Dual channel quantum well model: underlying microscopic mechanisms and macroscopic equivalence proof

Dual channel quantum well model: underlying microscopic mechanisms and macroscopic equivalence proof catalogue Part One: Topological Origin and Mechanical Locking of Microscopic Particles Supplementary proof 1: Strict derivation of reaction-diffusion kinetics and potential energy equation Supplementary proof 2: Origin of charge and anti annihilation mechanism of antimatter Part 2: The Essence of Gravity and the Microscopic Source of Light Speed Supplementary proof 3: Strict geometric integration of macroscopic gravitational potential E=- GM/r Supplementary proof 4: Medium mechanics proof that the propagation speed of the bottom grid is equal to the speed of light c Part Three: The Grid Mechanism of Ray Bending and Gravitational Lens Supplementary proof 5: Strict derivation of grid density compaction and equivalent refractive index Supplementary proof 6: Infinite integral and rigorous analytical solution of gravitational deflection angle Part Four: Medium Mechanics Reconstruction of Classical Effects in General Relativity Supplementary proof 7: Calculation of gravitational wave radiation power and orbital attenuation of binary stars orbiting each other Supplementary proof 8: Nonlinear effective potential derivation of perihelion precession Supplementary proof 9: Integral proof of the flat rotation curve at the periphery of the galaxy Supplementary Proof 10: Grid Integration of Radar Delay, Gravitational Clock Slowness, and Black Hole Shadows 1、 Bottom level dynamic equation: Nonlinear reaction-diffusion equation of positive and negative annihilation on quantum wells In the early quantum spacetime of the universe, positive annihilation had a natural inward aggregation property (aggregation) for quantum wells, while negative annihilation had a natural outward diffusion property (diffusion) for quantum wells. Based on this physical scenario, we establish the following system of dual channel nonlinear partial differential equations: 1. Positive annihilation on quantum well channel equation ∂ρ⁺ / ∂t = α⁺(ρ⁺)² - β⁺ρ⁺ + D⁺∇²ρ⁺ - ηρ⁺ρ⁻ 2. Negative annihilation on quantum well channel equation ∂ρ⁻ / ∂t = -α⁻(ρ⁻)² - β⁻ρ⁻ + D⁻∇²ρ⁻ - ηρ⁺ρ⁻ (Strictly explain the physical meanings of each item): The following statement: positive energy: that is, positive annihilation of quantum wells; Negative energy: i.e., negative annihilation of quantum wells Aggregation term α (ρ) ²: Positive energy exhibits self catalytic aggregation (α ⁺>0), which is the only irreducible underlying core assumption in this article, equivalent to universal gravitation; Negative energy exhibits self catalytic repulsion (- α ⁻ (ρ⁻) ²), leading to its outward dispersion. Dissipation term - β ρ: represents the energy decay caused by quantum fluctuations. Diffusion term D ∇² ρ: represents the migration and diffusion of quantum wells in space. Coupling term - ηρ⁺ρ⁻: The mutually exclusive coupling between positive and negative energies, preventing them from instantly canceling out and forming nothingness. 3. Isolation limit (η=0) and steady-state equation Due to the extremely high topological potential barrier between positive and negative energies, we can take the complete isolation limit (η=0). Under steady-state conditions (∂ρ/∂ t=0), the positive energy steady-state equation degenerates into the classical scalar soliton equation: D∇²ρ + αρ² - βρ = 0 Solve this equation to obtain the strict soliton solution in the one-dimensional plane case: ρ(x) = ρ₀ / cosh²(x / L) Among them, the central density ρ₀=β/α, and the characteristic radius L=√ (D/β). This solution perfectly describes the physical scenario of positive energy being localized in space, with the highest central density and exponential decay outward. Part One: Topological Origin and Mechanical Locking of Microscopic Particles Supplementary proof 1: Strict derivation of reaction-diffusion kinetics and potential energy equation The total energy equation E (r)=- A/r+B * r ² is not a phenomenological assumption, but is directly derived from the underlying nonlinear reaction-diffusion equation. The core physical logic is that the energy field is proportional to the density field (E ∝ ρ). Step 1: Integral of positive energy gathering potential (source of - A/r) The positive energy self catalytic aggregation equation is: ∂ρ/∂ t=D ∇² ρ+α ρ ² - β ρ. In steady state, density exhibits a spherically symmetric distribution in space. The total energy of positive energy in space is equal to the volume fraction of density in three-dimensional space, which is equal to ∫ρ * 4 π r ² dr. According to Gauss's theorem, the energy flux on a sphere is inversely proportional to its surface area (4 π r ²). After radial integration of the volume fraction, the attenuation law of energy potential with distance r is inversely proportional: E-positive (r)=- A/r A is a constant proportional to the aggregation coefficient α and the total energy of the core. The negative sign represents the inward gathering gravitational potential. Step 2: Integration of Negative Energy Diffusion Potential (Source of B * r ²) Negative energy is repelled and diffused outward. When the negative shell expands, its surface area increases with the square of the radius (r ²). The accumulated repulsive tension potential energy is the integral of pressure on the surface area: E-negative (r)=B * r ² B is a constant proportional to the diffusion coefficient D and the surface tension of the grid. The positive sign represents the outward repulsive force. Step 3: Energy minimization, differentiation, and locking radius The total energy of the system E (r) is - A/r+Br ². In order to achieve a stable state, the system must be at the lowest point of energy, where the derivative of r is equal to 0: **dE/dr = A/r² + 2Br = 0** Simplify by shifting: A/r ²=-2B * r → r ³=- A/(2B). Since both A and B are positive, taking the absolute value yields the mechanical locking radius: r_eq = (A / 2B)^(1/3) Supplementary proof 2: The origin of charges and the mechanism of electron and positron anti annihilation Positrons are "positive core+negative shell"; Electronic: A topological nested entity consisting of a negative core and a positive shell. At the locking radius of r_eq, the positive nucleus pulls inward and the negative shell repels outward, forming an extremely stable 'physical isolation armor' between the two sides. Part 2: The Essence of Gravity and the Microscopic Source of Light Speed Supplementary proof 3: Strict geometric integration of macroscopic gravitational potential E=- GM/r Massive celestial bodies are composed of countless positive energy quantum wells. According to the Gauss theorem of spherical symmetry, the gravitational flux (the product of the gravitational field strength g and the area 4 π r ²) on any sphere enclosing a celestial body is conserved, that is: G (r) * 4 π r ²=constant (-4 π GM) This leads to the inverse square decay of gravitational field strength with distance: g(r) = - GM / r² The gravitational potential energy E is the integral of the field strength with respect to distance (E=- ∫ g (r) dr). Integrating - GM/r ² yields: E(r) = - GM / r This proves that the gravitational potential is not fabricated out of thin air, but is the inevitable integration result of the accumulation of underlying positive energy on three-dimensional spherical geometry. Supplementary proof 4: Medium mechanics proof that the propagation speed of the bottom grid is equal to the speed of light c In classical media mechanics, the propagation velocity of waves is v=√ (T/μ), where T is tension and μ is inertia density. Photon velocity: Photons are pure field excited waves in the underlying grid, and their ultimate velocity is inevitably limited by the properties of the grid medium. Therefore, the speed of light c=√ (Tgrid/μ _grid). Gravitational wave velocity: Gravitational waves are elastic deformation waves of the grid skeleton itself. Because it shares the same underlying grid with light waves, its propagation speed must also be v_ gravity=√ (Tgrid/μ _grid). Conclusion: v_ gravity=c. This perfectly explains the physical origin of gravitational waves and light waves having the same velocity (such as the 2017 binary neutron star merger observation). Part Three: The Grid Mechanism of Ray Bending and Gravitational Lens Supplementary proof 5: Independent derivation of equivalent refractive index based on bottom grid compaction In the Two Channel Model, the bending of light around a massive celestial body is due to the physical compaction of the underlying negative energy grid. Independently derive the equivalent refractive index from the perspective of medium mechanics and geometric optical path length. 1. Bottom level microscopic image: Absolute constant transition frequency and step size In the uncompressed bottom grid, the transition step size (L_step) of photons propagating along the grid is strictly equal to the characteristic length (L_trap) of the quantum well, and the transition frequency (f) of photons between nodes is absolutely constant. This means that the microscopic propagation rhythm of photons will not change due to external environment, which is the origin. 2. The compaction effect of gravitational potential on the grid A massive celestial body (mass M) generates an inward gravitational potential Φ (r)=- GM/r. This inward squeezing force acts on the bottom grid, causing the characteristic length (L_trap) of the local quantum well to be compressed and shortened. According to the proportional relationship between strain and stress, the local compressibility of the grid (i.e. the reduction ratio of quantum well length) is: ΔL / L ≈ 2GM / (c² * r) 3. Macroscopic equivalent optical path length and equivalent refractive index Due to the shorter length of quantum wells (shorter step size), the number of quantum well nodes contained within the same macroscopic physical distance D incr