UNITSTATE: THE BLACK/WHITE HOLE AS A SINGLE OBJECT The Method of Opposite Equations Andrzej Statowski ORCID: 0009-0002-2094-5450 14 July 2026 ─── 1. Definition of the Unitstate I define the Unitstate as the common solution of two equations related by the time-reversal transformation [ t \rightarrow -t. ] Two equations are “opposite” if they describe physical processes that are reversals of each other in terms of field dynamics — such as collapse and emission, absorption and de-collapse, expansion and contraction. If such equations share a common value, a common state, or a common boundary, they describe a directionless state, the Unitstate. Properties of the Unitstate: • absence of a time arrow, • absence of directional choice, • spontaneity, • compensation, • no privileged dynamical orientation. Example 1: common modulus Consider: [ x^2 = 4,\qquad x^2 = -4. ] The first equation has real solutions ±2, the second has complex solutions ±2i. The shared property of all solutions is: [ |x| = 2. ] The modulus is the Unitstate — the common value of two opposite equations. Example 2: trigonometry [ \sin^2 x + \cos^2 x = 1. ] This can be written as two opposite forms: [ \sin^2 x = 1 - \cos^2 x,\qquad \cos^2 x = 1 - \sin^2 x. ] Two opposite expressions, one result — the Unitstate. Example 3 (central): time Equation A: [ +t ] Equation B: [ -t ] Adding the equations is a test operation: we check whether a process and its reverse compensate into a directionless state. [ (+t) + (-t) = 0. ] The result t = 0 is the Unitstate — the state without temporal direction. This is the foundation of the method. ─── 1. Proof: the black hole and the white hole are one object In general relativity: • the black hole metric is the Schwarzschild solution for the +t direction, • the white hole metric is the Schwarzschild solution for the –t direction. Formally: [ g{\mu\nu}^{\text{white}} = g{\mu\nu}^{\text{black}}(t \rightarrow -t). ] This is a mathematical transformation, not a metaphor. The black hole and the white hole are one solution with reversed time. Conclusion: The black hole and the white hole are one object in two spontaneous states — the Unitstate. ─── 1. Hawking radiation as the mechanism of the white hole Hawking radiation: • is spontaneous, • directionless, • occurs “outwards”, • does not require a chosen orientation. Gravitational collapse: • is spontaneous, • occurs “inwards”. Thesis: Hawking radiation is the physical manifestation of the –t process, i.e., the white-hole mechanism. Collapse is the physical manifestation of the +t process, i.e., the black-hole mechanism. I do not claim that Hawking interpreted his equations as a white hole. I claim that Hawking radiation satisfies the mathematical criterion of being the reverse process of collapse. Together they form a cycle of spontaneous compensation. ─── 1. The horizon as the boundary of the Unitstate If: • collapse = +t, • Hawking radiation = –t, then the boundary between them is the event horizon. The horizon is not the surface of the black hole. The horizon is the boundary of the Unitstate — the place where the transformation [ t \rightarrow -t ] changes sign. On one side: +t (inwards). On the other side: –t (outwards). On the horizon: [ t = 0. ] Therefore compensation occurs at the horizon. ─── 1. Double debit — a mechanism without negative energy In the classical description of Hawking radiation, the concept of negative energy is introduced. In the Unitstate framework: • there is no negative energy, • the singularity has real energy, • two opposite processes compensate spontaneously. Mechanism: The singularity possesses energy. Two processes occur: • one inward (+t), • one outward (–t). Directional balance = 0. Formally: [ g{\mu\nu}^{\text{black}} + g{\mu\nu}^{\text{white}} = 0. ] “0” does not mean absence of a metric — it means absence of temporal direction. ─── 1. A new method: equation reversal as a test of physical theories I propose a method for testing physical theories: Step 1: Write the equation for the +t process. Step 2: Write the equation for the –t reverse process. Step 3: Add them. Step 4: • If the result = 0 → the theory is spontaneously consistent. • If the result ≠ 0 → the theory contains a hidden directional choice, i.e., a systematic bias. Examples: • Electrodynamics: photon emission and absorption — does the sum = 0? • Cosmology: expansion and contraction — does ΛCDM satisfy the Unitstate condition? Hypothesis: it does not, hence it requires dark energy as a prosthesis. • Quantum mechanics: collapse and de-collapse of the wavefunction — does the sum = 0? The method reveals errors in one-direction theories. This method does not claim to replace existing theories, but to provide an additional consistency check for hidden time-direction bias. ─── 1. Conclusion 1. The black hole and the white hole are one object described by the transformation t → –t. 2. Hawking radiation is the physical manifestation of the white-hole mechanism. 3. The event horizon is the boundary of the Unitstate. 4. The Unitstate is the directionless state: [ g{\text{black}}(t) + g{\text{white}}(-t) = 0. ] 5. The method of opposite equations is a tool for detecting hidden directional biases in physical theories. Proposed experimental verification: Vacuum fluctuations at the horizon should be directionless and spontaneous. The experiment by Procopio & Landhardt (preprint, 01 July 2026), showing spontaneous vacuum energy, is interpreted as supporting evidence. ─── References • Schwarzschild K. (1916) • Hawking S. (1974) Black hole explosions? • Labini & Vasilyev – large scale structures • Statowski A. (29 May 2026) Reflection Cosmology v2.3, Zenodo • Statowski A. (31 May 2026) UGNFT Signal Extraction Protocol, Zenodo • Procopio & Landhardt (01 July 2026), preprint
UNITSTATE: THE BLACK/WHITE HOLE AS A SINGLE OBJECT The Method of Opposite Equations Andrzej Statowski ORCID: 0009-0002-2094-5450 14 July 2026 ─── 1. Definition of the Unitstate I define the Unitstate as the common solution of two equations related by the time-reversal transformation [ t \rightarrow -t. ] Two equations are “opposite” if they describe physical processes that are reversals of each other in terms of field dynamics — such as collapse and emission, absorption and de-collapse, expansion and contraction. If such equations share a common value, a common state, or a common boundary, they describe a directionless state, the Unitstate. Properties of the Unitstate: • absence of a time arrow, • absence of directional choice, • spontaneity, • compensation, • no privileged dynamical orientation. Example 1: common modulus Consider: [ x^2 = 4,\qquad x^2 = -4. ] The first equation has real solutions ±2, the second has complex solutions ±2i. The shared property of all solutions is: [ |x| = 2. ] The modulus is the Unitstate — the common value of two opposite equations. Example 2: trigonometry [ \sin^2 x + \cos^2 x = 1. ] This can be written as two opposite forms: [ \sin^2 x = 1 - \cos^2 x,\qquad \cos^2 x = 1 - \sin^2 x. ] Two opposite expressions, one result — the Unitstate. Example 3 (central): time Equation A: [ +t ] Equation B: [ -t ] Adding the equations is a test operation: we check whether a process and its reverse compensate into a directionless state. [ (+t) + (-t) = 0. ] The result t = 0 is the Unitstate — the state without temporal direction. This is the foundation of the method. ─── 1. Proof: the black hole and the white hole are one object In general relativity: • the black hole metric is the Schwarzschild solution for the +t direction, • the white hole metric is the Schwarzschild solution for the –t direction. Formally: [ g{\mu\nu}^{\text{white}} = g{\mu\nu}^{\text{black}}(t \rightarrow -t). ] This is a mathematical transformation, not a metaphor. The black hole and the white hole are one solution with reversed time. Conclusion: The black hole and the white hole are one object in two spontaneous states — the Unitstate. ─── 1. Hawking radiation as the mechanism of the white hole Hawking radiation: • is spontaneous, • directionless, • occurs “outwards”, • does not require a chosen orientation. Gravitational collapse: • is spontaneous, • occurs “inwards”. Thesis: Hawking radiation is the physical manifestation of the –t process, i.e., the white-hole mechanism. Collapse is the physical manifestation of the +t process, i.e., the black-hole mechanism. I do not claim that Hawking interpreted his equations as a white hole. I claim that Hawking radiation satisfies the mathematical criterion of being the reverse process of collapse. Together they form a cycle of spontaneous compensation. ─── 1. The horizon as the boundary of the Unitstate If: • collapse = +t, • Hawking radiation = –t, then the boundary between them is the event horizon. The horizon is not the surface of the black hole. The horizon is the boundary of the Unitstate — the place where the transformation [ t \rightarrow -t ] changes sign. On one side: +t (inwards). On the other side: –t (outwards). On the horizon: [ t = 0. ] Therefore compensation occurs at the horizon. ─── 1. Double debit — a mechanism without negative energy In the classical description of Hawking radiation, the concept of negative energy is introduced. In the Unitstate framework: • there is no negative energy, • the singularity has real energy, • two opposite processes compensate spontaneously. Mechanism: The singularity possesses energy. Two processes occur: • one inward (+t), • one outward (–t). Directional balance = 0. Formally: [ g{\mu\nu}^{\text{black}} + g{\mu\nu}^{\text{white}} = 0. ] “0” does not mean absence of a metric — it means absence of temporal direction. ─── 1. A new method: equation reversal as a test of physical theories I propose a method for testing physical theories: Step 1: Write the equation for the +t process. Step 2: Write the equation for the –t reverse process. Step 3: Add them. Step 4: • If the result = 0 → the theory is spontaneously consistent. • If the result ≠ 0 → the theory contains a hidden directional choice, i.e., a systematic bias. Examples: • Electrodynamics: photon emission and absorption — does the sum = 0? • Cosmology: expansion and contraction — does ΛCDM satisfy the Unitstate condition? Hypothesis: it does not, hence it requires dark energy as a prosthesis. • Quantum mechanics: collapse and de-collapse of the wavefunction — does the sum = 0? The method reveals errors in one-direction theories. This method does not claim to replace existing theories, but to provide an additional consistency check for hidden time-direction bias. ─── 1. Conclusion 1. The black hole and the white hole are one object described by the transformation t → –t. 2. Hawking radiation is the physical manifestation of the white-hole mechanism. 3. The event horizon is the boundary of the Unitstate. 4. The Unitstate is the directionless state: [ g{\text{black}}(t) + g{\text{white}}(-t) = 0. ] 5. The method of opposite equations is a tool for detecting hidden directional biases in physical theories. Proposed experimental verification: Vacuum fluctuations at the horizon should be directionless and spontaneous. The experiment by Procopio & Landhardt (preprint, 01 July 2026), showing spontaneous vacuum energy, is interpreted as supporting evidence. ─── References • Schwarzschild K. (1916) • Hawking S. (1974) Black hole explosions? • Labini & Vasilyev – large scale structures • Statowski A. (29 May 2026) Reflection Cosmology v2.3, Zenodo • Statowski A. (31 May 2026) UGNFT Signal Extraction Protocol, Zenodo • Procopio & Landhardt (01 July 2026), preprint
It is shown that the internal stationary state of the Schwarzschild black hole can be represented by a maximally entangled two-mode squeezed state of collapsing matter and infalling Hawking radiation. The final boundary condition at the singularity is then described by the random unitary transformation acting on the collapsing matter field. The outgoing Hawking radiation is obtained by the final-state projection on the total wave function, which looks like a quantum teleportation process without the classical information transmitted. The black hole evaporation process as seen by the observer outside the black hole is now a unitary process but nonlocal physics is required to transmit the information outside the black hole. It is also shown that the final-state projection by the evaporation process is strongly affected by the quantum state outside the event horizon, which clearly violates the locality principle.
This paper proposes that black holes can be described as globally coherent quantum systems, despite the emission of Hawking radiation. While the Hawking process induces a weak coupling between the interior and exterior regions, this interaction is demonstrated to be too small to produce significant decoherence for macroscopic and supermassive black holes. By treating the event horizon as a quasi-isolated, semi-transparent quantum interface, this framework reconciles semiclassical gravity with quantum unitarity. The black hole and its emitted radiation are modeled as a single, entangled pure quantum state that evolves unitarily. Key contributions of this work include: Formalization of the black hole-radiation system as a coherent quantum cavity. Mathematical derivation showing that the decoherence rate scales as Γ_dec ∝ M⁻⁸. Demonstration that for astrophysical black holes, the decoherence time τ_D is astronomically larger than the evaporation time t_evap, implying that global quantum coherence is effectively preserved. A potential resolution to the black hole information paradox within a unitary evolution framework. This research contributes to the ongoing theoretical discussion on the nature of event horizons, quantum information, and the holographic principle. Acknowledgements: I would like to express my deepest gratitude to my beloved dog, Milo, and my father, for their constant support, encouragement, and companionship throughout the development of this research.
The Problem of Time — the apparent incompatibility between quantum mechanics' time parameter t and general relativity's proper time τ — has been treated as a fundamental obstacle to quantum gravity. I argue that this is a level-confusion: both theories define time within their own entity-theoretic frameworks, generating an apparent opposition that disappears when time is relocated to a meta-structural layer. Drawing on the meta-level temporal structure T_SDT := (V_G, ≺) (Zeng 2026), the paper formalizes QM and GR time as two projections of a single Universal Time Formula, parameterized by entity-theoretic tokens. Four token specializations recover the corresponding time concepts. A Bridge Theorem establishes mutual compatibility of all token projections. A bidirectional QM-GR channel is constructed (forward: GR geometry → QM phase; reverse: discrete chain-pair → Lorentzian spacetime), composing into a self-consistent closed loop. Three further results move beyond dissolution toward unification. First, a Madelung-bridge construction derives the Schrödinger equation — Hilbert space, unitarity, Hamiltonian, quantum potential, Born rule — from the determination-graph axioms together with the continuum embedding (CL1), with no independent QM input; the derivation extends to Klein-Gordon. Second, a Planck-scale token μ = t_P / |A_{v'}| is constructed on the discrete 𝒢_D, where |A_{v'}| is the admissibility cardinality; under CL1, this single step-measure recovers QM (|A| ≫ 1) and GR (|A| = 1) as its two continuum limits, with Einstein's equation proved in the classical regime (|A| = 1) via a two-stage argument (Benincasa-Dowker action convergence + standard scalar-tensor variation). The exact quantum-gravitational correction is derived as a scalar-tensor equation with φ = 1/|A|. Third, the discrete determination graph without continuum embedding constitutes a self-contained Planck-scale quantum gravity theory: it has a well-defined path integral, a finite-dimensional S-matrix (unitary in the κ = 0 sector, isometric in the κ > 0 sector; no UV divergences), preserves information across black hole horizons, resolves singularities, derives the Past Hypothesis, and provides structural origins for gauge symmetry, Hawking radiation, spectral dimensionality, and the spin-statistics connection. The Problem of Time is not merely dissolved — it is replaced by a constructive framework in which QM and GR are unified as two limits of a single discrete structure.
Programme status: This record belongs to an active research programme. It is not a finished or final theory. The framework is currently undergoing a systematic audit, together with corrections and improvements prompted by reader feedback. Several significant issues have been identified, and some claims, derivations and interpretations in the current version may be revised, qualified or withdrawn in future versions. Revised versions will be released when this work is complete. See PROGRAM_STATUS.md. Status as of October 2026. We examine black hole information unitarity within the Mathematical Foundations of Reflexive Reality (MFRR) framework SpivackNEMSMFRRBridge. The MFRR Transputation operator PT^-1 provides a physical mechanism for unitary information recovery consistent with the Perfect Self-Containment condition. Since every CPTP map admits a Stinespring dilation (a mathematical theorem independent of gravitational modelling), Hawking evaporation is unitarily equivalent to a reversible evolution on a system-plus-environment space. We construct the evaporation channel _ t = e^ tL as a GKSL master equation for a JT-like system of n=3 bosonic modes (d_H = 8) at T_H 0.004. Stinespring dilation achieves fidelity F 1 - 10^-8; detailed balance holds to 0.00\%; the entropy curve reaches 97.2\% of the ideal thermal value. In two complementary closed unitary models - a pure initial BH state and a purified thermal initial state with reference system R - the full Page curve with turnover is reproduced and PT^-1 recovery achieves F = 1.000000. Open fronts are stated explicitly: extension to 3+1 quantum gravity, the firewall tension, and a first-principles derivation of PT^-1.
Black holes are usually described either as classical geometries on which quantum fields propagate or as objects whose complete quantum description lies beyond semiclassical gravity. Hawking radiation already shows, however, that a black hole is not an exactly isolated system: the horizon carries a thermal scale and, after coarse graining, may be associated with fluctuations, dissipation and environmental correlations. Whether this horizon-scale openness can be encoded directly in the gravitational background, before any stochastic treatment of the metric, remains an open question. Here we answer it by introducing, for the first time, a conserved Hawking-scale anisotropic source representing the coarse-grained near-horizon atmosphere and solving the Einstein equations for a static black hole at fixed ADM mass. The dressed geometry has a renormalized mass function, a self-consistently shifted horizon and a modified surface gravity. Because the same lapse governs the horizon and timelike geodesics, the thermal response is accompanied by a correlated orbital restructuring: the effective potential is renormalized, the ISCO moves inward and the near-horizon residence time increases. The construction adds no Langevin force to a fixed Schwarzschild background; rather, it defines a deterministic dressed mean geometry, providing a controlled background on which stochastic metric fluctuations may subsequently be treated in the Einstein-Langevin framework. This establishes the deterministic mean-field sector as the necessary first step for future stochastic gravity programs addressing horizon-scale openness.
Essentially, in this paper we propose a new description of the quantum dynamics by two relativistic propagation wave packets, in the two conjugated spaces, of the coordinates and of the momentum. Compared to the Schrödinger-Dirac equation, which describes a free particle by a wave function continuously expanding in time, considered as the amplitude of a probabilistic distribution of this particle, the new equations describe a free particle as an invariant distribution of matter propagating in the two spaces, as it should be. Matter quantization arises from the equality of the integral of the matter density with the mass describing the dynamics of this density in the phases of the wave packets. In this description, the classical Lagrange and Hamilton equations are obtained as the group velocities of the two wave packets in the coordinate and momentum spaces. When to the relativistic Lagrangian we add terms with a vector potential conjugated to coordinates, as in the Aharonov-Bohm effect, and a scalar potential conjugated to time, we obtain the Lorentz force and the Maxwell equations as characteristics of the quantum dynamics. In this framework, the conventional Schrödinger-Dirac equations of a quantum particle in an electromagnetic field obtain additional terms explicitly depending on velocity, as is expected in the framework of relativistic theory. Such a particle wave function takes the form of a rapidly varying wave, with the frequency corresponding to the rest energy, modulated by the electric potential where this particle is confined, the vector potential of the electromagnetic radiation, and the gravitational potential. In the gravitational field of a black hole, the velocity of a quantum particle is directed to the Schwarzschild boundary from the outside and from the inside, without passing this boundary. When the very strong gravitational field of a large agglomeration of matter has spherical symmetry with a radius smaller than the Schwarzschild radius, the quantum particles in the central region are crushed up to quasi-uniform distributions of matter, exploding with velocities much larger than the light velocity, towards the Schwarzschild boundary, but decelerating in the gravitational field and reaching this boundary only in an infinite time. The matter of a black hole has the tendency to concentrate around the Schwarzschild boundary, with a radius proportional to its mass. In this framework, we believe that our universe is a huge black hole, with the explosion of its central matter as the Big Bang-inflation process, and an increasing radius on account of an absorption of matter from the Total Universe, where our universe has been formed. In a gravitational wave, the time-space interval in the first-order approximation is the product of a gravitational invariant including the amplitude tensor, and a coordinate invariant including the polarization vector. The gravitational invariant describes a rotation of the metric tensor around the direction of the propagation of this wave, with spin 2, as a gravitational quantum called graviton. The coordinate invariant describes a matter rotation with the spins ½ for Fermions, and 1 for Bosons. From the new dynamic equations, for a free particle in the coordinate and momentum spaces, we reobtain the two basic equations of the quantum field theory, but with a change of sign, and an additional term depending on momentum, to the rest mass as the eigenvalue of these equations. However, when these eigenvalues are eliminated, the wave function takes the form of a wave packet of spinors of the same form as in the conventional quantum field theory, with a normalization volume as the integral of the ratio of the energy to the rest energy, over the momentum domain which gives finite dimensions to the quantum particle, as a finite distribution of matter in the coordinate space.