The Dual-Channel Quantum-Well Model: A Self-Consistent Derivation of the Black-Hole Horizon and Shadow Diameter (Complete Version) — Step-by-Step Derivation from the Underlying Field Equations, the Centre-Balance Mechanism, and the Background-Radiation Frequency, with a Quantitative Check against EHT Observations (Special Topic)
Year: 2026
Type: article
Abstract: The Dual-Channel Quantum-Well Model: A Self-Consistent Derivation of the Black-Hole Horizon and Shadow Diameter (Complete Version) — Step-by-Step Derivation from the Underlying Field Equations, the Centre-Balance Mechanism, and the Background-Radiation Frequency, with a Quantitative Check against EHT Observations (Special Topic) Contents Abstract Introduction: Why the Derivation Must Start from This Model’s Own Underlying Formulas Part One The Model’s Basic Formulas 1.1 The Reaction–Diffusion Equations of the Positive and Negative Channels 1.2 Spherical Steady State and the Soliton Solution 1.3 Characteristic Length and Central Density Part Two The Centre-Balance Mechanism: Why the Centre is Stable at a Very High Density 2.1 Definition of Attack and Defence, and “Filling on the Spot” 2.2 The Central Mechanism: the Repeated Alternation of Squeeze-out, Spill-out, Eruption and Replenishment 2.3 Quantitative Criteria: the Attack–Defence Ratio and the Conservation Laws 2.4 The Eruption Criterion and the Fast–Slow Relation 2.5 Action Laws of the Two Wells and Comparison of Their Reaction Speeds 2.6 Proof of Central Stability: the Black-Hole-Style Eruption–Replenishment Bounded Oscillation Part Three The Physical Definition of the Horizon and Its Proof: The Density Critical Point 3.1 Proof One: Viewed from Outside In, the Positive-Energy Density Grows from Small to Large and Rises Steeply to the Critical Value 3.2 Proof Two: The Inward Aggregation and the Outward Support Cancel Each Other 3.3 Proof Three: The Dissipation of the Underlying Grid Prevents a Photon from Escaping 3.4 Proof Four: The Critical-Point Criterion and the Determination of ρ_crit 3.5 Summary: The Uniqueness of the Horizon Part Four Independent Calculation of This Universe’s Background-Radiation Frequency and Anchoring of the Lattice Spacing 4.1 Frequency Determined by the Intrinsic Lattice Spacing 4.2 Anchoring the Microscopic Lattice Spacing of This Universe from the CMB Peak Frequency 4.3 The Thermodynamic Origin of the Microwave Background 4.4 The Spacing and Frequency of the Outer Quasi-Super Quantum Well 4.5 Independent Locking by Observation Part Five Step-by-Step Derivation of the Horizon Radius 5.1 Step 1: Setting the Critical Condition 5.2 Step 2: Substituting the Soliton Solution 5.3 Step 3: Solving for the Horizon Radius 5.4 Physical Meaning of the Horizon-Radius Formula Part Six From the Horizon Radius to the Shadow Diameter: How the Shadow Forms, and the Photosphere 6.1 How the Shadow Forms 6.2 The Photosphere: the Radius at Which Light Can Travel, outside the Horizon 6.3 The Relation between the Horizon and the Shadow: One and the Same Place, One and the Same Diameter Part Seven Quantitative Check against EHT Observations 7.1 Selection of the Observational Data 7.2 Converting the Angular Diameter into a Physical Shadow Radius 7.3 Checking the Dimensionless Shadow Diameter 7.4 Observational Constraint on the Underlying Parameter (the Microscopic Lattice Spacing of This Universe) 7.5 Characteristic Thickness of the Negative Quantum-Well Layer outside the Horizon 7.6 Summary of the Inversion Results Part Eight The Multi-Layer (Onion) Structure inside a Black Hole and Its Detection 8.1 The Multi-Layer (Onion) Structure inside a Black Hole 8.2 Calculation of the Structural Parameters and Detection Methods 8.3 Predictions Unique to This Model Part Nine Conclusion Appendix A Note on Terminology: the “Ruler” and “Borrowed Formulas” References Abstract This paper starts from the underlying field equations of the Dual-Channel Quantum-Well Annihilation-Field Model and derives, step by step, the black-hole horizon (shadow) diameter, then checks it against the observational data already released by the Event Horizon Telescope (EHT). The derivation uses only formulas belonging to this system itself: first, the centre-balance mechanism explains why the centre is stable at a very high density (Part Two); then, from the reaction–diffusion equation of the positive-energy channel and its spherically symmetric steady-state soliton solution ρ₊(r) = ρ₀ / cosh²(r/L), the density is required to fall to the critical value ρ_crit at which a photon can no longer escape, which yields the horizon radius directly as R_horizon = L · ln( √(ρ₀/ρ_crit) + √(ρ₀/ρ_crit − 1) ); here the microscopic lattice spacing l₀ of this universe is independently anchored to 0.3 mm by the CMB peak frequency of 160.2 GHz (Part Four). The resulting shadow, slightly enlarged by the effective refraction of the negative quantum-well layer outside the horizon, comes out a little larger than the purely geometric value; substituting the observed values for M87* (dimensionless shadow diameter 11.0 ± 1.5, angular diameter 42 ± 3 μas, mass 6.5 ± 0.7 × 10⁹ M_⊙, distance 16.8 Mpc) and Sgr A* (9.5 ± 1.4), the model agrees with the observations within 1σ. The derivation borrows no external formula; in the observational check, generally accepted conversion yardsticks are used only for comparison and take no part in the model’s derivation. Introduction: Why the Derivation Must Start from This Model’s Own Underlying Formulas The size of a standard black-hole horizon is usually defined by the Schwarzschild radius R_s = 2GM/c². But this system does not take the geometric radius of general relativity as its starting point: in this model the horizon is not a purely geometric boundary, but a physical boundary at which the inward aggregation of positive annihilation pairs and the outward support of negative quantum wells come into balance — equivalently, the place at which the boundary advances from outside in, the positive-energy density grows from small to large and stops when reaching a critical value. To speak of the horizon and the shadow, therefore, one must return to this system’s own underlying equations and derive them afresh, rather than borrow an external formula. The derivation in this paper relies on only two things: first, the reaction–diffusion equation of the positive-energy channel and its spherically symmetric steady-state soliton solution; second, the criterion that every physical effect is determined by the nesting level, and that a photon can travel only when it matches the quantum-well conditions of the spacetime it is in. Constants such as the gravitational constant and the speed of light appear only when the result is converted into commonly accepted units for comparison with observations; they take no part in the model’s derivation. So that the derivation is self-consistent and leaves no premise dangling, two further chains are supplied. First, the definition of the horizon (Part Three) rests on the premise that the centre can be stable at a very high density; Part Two proves this in full by the black-hole-style eruption–replenishment mechanism: the centre is a self-regulating balance centre of attack and defence, and whatever its size it has a centre and a similar balance. Second, the microscopic lattice spacing l₀ of this universe that appears in the horizon formula (Part Five) is not a freely chosen number: Part Four independently anchors it to 0.3 mm by the CMB peak frequency. Both chains use only criteria belonging to this system. One point of terminology must be stated in advance: in the observational check below, the Schwarzschild radius R_s = 2GM/c² and the classical general-relativistic shadow value 6√3 will appear. Here they serve only as a “ruler” — to convert an observed angular diameter into the dimensionless ratio customary in astronomy, so that it can be compared directly with published results. They take no part in any step of this model’s derivation; the model’s own horizon formula is independent of both. Part One The Model’s Basic Formulas 1.1 The Reaction–Diffusion Equations of the Positive and Negative Channels The basic objects of this system are two kinds of annihilation pairs and three kinds of quantum wells. Positive annihilation pairs undergo autocatalytic aggregation, negative annihilation pairs undergo autocatalytic repulsion, and neutral quantum wells are externally neutral. Writing the positive and negative channels separately as reaction–diffusion equations gives the model’s underlying field equations: ∂ρ₊/∂t = D₊∇²ρ₊ + α₊ρ₊² − β₊ρ₊ (1-1) ∂ρ₋/∂t = D₋∇²ρ₋ − α₋ρ₋² − β₋ρ₋ (1-2) Here ρ₊ and ρ₋ are the number densities of positive and negative annihilation pairs; D is the diffusion coefficient; α is the coefficient of the autocatalytic term, with α₊ > 0 in the positive channel meaning aggregation and −α₋ < 0 in the negative channel meaning repulsion; β is the dissipation coefficient. The two channels are held in mutually exclusive balance by a coupling term; writing it out explicitly gives: ∂ρ₊/∂t = D₊∇²ρ₊ + α₊ρ₊² − β₊ρ₊ − ηρ₊ρ₋ (1-3) ∂ρ₋/∂t = D₋∇²ρ₋ − α₋ρ₋² − β₋ρ₋ + ηρ₊ρ₋ (1-4) Here η is the coupling coefficient, representing the strength of the topological barrier between positive and negative wells. 1.2 Spherical Steady State and the Soliton Solution A black hole is an approximately spherically symmetric steady structure. Setting ∂/∂t = 0 for the steady state and taking spherical symmetry, the Laplacian is written (1/r²)(d/dr)(r² d/dr). Looking first at the positive-energy channel alone in the weak-coupling limit (η → 0), its steady-state equation is: D₊·(1/r²)·(d/dr)( r²·dρ₊/dr ) + α₊ρ₊² − β₊ρ₊ = 0 (1-5) This equation possesses a spherically symmetric steady-state soliton solution, whose radial density profile is: ρ₊(r) = ρ₀ / cosh²(r/L) (1-6) The meaning of this solution is that the positive annihilation pairs are densest at the centre and decay rapidly outward as the inverse square of the hyperbolic cosine, tending to zero at infinity. Neither the central density ρ₀ nor the characteristic length L is a free parameter; both are fixed by the coefficients of the equation itself. 1.3 Characteristic Length and
Author: weihui, lu
Language: English
Subfield: Nuclear and High Energy Physics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225460748