The universe began with the Big Bang, which is a current prevailing scientific opinion. But this Big Bang was not an explosion of 4-dimensional spacetime, which also is a current prevailing scientific opinion, but an explosion of dimensions. More precisely, in the $-1$-dimensional void, a 0-dimensional point appeared, inducing the appearance of countably infinitely other points indistinguishable from the first one. 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$1, $a\in \mathbb{I}\iff a=ib,b\in \mathbb{R}$). Then out of 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 ${\mathbb{R}}^2$. And so on, 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, $n≔a+b$, and the scalar product defined by where $\mathbf{x},\mathbf{y}\in {\mathbb{R}}^a\times {\mathbb{I}}^b$. With the onset of the first 0-dimensional point, information began to evolve [1,2,3,4,5,6].

However, dimensional properties are not uniform. Concerning regular convex n-polytopes in natural dimensions, 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 $n>3$ [7]. In particular, 4-dimensional Euclidean space is endowed with a peculiar property known as exotic ${\mathbb{R}}^4$ [8], absent in other dimensionalities. Thanks to this property, ${\mathbb{R}}^3\times \mathbb{I}$ 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 allowed for variation of phenotypic traits within populations of individuals [9] and extended the evolution of information into biological evolution. Exotic ${\mathbb{R}}^4$ solves the problem of extra dimensions of nature and perceived space requires a natural number of dimensions [10]. Each biological cell perceives emergent space of three real and one imaginary (time) dimension observer-dependently [11] 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 emergent dimensionality (ED) [5,9,12,13,14].

Each dimension requires certain units of measure. In real dimensions, the natural units of measure were derived by Max Planck in 1899 as "independent of special bodies or substances, thereby necessarily retaining their meaning for all times and for all civilizations, including extraterrestrial and non-human ones" [15]. Planck units utilize the Planck constant h that he introduced in his black-body radiation formula. However, already in 1881, George Stoney derived a system of natural units [16] based on the elementary charge e (Planck’s constant was unknown at that time). The ratio of Stoney units to Planck units is $\sqrt{\alpha }$, where $\alpha$ is the fine-structure constant. This study derives the complementary set of natural units applicable for imaginary dimensions, including the imaginary units, based on the discovered negative fine-structure constant ${\alpha }_2$ leading to the negative speed of light in vacuum $c_n$ greater in modulus than the speed of light c. Thus, the imaginary Planck energy $E_{\mathrm{P}i}$ and temperature $T_{\mathrm{P}i}$ are larger in moduli than the Planck energy $E_{\mathrm{P}}$ and temperature $T_{\mathrm{P}}$ setting more favorable conditions for biological evolution to emerge in ${\mathbb{R}}^3\times \mathbb{I}$ Euclidean space than in ${\mathbb{I}}^3\times \mathbb{R}$ Euclidean one due to the minimum energy principle.

The study shows that the energies of neutron stars and white dwarfs exceed their mass–energy equivalences and that excess energy is stored in imaginary dimensions and is inaccessible to direct observations. This corrects the value of the photon sphere radius and results in the upper bound on the size-to-mass ratio of their cores, where the Schwarzschild radius sets the lower bound.

The paper is structured as follows. Section II shows that Fresnel coefficients for the normal incidence of electromagnetic radiation on monolayer graphene include the second, negative fine-structure constant ${\alpha }_2$ as a fundamental constant of nature. Section III shows that by this second fine-structure constant nature endows us with the complementary set of ${\alpha }_2$-natural units. Section IV introduces the concept of a black-body object in thermodynamic equilibrium, emitting perfect black-body radiation, and reviews its necessary properties. Section V introduces complex mass, charge, and Compton energies expressed in terms of real and imaginary Planck units introduced in Section III and discusses equilibria of their moduli. Also, the equilibrium of all their moduli is applied to black-body objects to derive the range of their size-to-mass ratios and the equilibrium size-to-mass ratio. Section VI applies this range to the observed mergers of black-body objects to show that the observed data is explainable with no need to introduce hypothetical exotic stellar objects. Section VII discusses fluctuations of black-body objects. Section VIII defines complex forces to derive a black-body object surface gravity and the generalized Hawking radiation temperature. Section IX summarizes the findings of this study. Certain prospects for further research are given 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 $\alpha$ where $q_{\mathrm{P}}$ is the Planck charge, ℏ is the reduced Planck constant, ${ϵ}_0\approx 8.8542\times {10}^{-12}\left[{\mathrm{kg}}^{-1}·{\mathrm{m}}^{-3}·{\mathrm{s}}^2·{\mathrm{C}}^2\right]$ is vacuum permittivity (the electric constant), and e is the elementary charge. Transmittance (T) of MLG for normal EMR incidence was derived from the Fresnel equation in the thin-film limit [17] (Eq. 3), whereas spectrally flat absorptance (A) $\mathrm{A}\approx \pi \alpha \approx 2.3\%$ was reported [18,19] for photon energies between about $0.5$ and $2.5$ [eV]. T was related to reflectance (R) [20] (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 [21] (Eqs. 29-31) based on the thin film model (setting $n_s=1$ for substrate). The sum of transmittance (3) and the reflectance (4) at normal EMR incidence on MLG was derived [22] (Eq. 4a) as where $\eta \approx 376.73[\Omega ]$ is the vacuum impedance, $\sigma =e^2/(4\hslash )=\pi \alpha /\eta \approx 6.0853\times {10}^{-5}\left[{\Omega }^{-1}\right]$ is the MLG conductivity [23], k is the wave vector of light in vacuum, and $\chi =0$ is the electric susceptibility of vacuum. These coefficients are thus well-established theoretically and experimentally confirmed [17,18,19,22,24,25].

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) of MLG, modified to include its absorption.

The reflectance $\mathrm{R}=0.013\%$ (4) of MLG can be expressed as a quadratic equation with respect to $\alpha$This quadratic equation (8) has two roots with reciprocals Therefore, the equation (8) includes the second, negative fine-structure constant ${\alpha }_2$. It turns out that the sum of the reciprocals of these fine-structure constants (9) and (10) is remarkably independent of the value of the reflectance R. The same result can only be obtained for $\mathrm{T}+\mathrm{A}$ (cf. Appendix B). This result is intriguing in the context of a peculiar algebraic expression for the fine-structure constant [26] that contains a free $\pi$ term and is very close to the physical definition (2) of ${\alpha }^{-1}$, which according to the CODATA 2018 value is $137.035999084$. Notably, the value of the fine-structure constant is not constant but increases with time [27,28,29,30,31]. Thus, the algebraic value given by (12) can be interpreted as the initial Big Bang geometric ${\alpha }^{-1}$.

Using relations (11) and (12), we can express the negative reciprocal of the 2${}^{\mathrm{nd}}$ fine-structure constant ${\alpha }_2^{-1}$ that emerged in the quadratic equation (8) also as a function of $\pi$ only and this value can also be interpreted as the initial ${\alpha }_2^{-1}$, where the current value would amount to ${\alpha }_2^{-1}\approx -140.177591737$, assuming the rate of change is the same for $\alpha$ and ${\alpha }_2$.

The floor function of the inverse of the fine-structure constant $\alpha$ represents the threshold on the atomic number (137) of a hypothetical element feynmanium that, in the Bohr model of the atom, still allows the 1s orbital electrons to travel slower than the speed of light c. This raises the question of whether the fine-structure constants’ inverses correspond to the number of bits. Furthermore, the fine-structure constant has been reported as the quantum of rotation [32].

Using relations (12) and (13), T (3), R (4), and A (5) of MLG for normal EMR incidence can be expressed just by $\pi$. Moreover, equation (8) includes two $\pi$-like constants for two surfaces with positive and negative Gaussian curvatures (cf. Appendix C).

In this section, we shall derive the complementary set of ${\alpha }_2$-Planck units based on the second fine-structure constant ${\alpha }_2$, which are mostly bivalued and imaginary. Real Planck units are also bivalued with negative values provided by negative non-principal square roots. By choosing complex analysis, within the framework of ED, we enter into bivalence by the very nature of this analysis ($d=d^{2/2}=\sqrt{d^2}\stackrel{?}{=}\pm d)$ [14]. On the other hand, imaginary and negative physical quantities are the subject of research. In particular, the subject of scientific research is thermodynamics in the complex plane. For example, Lee–Yang zeros [33,34] and photon-photon thermodynamic processes under negative optical temperature conditions [35] have been experimentally observed. Nonetheless, physical quantities accessible for direct, everyday observation are mostly real and positive with the negativity of distances, velocities, accelerations, etc., induced by the assumed orientation of space.

Natural units can be derived from numerous starting points [5,36] (cf. Appendices Appendix D and 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 - let us call it c - $c={\ell }_{*}/t_{*}$. 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 $c\alpha /2$. On the other hand, c as the velocity of the electromagnetic wave is derivable from Maxwell’s Equations in vacuum where $\mathbf{E}$ is the electric field, ${ϵ}_0$ is vacuum permittivity (the electric constant) and ${\mu }_0$ is vacum permeability (the magnetic constant). Without postulating any solution to this equation but by simple substitution $\partial x={\ell }_{*}$ and $\partial t=t_{*}$, ${\partial }^{2}E=E_{*}$ factors out, and we obtain well known symmetric in its electric and magnetic parts [37] from which the value of $c^2$ can be obtained, knowing the values of ${\mu }_0$ and ${ϵ}_0$, yielding bivalued $c=\pm 1/\sqrt{{\mu }_0{ϵ}_0}$. We note that it is $c^2$, not c, present in mass-energy equivalence, the Lorentz factor, the BH potential, etc. We further note that Maxwell’s Equations in vacuum are not directly dependent on the fine-structure constant(s). It is sewn into the magnetic constant ${\mu }_0$.

In the following, we assume the universality of the real elementary electric charge e defining both matter and antimatter, the Planck constant h, the uncertainty principle parameter, and the gravitational constant G; i.e., we assume that there are no counterparts to these physical constants in other physical dimensions in our model. The last two assumptions are probably too far-reaching, given that we don’t need to know the gravitational constant G, the Planck constant h, or the speed of light c to find the product of the Planck length ${\ell }_{\mathrm{P}}$ and the speed of light c [38]. The fine-structure constant can be defined as the quotient (2) of the squared (and thus positive) elementary charge e and the squared (and thus also positive) Planck charge, $\alpha =e^2/q_{\mathrm{P}}^2$. We chose Planck units over other systems of natural units 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. The reason is that only the Planck area defines one bit of information on a patternless black hole surface given by the Bekenstein bound (47) and the binary entropy variation [5].

To accommodate a negative fine-structure constant discovered in the preceding section, we must introduce the imaginary Planck charge $q_{\mathrm{P}i}$ so that its square would yield a negative value of ${\alpha }_2$. Planck charge relation (16) and the charge conservation principle imply that the elementary charge e is the same in real and imaginary dimensions. Next, we note that an imaginary $q_{\mathrm{P}i}$, that must have a physical definition analogous to $q_{\mathrm{P}}$, requires either real and negative speed of light parameter or real and negative electric constant. Let us call them $c_n$ and $\widetilde{{ϵ}_0}$From this equation, we can find the value of the product $\widetilde{{ϵ}_0}{c}_n<0$, as the values of the other constants are known. Next, we assume that the solution (15) of Maxwell’s Equations in vacuum is valid also for other values of the constants involved. Let us call the unknown magnetic constant ${\mu }_2$, so From that and from $\widetilde{{ϵ}_0}{c}_n<0$, we conclude that also the product ${\mu }_2{c}_n<0$. We note that the quotient of the squared Planck charge and mass introduces the imaginary Planck mass $m_{\mathrm{P}i}$ the value of which can be calculated, knowing the value of the imaginary Planck charge $q_{\mathrm{P}i}$ from the relation (16). From (19) we also conclude that $\widetilde{{ϵ}_0}={ϵ}_0>0$ and then by (18) ${\mu }_2>0$. Finally, knowing $m_{\mathrm{P}i}$ we can determine the value of the negative, non-principal square root of $c_n=\pm 1/\sqrt{{\mu }_2{ϵ}_0}$ in (18) as which is greater than the speed of light in vacuum c in modulus2. Mass, length, time, and charge units can express all electrical units. Therefore, along with temperature, they can be considered base units. We further conclude that the magnetic constants are Contrary to the electric constant ${ϵ}_0$, the magnetic constants $\mu$ are time-independent. Furthermore, both ${\alpha }_2$ and $c_n$ lead to the second, also time-dependent but negative vacuum impedance Finally, relations (16) and (17) yield the following important relation between the speed of light in vacuum c, negative parameter $c_n$, and the fine-structure constants $\alpha$, ${\alpha }_2$Notably, $c\alpha$ is the electron’s velocity at the first circular orbit in the Bohr model of the hydrogen atom and the unit of speed in Hatree and Schrodinger natural units. This is not the only $\alpha$ to ${\alpha }_2$ relation. Along with the two $\pi$-like constants ${\pi }_1$, ${\pi }_2$ (relations (C8) and (C10), cf. Appendix C) The negative parameter $c_n$ (20) leads to the imaginary Planck charge $q_{\mathrm{P}i}$, length ${\ell }_{\mathrm{P}i}$, mass $m_{\mathrm{P}i}$, time $t_{\mathrm{P}i}$, and temperature $T_{\mathrm{P}i}$ that redefined by square roots containing $c_n$ raised to odd (1, 3, 5) powers become imaginary and bivalued and furthermore can be expressed, using the relation (24), in terms of base Planck units $q_{\mathrm{P}}$, ${\ell }_{\mathrm{P}}$, $m_{\mathrm{P}}$, $t_{\mathrm{P}}$, and $T_{\mathrm{P}}$.

Planck units derived from the imaginary base units (26)-(30) are mostly also imaginary. The ${\alpha }_2$ Planck volume the ${\alpha }_2$ Planck momentum the ${\alpha }_2$ Planck energy and the ${\alpha }_2$ Planck acceleration are imaginary and bivalued. The ${\alpha }_2$-Planck density and the ${\alpha }_2$-Planck area are strictly negative, while the Planck density ${\rho }_{\mathrm{P}}$ and area ${\ell }_{\mathrm{P}}^2$ are strictly positive. However, both Planck forces are strictly positive. We note that Coulomb’s law for elementary charges and Newton’s law of gravity for Planck masses define the fine-structure constants where $R_{*}$ is some real or imaginary distance. The area of a disk in the denominator of the Coulomb force invites further research.

Notably, the imaginary Planck Units are not imaginary due to being multiplied by the imaginary unit i. They are imaginary due to the negativity of odd powers of negative $c_n$ being the square root argument; thus, they define imaginary physical quantities inaccessible to direct measurements3. They do not apply only to the time dimension but to any imaginary dimension. However, in our four-dimensional Euclidean ${\mathbb{R}}^3\times \mathbb{I}$ space-time, Planck units apply in general to the spatial dimensions, while the imaginary ones in general to the imaginary temporal dimension. All the ${\alpha }_2$-Planck units have physical meanings. However, some are elusive, like the negative area or imaginary volume, which require two or three orthogonal imaginary dimensions. The speed of electromagnetic radiation is the product of its wavelength and frequency, and these quantities would be imaginary if factored by imaginary Planck units; the negative speed of light is necessary to accommodate it as $i^2=-1$. Therefore, non-principal square root of $c=\pm 1/\sqrt{{\mu }_0{ϵ}_0}$ and principal square root of $c_n=\pm 1/\sqrt{{\mu }_2{ϵ}_0}$ in (18) also introduce, respectively, imaginary $-c$-Planck units and real $-c_n$-Planck units. In particular, the imaginary $-c$-Planck time parametrizes the real to imaginary time relations [5,12]. However, these symmetric systems of units seem more appropriate for factoring physical quantities of ${\mathbb{I}}^3\times \mathbb{R}$ Euclidean space rather than ${\mathbb{R}}^3\times \mathbb{I}$ Euclidean one, that we perceive due to the minimum energy principle ($|E_{\mathrm{P}i}|>|E_{\mathrm{P}}|$). Furthermore, the relation (24) introduces an interesting interplay between $\alpha$ vs. ${\alpha }_2$ and c vs. $c_n$ that, as we conjecture, should be able to explain $\nu =5/2$ state in the fractional quantum Hall effect in 2D system of electrons, as well as other fractional states with an even denominator [39].

The relations between time (29) and temperature (30) ${\alpha }_2$-Planck units are inverted, ${\alpha }^5{t}_{\mathrm{P}i}^2={\alpha }_2^5{t}_{\mathrm{P}}^2$, ${\alpha }_2^5{T}_{\mathrm{P}i}^2={\alpha }^5{T}_{\mathrm{P}}^2$, and saturate the energy-time version of Heisenberg’s uncertainty principle (HUP) taking energy from the equipartition theorem for one degree of freedom (or one bit of information [5,40]4) Furthermore, eliminating $\alpha$ and ${\alpha }_2$ from the relations (27)-(28), yields Contrary to the elementary charge e (16), there is no physically meaningful elementary mass $M_e=\pm 1.8592\times {10}^{-9}\left[\mathrm{kg}\right]$ that would satisfy the relation (28) Neither is there a physically meaningful elementary (and imaginary) length $L_e\approx \pm i9.7382\times {10}^{-39}\left[\mathrm{m}\right]$ satisfying the relation (36) (which in modulus is almost 1660 times smaller than the Planck length), or an elementary temperature $T_e\approx \pm 6.4450\times {10}^{26}\left[\mathrm{K}\right]$ abiding to (30) and close to the Hagedorn temperature of grand unified string models. Thus, as to the modulus, charges are the same in real and imaginary dimensions, while masses, lengths, temperatures, and other derived quantities that can vary with time, may differ (the dimensional character of the charges is additionally emphasized by the real $\sqrt{\alpha }$ multiplied by i in the imaginary charge energy (70) and imaginary $\sqrt{{\alpha }_2}$ in the real charge energy (71)). We note that the same form of the relations (16) and (41) reflect the same form of Coulomb’s law and Newton’s law of gravity, which are inverse-square laws.

In the following, where deemed appropriate, we shall express the physical quantities by Planck units where uppercase letters M, Q, $\lambda$, R, D, E, and T denote respectively masses, charges, Compton wavelengths, diameters, radii (or lengths), Compton energies, and temperatures, lowercase letters (except temperatures, where "hats" are used) denote multipliers of the positive (principal square roots) Planck units, and the subscripts i refer to imaginary quantities. We note that the discretization of charges by integer multipliers q of the elementary charge e seems too far-reaching, considering the fractional charges of quasiparticles, in particular in the open research problem of the fractional quantum Hall effect.

Coulomb’s force $F_C$ is positive or negative, depending on the sign and type (real or imaginary) of charges, as summarized below Newton’s law of universal gravitation is also positive or negative, depending on the sign and type of masses, as summarized below However, it is larger in modulus in the case of imaginary masses. Unlike charges, negative, real masses are generally inaccessible for direct observation. We note that dissipative coupling between excitons and photons in an optical microcavity leads to the formation of exciton polaritons with negative mass [41]. Bose-Einstein exciton condensate is present in photosynthetic light-harvesting complexes already in ambient conditions [42]. In Section VI we will show that negative, real masses also result from merging black-body objects.

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 it is accepted, by neutron degeneracy pressure, and white dwarfs (WDs), supported, as it is accepted, by electron degeneracy pressure (the least dense). We shall collectively call them black-body objects (BBOs). It was also shown that the spectral density in sonoluminescence, light emission by sound-induced collapsing gas bubbles in fluids has the same frequency dependence as black-body radiation [43,44]. Thus, the sonoluminescence, and in particular shrimpoluminescence [45], is emitted by collapsing micro-BBOs. A micro-BH induced in glycerin by modulating acoustic waves was reported [46].

The term "black-body object" is not used in 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 [47]. Entropic gravity [40] explains galaxy rotation curves without resorting to dark matter, has been experimentally confirmed [48], and is decoherence-free [49]. It has been experimentally confirmed that the so-called accretion instability is a fundamental physical process [50]. We conjecture that this process, already recreated in laboratory conditions [51], is common for all BBOs. Also, 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 $\mathsf{\Lambda }$ cold dark matter model ($\mathsf{\Lambda }$CDM) expectations on how galaxies formed at early times at both redshifts, even when considering observational uncertainties [52]. This is an important unresolved issue indicating that fundamental changes to the reigning $\mathsf{\Lambda }$CDM model of cosmology are needed [52]. Therefore, the term object as a collection of matter is a misnomer, as it neglects (quantum) nonlocality [53] that is independent of the entanglement among the particles [54], as well as of Kochen-Specker contextuality [55], and increases as the number of particles grows [56]. Thus we use emphasis for (perceivably indistinguishable) particle and (perceivably distinguishable) object, as well as for matter and distance. The ugly duckling theorem [57,58] asserts that every two objects we perceive are equally similar (or equally dissimilar), however ridiculous and contrary to common sense5 that may sound. Therefore, these terms have no absolute meaning in ED. In particular, given the observation of quasiparticles in classical systems [59]. Within the framework of ED no object is enclosed in space.

As black-body radiation is radiation of global thermodynamic equilibrium, it is patternless [60] (thermal noise) radiation that depends only on one parameter. In the case of BHs, this is known as Hawking [61] 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 [5] $D_{\mathrm{BH}}=d_{\mathrm{BH}}{\ell }_{\mathrm{P}}$, where $d_{\mathrm{BH}}\in \mathbb{R}$. Furthermore, BHs absorb patternless information [5,62]. Therefore, since Hawking radiation depends only on the diameter of a BH, it is the same for a given BH, even though it is momentary as the BH fluctuates (cf. Section VII).

As black-body radiation is patternless, the triangulated [5] BBOs contain a balanced number of Planck area triangles, each carrying binary potential $\delta {\phi }_k=-c^2·\{0,1\}$, as it has been shown for BHs [5], based on Bekenstein-Hawking (BH) entropy [63] $S_{\mathrm{BH}}=k_{\mathrm{B}}{N}_{\mathrm{BH}}/4$, where $N_{\mathrm{BH}}=\pi d_{\mathrm{BH}}^2$ is the BH information capacity.

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. After plugging the 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 (47), it reduces to the BH entropy. In other words, the BH entropy saturates the Bekenstein bound (47).

The patternless nature of the perfect black-body radiation was derived [5] by comparing BH entropy with the binary entropy variation $\delta S=k_{\mathrm{B}}{N}_1/2$ ([5] Eq. (55)), valid for any entropy variation sphere (EVS), where $N_1\in \mathbb{N}$ denotes the number of active Planck triangles with binary potential $\delta {\phi }_k=-c^2$. Thus, the entropy of all BBOs is where $N_{\mathrm{BBO}}1234567894\pi R_{\mathrm{BBO}}^2/{\ell }_{\mathrm{P}}^2=\pi d_{\mathrm{BBO}}^2$ is the information capacity of the BBO surface, i.e., the $⌊N_{\mathrm{BBO}}⌋\in \mathbb{N}$ Planck triangles6 corresponding to bits of information [5,12,40,63,64], and the fractional part triangle(s) having the area $\left\{N_{\mathrm{BBO}}\right\}{\ell }_{\mathrm{P}}^2=(N_{\mathrm{BBO}}-⌊N_{\mathrm{BBO}}⌋){\ell }_{\mathrm{P}}^2$ to small to carry a single bit of information [sic!]. Furthermore, $N_1=N_{\mathrm{BBO}}/2$ confirms the patternless thermodynamic equilibrium of the BBOs by maximizing Shannon entropy [5].

We shall define the generalized radius of a BBO (this definition applies to all EVSs) having mass $M_{\mathrm{BBO}}$ as a function of $G{M}_{\mathrm{BBO}}/c^2$ multiplier $k\in \mathbb{R},k>0$ and the generalized BBO energy $E_{\mathrm{BBO}}$ as a function of $M_{\mathrm{BBO}}{c}^2$ multiplier $a\in \mathbb{R}$ (this definition also applies to all EVSs) Plugging $M_{\mathrm{BBO}}$ from (49) into (50) and the latter into the Bekenstein bound (47) it becomes and equals the BBO entropy (48) if $\frac{a}{2k}=\frac{1}{4}\Rightarrow a=\frac{k}{2}$. Thus, the energy of all BBOs having a radius (49) is with $k\ge 2$ and $k=2$ in the case of BHs, setting the lower bound for other BBOs. We shall further call the coefficient k the size-to-mass ratio (STM). It is similar to the specific volume (the reciprocal of density) of the BBO. We shall derive the upper STM bound in Section V D.

According to the no-hair theorem, all BHs general relativity (GR) solutions are characterized only by three parameters: mass, electric charge, and angular momentum. However, BHs are fundamentally uncharged since the parameters of any conceivable BH, in particular, charged (Reissner–Nordström) and charged-rotating (Kerr–Newman) BH, can be altered arbitrarily, provided that the BH area does not decrease [65] by means of Penrose processes [66,67] to extract BH electrostatic and/or rotational energy [68]. Thus any BH is defined by only one real parameter: its diameter (cf. [5] Figure 2b), mass, temperature, energy, etc., each corresponding to the other. We note that in the complex Euclidean ${\mathbb{R}}^3\times \mathbb{I}$ space, an n-ball ($n\in \mathbb{C}$) is spherical only for a vanishing imaginary dimension [14]. As the interiors of the BBOs are inaccessible to an exterior observer [63], BBOs do not have interiors7, which makes them similar to interior-less mathematical points representing the real numbers on a number line. Yet, a BH can embrace this defining parameter. That means that three points forming a Planck triangle corresponding to a bit of information on a BH surface can store this parameter, and this is intuitively comprehensible: the area of a spherical triangle is larger than that of a flat triangle defined by the same vertices, providing the curvature is nonvanishing, and depends on this curvature, i.e., this additional parameter defines it. Thus, the only meaningful spatial notion is the Planck area triangle, encoding one bit of classical information and its curvature.

On the other hand, it is accepted that in the case of NSs, electrons combine with protons to form neutrons so that NSs are composed almost entirely of neutrons. But it is never the case that all electrons and all protons of an NS become neutrons. WDs are charged by definition as they are accepted to be composed mostly of electron-degenerate matter. But how can a charged BBO store both the curvature and an additional parameter corresponding to its charge? Fortunately, the relation (16) ensures that charges are the same in real and imaginary dimensions. Therefore each charged Planck triangle of a BBO surface is associated with at least three $\mathbb{R}\times \mathbb{I}$ Planck triangles, each sharing a vertex or two vertices with this triangle in ${\mathbb{R}}^2$. And this configuration is capable of storing both the curvature and the charge. The Planck area ${\ell }_{\mathrm{P}}^2$ and the $\mathbb{R}\times \mathbb{I}$ imaginary Planck area ${\ell }_{\mathrm{P}}{\ell }_{\mathrm{P}i}={\ell }_{\mathrm{P}}^2\sqrt{{\alpha }_2^3/{\alpha }^3}\approx \pm 0.9666i{\ell }_{\mathrm{P}}^2$, which is smaller in modulus, 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) [5], while electrostatics act in a tangential direction. We note, however, that a triangle has a bivalued complex volume and surface in purely imaginary and complex dimensions even if its edge length is real [14]. Contrary to the no-hair theorem, we characterize BBOs only by mass and charge, neglecting the angular momentum since the latter introduces the notion of time, which we find redundant in the BBO description of a patternless thermodynamical equilibrium.

Not only BBOs are perfectly spherical. Also, their mergers, to which we shall return in Section VI, are perfectly spherical, as it has been experimentally confirmed [69] based on the registered gravitational event GW170817. One can hardly expect a collision of two perfectly spherical, patternless thermal noises to produce some aspherical pattern instead of another perfectly spherical patternless noise. Where would the information about this pattern come from at the moment of the collision? From the point of impact? No point of impact is distinct on a patternless surface.

The hitherto considerations may be unsettling for the reader, as the energy (52) of BBOs other than BHs (i.e., for $k>2$) exceeds mass-energy equivalence $E=M{c}^2$, which is the limit of the maximum real energy. We note that mass-energy equivalence stems from Taylor expansion of the Lorentz factor $\gamma =1/\sqrt{1-v^2/c^2}$ around the vanishing velocity $v=0$ which if multiplied by $M{c}^2$ and truncated to the first two terms yields the 1${}^{\mathrm{st}}$ timeless term corresponding to energy in a system’s rest frame, and the 2${}^{\mathrm{nd}}$ corresponding to the kinetic energy of mass Mmoving at the speed v. Thus, the notion of time is included in the 2${}^{\mathrm{nd}}$ and the remaining countably infinite fractions of Taylor expansion (53). But $M{c}^2$ is time-independent. In the subsequent section, we shall model a part of the energy of NSs and WDs, exceeding $M{c}^2$ as imaginary and thus unmeasurable.

A complex energy formula where $E_{M_R}$ and $i{E}_{Q_R}$ represent respectively real and imaginary energy of an object having mass $M_R$ and charge $Q_R$8 was proposed in [70]. Equation (54) considers real (i.e., physically measurable) masses $M_R$ and charges $Q_R$. We shall modify it to a form involving real and imaginary physical quantities using Planck units, relations (24), (28), (33), (44), and (23) To this end, we define the following six complex energies, the complex energy of real mass and imaginary charge of real charge and imaginary mass of real Compton energy and imaginary mass of real mass and imaginary Compton energy (frequency ${\nu }_i=c_n/{\lambda }_i$) of real Compton energy and imaginary charge and of real charge and imaginary Compton energy where $h\nu =2\pi \hslash \frac{c}{\lambda }=\frac{2\pi }{l}{E}_{\mathrm{P}}≔f{E}_{\mathrm{P}}$, $h{\nu }_i≔f_i{E}_{\mathrm{P}i}$, $f,f_i\in \mathbb{R}$. We note that using different speeds of light c or $c_n$ in energies (56), (57), (60), and (61) yields a contradiction (cf. Appendix F). Therefore, the fundamental unit of energy is mass, not a product of mass and squared velocity.

Complex energies (56)–(61) link mass, charge, and Compton energies within the framework of ED. Their squared moduli are Complex energies (56), (57), (60), and (61) are real-to-imaginary equilibrium if their real and imaginary parts are equal in modulus. This holds for However, they cannot be simultaneously in equilibrium with the energies (58) and (59), as We note that real and imaginary mass and Compton energies vanish in (68) for $q=0$, i.e., for the equilibrium case of an uncharged BH. This implies that BHs cannot be in real-to-imaginary equilibrium and lead to their fluctuations.

Energies (56), (57), (60), and (61) yield two different charge energies corresponding to the elementary charge, the imaginary quantum and the - larger in modulus - real quantum Furthermore, $\forall q$, ${\alpha }^2{E}_{Qi}=i{\alpha }_2^2{E}_Q$.