values of the constants involved. Let us call the unknown magnetic constant, corresponding to $c_{*}$, ${\mu }_{*}$. Therefore, from which the bivalued $c_{*}=\pm 1/\sqrt{{\mu }_{*}{ϵ}_0}$ can be obtained. Unlike the electric constant ${ϵ}_0$, magnetic constants ${\mu }_{*}$ are independent of the unit of time. Furthermore, Maxwell’s equations in vacuum are not directly dependent on the fine-structure constant(s), which is included in the magnetic constant ${\mu }_0$. Finally, combining relations (26) and (27) yields which establishes the universality of the elementary charge e that defines both matter and antimatter and leads to the following important relation between the speeds of light in vacuum $c_{*}$, and the fine-structure constants ${\alpha }_{*}$ $c\alpha$ is also the velocity of the electron in the first circular orbit in the Bohr hydrogen atom model to which we shall return in Section 9. Because $c_{*}$ derivable from the final Maxwell equation (30) are bivalued, all sets of ${\alpha }_{*}$-Planck units have four forms equal in modulus: real positive, real negative, imaginary positive and imaginary negative. However, due to the relation (32), we consider mostly real, positive ${\alpha }_{+}$-Planck units and imaginary, positive ${\alpha }_{-}$-Planck units. We note that switching the signs of $c_{*}$ in the relation (32) would require the imaginary charge $e_i=ie$. Therefore, we consider real Planck units associated with the positive speed of light $c_{2+}$ associated with ${\alpha }_2$ in the Appendix D only.

Complementary speeds of light $c_{*}$ (32) introduce complementary sets of Planck units, wherein basic ${\alpha }_{-}$-Planck units are imaginary since they are defined as square roots containing $c_{-}$ raised to odd powers (1, 3, 5). Furthermore, the speed of electromagnetic radiation is the product of its wavelength and frequency, and these quantities would be imaginary in terms of imaginary Planck units; a negative speed of light is necessary to accommodate this.

${\alpha }_2$-Planck units that can be expressed, using the relation (32), in terms of base Planck units $q_{\mathrm{P}}$, ${\ell }_{\mathrm{P}}$, $m_{\mathrm{P}}$, $t_{\mathrm{P}}$, and $T_{\mathrm{P}}$ are Most Planck units derived from the ${\alpha }_2$-Planck base units (33)-(37) are also imaginary. These include the ${\alpha }_2$ Planck volume the ${\alpha }_2$ Planck momentum the ${\alpha }_2$ Planck energy and the ${\alpha }_2$ Planck acceleration However, the ${\alpha }_2$-Planck density and the ${\alpha }_2$-Planck area are real and negative. Interestingly, both Planck forces $F_{\mathrm{P}}$ and are strictly positive. The remaining sets of Planck units are listed in Table 1.

Contrary to the elementary charge e (26), there is no physically meaningful elementary mass $M_e=\pm 1.8592e-9\left[\mathrm{kg}\right]$ satisfying the relation (35) There is no physically meaningful elementary length $L_e\approx \pm 2.5927e-32\left[\mathrm{m}\right]$ satisfying the relation (34) or an elementary temperature $T_e\approx \pm 6.4450\times {10}^{26}\left[\mathrm{K}\right]$ satisfying the relation (37) and close to the Hagedorn temperature of grand unified string models, or an elementary period $t_e=\pm 1.1851e-38$ satisfying the relation (36) However, the relations between periods (48) and temperatures (47) are inverted. Hence, the energy-time version of Heisenberg’s uncertainty principle (HUP) is saturated using energy from the equipartition theorem for one bit of information [5,6,30] both by Planck temperatures and times and elementary temperatures and periods (48), (47)

The Planck charge relation (26) and charge conservation principle imply that the elementary charge e, unlike mass $M_e$ is the same in all systems of Planck units, even though the same forms of the relations (26) and (45) reflect the same forms of Coulomb’s law and Newton’s law of gravity, which are inverse-square laws.

In the following, where deemed appropriate, we express the physical quantities in Planck units: where uppercase letters M, E, Q, $\lambda$, R, and D denote masses, energies, charges, Compton wavelengths, radii, and diameters (or lengths), lowercase letters m, l, etc. denote multipliers of the real ${\alpha }_{+}$ and imaginary ${\alpha }_{-}$ Planck units, respectively, and q is an integer multiplier of the elementary charge e. The latter assumption is most likely too far-reaching, considering the fractional charges of quasiparticles, particularly in the open research problem of the fractional quantum Hall effect (cf. Appendix G) and energy-dependent fractional charges in electron pairing [31].

There are only three observable objects in nature that emit perfect black-body radiation: unsupported black holes (BHs, the densest), neutron stars (NSs), supported, as believed, by neutron degeneracy pressure, and white dwarfs (WDs), supported, as believed, by electron degeneracy pressure (the least dense). We collectively refer to these black-body objects (BBs). The spectral density in sonoluminescence, that is light emission by sound-induced collapsing gas bubbles in fluids, was also shown to have the same frequency dependence as black-body radiation [32,33]. Thus, sonoluminescence, particularly shrimpoluminescence [34], is probably emitted by collapsing micro-BBs. Micro-BH induced in glycerin by modulating acoustic waves has also been reported [35].

The term black-body object is not used in general relativity (GR) and standard cosmology, but standard cosmology scrunches under embarrassingly significant failings, not just tensions as is sometimes described, as if to somehow imply that a resolution will eventually be found [36]. In addition, James Webb Space Telescope data show multiple galaxies that grew too massive too soon after the Big Bang, which is a strong discrepancy with the $\Lambda$ cold dark matter model ($\Lambda$CDM) expectations of how galaxies formed at early times at both redshifts, even when considering observational uncertainties [37]. For example, the supermassive BH of J1120+0641 quasar with mass $M_{\mathrm{BH}}=1.52\pm 0.17\times {10}^9{M}_{\odot }$ assembled in less than 0.77 billion years after the Big Bang [38,39]. This is an important unresolved issue, indicating that fundamental changes to the reigning $\Lambda$CDM model of cosmology are required [37]. In particular, it is well known that entropic gravity [30] explains the galaxy rotation curves without resorting to dark matter (dark matter is not required to explain the rotation curves of certain galaxies, such as the massive relic galaxy NGC 1277 [40]), has been experimentally confirmed [41], and is decoherence-free [42].

The term object as a collection of matter is a misnomer because it neglects the (quantum) nonlocality [9,43] that is independent of the entanglement among particles [44], as well as the Kochen-Specker contextuality [45], and increases as the number of particles increases [46,47]. Macro-realistic theories are false [48]. Thus, we use emphasis for (perceivably indistinguishable) particle and (perceivably distinguishable) object, as well as matter and distance. The ugly duckling theorem [49,50] asserts that every two objects we perceive are equally similar (or equally dissimilar). These terms do not have an absolute meaning in the ED. In particular, given the observation of quasiparticles in classical systems [51]. Within the ED framework, no object is enclosed in space. The interiors of the BBs are inaccessible to an exterior observer [52], which makes them similar to interior-less mathematical points representing real numbers on a number line. Thus, the term object is a particularly staring misnomer if applied to BBs.

It has been experimentally confirmed that (so-called) accretion instability is a fundamental physical process [53]. We conjecture that this process, which has already been recreated under laboratory conditions [54], is common for all BBs. As black-body radiation is a radiation of global thermodynamic equilibrium, it is patternless [55] (thermal noise) radiation that depends only on one parameter. In the case of BHs, this is known as Hawking [56] radiation, and this parameter is the BH temperature $T_{\mathrm{BH}}=T_{\mathrm{P}}/\left(2\pi d_{\mathrm{BH}}\right)$ corresponding to the BH diameter [6] $D_{\mathrm{BH}}=d_{\mathrm{BH}}{\ell }_{\mathrm{P}}$. Furthermore, BHs absorb patternless information [6,57]. Therefore, because Hawking radiation depends only on the diameter of a BH, it is the same for a given BH, even though it fluctuates (cf. Section 7).

As black-body radiation is patternless, triangulated [6] BBs contain a balanced number of Planck area triangles, each having binary potential $\delta {\phi }_k=-c^2·\{0,1\}$, as has been shown for BHs [6], based on the Bekenstein-Hawking (BH) entropy [52] $S_{\mathrm{BH}}=k_{\mathrm{B}}{N}_{\mathrm{BH}}/4$, where $N_{\mathrm{BH}}4\pi R_{\mathrm{BH}}^2/{\ell }_{\mathrm{P}}^2=\pi d_{\mathrm{BH}}^2$ is the information capacity of the BH surface, i.e., the $⌊N_{\mathrm{BH}}⌋\in {\mathbb{N}}_0$ Planck triangles corresponding to bits of information [5,6,30,52,58], and the fractional part triangle(s) having the area $\left\{N_{\mathrm{BH}}\right\}{\ell }_{\mathrm{P}}^2=(N_{\mathrm{BH}}-⌊N_{\mathrm{BH}}⌋){\ell }_{\mathrm{P}}^2$ too small to carry a single bit of information [5,6], where "$⌊x⌋$" is the floor function that yields the greatest integer less than or equal to its argument x.

BH entropy can be derived from the Bekenstein bound which defines an upper limit on the thermodynamic entropy S that can be contained within a sphere of radius R and energy E. Substituting BH (Schwarzschild) radius $R_{\mathrm{BH}}=2G{M}_{\mathrm{BH}}/c^2$ and mass-energy equivalence $E_{\mathrm{BH}}=M_{\mathrm{BH}}{c}^2$, where $M_{\mathrm{BH}}$ is the BH mass, into the bound (51), it reduces to the BH entropy. In other words, the BH entropy saturates the Bekenstein bound (51). Furthermore, the Bekenstein bound can be derived from the BH entropy where we used $M\le {\displaystyle \frac{R{c}^2}{2G}}$ and $E=M{c}^2$.

The patternless nature of perfect black-body radiation was derived [6] by comparing the BH entropy with the binary entropy variation $\delta S=k_{\mathrm{B}}{N}_1/2$ ([6] Eq. (55)), which is valid for any HS, where $N_1\in \mathbb{N}$ denotes the number of active Planck triangles with a binary potential $\delta {\phi }_k=-c^2$. Thus, the entropy of all BBs is Furthermore, $N_1=N_{\mathrm{BB}}/2$ confirms the patternless thermodynamic equilibrium of BBs by maximizing Shannon entropy [6]. In complex Euclidean ${\mathbb{R}}^a\times {\mathbb{I}}^b$ space, a n-ball ($n=a+bi\in \mathbb{C}$) is spherical only for $b=0$, i.e. when perceived. Not only BBs are perfectly spherical, when perceived; their mergers (cf. Section 6) are also perfectly spherical. Furthermore, the trigonometric member of its volume and surface formulas vanishes for the radius $r=1/\sqrt{\pi }$ ($R={\ell }_{\mathrm{P}}/\sqrt{\pi }$) [5,16], resulting in its information capacity $N=4$, one unit of BH entropy [52]. This corroborates the universality and applicability of BH entropy (53) to all BBs. Furthermore, some binary strings of of length $N_{\mathrm{BB}}\ge 4$ can be assembled in less than $N_{\mathrm{BB}}-1$ steps. There is no disorder or uncertainty in a binary string of length $N_{\mathrm{BB}}\le 3$ [9].

We shall define the generalized radius of a BB (this definition applies to all HSs) having mass $M_{\mathrm{BB}}$ as a function of $G{M}_{\mathrm{BB}}/c^2$ multiplier $k\in \mathbb{R},k\ge 2$ and the generalized BB energy $E_{\mathrm{BB}}$ (this definition also applies to all HSs) as a function of $M_{\mathrm{BB}}{c}^2$ multiplier $a\in \mathbb{R}$ Substituting $M_{\mathrm{BB}}$ from the generalized radius definition (54) into the generalized BB energy definition (55) and the latter into the Bekenstein bound (51), we obtain which equals BB entropy (53) if ${\displaystyle \frac{a}{2k}}={\displaystyle \frac{1}{4}}\iff a={\displaystyle \frac{k}{2}}$. Thus, the energy (55) of all BBs with a generalized radius (54) turns into with $k=2$ in the case of the BHs, setting the lower bound for the other BBs. We further call coefficient k the size-to-mass ratio (STM). This is similar to the specific volume (reciprocal of density) of BB. We derive the upper STM bound in Section 5. The energy (57) of BBs other than BHs (i.e., for $k>2$) exceeds the mass-energy equivalence $E=M{c}^2$, which is the limit of the maximum real energy. Therefore, we consider this surplus energy that exceeds $M_{\mathrm{BB}}{c}^2$ as related to charge and we shall model it as imaginary and thus unmeasurable.

According to the no-hair theorem, all BHs general relativity (GR) solutions are characterized by only three parameters: mass, electric charge, and angular momentum. However, BHs are fundamentally uncharged, because the parameters of any conceivable BH, in particular, charged (Reissner–Nordström) and charged-rotating (Kerr–Newman) BH, can be arbitrarily altered, provided that the BH area does not decrease [59] using Penrose processes [60,61] to extract the electrostatic and/or rotational energy of BH [62]. Thus, any BH is defined by only one real parameter: its diameter, mass, temperature, energy, etc., each differing from each other by a multiplicative constant. A BH can embrace the real number that defines it as a curvature of a spherical triangle corresponding to one bit of classical information. The area of a spherical triangle is larger than that of a flat triangle defined by the same vertices and depends on its curvature.

However, it is accepted that in the case of NSs, electrons combine with protons to form neutrons, such that NSs are composed almost entirely of neutrons. However, it is never the case that all electrons and protons of an NS become neutrons. WDs are charged by definition because they are believed to be mostly composed of electron-degenerate matter. But how can a charged BB store both its curvature corresponding to its mass and an additional parameter corresponding to its charge? Fortunately, the relation (26) ensures that the elementary charge is the same in all systems of Planck units. Therefore, charge of a spherical Planck triangle of a BB surface can link the perceivable Euclidean space ${\mathbb{R}}_{\alpha }^3\times {\mathbb{I}}_{\alpha }$ parameterized by Planck units with the parameterization provided by one of the remaining negative fine-structure constants ${\alpha }_{-}$. This can be considered in a polyspherical coordinate system, in which gravitation/acceleration acts in a radial direction (with the entropic gravitation acting inwardly and acceleration acting in both radial directions) [6], while electrostatics act in a tangential direction. Contrary to the no-hair theorem, we characterize BBs only by mass and charge, neglecting the angular momentum because the latter introduces the notion of time, which we find redundant in the BB description of a patternless thermodynamical equilibrium. Time is required for directed exploration only [8,9].

A complex energy formula where $E_{M_R}$ and $i{E}_{Q_R}$ represent the real and imaginary energies of an object having mass $M_R$ and charge $Q_R$ was proposed in ref. [63]. Equation (58) considers the real masses $M_R$ and charges $Q_R$ which in ref. [63] are defined in CGS units. We adopted SI units and modified this formula to a form involving real physical expressed in terms of real ${\alpha }_{+}$-Planck units and imaginary quantities expressed in terms of the imaginary ${\alpha }_{-}$-Planck units using relations (32), (35), (40), (50), and (31) as To this end, we defined the following three complex energies linking mass, imaginary mass, and charge: the complex energy of real mass and imaginary charge of real charge and imaginary mass and of real mass and imaginary mass We neglected the energy of real and imaginary charges $E_{Q{Q}_i}$, because by equation (26), the elementary charge is the same in all systems of measurement units, and hence we use the same elementary charge multiplier q in (50). Furthermore, the mass-energy equivalence relates the mass $M_{+}$ or $M_{-}$ to the speed of light $c_{+}$ or $c_{-}$, which subsequently parameterizes both parts of the energies $E_{M{Q}_i}$ and $E_{Q{M}_i}$ (cf. Appendix F). We express all energies (60)-(62) in terms of the same Planck energy $E_{\mathrm{P}+}$ in order to be able to compare them, as we assume that they are fundamental to any object and particle. We also note that the complex conjugates $E_{M{Q}_i}$ and $\overline{E_{M{Q}_i}}$ of the energy (60) represent respectively particle and antiparticle. We note that antimatter obeys gravity [64], which is consistent with the definition (60) and the findings of this study.

In the remainder of this section we shall analyze the energies (60)-(62) for different pairs of $\{{\alpha }_{+},{\alpha }_{-}\}$ in order $\{{\alpha }_1,{\alpha }_2\}$, $\{{\alpha }_4,{\alpha }_2\}$, $\{{\alpha }_1,{\alpha }_3\}$, and $\{{\alpha }_4,{\alpha }_3\}$. Our aim is to determine which pair is the most plausible physically.

Energies (60) and (61) yield two different charge energies corresponding to the elementary charge, imaginary quantum and the - larger in modulus - real quantum Furthermore, $\forall q$, ${\alpha }_{+}^2{E}_{Q_{-}}=i{\alpha }_{-}^2{E}_{Q_{+}}$.

The universal character of the charges is additionally emphasized by the real $\sqrt{\alpha }$ multiplied by i in the imaginary charge energy (60) and imaginary $\sqrt{{\alpha }_2}$ in the real charge energy (61).

The squared moduli of the complex energies (60)-(62), expressed in terms of the Planck energy, are ad shown in Figure 2.

Equalities of the squared moduli of the complex energies lead to the following results Because the mass multiplier $m_{-}\in \mathbb{R}$, the square root argument must be nonnegative in equation (71), which leads to (the sign of this inequality changes, as ${\alpha }_{-}<0$) which is imaginary for $\sqrt{\alpha ({\alpha }^4/{\alpha }_3^4-1)}$ ruling out the third pair of alphas $\{{\alpha }_1,{\alpha }_3\}$. Otherwise, we would have to exclude objects having the same energies $E_{M{Q}_i}$ and $E_{Q{M}_i}$ from our model. Furthermore, for $q=0$ equation (71) yields which corresponds to equation (66).

Furthermore, the relation (74) means that the masses of the uncharged micro-BHs ($q=0$) can be arbitrary ($m\ge 0$). However, micro NSs and micro WDs, which are also in thermodynamic equilibrium, are charged. Thus, even a single elementary charge ($q=1$) of a white dwarf renders its minimum allowable mass (74) $M_{\mathrm{WD}}\ge 5.7275\times {10}^{-10}\left[\mathrm{kg}\right]$ comparable to the mass of a sand grain.

We can interpret the squared generalized energy of the BBs (57) as the squared modulus of each of the complex energies (68)-(70). That is, for $E_{M{Q}_i}$ (60) and for $E_{M{M}_i}$ (62), we assume that the real mass energy $E_{\mathrm{BB}}=M_{\mathrm{BB}}{c}^2$ of the BB is the observable real part of this complex energy. For $|E_{M{Q}_i}{|}^2$ (68) we have where $q_{\mathrm{BB}}^2{\alpha }_{+}$ represents the charge surplus energy exceeding $M_{\mathrm{BB}}{c}^2$. We note that $k=2$ implies $q_{\mathrm{BB}}=0$, confirming the vanishing charge of BHs. For $|E_{M{M}_i}{|}^2$ (70) we have where the square root argument must be nonnegative, which implies (along with (82)), as expected, $k\ge 2$, where $k=2$ implies $m_{i\mathrm{BB}}=0$. Finally, for $|E_{Q{M}_i}{|}^2$ (69) we have where we substituted $q_{\mathrm{BB}}^2{\alpha }_{+}$ as a function of $m_{\mathrm{BB}}^2$ from equation (82). Comparison of $m_{i\mathrm{BB}}$ given by (84) with $m_i$ given by (71) also leads to $k\ge 2$, and for $k=2$ or $q=0$ corresponds to the balance of uncharged masses (66), (75) which are unrelated to any assumptions regarding BB energy and independent of STM. Comparing equations (71), as a function of charge q, and (84), as a function of the STM k, leads, as expected, to the relation (82).

Furthermore, the square root argument in equation (84) must be nonnegative, because $m_{\mathrm{BB}},m_{i\mathrm{BB}}\in \mathbb{R}$. This leads to the maximum STM-bound where again third pair of examined alphas $\{{\alpha }_1,{\alpha }_3\}$ must be ruled out, while pairs $\{{\alpha }_2,{\alpha }_4\}$ and $\{{\alpha }_3,{\alpha }_4\}$ do not allow to extend our model above the STM of BH. The relations (82) and (84) are shown in Figure 3 for $\alpha$ and ${\alpha }_2$.

The maximum STM-bound $k_{\max }$ (85) sets the bounds on BB energy (57), mass, and radius (54) In particular, using the relations (50), $2m_{\mathrm{BB}}\le r_{\mathrm{BB}}\le k_{\max }{m}_{\mathrm{BB}}$ or $r_{\mathrm{BB}}/k_{\max }\le m_{\mathrm{BB}}\le r_{\mathrm{BB}}/2$.

From the relation (77) we see that uncharged BHs ($q=0$) cannot remain in a nonvanishing equilibrium of complex energies (76), even though they are in thermodynamic equilibrium. It was shown [5] based on the Mandelstam-Tamm [65], Margolus–Levitin [66], and Levitin-Toffoli [67] theorems on the quantum orthogonalization interval that a BH is represented by a qubit $|{\psi }_{\mathrm{BH}}\rangle ={\displaystyle \frac{1}{\sqrt{2}}}\left(|0\rangle +|E_{\mathrm{BH}}\rangle \right)$ in an equal superposition of the eigenstates corresponding to the BH energy $E_{\mathrm{BH}}=M_{\mathrm{BH}}{c}^2$ and vanishing ground state energy. However, such a nonvanishing equilibrium of complex energies (76) is possible for charged BBs. We obtain it by substituting squared generalized energy of the BBs (57) into equation (76) as $A^2={\displaystyle \frac{1}{4}}{k}^2{m}_{\mathrm{BB}}^2$ and solving for $\left(1+{\alpha }_{-}^4/{\alpha }_{+}^4\right)m_{\mathrm{BB}}^2$. This yields where all three energies are equal. The equilibrium $k_{\mathrm{eq}}$ (87) and maximum $k_{\max }$ (85) STMa satisfy $k_{\mathrm{eq}}^2+16/k_{\max }^2=8$ (which resolves to $2\le |k_{\mathrm{eq}}|<2\sqrt{2},\left|k_{\max }\right|\ge \sqrt{2},\forall k_{\mathrm{eq}},k_{\max }\ne 0\in \mathbb{R}$).

The relations (74), (85), and (87) show that the negative fine-structure constant ${\alpha }_{-}$ corresponding to $\alpha$ is ${\alpha }_2$ not ${\alpha }_3$. As we have seen in Section 2.3, $\alpha$ and ${\alpha }_2$ are dual to each other. The BB having the STM $k_{\mathrm{eq}}\approx 2.7665$ (87) and the elementary charge ($q^2=1$) would have mass $M_{\mathrm{BB}}\left(k_{\mathrm{eq}}\right)\approx \pm 1.9455e-9\left[\mathrm{kg}\right]$, imaginary mass $M_{i\mathrm{BB}}\left(k_{\mathrm{eq}}\right)\approx \pm i1.7768e-9\left[\mathrm{kg}\right]$, corresponding to the Compton wavelength ${\lambda }_{\mathrm{BB}}\left(k_{\mathrm{eq}}\right)\approx \pm 1.1361e-33\left[\mathrm{m}\right]$, and the imaginary Compton wavelength ${\lambda }_{i\mathrm{BB}}\left(k_{\mathrm{eq}}\right)\approx \pm i1.2160e-33\left[\mathrm{m}\right]$. On the other hand, equation (82) provides the BB charge in equilibrium (76) as $q_{\mathrm{BB}}\left(k_{\mathrm{eq}}\right)\approx 11.1874m_{\mathrm{BB}}$ and the limit of the BB charge $q_{\mathrm{BB}}\left(k_{\max }\right)\approx 37.9995m_{\mathrm{BB}}$.