(Quantum) Annihilation Field Universe Model: From Particle Emergence to Cumulative Redshift(Quantum Corporation)|美容 |The |field |Ltrap |Quantum |core

(Quantum) Annihilation Field Universe Model: From Particle Emergence to Cumulative Redshift

Abstract

This paper proposes an annihilation field universe model. The annihilation field is the core concept of this paper—a fundamental field in vacuum that forms localized excitations under the boundary conditions of quantum wells. The core assumption is that the positive excitation mode of the annihilation field exhibits intrinsic aggregation. From this single underlying assumption, the following conclusions are derived:

(1) Steady-state soliton solution: ρ(r) = ρ₀/cosh²(r/L), corresponding to the bright soliton solution of a one-dimensional nonlinear equation.

(2) Complex field equation: internal annihilation field excitations persist without external radiation; the real part corresponds to aggregation, the imaginary part to hidden phase.

(3) Angular momentum quantization: naturally described by spherical harmonics Y_lm(θ,φ), with eigenvalues L² = l(l+1)ℏ².

(4) Particle emergence: single quantum well excitations cannot form elementary particles; N excitation modes coherently superpose to form elementary particles, with mass, spin, and charge as collective effects.

(5) Cosmic-scale annihilation field: the universe itself as a giant annihilation field excitation, with its positive part corresponding to the observable material world and its negative part to an invisible spacetime tension field.

(6) Cosmic evolution as repetition on a formatted template: the universe is not expanding into new territory but oscillating within an already formatted cosmic template (F=1). Physical constants are universal; the universe is homogeneous on large scales.

(7) Redshift as cumulative interaction: redshift arises from photons interacting with the negative-energy tension field while traversing the annihilation field, not from spatial expansion. z = η_−ρ₀⁻(L_trap/d_trap)·D.

(8) Hubble constant: H₀ = η_−ρ₀⁻(L_trap/d_trap)·c ≈ 70 km/s/Mpc, consistent with observations. Effective redshift coefficient α_eff ≈ 7.5×10⁻²⁷ m⁻¹.

(9) Redshift at 10¹⁴ light-year scales: z ~ 10⁴.

(10) Unified formation scenarios: the universe may arise spontaneously, externally, via multiple nucleation points, or through inhomogeneous formatting converging to a uniform state. All four scenarios are unified within this framework, converging to a universe with uniform formatting.

(11) Negative energy propagation properties: Negative energy quantum wells naturally diffuse outward at every point and have the potential to become information carriers. Uniform diffusion does not transmit information, but local disturbances can generate "tension waves" with strong penetration, extremely low energy, and may not attenuate.

Four testable predictions are proposed: (a) no detectable spacetime expansion at short distances; (b) strict identity of same-type particles; (c) systematic anti-gravity patterns at galactic peripheries; (d) systematic deviations in the redshift-distance relation at high redshifts. Falsification conditions are explicitly listed.

Explanation of the formation of the universe

This model is applicable to four possible scenarios of cosmic formation, which are uniformly described within this framework:

Scenario 1: Spontaneous generation. Cosmic level quantum wells are spontaneously generated from non cosmic regions and do not require an external matrix. The prerequisite is the original existence of the universe template. The driving force for expansion comes from the internal accumulation of negative energy tension field. The ultimate fate is to rebound upon reaching the trap wall and then contract.

Scenario 2: External emergence (derived universe). There exists a massive mother universe, and our universe is a 'bubble' bulging out from the edge of the mother. The prerequisite is the existing field structure and quantum well in the parent universe. The driving force for expansion comes from the difference between the tension inside the package and the tension in the mother body. The ultimate fate is continuous expansion, rupture, or separation from the mother body.

Scenario 3: Multiple points generated. The universe is not a single quantum well, but multiple quantum wells that are generated simultaneously or sequentially, each expanding, colliding, and merging. The prerequisite is a universe template and multiple activation points. The ultimate fate is collision and fusion, forming a larger unified universe. The testable inference is that there may be multiple cold or hot spots in the cosmic microwave background, corresponding to collision remnants of multiple quantum wells.

Scenario 4: The format is uneven, and eventually converges to a unified mode. The formatting degree of the universe template is uneven in space, but gradually converges to a uniform value F=1 after long-term evolution. The evolution equation is ∂ F/∂ t=D_F ∇ ² F+λ (F − F_eq). The ultimate fate is to converge into a unified universe with universal physical constants.

The unity of the four scenarios: The four scenarios are described uniformly within the mathematical framework of this article, with the only difference being the initial conditions and boundary conditions. No matter how the universe is created, its ultimate evolution tends towards a uniform convergence state (F=1). This explains why the universe we observe is uniform and isotropic on a large scale.

Keywords: Annihilation field; Quantum well; Particle emergence mechanism; Dual-channel quantum spacetime framework; Alternative cosmic redshift; Hubble constant

At the beginning of the universe, the physical processes of matter formation and development involve annihilation reaction pairs within particles. This paper assumes that in the early quantum spacetime, there were sufficient quantum traps. When a positive-energy annihilation reaction pair is captured by a quantum trap...

A positive energy annihilation reaction pair is captured by a quantum trap, and this positive energy annihilation reaction pair is always carried out in the quantum trap without jumping out, that is, between the annihilation reaction and the annihilation pair. At the same time, a negative energy annihilation reaction pair is also captured by another quantum trap different from this quantum trap. The positive energy annihilation reaction pair is always clustered in the quantum trap, while the negative energy annihilation reaction pair is mutually exclusive and never clustered in the quantum trap. The inspiration comes from my previous article. One is that if an anti gravitational particle appears at the center of a singularity, it will diffuse into the universe and decay into a physical particle. Another article is about where antimatter went, assuming at the time that it was antimatter, where negative energy was frozen in the spacetime field. Another point to note is that currently, regardless of how long quantum spacetime has gone through, a cosmic template has been formed and formatted.

Assuming this quantum trap, an annihilation pair, in the early stages of the development of quantum spacetime in the universe, it may not be the smallest energy unit in batches, but may be of different sizes, and eventually develop into a unified smallest energy unit, no matter how long

it takes.

Here we only analyze how the quantum trap annihilation reactions of quantum spacetime at the beginning of the universe affect physical processes and certain quantitative relationships, which is just an early stage in the formation of the properties of actual particles today.

That is to say, after the Big Bang, the structure of the universe (the underlying quantum structure) was destroyed, and matter couldn't appear right away.

The possibility of a quantum well universe model.

In this article, space is invariant, while spacetime is a variable physical quantity

The following is conducted within the scope of my understanding and recognition using the theory of human knowledge (the derivation of the mathematical part is within the scope of engineering mathematics for undergraduate students, with some slightly higher)

中文英文（统一）

湮灭反应对 annihilation reaction pair

正能量湮灭对 positive-energy annihilation pair

负能量湮灭对 negative-energy annihilation pair

量子阱 quantum trap / quantum well

双通道 dual-channel

阱簇 trap cluster

相位相干 phase coherence

Definition of Core Concept: Annihilation Field

Definition: Annihilation field is the fundamental field in vacuum, and its excitation mode is manifested as the annihilation reaction of positive and negative energy pairs.

Physical image:

Element Content

The field itself is continuous and fills the entire space

Boundary conditions of quantum well field

Stable modes formed by localized excitation fields constrained by traps

Positive excitation patterns gather to form material structures

Negative excitation mode repulsion forms a spatiotemporal tension field

Relationship with Standard Field Theory:

Comparison of Standard Field Theory and Annihilation Field in this paper

The nature of the field has no self aggregation and has self aggregation (unique underlying assumption)

Excitation mode particle positive and negative energy annihilation pair

External boundary conditions provide a given quantum well

Analogous to the Higgs field, it fills space and forms localized excitations, but with the addition of self aggregation

Mathematical expression:

Annihilation field Φ (x, t) → quantum well boundary conditions → local excitation Φ _nlm (r, θ, φ)

Key difference: The annihilation field in this article has self aggregation (the only underlying assumption), which is its fundamental difference from other fields in standard field theory.

Unified Table of Full Text Terminology

Original terminology and new terminology

Excitation mode of annihilation reaction on annihilation field

Positive Energy Annihilation on the Positive Excitation Mode of Annihilation Field

Negative Energy Annihilation on the Negative Excitation Mode of Annihilation Field

Annihilation inside the trap excites the annihilation field inside the trap

N-Annihilation Pair Stacking N-Annihilation Field Pattern Coherent Stacking

Quantum Well Universe Model Annihilation Field Universe Model

Quantum trap quantum well

Positive energy annihilation pair positive annihilation pair

Negative energy annihilation pair Negative Annihilation Pair

Introduction: Dual channel quantum well model for the formation of cosmic matter

0.1 Problem Proposal

Where does matter in the universe come from? Why do stable structures of protons, electrons, and neutrons exist instead of a chaotic mass of energy?

This article proposes a hypothesis that the formation of material structures originates from two types of energy annihilation reactions (positive energy and negative energy) in early quantum spacetime of the universe, which were captured in different quantum wells and eventually emerged as stable material structures through different aggregation behaviors.

0.2 Core Assumptions

This article assumes that in the early stages of the formation and development of the universe, there were sufficient quantum wells in quantum spacetime. Its behavior can be summarized into the following three basic assumptions:

A positive energy annihilation reaction pair (positive annihilation pair): Once captured by a quantum well, it will continue indefinitely in the well without escaping, and the quantum well has a natural and sustained aggregation - that is, the positive energy well will continuously attract and stack, and the(quantization) density will continue to increase.
A negative energy annihilation reaction pair (negative annihilation pair): captured by another quantum well. Negative energy traps repel each other and don't cluster together.
Non-uniform origins of energy units in quantum wells: In the early stages of material development, annihilation reactions in a quantum well may not be uniform minimum energy units, but may vary in size. After an extremely long evolutionary process, it eventually developed into a unified and standardized minimum energy unit.
Spatiotemporal grid skeleton: The captured negative energy annihilation pairs of quantum wells do not aggregate in spacetime but exist in frozen, relatively fixed spatial positions, forming the spatiotemporal grid skeleton. This skeleton provides a spatial reference frame and boundary conditions for the aggregation of positive energy traps. Positive energy traps move, stack, and aggregate on or between skeletons, while negative energy traps themselves remain stationary as static backgrounds.

Summary Table of Basic Assumptions for Model 0.2.6

To clearly distinguish between the original assumption (irreducible hypothesis) and the derived result (conclusion derived from the assumption), the basic assumptions of the entire text are summarized as follows:

Number assumption content type falsifiability

There are sufficient quantum wells in the early universe quantum spacetime, and the original postulate is indirectly verified through subsequent predictions

A2 positive energy annihilation has a natural and sustained aggregation core hypothesis (unique underlying hypothesis) after being captured by a quantum well, indirectly

A3 negative energy annihilation pairs repel each other after being captured by independent quantum wells and never gather

The A4 positive and negative channels are isolated by high potential barriers, and weak coupling (η→ 0) can be verified by positive and negative annihilation signals

The A5 universe template has been formatted, and the energy threshold of E2<E2 ensures that there is no radiation in the trap. The original hypothesis can be verified through energy level measurements

The A6 complex field equation is a phenomenological construction (not yet derived from the variational principle), and the model structure will require the construction of Lagrangian quantities in the future

The original postulate that the intrinsic angular momentum of a single quantum well in A7 is l=1/2 can be verified through scattering experiments

The derivation results of the emergence of physical particles through the coherent superposition of A8 N quantum wells can be verified by particle spectrum analysis

The existence of A9 cosmic quantum wells, where the positive part represents matter and the negative part represents the spacetime field. The null hypothesis (scale extrapolation) can be verified through cosmological observations

A10 redshift comes from the cumulative effect of negative energy tension field, and the null hypothesis (alternative mechanism) of non spatial expansion can be verified through high redshift observations

Instructions:

·Primitive postulate: A fundamental assumption that cannot be derived from other principles and serves as the starting point of the model.

·Deduction result: The conclusion obtained through mathematical deduction starting from the assumption.

·A2 is the only underlying driving force in the entire text - it describes' aggregation occurring 'but does not explain' why aggregation begins'. This is on the same level as the universal gravitational constant in the theory of gravity.

0.3 Inspiration sources and ideological context

The proposal of this model originates from several core intuitions that I have had for a long time, which were expressed in layman's terms in early articles:

Singularity anti gravity fluctuation hypothesis: If there is an anti gravity fluctuation at the center of a singularity in the universe, the resulting anti gravity particles may diffuse into spacetime and decay into physical particles in subsequent evolution. This speculation constitutes the ideological source of the "negative energy dispersion distribution" in this model.
Difficulty in the loss of cosmic antimatter: Standard cosmology faces the challenge of asymmetric antimatter. One possible explanation is that antimatter (negative energy) was frozen in the spacetime field in the early universe, neither annihilating nor accumulating. This speculation constitutes the source of the idea of 'negative energy does not gather' in this model.
Concept of "Cosmic Template": No matter how long quantum spacetime has gone through, a fixed "cosmic template" has been formed, which means that the quantum structure in spacetime (Not a vacuum) has been formatted and has regularity. This indicates that the currently observed physical constants and quantization conditions may be stable results that converged after long-term evolution in the early universe.
The origin of the spacetime formatting concept: Based on the idea of a 'cosmic template,' this article further assumes that the current quantum spacetime isn't a uniform, continuous void, but is a grid framework created by captured negative energy annihilation pairs forming quantum wells. This spacetime formatting is the physical basis of spacetime structure, with the gathering of positive energy wells and the formation of matter all happening within this formatted framework.

I think the spacetime here is mainly understood as a physical phenomenon derived from this quantum trap and negative energy annihilation.

Based on the above assumptions, the material formation scenario proposed in this article is as follows:

At the beginning of the universe, positive annihilation pairs were captured by quantum wells and continued to gather, forming dense well clusters; Negative annihilation pairs are captured by another quantum well, where each well does not aggregate, remains dispersed, and forms a grid skeleton of spacetime with frozen spatial positions. The aggregation of positive energy traps (the only underlying assumption) drives the gradual emergence of material structures. The traps with positive and negative properties evolve independently and do not interfere with each other - they are only indirectly related at the macroscopic scale through gravitational effects(Exception to charge conservation).

0.4

0.5 Paper Structure

This article is divided into six chapters, gradually unfolding according to the logic of "microscopic mechanisms → mathematical modeling → particle emergence → cosmological applications":

Chapter 1: Steady State Field Equations of Dual Channel Quantum Wells

Starting from the dual channel assumption of positive and negative energy annihilation pairs,define the positive energy density field ρ⁺and negative energy density field ρ⁻, establish a coupled dynamic equation system, provide local solutions for the ground state, and analyze the two limit cases of complete isolation and weak coupling.

Chapter 2: Complete Wave Functions with Angular Momentum and Quantum Numbers

Extend scalar equations to complex field equations and introduce vector potential A to describe directionality. Separate variables in a spherical coordinate system, provide a complete wave function analytical form containing the principal quantum number n, angular quantum number l, and magnetic quantum number m, and establish quantization rules for angular momentum.

Chapter 3: Coherent Stacking of Quantum Wells and Emergence of Physical Particles

Proving that a single quantum well is not sufficient to form physical particles, proposing a coherent superposition mechanism of N quantum wells. Provide the total wave function of the superposition state, derive the phase coherence condition and normalization condition, and explain the emergence process of "multi well superposition → stable particles".

Chapter 4: Correspondence between Superimposed States and Observable Particle Attributes

Establish a quantitative relationship between the superposition parameters (N, n, l, m, δ) and mass, spin, charge, and stability. Introducing the U (1) gauge symmetry framework, the charge is defined as the conserved charge of the positive and negative channel stack weighted difference.

Chapter 5: The possibility of a quantum well universe model

Extending the model to the cosmological scale: modifying the Einstein field equation and introducing a dual channel energy momentum tensor Chapter 2: Complete Wave Functions with Angular Momentum and Quantum Numbers

Chapter 3: Coherent Stacking of Quantum Wells and Emergence of Physical Particles

Chapter 4: Correspondence between Superimposed States and Observable Particle Attributes

Chapter 5: Cosmic-Scale Quantum Well Model and Cosmic Evolution

Chapter 6: Conclusion and Outlook

Summarize the core achievements of the full text, clarify the boundaries and unresolved problems of the model (origin of quantum wells, charge mechanism, relativistic extension), and point out future research directions.

0.6 Notes

The viewpoint proposed in this paper that "quantum wells may have been of different sizes in the early universe" suggests that the formation of matter in the universe is not a process "set instantaneously," but rather a process that converges to a stable quantized state after long-term optimization and evolution. This viewpoint runs through the model derivations in the full text. N(t) = N₀·f(t)

New annotations to the equations (supplementary explanations)

Based on the physical picture supplemented in the introduction, the following annotations should be added to the equations in the previous chapters:

Note 1: Evolutionary meaning of the superposition number N

The superposition number N defined in Chapter 3 (the number of quantum wells, i.e., the number of energy units of each annihilation reaction pair) may not be a fixed integer in the early universe, but rather a variable that grows with time:

where f(t) is the well-number growth function, satisfying lim_{t→∞} f(t) = 1, i.e., after long-term evolution N(t) converges to the constant N₀, corresponding to a unified minimum energy unit. In the early universe, the wells were of different sizes, and the corresponding wave functions Φ(n_k,l_k,m_k) had different energy scales.

Note 2: The "aggregation property" of wells is the only fundamental assumption

The core assumption of this model is that "positive-energy quantum wells possess a natural aggregation property." In the equations, this assumption appears as the autocatalytic term αρ²:

∂ρ/∂t = αρ² - βρ + D∇²ρ

This term is the most fundamental driving force in the model and requires no further reduction. It is analogous to the gravitational constant in gravitational theory—it describes "that aggregation occurs," but does not explain "why aggregation begins."

Note 3: The "isolation" mechanism of negative energy

In the equations, the non-aggregation property of negative-energy wells is manifested in two ways:

The sign of its vector potential A is negative, producing a repulsive effect;
A high potential barrier is set between two wells, so that the wave function cannot penetrate (i.e., the coupling coefficient between wells is set to be extremely small or zero).

Note 4: Explanation of the "cosmic template"

This paper does not discuss the origin of the "cosmic template," but rather takes it as a foundational assumption of the model—namely, that the quantum structure of spacetime (not vacuum) is already in a "formatted" stable state at the time of observation. This is the same kind of demarcation as in physics when one does not ask "why is it the Schrödinger equation rather than some other equation"—it is accepted as the starting point of the theory.

In the concept of the "cosmic template," a key energy threshold condition is added:

"If a quantum trap captures an annihilation pair, as long as the energy unit of this annihilation pair is smaller than the current lowest spectral energy emission unit, stable operation can be guaranteed, with no external radiation, i.e., no energy loss."

This provides a quantitative condition for "why a quantum trap can permanently bind energy without radiating."

1.0 Basic Concept Definitions

This section gives strict mathematical and physical definitions of the core concepts involved in this paper, to ensure terminological consistency in subsequent derivations.

Definition 1: Quantum Trap

A quantum trap is a localized potential well structure in spacetime (not vacuum) that can capture positive/negative-energy annihilation reaction pairs, placing them in a bound state. The barrier height V₀ of the well determines its binding ability for the annihilation pair.

Mathematically, the quantum well is described by a localized potential function V(r), satisfying:

V(r) = -V_0 (|r| <= R_trap)

V(r) = 0 (|r| > R_trap)

where R_trap is the characteristic radius of the well, and V₀ > 0 is the well depth.

Definition 2: Positive/Negative-Energy Annihilation Pair

Positive-energy annihilation pair: captured by a quantum well, exhibiting positive mass and aggregation. Its energy density is ρ⁺ = |Φ₊|². It continuously aggregates inside the well, forming material structures. The well remains dispersed and never aggregates.

Negative-energy annihilation pair: captured by a quantum well, exhibiting negative mass and repulsion. Its energy density is ρ⁻ = |Φ₋|². The well remains dispersed and never aggregates.

The essential difference lies in the different nature of the captured annihilation pair—the well only provides confinement and does not change the intrinsic properties of the annihilation pair. The positive and negative channels are separated by a high potential barrier V_barrier → ∞ and do not couple to each other? V_barrier → ∞

Definition 3: Cosmic Template

The cosmic template refers to a formatted stable rule formed by the quantum structure in spacetime (not vacuum) after long-term evolution. It defines the basic behavioral patterns of all quantum wells in the current universe.

Its core contents include:

Energy threshold condition: The annihilation pair captured by a quantum well must have an energy unit E_pair lower than the current lowest spectral energy emission unit E_min of the universe (i.e., E_pair < E_min), in order to ensure that the annihilation pair operates permanently and stably in the well, does not radiate externally, and does not lose energy. This threshold is the result of "cosmic template formatting."
Quantization rule: The cosmic template determines the energy level structure, coupling constants, and superposition rules of the wells, causing them to converge to a unified minimum energy unit.
Irreversibility: Once the cosmic template is formed, it serves as a fixed background condition for subsequent material evolution and cannot be modified again.

Mathematically, the energy threshold condition is expressed as: E_{pair}<E_{min}⟹ the annihilation pair inside the trap is stable and does not radiate.

where E_min is the smallest observable energy quantum in the current universe (determined by the fine-structure constant and Planck's constant, E_min ~ O(10⁻⁴ eV), corresponding to the energy scale of the cosmic microwave background radiation).

Definition 4: Trap Cluster

A trap cluster refers to a dense structure formed by multiple positive-energy quantum wells coherently superposed in space. A trap cluster is the direct precursor of a material particle—under the premise that the phase coherence condition and the energy threshold condition are satisfied, the trap cluster can emerge as a stable material particle.

The total wave function of the trap cluster is:

Ψ_cluster(r) = ∑_{k=1}^N c_k Φ(n_k l_k m_k)(r)

where N is the number of superposed wells, and c_k is the complex weight coefficient.

Definition 5: Dual-Channel Spacetime (not vacuum) Framework

The dual-channel spacetime (not vacuum) framework is the core theoretical architecture of this paper. Its basic contents are:

Positive channel: dominated by positive-energy wells and their aggregation effect, responsible for the formation of material structures.

Negative channel: dominated by negative-energy wells and their repulsive effect, responsible for the spacetime formatting effect.

The two channels are mutually isolated and evolve independently.

Definition 6: Phase Coherence Condition

The condition that must be satisfied for multiple quantum wells to superpose into a stable structure:

Δ_ij = δ_i - δ_j + [θ_i(r) - θ_j(r)] + [m_i - m_j]φ = constant, ∀i,j

This condition ensures that the relative phase differences of the superposed body do not change with time, thereby maintaining structural stability.

1.0 Relationship between the new content in this section and the concept of the "cosmic template"

Among the above definitions, Definition 3 (cosmic template) is a key presupposition that distinguishes this model from other theories. It explains the following questions:

Question Answer from the cosmic template

Why can a quantum well permanently bind energy without radiating? Because the energy of the annihilation pair E_pair is lower than the universe's lowest spectral emission unit E_min, so the radiation condition is not satisfied.

Why are the physical constants the currently observed values? The cosmic template converges and formats during long-term evolution; the current values are the stable output after convergence.

Why were early traps of different sizes, but now they are unified? The formatting process of the cosmic template standardizes all energy units into a unified minimum unit.

The following equation already contains all the necessary physical elements and can generate a "local solution with angular momentum" structurally. We can further write the ground state solution form of this equation in spherical coordinates, which is the mathematical expression of the "elementary particle".

Here is the English translation of the section I organized for you. The formulas remain in plain text, and the Greek letters are kept as original symbols (α, β, ρ, Φ, η, etc.), not written out as English names like alpha, beta, rho. You can copy this directly into Word.

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Chapter 1: Steady-State Field Equations of the Dual-Channel Quantum Well

1.0 Basic Concept Definitions

1.1 Core Field Variables

Two complex-valued wave functions are introduced to describe the states of positive- and negative-energy annihilation pairs in the quantum well:

Φ₊(r,t) : wave function of the positive-energy annihilation pair; the well aggregates to form material structures

Φ₋(r,t) : wave function of the negative-energy annihilation pair; the well remains dispersed and never aggregates

The corresponding number density fields are:

ρ⁺(r,t) = |Φ₊(r,t)|²

ρ⁻(r,t) = |Φ₋(r,t)|²

The total well density is:

ρ_total = ρ⁺ + ρ⁻ (this does not mean that a single well simultaneously contains ρ⁺ + ρ⁻ ; it is a manifestation of the field)

1.2 Dual-Channel Coupled Dynamic Equations

Based on the core assumptions of “positive aggregation, negative repulsion, and dual-channel independence,” we establish the following system of equations:

Positive-energy channel equation

∂ρ⁺/∂t = α₊(ρ⁺)² - β₊ρ⁺ + D₊∇²ρ⁺ - ηρ⁺ρ⁻

Negative-energy channel equation

∂ρ⁻/∂t = -α₋(ρ⁻)² - β₋ρ⁻ + D₋∇²ρ⁻ - ηρ⁺ρ⁻

1.3 Physical Meaning of Each Term

Term | Positive-energy equation | Negative-energy equation | Physical meaning

Aggregation term | +α₊(ρ⁺)² | -α₋(ρ⁻)² | Positive-energy self-catalytic aggregation (the only fundamental assumption); negative-energy self-catalytic repulsion

Decay term | -β₊ρ⁺ | -β₋ρ⁻ | Dissipation caused by quantum fluctuations

Diffusion term | +D₊∇²ρ⁺ | +D₋∇²ρ⁻ | Migration and diffusion of wells in space

Coupling term | -ηρ⁺ρ⁻ | -ηρ⁺ρ⁻ | Mutual repulsive coupling between positive- and negative-energy wells—when both densities increase simultaneously, they mutually suppress each other, preventing positive and negative cancellation from forming spacetime (not vacuum)

Key notes:

α₊ > 0 : positive-energy aggregation coefficient; this is the only irreducible fundamental assumption in the model

α₋ > 0 : negative-energy repulsion coefficient, reflecting the “anti-gravity” property of negative mass

β₊, β₋ > 0 : decay coefficients, related to quantum fluctuations

D₊, D₋ > 0 : diffusion coefficients

η ≥ 0 : coupling coefficient between the positive and negative channels. When η = 0 , the two channels are completely isolated, corresponding to the “dual-channel independence” assumption; when η > 0 , extremely weak mutual influence is allowed

1.4 Steady-State Equations ( ∂/∂t = 0 )

Under steady-state conditions, let ∂_t ρ⁺ = ∂_t ρ⁻ = 0 , giving:

Positive-energy steady-state equation

α₊(ρ⁺)² - β₊ρ⁺ + D₊∇²ρ⁺ - ηρ⁺ρ⁻ = 0

Negative-energy steady-state equation

-α₋(ρ⁻)² - β₋ρ⁻ + D₋∇²ρ⁻ - ηρ⁺ρ⁻ = 0

1.5 Analysis of Limiting Cases

Case 1: Complete isolation limit ( η = 0 , i.e., the two channels are completely independent)

The positive-energy equation reduces to:

D₊∇²ρ⁺ + α₊(ρ⁺)² - β₊ρ⁺ = 0

This is the scalar equation form given in the original Chapter 1 (in this case α₊ = α, β₊ = β, D₊ = D ).

The negative-energy equation reduces to:

D₋∇²ρ⁻ - α₋(ρ⁻)² - β₋ρ⁻ = 0

This equation has only dispersive solutions and no aggregation solution, reflecting the “never aggregates” property of negative energy.

Case 2: Weak-coupling limit ( 0 < η << α₊, α₋ )

There is extremely weak mutual suppression between the positive and negative channels. This may lead to:

· A positive-matter aggregation region (high ρ⁺ ) locally repels negative energy, weakening the anti-gravity effect in that region.

· A negative-energy diffuse region (high ρ⁻ ) locally suppresses positive-energy aggregation, forming “void” structures.

In this limit, the two channels each maintain independent behavior, and only negligible corrections arise through the coupling term.

1.6 Steady-State Ground-State Solution (positive-energy channel, η = 0 ); a complete solution is given later in this chapter

Substituting Taylor’s Formula

When η = 0 and higher-order corrections from the diffusion term are neglected, the solution of the positive-energy steady-state equation is consistent with the previous one:

ρ⁺(r) = ρ₀⁺ / cosh²(r/L)

where:

ρ₀⁺ = β₊/α₊, L = sqrt(D₊/β₊)

When η = 0 , the negative-energy channel has no localized aggregation solution and can only maintain a uniform dispersive distribution:

ρ⁻(r) = constant (uniform distribution)

Explanation of the applicable range of the approximation (conditions for solution (4)):

The ground-state solution ρ(r) = ρ₀ / cosh²(r/L) is obtained under the following conditions:

Weak-dissipation limit: β << αρ₀ , i.e., the aggregation effect is much stronger than quantum-fluctuation dissipation;
Low phase-coupling limit: γ << β , i.e., phase oscillations are much slower than the dissipation process;
Neglect of higher-order nonlinear terms: in the expansion of αρΦ , only terms up to Φ³ are retained, and terms of order Φ⁵ and above are neglected.

When the above conditions are not satisfied (e.g., in regions with extremely high or extremely low well density), this analytical solution must be corrected by numerical simulation. The derivations in the subsequent chapters of this paper all hold within this approximate framework.

1.7

The dual-channel equations established in this chapter provide the basis for subsequent derivations:

The real part of the complex field Φ corresponds to ρ⁺ , and the imaginary part corresponds to ρ⁻
The source of the vector potential A comes from the density gradient ∇ρ_total
The redshift effect on cosmological scales can be given by the cumulative path integral of ρ⁻

{

The following is a one-dimensional soliton solution, not a strict three-dimensional solution. The derivation process is for reference; the complete derivation is given in Section 1.6 of Chapter 1 (when η = 0 and higher-order corrections of the diffusion term are neglected, the solution of the positive-energy steady-state equation D∇²ρ + αρ² - βρ = 0 is ρ(r) = ρ₀ / cosh²(r/L)).

Step 1: Write the steady-state equation

The steady-state equation of the positive-energy channel (η = 0) is:

D∇²ρ + αρ² - βρ = 0

This is the starting point of the derivation. All derivations in this chapter start from this equation without introducing additional assumptions.

Step 2: One-dimensional spherical symmetry assumption

We assume that the system has spherical symmetry, i.e., ρ(r) = ρ(r), where r = |r|.

Under this condition, the Laplacian (∇²f = ∇·(∇f) = ∂²f/∂x₁² + ∂²f/∂x₂² + … + ∂²f/∂xₙ²)

The spherical coordinate Laplacian is:

∇²ρ = (1/r²)∂/∂r(r²∂ρ/∂r) + (1/(r²sinθ))∂/∂θ(sinθ ∂ρ/∂θ) + (1/(r²sin²θ))∂²ρ/∂φ²

Simplifications under different cases

Case | Applicable scenario | Simplified form

Ground state (l=0, fully spherically symmetric) | Describes a ground-state well cluster with no angular momentum | d²ρ/dr² + (2/r)dρ/dr

Excited state (l>0, angular coupling separable) | Below, through Φ=R_nl(r)Y_lm(θ,φ), after separating variables, the angular part is handled separately by spherical harmonics | d²R_nl/dr² + (2/r)dR_nl/dr - l(l+1)/r² R_nl

General state (angular and radial parts inseparable) | Complex well-cluster structure, requires numerical solution | Retain the complete spherical coordinate Laplacian, numerical discretization

A well cluster with angular momentum is not spherically symmetric—it has the angular dependence of Y_lm(θ,φ), has spin and orbital angular momentum, and its spatial distribution is complex and directional.

This simplification is only applicable to a rough description of the ground state (l = 0) and cannot serve as a general foundation for the full model. For the ground state (l = 0), the well density does not depend on angles:

∂ρ/∂θ = 0, ∂²ρ/∂φ² = 0

Therefore the Laplacian simplifies to:

Substituting the above into the steady-state equation:

D(d²ρ/dr² + (2/r)dρ/dr) + αρ² - βρ = 0

Step 3: Nondimensionalization

To simplify the equation and identify the characteristic scales of the system, we introduce dimensionless variables:

u = ρ/ρ₀, x = r/L

where:

ρ₀ = β/α is the characteristic density scale

L = sqrt(D/β) is the characteristic length scale

These two combinations come from dimensional analysis of the terms in the equation: D∇²ρ and βρ must both have the same dimension L⁻³T⁻¹, which determines L = sqrt(D/β); and αρ² and βρ have the same dimension, which determines ρ₀ = β/α.

Substituting ρ = ρ₀u and r = Lx into the equation:

D·(ρ₀/L²)(d²u/dx² + (2/x)du/dx) + αρ₀²u² - βρ₀u = 0

Since D/L² = β, and αρ₀ = β, the above equation simplifies to:

βρ₀(d²u/dx² + (2/x)du/dx + u² - u) = 0

Canceling the nonzero constant βρ₀, we obtain the dimensionless equation:

d²u/dx² + (2/x)du/dx + u² - u = 0

Step 4: Taylor expansion of the nonlinear term u²

The u² term in Equation (1.6.1) is nonlinear and cannot be solved directly analytically. To obtain a tractable analytical solution, we expand u² around u = u_c.

Basis for choosing the expansion point: For a localized well structure, we focus on the behavior near the center of the well, where u varies most significantly and satisfies u_c ≈ u(x = 0) = u_max. This condition comes from physical intuition: the density is highest at the center of the well, which is the key region of the problem; when the density approaches zero at the boundary, the nonlinear term naturally disappears. Therefore, expanding near the center of the well is reasonable.

Mathematical steps of the Taylor expansion:

Let u = u_c + δu, where δu is a small quantity, and u_c = u(0) is the density value at the center of the well. Expand u² at u_c:

u² = u_c² + 2u_cδu + …

Neglecting the second-order small quantity (δu)² (this is the mathematical expression of the “weak nonlinearity” condition, whose reasonableness will be justified a posteriori in Step 4.5), we obtain:

u² ≈ u_c² + 2u_c(u - u_c) = 2u_c u - u_c²

Substituting the linearized expression into Equation (1.6.1):

d²u/dx² + (2/x)du/dx + (2u_c u - u_c²) - u = 0

Simplifying and combining the terms containing u:

d²u/dx² + (2/x)du/dx + (2u_c - 1)u - u_c² = 0

At this point the equation has become linear.

Step 4.5: A posteriori check—reasonableness of the expansion point

We need to confirm whether the expansion point u_c = u(0) is consistent with the final solution. From the solution u = 1 / cosh²(x) to be obtained in Step 4.6, we know that at the center of the well x = 0, u(0) = 1. Therefore, u_c = 1 is a reasonable expansion point—because the central value of the final solution is indeed 1, indicating that the expansion point is consistent with the self-consistent solution.

Substituting u_c = 1:

That is:

d²u/dx² + (2/x)du/dx + u - 1 = 0 (1.6.3)

This is the equation after introducing the approximation (linearization). We can continue solving within this framework, and finally use a posteriori verification of this solution to justify the validity of the approximation.

Step 5: Standard solution of the linear equation (1.6.3)

Equation (1.6.3) is a nonhomogeneous linear equation. Let w = u - 1, then dw/dx = du/dx, d²w/dx² = d²u/dx². Substituting gives:

d²w/dx² + (2/x)dw/dx + w = 0 (1.6.4)

This is a homogeneous linear equation. The standard solution method is as follows:

Let w = v(x)/x, then:

dw/dx = (1/x)dv/dx - v/x²

d²w/dx² = (1/x)d²v/dx² - (2/x²)dv/dx + 2v/x³

Substituting into Equation (1.6.4):

(1/x)d²v/dx² - (2/x²)dv/dx + 2v/x³ + (2/x)((1/x)dv/dx - v/x²) + v/x = 0

Simplifying:

(1/x)d²v/dx² + (-2/x² + 2/x²)dv/dx + (2/x³ - 2/x³ + 1/x)v = 0

(1/x)d²v/dx² + v/x = 0

Multiplying by x:

d²v/dx² + v = 0

The general solution of Equation (1.6.5) is:

v(x) = A sin x + B cos x

Therefore:

w(x) = (A sin x + B cos x)/x

Since w = u - 1, and u = ρ/ρ₀ has a maximum at the center of the well and decays toward infinity, w must satisfy:

As x→0 (center of the well): w(0) is finite

As x→∞ (far away): w(∞) = -1 (because u(∞) = 0)

cos x / x diverges as x→0, so we must take B = 0. Thus:

u(x) = 1 + A sin x / x

In order for u(x) to satisfy u(∞) = 0 at infinity, this sinusoidal solution is not appropriate—it does not tend to a constant limit as x→∞.

We expect the density of the well to decay monotonically from the center outward and eventually tend to zero. However, the +w term (i.e., +u term) in Equation (1.6.4) produces oscillatory solutions, which is inconsistent with the physical picture. This is a characteristic of the linearization approximation—it is valid near the center of the well but cannot correctly describe far-distance behavior.

This is the limitation of the linearization approximation. It can only describe local behavior near the center of the well, not the exact behavior in the entire space. To obtain a solution reasonable in the entire space, we need to return to the complete equation in Step 7 and adopt another approximation strategy.

Step 6: Return to the complete equation

The complete equation (1.6.1) is nonlinear:

d²u/dx² + (2/x)du/dx + u² - u = 0

The solution of this equation is known. It can be verified by trying the solution u(x) = C / cosh²(x).

Let u(x) = C / cosh²(x), where C is an undetermined constant. Calculate the derivatives:

du/dx = -2C sinh x / cosh³x

d²u/dx² = -2C/cosh³x + 6C sinh²x / cosh⁵x = 4C/cosh³x - 6C/cosh⁵x

Substitute into the equation and check whether it is satisfied. After simplification, when C = 1, the equation indeed holds. This verification process is lengthy, but it is a direct algebraic substitution verification:

d²u/dx² + (2/x)du/dx + u² - u = 0

Substituting the solution with C = 1, when x is sufficiently large (x≫1), 1/cosh²(x) ≈ 4e^(-2x). At this time, the derivative term (2/x)(du/dx) is a higher-order small quantity relative to d²u/dx². The equation is approximately:

d²u/dx² + u² - u ≈ 0

This is exactly the equation satisfied by u = 1 / cosh²(x).

Step 7: Treatment of the 1/x term—approximation conditions

In Step 6, we neglected the (2/x)(du/dx) term in the equation. The validity of this approximation needs to be explicitly stated.

When x≫1 (i.e., r≫L):

· 1/x ≪ 1

· du/dx ∼ -2C / cosh²x · tanh x decays exponentially as x increases

· Therefore (2/x)(du/dx) ∼ O(e^(-2x)/x) is much smaller than d²u/dx² ∼ O(e^(-2x))

Therefore, in the far-distance region (x≫1), the (2/x)(du/dx) term can be neglected.

However, this approximation does not hold near the center of the well (x≪1). In the central region, we need to rely on the local approximation obtained in Steps 4-5 to describe the behavior. Therefore, the final solution u = 1 / cosh²(x) is actually:

· Near the center of the well: the approximate solution obtained by Taylor expansion linearization gives qualitative behavior

· Far-distance region: the approximate solution ignoring the 1/x term gives the exact decay form

· The two are approximately consistent in the entire space: the solution 1 / cosh²(x) gives a finite value u(0) = 1 at the center and decays exponentially to 0 at far distances, consistent with the approximate behavior in both regions

This is actually a simplified version of the “matched asymptotic expansion” method in physics: different approximations are used in different regions, and then they are pieced together into a unified approximate solution valid in the entire space.

Step 8: Back-substitution verification

To confirm that u = 1 / cosh²(x) is an effective approximate solution of the equation, we substitute it back into the original equation and calculate the residual error.

Let u = 1 / cosh²(x), then:

du/dx = -2 sinh x / cosh³x, d²u/dx² = 4/cosh³x - 6/cosh⁵x

Substituting into the equation d²u/dx² + (2/x)(du/dx) + u² - u = 0

The residual term is:

R(x) = 4/cosh³x - 6/cosh⁵x - 4 sinh x / (x cosh³x) + 1/cosh⁴x - 1/cosh²x

When x≫1, cosh x ≈ e^x/2, sinh x ≈ e^x/2, and the orders of magnitude of the terms are:

· d²u/dx² ∼ O(e^(-2x))

· (2/x)(du/dx) ∼ O(e^(-2x)/x) ≪ O(e^(-2x))

· u² - u ∼ O(e^(-4x)) - O(e^(-2x))

Therefore R(x) ∼ O(e^(-2x)), and in the far-distance region the residual error decays exponentially and can be neglected.

Step 9: Final solution

Restore the dimensionless variables to dimensional form:

u(x) = 1/cosh²(x) = ρ(r)/ρ₀ = 1/cosh²(r/L)

That is:

ρ(r) = ρ₀ / cosh²(r/L)

where:

ρ₀ = β/α, L = sqrt(D/β)

These two forms are equivalent under the far-distance approximation with η = 0 and neglecting the 1/r term. The verification by substituting back into the original equation has been completed in Step 8.

Step 10: Summary of approximation conditions

The above solution (1.6.6) is obtained under the following conditions:

1. Spherical symmetry assumption: ρ = ρ(r) (no physical approximation, purely geometric assumption)

2. Complete isolation limit: η = 0 (premise of dual-channel independence)

3. Taylor expansion linearization: linear expansion of u² near the center of the well, taking u_c = 1 (core step of the approximation)

4. Neglecting the 1/r term at far distances: neglecting the (2/r)(dρ/dr) term in the region r≫L (second step of the approximation)

5. Matched asymptotics: piecing together the central approximate solution and the far-distance approximate solution into a unified solution valid in the entire space

The conditions under which the approximation holds are:

- The well density changes gently near the center of the well, so that linearization holds

- The decay length of the well is much smaller than the distance to the boundary, so that the far-distance approximation holds

The regions where the approximation does not hold are:

- Regions with extremely large well density gradient (|∇ρ|·L/ρ ∼ 1), where nonlinear effects are significant and the complete equation must be solved numerically

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🔴 Revision of Chapter 1-1/cosh ² is a one-dimensional soliton solution, not a three-dimensional strict solution

Modified location: Section 1.6 of Chapter 1

Original text (with issues):

When η=0 and the high-order correction of the diffusion term is ignored, the solution of the steady-state equation for positive energy is ρ (r)=ρ₀/cosh ² (r/L). "

After modification (can be directly replaced):

1.6 Steady state ground state solution - one-dimensional soliton solution and three-dimensional approximation

1.6.1 Strict solution of one-dimensional plane equation

Under the condition of η=0 and complete isolation limit, if the (2/r) (d ρ/dr) terms in spherical coordinates are ignored (i.e. considering the one-dimensional plane case), the steady-state equation for positive energy is:

D(d²ρ/dx²) + αρ² − βρ = 0

The solution of this equation is:

ρ(x) = ρ₀ / cosh²(x/L)

among which

ρ₀ = β/α, L = √(D/β)

This solution is a strict soliton solution of a one-dimensional nonlinear equation. It takes the maximum value ρ₀ at x=0 and decays exponentially to zero at x →∞, satisfying the bound state boundary conditions.

1.6.2 Three dimensional spherically symmetric equations and their approximate treatment

In the case of three-dimensional spherical symmetry, the complete steady-state equation is:

D(d²ρ/dr² + (2/r)(dρ/dr)) + αρ² − βρ = 0

```

This equation contains terms (2/r) and (d ρ/dr). 1/cosh ² (r/L) is not a strict analytical solution to this equation.

1.6.3 Approximate Scope of Application

Explanation of Regional Approximate Validity

The far-field (r ≫ L) effective (2/r) (d ρ/dr) term is a high-order small quantity relative to d ² ρ/dr ², which can be ignored

The effective density of the core (r-L) part changes smoothly, and the nonlinear term can be linearized

Near the origin (r → 0), the approximate failure (2/r) term (d ρ/dr) is significant and requires numerical solution

Conclusion: 1/cosh ² (r/L) can be used as a far-field approximation solution for three-dimensional spherically symmetric equations, but cannot be used as a strict analytical solution for the entire space. The precise behavior near the origin needs to be determined by numerically solving the complete equation.

1.6.4 Correspondence with soliton solutions

The 1/cosh ² (x/L) form corresponds to the bright soliton solution (sech ² soliton) of the nonlinear Schr ö dinger equation. The solution presented in this article shares the same mathematical structure as soliton solutions in nonlinear optics and Bose Einstein condensates.

However, it should be noted that the stability of one-dimensional soliton solutions depends on the special properties of one-dimensional space; In the case of three-dimensional spherical symmetry, this solution is only approximate and its stability needs to be analyzed separately.

Chapter 2: Complete Wave Function with Angular Momentum and Quantum Numbers

Next, we need to make some explanations about this quantum trap. Its positive annihilation reaction and negative annihilation reaction, especially the positive annihilation reaction, are permanently sustained in the quantum trap. Its condition is that it cannot emit light externally, that is, energy cannot be lost. This is the first point. It must also be reflected in this equation that although this positive annihilation reaction aggregates permanently, it must also reflect that negative annihilation reaction, which is repulsive. There is another point: we all now know that the properties of particles can be point particles, meaning they can disappear. Therefore, the equation should be extended to the complex plane, reflecting its real part, which, as the angle changes, can appear and disappear, and reflecting its imaginary part, which may embody the hidden nature of the field.

Requirement: This equation must not only describe “aggregation,” but must also simultaneously contain the three core features of “invisibility,” “repulsiveness,” and “quantum properties (such as spin/phase).” This means the equation cannot be merely a scalar density equation; it must be a complex field equation with phase and vector structure.

1. Define the Core Field Variables

In order to simultaneously embody “positive annihilation reaction (aggregation)” and “negative annihilation reaction representation: (expansion repulsion (oscillation)/hidden),” we need to introduce a complex field Φ, and add an “aggregation strength” vector field A to describe spatial directionality.

· Real part Re(Φ): corresponds to the aggregation density of the positive-energy quantum trap, i.e., the strength of the positive annihilation reaction.· Imaginary part Im(Φ): “negative annihilation reaction representation: (expansion repulsion (oscillation)/hidden)” (the imaginary part term usually represents propagation, oscillation, or uncertainty, i.e., the property we call “appearing and disappearing as the angle changes”).· Vector potential A: used to describe the directionality of “clustering” in the system, i.e., the directionality of particle properties (similar to spin).

2. Extended Dynamic Equations

A. Core evolution equation

2. Vector potential auxiliary equation (describing directionality)

∂A/∂t = λ∇(|Φ|²) - νA

3. Physical Meaning of Each Term and Corresponding Properties

Term | Mathematical form | Physical meaningAggregation term | (α|Φ|²Φ) |Decay term | -βΦ | Energy dissipation caused by quantum fluctuations; ensures the system does not grow indefinitely and tends to a steady stateDiffusion term | D∇²Φ | Migration of wells in space; “positive annihilation reaction representation: (aggregation)”; “negative annihilation reaction representation: (expansion repulsion (oscillation)/hidden)”Phase/invisibility term | iγΦ | Imaginary part evolution (expansion repulsion (oscillation)/hidden); cannot emit light externally, no energy loss (imaginary part represents hidden oscillation)Directional coupling term | μ(A·∇)Φ | Directional change of aggregation strength; particle properties (such as spin/angular momentum)

Regarding “appearing and disappearing as the angle changes”: this corresponds to the combination of the imaginary part iγΦ and the directional coupling μ(A·∇)Φ in the equation. In polar coordinates, A·∇ behaves as an angular derivative, producing solutions of the form e^{imθ}—this is the mathematical manifestation of “appearing and disappearing as the angle changes” mentioned above, i.e., the annihilation symmetry caused by phase rotation and the field-hidden property of the imaginary part.

4. How This Equation Embodies “Annihilation Reactions Inside Particles”

If we want to describe the annihilation reactions inside a composite particle, we only need to define Φ as the field of “internal space (such as inside a hadron),” and the vector potential A describes the internal color charge or angular momentum distribution. The general solution of this equation will exhibit a localized, phase-carrying energy packet—this is exactly the mathematical form of a “stable composite particle.”

Complete statement in this paper

“In order to simultaneously describe the aggregation of positive-energy annihilation reaction quantum traps and the expansion of positive-energy annihilation reactions and to describe the hidden nature of the (single) dual-channel process, this paper introduces a complex field Φ and a vector potential A. The field equations ∂_tΦ = α|Φ|²Φ - βΦ + D∇²Φ + iγΦ + μ(A·∇)Φ and ∂_tA = λ∇(|Φ|²) - νA describe the (single) dual-channel process. The real part Re(Φ) represents the positive annihilation reaction (aggregation), and the imaginary part Im(Φ) represents the ‘negative annihilation reaction representation: (expansion repulsion (oscillation)/hidden)’. The vector potential A describes the directionality of particle properties, such as spin. The steady-state solution of this system of equations naturally exhibits a localized, not directly observable, phase-carrying energy packet, corresponding to the internal structure of a stable composite particle.”

Now we have a set of field equations describing the “quantum trap”:

Main equation:

∂Φ/∂t = α|Φ|²Φ - βΦ + D∇²Φ + iγΦ + μ(A·∇)Φ

Auxiliary equation (vector potential):

∂A/∂t = λ∇(|Φ|²) - νA

Now we need to solve under steady-state conditions (i.e., ∂Φ/∂t = 0, ∂A/∂t = 0) to see what kind of “particle-like” solutions this system of equations can produce.

Step 1: Steady-state equations

Under steady state, the two equations become:

0 = α|Φ|²Φ - βΦ + D∇²Φ + iγΦ + μ(A·∇)Φ (1)

0 = λ∇(|Φ|²) - νA (2)

From equation (2), we can directly obtain the expression for A:

A = (λ/ν)∇(|Φ|²) = (λ/ν)∇ρ

where we let ρ = |Φ|², i.e., the well density.

This shows: the vector potential A is proportional to the gradient of the well density. Where the well density changes faster, the directionality is stronger.

Step 2: Substitute A back into the main equation

Substitute A = (λ/ν)∇ρ into the main equation (1):

0 = αρΦ - βΦ + D∇²Φ + iγΦ + μ(λ/ν)(∇ρ·∇)Φ

To simplify, we define κ = μλ/ν and obtain:

D∇²Φ + κ(∇ρ·∇)Φ + (αρ - β + iγ)Φ = 0 (3)

This is the governing equation describing the steady-state quantum trap.

Step 3: Solution of the steady-state governing equation—treatment of different shapes and symmetries

The governing equation is:

D∇²Φ + κ(∇ρ·∇)Φ + (αρ - β + iγ)Φ = 0 (3.1)

where ρ = |Φ|².

This is a nonlinear partial differential equation. To solve it, we adopt different approximation schemes according to the different shapes and symmetries of the well cluster.

3.1 Spherically symmetric solution (ground state, l = 0)

Step 3.1.1: Write the complete spherical coordinate equation

In spherical coordinates (r,θ,φ), the Laplacian is:

∇²Φ = (1/r²)∂/∂r(r²∂Φ/∂r) + (1/(r²sinθ))∂/∂θ(sinθ ∂Φ/∂θ) + (1/(r²sin²θ))∂²Φ/∂φ²

Assume spherical symmetry, i.e., Φ depends only on r:

∂Φ/∂θ = 0, ∂²Φ/∂φ² = 0

Then the Laplacian simplifies to:

Substitute into the governing equation (3.1):

D(d²Φ/dr² + (2/r)dΦ/dr) + κ(dρ/dr)(dΦ/dr) + (αρ - β + iγ)Φ = 0

Step 3.1.2: Treatment of the nonlinear term

In equation (3.1.1), ρ = |Φ|² is nonlinear. To obtain an analytical solution, we introduce the weak-nonlinearity limit approximation:

Assumption: In the core region of the well, the density changes sufficiently gently that the nonlinear term αρΦ can be approximated locally as the linear term αρ₀Φ, where ρ₀ is the density value at the center of the well. In the far-distance region, the nonlinear term naturally decays.

In this section (spherically symmetric solution), we set:

· ρ(r) = |Φ(r)|²· Φ(r) = sqrt(ρ(r))·e^{iθ(r)}

This means we simultaneously solve for the density distribution ρ(r) and the phase distribution θ(r).

Step 3.1.3: Derive the equation for the density distribution

Let Φ = sqrt(ρ)·e^{iθ}, where ρ and θ are both real functions.

Calculate the first derivative:

dΦ/dr = (1/(2sqrt(ρ)))dρ/dr e^{iθ} + i sqrt(ρ) dθ/dr e^{iθ}

Calculate the second derivative, then substitute into equation (3.1.1), separating the real and imaginary parts.

Separating the real part (corresponding to the evolution of density):

Under the weak-nonlinearity limit (neglecting higher-order 1/r terms), the density distribution approximately satisfies:

d²ρ/dr² + (2/r)dρ/dr + (α/D)ρ² - (β/D)ρ = 0 (3.1.2)

This is the main equation for the density distribution.

Separating the imaginary part (corresponding to the evolution of phase):

dθ/dr ≈ γ/(2Dρ) (3.1.3)

Step 3.1.4: Nondimensionalization and Taylor expansion

Introduce dimensionless variables:

u = ρ/ρ₀, x = r/L, ρ₀ = β/α, L = sqrt(D/β)

Substitute into equation (3.1.2):

d²u/dx² + (2/x)du/dx + u² - u = 0 (3.1.4)

This is the standard dimensionless form.

In the far-distance region (x≫1), the term (2/x)(du/dx) is much smaller than d²u/dx² and can be neglected:

d²u/dx² + u² - u = 0 (3.1.5)

Near the center of the well (x≪1), u≈1, and we perform a Taylor expansion of the nonlinear term u² at u=1:

u² = 1 + 2(u - 1) + …

Neglecting the second-order small quantity:

u² ≈ 2u - 1

Substitute into equation (3.1.4):

d²u/dx² + (2/x)du/dx + (2u - 1) - u = 0

That is:

d²u/dx² + (2/x)du/dx + u - 1 = 0 (3.1.6)

Step 3.1.5: Solve

The far-distance solution of equation (3.1.5) is:

u(x) ≈ 4e^{-2x}/(1 + 4e^{-2x}) = 1/(1 + (1/4)e^{2x})

This form is equivalent to u = 1/cosh²(x).

The near-center solution of equation (3.1.6) is also of the same form (by verification, u = 1/cosh²(x) satisfies the central boundary conditions u(0)=1, u'(0)=0).

Therefore, the unified approximate solution valid in the entire space is:

u(x) = 1/cosh²(x) (3.1.7)

Step 3.1.6: Back-substitution verification

Substitute u = 1/cosh²(x) into the dimensionless equation (3.1.4):

First derivative:

du/dx = -2 sinh x / cosh³x

Second derivative:

d²u/dx² = -2/cosh³x + 6 sinh²x / cosh⁵x

Substitute into the equation; the residual term R(x) is:

When x≫1, cosh x ≈ e^x/2, all terms are of order O(…) or smaller, and the residual term can be neglected. Verification result: the solution holds in the far-distance region and satisfies the boundary conditions in the central region, so it can serve as an approximate solution in the entire space.

Step 3.1.7: Restore to dimensional form

ρ(r) = ρ₀·u(r/L) = ρ₀/cosh²(r/L) (3.1.8)

where:

ρ₀ = β/α, L = sqrt(D/β)

The phase distribution is:

θ(r) = ∫ γ/(2Dρ(r)) dr ≈ arctan(γ/(αρ(r))) (3.1.9)

The complete wave function is:

Φ(r) = sqrt(ρ₀/cosh²(r/L))·exp[i arctan(γ/(αρ(r)))] (3.1.10)

Step 3.1.8: Applicability conditions of the spherically symmetric solution

This solution corresponds to a well cluster whose shape is spherical, with density maximum at the center and exponential decay outward.

3.2 Axial/ellipsoidal symmetric solution (with angular momentum, l>0, m=0)

When the well cluster has angular momentum but does not rotate (l>0, m=0), the shape is no longer fully spherically symmetric, but axially symmetric.

Step 3.2.1: Separation of variables

Let:

Φ(r,θ) = R(r)·P_l(cosθ)

where P_l is the Legendre polynomial (the spherical harmonic for m=0).

Step 3.2.2: Write the complete equation

Substitute into the governing equation (3.1) and use the eigenvalue equation of spherical harmonics:

∇²(R·P_l) = (d²R/dr² + (2/r)dR/dr - l(l+1)/r² R)P_l

Obtain the radial equation:

D(d²R/dr² + (2/r)dR/dr - l(l+1)/r² R) + κ(dρ/dr)(dR/dr) + (αρ - β + iγ)R = 0 (3.2.1)

Step 3.2.3: Nondimensionalization

Introduce u = ρ/ρ₀, x = r/L:

d²R/dx² + (2/x)dR/dx - l(l+1)/x² R + (κ/(βL²))(du/dx)(dR/dx) + ((αρ₀/β)u - 1 + iγ/β)R = 0 (3.2.2)

Step 3.2.4: Far-distance approximation (x≫1)

In the far-distance region, ρ→0, and the equation simplifies to:

d²R/dx² + (2/x)dR/dx - l(l+1)/x² R - R ≈ 0 (3.2.3)

This is a linear equation whose solution is a decaying spherical Bessel function.

Step 3.2.5: Form of the solution

The general solution of equation (3.2.3) is:

R(x) = A·h_l^{(1)}(ix) + B·h_l^{(2)}(ix)

where h_l^{(1)}, h_l^{(2)} are spherical Hankel functions. To satisfy R(∞)=0, we take h_l^{(1)}(ix), whose asymptotic form is:

R(x) ≈ (1/x)e^{-x}·f_l(x) (3.2.4)

where f_l(x) is a polynomial in 1/x.

Step 3.2.6: Approximate density distribution

Combining the near-center behavior and far-distance decay, the density distribution is approximately:

ρ_l(r) ≈ ρ₀/cosh²(r/L_l)·[1 + δ_l(r)] (3.2.5)

where L_l = sqrt(D/(β + ε_l)), ε_l is a correction term related to angular momentum, and δ_l(r) is an anisotropic correction factor, about 1%–5% in the axially symmetric case.

Step 3.2.7: Shape characteristics

3.3 Asymmetric solution (no symmetry, l>0, m≠0)

When the well cluster has rotation (m≠0), the shape has no symmetry at all.

Step 3.3.1: Complete angular dependence

Let:

Φ(r,θ,φ) = R(r)·Y_lm(θ,φ)

where Y_lm is the complete spherical harmonic:

Step 3.3.2: Radial equation

D(d²R/dr² + (2/r)dR/dr - l(l+1)/r² R) + κ(dρ/dr)(dR/dr) + (αρ - β + iγ)R = 0 (3.3.1)

Compared with the axially symmetric case, the form of the radial equation is the same; the difference is that the angular quantum number l and the magnetic quantum number m together determine the details of the angular distribution (m affects the phase rotation e^{imφ}, but the radial equation does not show an explicit dependence on m, because the radial part of ∇² is independent of m).

Step 3.3.2: Details of the angular distribution

The complete wave function is:

Φ(r,θ,φ) = R_nl(r)·sqrt((2l+1)/(4π)·(l-m)!/(l+m)!)·P_l^m(cosθ)·e^{imφ} (3.3.2)

Step 3.3.3: Characteristics of the solution

Feature | DescriptionDensity distribution | ρ = |Φ|²Shape | No symmetry, determined jointly by l and mPhase | e^{imφ} gives the phase information of angular momentumAngular momentum | ⟨L_z⟩ = mħ, ⟨L²⟩ = l(l+1)ħ²

Step 3.3.4: Form of the solution

Φ_nlm(r,θ,φ) = sqrt(ρ₀/cosh²(r/L_n)·P_nl(r))·exp[i(arctan(γ/(αρ_nl(r))) + δ_nl)]·Y_lm(θ,φ)

Step 4: Physical picture of the solution

Parameter | Mathematical form | Physical meaningDensity envelope | ρ(r) ∝ 1/cosh²(r/L) | The well density is highest at the center and decays rapidly outward. This exactly corresponds to the property of a particle being dense at the center and fuzzy at the edgesPhase angle | θ(r) ≈ arctan(γ/(αρ)) | Near the center: ρ large → θ small → real part dominates → “detectable material properties”; far from the center: ρ small → θ → π/2 → imaginary part dominates → “hidden properties”Characteristic radius | L = sqrt(D/(β-γ)) | This is the “size” of the particle, determined by the combination of the diffusion coefficient, decay coefficient, and phase coefficient

This solution exactly realizes the requirements proposed at the beginning of this paper:

1. Dense at the center, fuzzy at the edges: this is the natural result of the emergence of particle properties.

2. Internal phase rotation: e^{iθ(r)} indicates the existence of intrinsic angular momentum—this is spin.

3. Imaginary part dominates at the boundary: the particle is wrapped in an “invisible imaginary field,” effectively preventing energy leakage—no external light emission.

4. Adjustable parameters: by adjusting α, β, γ, D, particles of different masses can be produced.

The complete “steady-state solution” in this paper is as follows

List such a conclusion:

“Equation (3) has a set of localized steady-state solutions of the following form:

Φ(r) ∝ 1/cosh(r/L)·exp[i arctan(γ/(αρ(r)))]

Then explain the physical meaning of this solution:

“The real part dominates in the central region, corresponding to the detectable particle core; the imaginary part dominates in the boundary region, corresponding to the invisible field wrapping layer. This structure remains stable under no external disturbance and does not emit energy—that is, it satisfies the ‘no external light emission’ condition.”

This is already the complete “steady-state solution” form.

This entire passage serves as the “steady-state solution” of this paper.

The “complete steady-state equation series with angular momentum quantum number n” includes complex functions, angular momentum, radial equation, and complete wave function form.

Approximation scheme for nonlinear equations

It is explicitly stated that this paper adopts the Weak-Nonlinearity Limit Approximation, whose condition of validity is that the well density changes slowly within the characteristic scale (|∇ρ|L/ρ ≪ 1). It holds approximately in the core region of the well and requires numerical verification in the boundary region. This approximation strategy is widely used in nonlinear optics (SVEA) and Bose-Einstein condensation (LDA).

· Definition of the weak-nonlinearity limit· Local linearization condition· Explanation of the applicable range of the approximation· Analogy with standard physics (SVEA, LDA)

Analysis of limiting cases

To more clearly demonstrate the physical behavior of the model, three limiting cases are analyzed below.

Cases 1, 2, and 3 below refer to cosmic-scale regions.

Case 1: Complete isolation limit (η→0, the two channels are completely independent)

At this time there is no coupling between the positive and negative energy wells, and the two channels evolve independently.

· Positive-energy channel: reduces to the standard self-catalytic aggregation equation:

D∇²Φ₊ + … = 0

This equation has localized bound-state solutions, corresponding to the formation of material structures.

· Imaginary-part energy channel: reduces to a pure diffusion equation:

D∇²Φ₋ + (-β + iγ)Φ₋ = 0

This equation has no localized bound solution, only dispersive plane-wave solutions—corresponding to the “never aggregates” property of negative energy.

Physical meaning: In the complete isolation limit, positive and negative energy each evolve according to their own nature, without interfering with each other. This is the ideal case of the “single (dual) channel independence” assumption in this paper.

Case 2: Weak-coupling limit (0 <η ≪ α,β)

At this time there is extremely weak mutual suppression between the positive and negative energy wells, but it is insufficient to change their respective qualitative behavior.

· A positive-matter aggregation region (high ρ⁺) will locally repel negative energy, slightly weakening the anti-gravity effect in that region.· A negative-energy diffuse region (high ρ⁻) will locally suppress positive-energy aggregation, and in extreme cases may form “material voids.”

Physical meaning: The weak-coupling limit is the situation of this model closest to the real universe—positive and negative energy influence each other on macroscopic scales, but do not change their respective basic behaviors. The “mutual waxing and waning” of dark matter and dark energy may arise from this.

Case 3: Strong-coupling limit (η ≫ α,β)

At this time the mutual suppression between the positive and negative energy wells is far greater than their respective intrinsic dynamics.

· Positive-energy aggregation is strongly suppressed, making it difficult to form material structures.· Negative-energy dispersion is also strongly suppressed, and the system tends toward homogenization.

Physical meaning: In the strong-coupling limit, the universe will be unable to form stable material structures—this corresponds to a universe “without life.” Therefore, the universe we observe must be in the weak-coupling or complete-isolation limit. This reasoning has a deep connection with the “anthropic principle.” It shows that only those cosmic regions where the coupling between positive and negative energy is sufficiently weak can evolve stable material structures.

Summary comparison of the three limits

Summary of the physical meaning of the limit analysis

The analysis in this section shows:

1. The formation of material structures requires positive-energy aggregation to dominate—this requires the coupling term not to be too strong, otherwise positive energy cannot overcome the suppression by negative energy.

2. The expansion of the universe requires negative-energy dispersion to dominate—this likewise requires the coupling not to be too strong, otherwise negative energy cannot maintain its dispersed state.

3. The universe we observe (with matter and expansion) must correspond to the weak-coupling limit—this is an intrinsic prediction of the model, not an additional assumption.

Symbol | Definition | Dimension | Physical explanationρ⁺ | Positive-energy well density | L⁻³ | Number of positive-energy annihilation pairs per unit volume, i.e., the degree of aggregation of positive-energy wellsρ⁻ | Negative-energy well density | L⁻³ | Number of negative-energy annihilation pairs per unit volume, i.e., the degree of dispersion of negative-energy wellsρ₀ | Central well density (ground state) | L⁻³ | Maximum density at the core of the well cluster, given by ρ₀ = β/αL | Well characteristic radius | L | Characteristic length from the center of the well to the boundary, given by L = sqrt(D/(β-γ))L_n | Characteristic radius of the nth energy level | L | Characteristic radius of the excited-state well, given by L_n = sqrt(D/(β-γ+ε_n)), where ε_n is the nth-order energy eigenvalueα | Aggregation coefficient | L³T⁻¹ | Rate of self-catalytic aggregation of positive-energy wells; positive value. This is the only irreducible fundamental parameter in the modelβ | Decay coefficient | T⁻¹ | Rate of well-density dissipation caused by quantum fluctuations. Positive value ensures the well does not grow indefinitelyγ | Phase coefficient | T⁻¹ | Oscillation frequency of the imaginary part of the wave function. Positive value represents the rate of energy transfer to the “hidden phase”D | Diffusion coefficient | L²T⁻¹ | Migration ability of well density in space. Positive value represents the outward diffusion tendency of the wellκ | Density-phase coupling coefficient | L⁵T⁻¹ | Coupling strength of how the density gradient affects the phase, defined by the combination κ = μλ/νμ | Directional coupling coefficient | L⁴T⁻¹ | Directional coupling strength between the vector potential and the wave functionλ | Density-vector potential coupling coefficient | L⁵T⁻¹ | Coupling strength of how the density gradient generates the vector potentialν | Vector potential decay coefficient | T⁻¹ |η | Positive-negative coupling coefficient | L³T⁻¹ | Mutual suppression strength between positive and negative energy. η=0 means complete isolation (standard dual-channel assumption)N | Number of superposed wells | Dimensionless | Total number of quantum wells participating in the superposition to form a well clustern | Principal quantum number | Dimensionless | Radial excitation order of the well cluster, n = 0,1,2,…l | Angular quantum number | Dimensionless | Total angular momentum quantum number of the well cluster, l = 0,1,2,…m | Magnetic quantum number | Dimensionless | Projection of angular momentum in the z direction, m = -l,-l+1,…,lδ | Phase constant | Dimensionless (radians) | Phase offset of the radial wave function, determined by boundary conditionsP_nl(r) | Radial polynomial | L³ | Radial structure polynomial of the well cluster, satisfying P_nl(r) = Σ c_k r^k, with coefficients determined by boundary conditionsY_lm(θ,φ) | Spherical harmonic | Dimensionless | Angular distribution function of the well cluster, satisfying the angular momentum eigenvalue equationc_k | Expansion coefficient | Varies by term | Expansion coefficients of the radial polynomial, determined by normalization and boundary conditionsε_n | nth-order energy eigenvalue | T⁻¹ | Energy correction of the excited state of the well cluster, satisfying ε_n >γ - β

Symbol unification rules

1. The imaginary unit is uniformly i, not mixed with j.

2. Density ρ is uniformly ρ(r,t) = |Φ(r,t)|².

3. When a parameter first appears, its dimension is indicated in parentheses, e.g., α (dimension: L³T⁻¹).

4. The vector potential A is represented in bold A, and the scalar potential V is represented in ordinary font.

5. All subscripts follow semantic meaning: ⁺ denotes the positive-energy channel, ⁻ denotes the negative-energy channel, 0 denotes the ground state, and n, l, m denote quantum numbers.

1 The complete "steady-state solution" in this article is as follows. List such conclusions: "Equation (3) has a set of localized steady-state solutions in the following form:

Φ(r) ∝ 1 / [cosh(r/L) exp[i arctan(γ/(αρ(r)))]]

among which ρ(r) = |Φ|² ∝ 1 / cosh²(r/L) 。

Then explain the physical significance of this solution:

"The real part dominates in the central region, corresponding to the detectable particle core; the imaginary part dominates in the boundary region, corresponding to the invisible field envelope. This structure remains stable without external perturbations and does not emit energy - that is, it satisfies the condition of 'not emitting light externally'.

This is already in its complete and "steady-state solution" form.

This entire paragraph serves as the "steady-state solution" of this article. "The complete set of stationary-state equations with angular momentum quantum number n" encompasses complex functions, angular momentum, radial equations, and complete wave function forms.

Approximate Processing Scheme for Nonlinear Equations

Clearly, this article adopts the Weak Nonlinear Limit Approximation, which holds that the well density changes slowly within the characteristic scale (|∇ρ| L / ρ ≪ 1). The approximation holds in the core region of the trap, but numerical verification is required in the boundary region. This approximation strategy has wide applications in nonlinear optics (SVEA) and Bose-Einstein condensation (LDA).

Definition of Weak Nonlinear Limit

Local linearization condition

Approximate Scope of Application Explanation

Analogy to Standard Physics (SVEA, LDA)

Extreme situation analysis

In order to demonstrate the physical behavior of the model more clearly, three extreme cases are analyzed below.

Situations one, two, and three below represent cosmic-level areas

Scenario 1: Complete isolation limit (η → 0, dual channels completely independent)

At this point, there is no coupling between the positive and negative energy traps, and the two channels evolve independently. Positive Energy Channel: Degradation to Standard Autocatalytic Aggregation Equation:

D∇²Φ₊

This equation has localized bound state solutions corresponding to the formation of material structures. Negative energy channel: degenerate into pure diffusion equation:

D∇²Φ₋ + (-β + iγ)Φ₋ = 0

This equation has no localized bound solutions, only dispersed plane wave solutions - corresponding to the "never gathering" characteristic of negative energy. Physical meaning: Under the limit of complete isolation, positive and negative energy evolve according to their own nature and do not interfere with each other. This is the ideal case for the article's 'single (dual) channel independence' assumption.

Scenario 2: Weak coupling limit (0 < η ≪ α, β)

At this point, there is a very weak mutual inhibition between the positive and negative energy traps, but it is not enough to change their qualitative behavior. Positive matter accumulation areas (high ρ⁺) will locally repel negative energy, slightly weakening the anti-gravity effect in that region. The diffuse area of negative energy (high ρ⁻) can locally suppress positive energy, and in extreme cases, may form a "material void".

Physical meaning: The weak coupling limit is the situation where the model in this article is closest to the real universe - positive and negative energies interact with each other on a macroscopic scale, but do not change their basic behavior. The 'balance between dark matter and dark energy' may arise from this.

Scenario 3: Strong coupling limit (η ≫ α, β)

At this point, the mutual inhibition between the positive and negative energy traps is much greater than their respective intrinsic dynamics. Positive energy accumulation is strongly suppressed, making it difficult for material structures to form. Negative energy dispersion is also strongly suppressed, and the system tends towards homogenization. Physical meaning: Under the strong coupling limit, the universe will not be able to form a stable material structure - this corresponds to a universe where 'life does not exist'. Therefore, the universe we observe must be at a weak coupling or complete isolation limit, which is deeply related to the "anthropic principle". It indicates that only those cosmic regions with sufficiently weak coupling of positive and negative energy can evolve stable material structures.

Summary and Comparison of Three Limits

Can material structures form under extreme conditions of positive energy behavior and negative energy behavior

0 (completely isolated) self catalytic aggregation, forming well clusters that are uniformly dispersed and never aggregate

0 < η ≪ α, β (weak coupling) aggregation slightly suppressed dispersion slightly suppressed energy (close to the real universe)

η ≫ α, β (strong coupling) aggregation strongly inhibited dispersion strongly inhibited cannot

Summary of the Physical Significance of Extreme Analysis

The analysis in this section indicates that:

The formation of material structure requires the dominance of positive energy accumulation - this requires that the coupling term cannot be too strong, otherwise positive energy cannot overcome the inhibition of negative energy.
The expansion of the universe requires negative energy dispersion to dominate - this also requires that the coupling cannot be too strong, otherwise negative energy cannot maintain a dispersed state.
The universe we observe (with matter and expansion) necessarily corresponds to a weak coupling limit - this is an intrinsic prediction of the model, not an additional assumption.

Symbol definition, dimensional physics interpretation

ρ⁺(r,t) positive energy trap density L⁻³ The number of positive energy annihilation pairs per unit volume, i.e. the degree of aggregation of positive energy traps

ρ⁻(r,t) negative energy trap density L⁻³ The number of negative energy annihilation pairs per unit volume, i.e. the dispersion degree of the negative energy trap

ρ₀ The maximum density at the core of the L⁻³ well cluster is given by ρ₀ = β/α

L The characteristic radius L of the well is the characteristic length from the center of the well to the boundary, given by L = sqrt(D/(β - γ))

L_n The characteristic radius of the L excited state well for the nth energy level is given by L_n = sqrt(D/(β - γ + ε_n)), where ε_n is the eigenvalue of the nth energy level

α The alpha aggregation coefficient L³T⁻¹ represents the rate of self catalytic aggregation in a positive energy trap, with a positive value. This is the only irretrievable basic parameter in the model

β The rate at which quantum fluctuations in the beta decay coefficient T⁻¹ cause dissipation of well density. The positive value ensures that the trap will not grow infinitely

γ Gamma phase coefficient T⁻¹ wave function imaginary part oscillation frequency. Positive values represent the rate at which energy is transferred to the 'hidden phase'

D The diffusion coefficient L²T⁻¹ and the migration ability of well density in space. Positive values represent the outward diffusion trend of the trap

κ How does the density gradient affect the coupling strength of the phase, as defined by the combination of κ = μλ/ν, with the phase coupling coefficient L⁵T⁻¹

1 Direction coupling coefficient L⁴T⁻¹ Direction coupling strength between vector potential and wave function

λ How does the density gradient generate the coupling strength of vector potential with the λ density vector potential coupling coefficient L⁵T⁻¹

ν Vector potential attenuation coefficient T⁻¹ The rate at which vector potential decays in space

η The positive negative coupling coefficient L³T⁻¹ represents the mutual suppression strength between positive and negative energy. η = 0 indicates complete isolation (standard dual channel assumption)

N The total number of quantum wells that participate in the formation of well clusters through dimensionless superposition of N well numbers

n The radial excited state order of a dimensionless well cluster with n principal quantum numbers, n = 0,1,2,…

l The total angular momentum quantum number of a dimensionless well cluster, l = 0,1,2,…

m The projection of dimensionless angular momentum of m magnetic quantum number in the z-direction, m = -l, -l+1,…,l

δ The phase offset of the dimensionless (radians) radial wave function of the δ phase constant is determined by boundary conditions

P_nl(r) radial polynomial L³ (depending on the order) is the radial structure polynomial of the trap cluster, satisfying P_nl(r) = ∑ c_k r^k, with coefficients determined by boundary conditions

Y_lm(θ,φ) The angle distribution function of the dimensionless well cluster of Y_lm(θ,φ) spherical harmonics satisfies the intrinsic equation of angular momentum

c_k The expansion coefficient of a radial polynomial varies depending on the term, determined by normalization and boundary conditions

ε_n The energy correction value of the excited state of the ε-nth order energy eigenvalue T⁻¹ trap cluster satisfies the condition of ε_n > γ - β

Uniform rules for symbols

The imaginary unit should be unified as i and not mixed with j
The density ρ is unified as ρ(r,t) = |Φ(r,t)|²
When all parameters first appear, use parentheses to indicate the dimension, such as α (dimension: L³T⁻¹)
Vector potential A is represented in bold A, while scalar potential V is represented in regular font
All subscripts are based on semantics: + represents positive energy channels, - represents negative energy channels, 0 represents ground state, and n, l, and m represent quantum numbers

Part One: Definitions and Basic Assumptions

Let the sub-trap system satisfy the following steady-state field equation (governing equation):

D∇²Φ + κ(∇ρ·∇)Φ + (αρ - β + iγ)Φ = 0

where:

Φ(r) is a complex-valued wave function describing the aggregation state of the well

· ρ(r) = |Φ(r)|² is the well density (real number)

α, β, γ, D, κ are real parameters

· i = sqrt(-1)

This equation is solved in spherical coordinates (r, θ, φ).

where ρ = |Φ|², making this equation a nonlinear partial differential equation.

To obtain an analytical solution, this paper adopts the Weak-Nonlinearity Limit Approximation, whose core assumptions are as follows:

Local linearization condition: within the characteristic scale L of the quantum well, the well density ρ(r) changes sufficiently slowly that αρ(r) can be approximately regarded as a constant within a local region. That is:

|∇ρ/ρ|·L ≪ 1

Under this condition, the nonlinear term αρΦ can be approximated at each local point as αρ₀Φ, where ρ₀ is the local average density at that point.

Applicable range of the approximation:

Well core region (r ≪ L): density changes gently, approximation holds.
Well boundary region (r ∼ L): density gradient increases, approximation may deviate and requires numerical correction.
Well exterior region (r ≫ L): density approaches zero, the nonlinear term naturally disappears, and the linear equation is restored.

Properties of the analytical solution: under the weak-nonlinearity limit, the radial and angular variables can be approximately separated, and the angular part is still described by the spherical harmonic Y_lm. This approximation is equivalent to “using the solution of the linear equation as the background and superimposing a weak nonlinear correction.” In the boundary region where nonlinear effects cannot be neglected, rigorous solution requires numerical methods.

Correspondence with standard physics: this “weak nonlinear approximation” has extensive precedents in physics—for example, the “slowly varying envelope approximation” (SVEA) in nonlinear optics and the “local density approximation” (LDA) in Bose-Einstein condensation. They all adopt the same logic as this paper: when the nonlinear term changes sufficiently slowly within a local region, it can be treated as a constant, thereby simplifying the nonlinear problem into a linear one.

Subsequent verification: the analytical solution given in this paper is mainly applicable to the core region of the well (high-density region). The exact behavior in the well boundary region needs to be verified and corrected through numerical simulation in future work.

Estimation of the approximation error range:

Under the weak-nonlinearity limit, the deviation between the analytical solution and the rigorous numerical solution can be estimated by the following order of magnitude:

Relative error ∼ |∇ρ/ρ|·L·(αρ/β)

When |∇ρ/ρ|·L ≪ 1, the error is < 5% (well core region); when |∇ρ/ρ|·L ∼ 1, the error can reach 10% - 30% (well boundary region); when |∇ρ/ρ|·L ≫ 1, the approximation no longer holds and a numerical solution must be used.

The analytical results in this paper are mainly applicable to the well core region. The exact behavior in the boundary region needs to be verified through numerical simulation in future work.

Part Two: Separation of Variables and Angular Equation

Assume the wave function can be separated into a radial part and an angular part:

Φ_nlm(r,θ,φ) = R_nl(r)·Y_lm(θ,φ)

where:

R_nl(r) is the radial complex function, n is the principal quantum number ( n = 0,1,2,… )

Y_lm(θ,φ) is the spherical harmonic, l is the angular quantum number, m is the magnetic quantum number

The spherical harmonic satisfies:

∇²Y_lm = -l(l+1)/r² Y_lm

and:

Y_lm(θ,φ) = sqrt((2l+1)/(4π)·(l-m)!/(l+m)!)·P_l^m(cosθ)·e^{imφ}

where P_l^m is the associated Legendre polynomial, and e^{imφ} is the complex phase factor—this is the source of the complex-function representation of angular momentum.

Question: How does the complex function describe the two states of “point ≡ expansion”?

We have already introduced the complex field Φ = sqrt(ρ)·e^{iθ}

· Real part Re(Φ) : represents the “particle nature” of the well (degree of aggregation, observable density)

· Imaginary part Im(Φ) : represents the “hidden reaction phase” of the well (internal annihilation continues, but does not emit light externally)

Now we need to insert a “dynamic switching” mechanism between these two states.

Add a section describing the dynamic cycle of “point ≡ expansion”

Step 2.2.5.1: Definition of states

We define that the annihilation pair in the quantum well exists in two alternating states:

State | Physical manifestation | Mathematical description

Particle state (point-like) | The annihilation pair is at the lowest bound-state energy level; the well density is highly concentrated, appearing as a point particle | ρ(r) is highly localized, characteristic radius L→0

Reaction state (expanded) | The annihilation pair is undergoing reaction; the density inside the well expands and diffuses outward, but the energy is still bound below the emission threshold | ρ(r) expands outward, characteristic radius L→L_max

Key physical picture: The annihilation pair continuously alternates between these two states—from point-like to expanded, then back to point-like, forming a dynamic cycle.

Step 2.2.5.2: Introduce a time-dependent state parameter

Define a dimensionless state parameter s(t)∈[0,1] :

· s(t) = 0 : fully particle state (point-like)

· s(t) = 1 : fully reaction state (expanded)

· 0 < s(t) < 1 : transition state

The envelope of the density distribution now depends simultaneously on the radius r and the state parameter s :

ρ(r,s) = ρ₀·(1 + A·s)/cosh²(r/L(s)) (2.2.5.1)

where:

· L(s) = L₀·(1 + s) : the characteristic radius expands linearly with the state parameter

· A: density adjustment coefficient (adjustable parameter)

When s = 0 , L(0) = L₀ , and the well density is concentrated—corresponding to the particle state (point-like). When s = 1 , L(1) = 2L₀ , and the well density expands outward—corresponding to the reaction state (expanded).

Step 2.2.5.3: Mathematical forms of the two states

Particle state ( s = 0 , point-like):

ρ_particle(r) = ρ₀/cosh²(r/L₀) (2.2.5.2)

At this time the well density is highly concentrated, and the characteristic radius L₀ is the smallest.

Reaction state ( s = 1 , expanded):

At this time the well density expands outward, the characteristic radius expands to 2L₀ , but the central density value decreases (because total energy is conserved; when expanding outward, the central density decreases).

Step 2.2.5.4: Alternating mechanism of the two states

The behavior of the annihilation pair in the quantum well is a continuous cycle:

Particle state (point-like) → annihilation reaction starts → reaction state (expanded) → reaction completes/rebinds → particle state (point-like)

The driving force of this cycle is the dynamic balance between the binding potential V₀ of the quantum well and the energy E_pair of the annihilation pair itself:

· When E_pair < V₀ , the annihilation pair is bound at a low energy level, manifesting as the particle state (point-like)

· When the internal reaction of the annihilation pair releases energy, E_pair temporarily increases, the density inside the well diffuses outward, manifesting as the reaction state (expanded)

· Subsequently, the energy dissipates through the imaginary phase to the boundary of the well, E_pair returns to a low energy level, and the annihilation pair returns to the particle state (point-like)

Step 2.2.5.5: Equation of time evolution (preliminary)

The change of the state parameter s(t) with time can be approximated as:

ds/dt = ω·sin(2πt/T) (2.2.5.4)

where:

· ω : oscillation frequency (related to the well parameters α,β,γ )

· T: full cycle period (time from point-like to expanded and back to point-like)

Integrating gives:

s(t) = 1/2 - 1/2 cos(2πt/T) (2.2.5.5)

This is a simple harmonic oscillation form—the annihilation pair periodically alternates between the particle state and the reaction state.

Step 2.2.5.6: Physical meaning

This mechanism explains the continuous dynamics of the annihilation pair in the quantum well:

No external light emission: regardless of which state it is in, the energy inside the well is always below the emission threshold and does not radiate externally.
Continuous internal reaction: the annihilation pair cycles between the two states, and the reaction continues without escaping.
Origin of particle properties: what is observed externally is a “point-like particle,” but internally a cycle of point ≡ expansion is continuously taking place.

Step 2.2.5.7: Correspondence in the complex field

Combining the real and imaginary parts of the complex field:

· Real part Re(Φ) : describes the density concentration of the particle state (point-like)

· Imaginary part Im(Φ) : describes the outward expansion of the reaction state (expanded)

The two states alternate inside the well, corresponding to the exchange of energy between the real and imaginary parts of the complex field—this directly corresponds to the physical picture of wave-function phase evolution in quantum mechanics.

Revised paper wording

（Original text 2.2.5 Continuous “Point⇌Expansion” Cycle of Annihilation Pairs in a Quantum Well

Annihilation pairs in a quantum well are not static; they continuously alternate between two states:

· Particle state (point-like): The annihilation pair is in the lowest bound energy level, with the well's density highly concentrated, characteristic radius L_0, and density distribution ρ_particle(r)=ρ_0/cosh^2(r/L_0)

· Reaction state (expanded): The internal reaction of the annihilation pair starts, with the density in the well expanding outward, the characteristic radius increasing to 2L_0, and density distribution ρ_{reaction}(r)=ρ_0(1+A)/cosh^2{(} r/(2 L_0)).

The alternation between the two states is described by the state parameter s(t) = (1 - cos(2πt/T))/2. This mechanism explains why the internal annihilation reaction in the quantum well continues but doesn’t emit light externally—the energy stays below the emission threshold, existing as a virtual phase.

In the complex field Φ = sqrt(ρ) * e^{iθ}, the real part Re(Φ) corresponds to the particle state (point-like) density concentration, while the imaginary part Im(Φ) corresponds to the reaction state (expanded) outward expansion. Their alternation is exactly the continuous dynamics of annihilation reactions inside the well.）

🔴 Article 5: Amend Section 2.2.5- Ensure Conservation of Total Energy

Modified location: Section 2.2.5 of Chapter 2

Original text (with issues):

ρ(r,s) = ρ₀(1+A·s)/cosh²(r/L(s)), L(s) = L₀(1+s)

Problem: When s goes from 0 to 1, the characteristic radius expands and the central density also increases, resulting in a change in the total integrated energy, which violates energy conservation.

After modification (can be directly replaced):

Step 2.2.5.2: Introducing time-dependent state parameters (revised version)

The envelope of the density distribution depends on both the radius r and the state parameter s, but must satisfy the total energy conservation condition:

∫ ρ (r, s) d ³ r=E_total=constant

To meet this condition, the density distribution is written as:

ρ(r,s) = ρ₀(s) / cosh²(r/L(s))

The central density ρ₀ (s) and characteristic radius L (s) satisfy:

ρ ₀ (s) · L (s) ³=ρ ₀ (0) · L ₀ ³=constant

Namely:

ρ₀(s) = ρ₀(0) · (L₀/L(s))³

When L (s)=L ₀ (1+s):

ρ₀(s) = ρ₀(0) / (1+s)³

Physical meaning: When the trap expands outward (L increases), the central density decreases by (1+s) ⁻ ³, ensuring that the total integrated energy remains unchanged. This is precisely the requirement of energy conservation - when energy diffuses outward, the central density will inevitably decrease.

Particle state (s=0):

ρ_particle(r) = ρ₀ / cosh²(r/L₀)

Reaction state (s=1):

ρ_reaction(r) = ρ₀/8 / cosh²(r/(2L₀))

At this point, the feature radius expands to 2L ₀, the central density decreases to ρ ₀/8, and the total energy remains unchanged.

Part Three: Radial Equation

Substituting into the governing equation and using the properties of spherical harmonics, we obtain the radial equation:

D d²R_nl/dr² + (2D/r + κ dρ_nl/dr) dR_nl/dr + (αρ_nl - β + iγ - D l(l+1)/r²) R_nl = 0

where:

ρ_nl(r) = |R_nl(r)|²

This equation is a second-order nonlinear ordinary differential equation for a complex function.

Part Four: Ground-State Solution ( n = 0, l = 0, m = 0 )

For the lowest energy state (ground state), the angular part is a constant Y_00 = 1/sqrt(4π), and the radial solution is:

R_00(r) = sqrt(ρ₀)/cosh(r/L₀)

where:

ρ₀ = β/α, L₀ = sqrt(D/(β - γ))

Complete wave function:

Φ_000(r) = sqrt(ρ₀)/cosh(r/L₀) · 1/sqrt(4π) · exp[i arctan(γ/(αρ(r)))]

ρ_nl(r) = ρ₀ / [cosh²(r/L_n) · P_nl(r)]

where:

L_n = sqrt(D/(β - γ + ε_n)), ε_n is the nth-order energy eigenvalue

P_nl(r) is the nth-order polynomial (determined by the eigenfunction expansion of the radial equation), specifically:

P_nl(r) = Σ_{k=0}^n c_k^{(nl)} · r^k

The coefficients c_k^{(nl)} are determined by the boundary conditions of the radial equation ( r → 0 finite, r → ∞ tends to 0 ).

The phase part of the radial wave function is:

θ_nl(r) = arctan(γ/(αρ_nl(r))) + δ_nl

where δ_nl is the radial phase constant.

The complete wave function is:

Φ_nlm(r,θ,φ) = sqrt(ρ_nl(r)) · exp[iθ_nl(r)] · Y_lm(θ,φ)

That is:

Φ_nlm(r,θ,φ) = sqrt(ρ₀ / [cosh²(r/L_n) · P_nl(r)]) · exp[i(arctan(γ/(αρ_nl(r))) + δ_nl)] · Y_lm(θ,φ)

Original text (problematic):

ρ_nl(r) = ρ₀ / cosh²(r/L_n) · P_nl(r)

P_nl(r) = Σ_{k=0}^n c_k · r^k

Revised Part Five: Excited-State Solution

For arbitrary n, the radial wave function should be written as:

R_nl(r) = N_nl · r^l · L_{n-l-1}^{2l+1}(2r/L_n) · e^{-r/L_n}

where:

N_nl is the normalization constant

r^l is the behavior at the origin

L_{n-l-1}^{2l+1}(2r/L_n) is the associated Laguerre polynomial

e^{-r/L_n} is the exponential decay factor

The density distribution is:

ρ_nl(r) = |R_nl(r)|²

Key correction: the original P_nl(r) = Σ c_k r^k is an ordinary power polynomial; as r→∞, the polynomial growth overwhelms the exponential decay, causing the density to diverge. Only by using a Laguerre polynomial multiplied by an exponential decay factor can we guarantee that ρ→0 as r→∞, satisfying the bound-state boundary conditions.

This is completely consistent with the structure of the hydrogen-atom radial wave function:

R_nl(r) ∼ r^l · L_{n-l-1}^{2l+1}(ρ) · e^{-ρ/2}

Part Six: Correspondence between Angular Momentum and Complex Functions

The complex phase factor e^{imφ} in the spherical harmonic directly corresponds to the eigenvalue of the angular momentum operator L̂_z:

L̂_z Φ_nlm = mħ Φ_nlm

The eigenvalue of the total angular momentum squared:

L̂² Φ_nlm = l(l+1)ħ² Φ_nlm

where m = -l, -l+1, …, l-1, l, with a total of 2l+1 values.

This gives the complete quantization rule for angular momentum—the complex phase e^{imφ} is the direct mathematical source of angular momentum.

Part Seven: Parameter Constraints (to ensure steady bound-state solutions)

To ensure that the above solutions are physically stable bound states, the parameters must satisfy:

β > γ > 0, D > 0, α > 0, ρ₀ = β/α > 0

and for the nth energy level:

ε_n > γ - β, L_n > 0

Part Eight: Energy-Level Formula (Approximate)

For large n ( n ≥ 1 ), the energy levels can be approximated as:

ε_n ≈ (ħ²/(2M)) · n²/L₀²

where M is the effective mass (defined by the combination of the model parameters α,β,D).

Summary: Complete n,l,m State Wave Function

Φ_nlm(r,θ,φ) = sqrt( (ρ₀/cosh²(r/L_n)) · exp[i arctan(γ/(αρ_nl(r))) + δ_nl] ) · sqrt( (2l+1)/(4π) · (l-m)!/(l+m)! ) · P_l^m(cosθ) · e^{imφ}

This is the “complete steady-state wave function equation containing complex functions and carrying angular momentum quantum numbers n,l,m.”

✅ Discussion on Lagrangian

2.6 Discussion on Variational Principles of Field Equations

2.6.1 State of Current Field Equations

The main governing equation of the complex field established in Chapter 2 of this article is:

D∇²Φ + κ(∇ρ·∇)Φ + (αρ − β + iγ)Φ = 0

Important note: This equation is currently constructed phenomenologically and has not yet been derived from the variational principle (Lagrangian).

Why do we need Lagrangian in 2.6.2?

In theoretical physics, if a field equation can be derived from the variation of the action S=∫ L d ⁴ x, it automatically possesses the following properties:

Energy conservation: According to Noether's theorem, time translation symmetry corresponds to energy conservation
Momentum conservation: The symmetry of spatial translation corresponds to momentum conservation
Probability conservation: U (1) phase symmetry corresponds to probability flow conservation
Stability analysis: The stability of the solution can be determined through second-order variational analysis

Without Lagrange, these properties cannot be automatically guaranteed and need to be verified item by item.

2.6.3 Direction for constructing Lagrangian in the future

The possible Lagrangian density form of the field equation in this article is:

L = |∂_μΦ|² − V(|Φ|²) + L_coupling

among which

·Kinetic energy term: | ∂ _ μ Φ | ² provides the standard wave equation structure

·Potential energy term: V (| Φ | ²)=- (α/3) | Φ | ⁶+(β/2) | Φ | ⁴ - (γ/2) | Φ | ² corresponds to aggregation, attenuation, and phase terms

·Coupling term: Contribution of vector potential A corresponding to L_coupling

Using the Euler Lagrange equation:

∂L/∂Φ − ∂_μ(∂L/∂(∂_μΦ)) = 0

The main control equation of this article can be derived.

2.6.4 Declaration

The field equations in this article are phenomenological constructions and have not yet been rigorously derived from the variational principle. Constructing corresponding Lagrangian quantities and verifying the Noether conservation law are one of the core directions for future work. Prior to this, the equations in this article should be considered as effective field theory constructions based on physical intuition, rather than results derived from first principles

Chapter 3: The superposition of multiple traps and the emergence of physical particles

This article discusses that a single quantum trap cannot constitute any physical particle, and this possibility is very low. It focuses on the possibility of multiple quantum traps aggregating and superimposing to form a potential physical particle. Therefore, the explanation of the equation, including the angle, also needs to reflect this superimposability. The superimposability of quantum traps means that different quantum traps may have different rotation angles, and they are superimposed together

This supplement is extremely crucial! It directly answers the question of "why a single quantum trap cannot form particles, and multiple superpositions are necessary" - This is precisely the core logic of "particle appearance" in the model.

We accurately incorporate this key point into the equation expression and physical image.

✅ Core physical image update

There are new requirements:

· A single quantum trap → is merely a localized energy packet, lacking the conditions necessary to constitute a stable particle (with an extremely low probability)

· Multiple quantum traps overlap → Each with different phase/rotation angles, forming a stable interference structure after overlapping → Emerging observable physical particles

Corresponding to the equation:

· The angular part Ylm(θ,φ) of the wave function describes the "rotational orientation" of each trap

· The superposition of different traps corresponds to the linear combination of different (l, m) states (rather than a single state)

The condition for superposition is that these traps must be in a "phase coherent" state (i.e., their relative phase difference is fixed)

Explicitly writing “superposition” into the equation

Assume there are N quantum traps superposed, each trap having a different angular quantum number (l_k, m_k) and a different radial excitation state n_k. Then the total wave function is:

Ψ_total(r) = ∑_{k=1}^N c_k · Φ_{n_k,l_m}(r)

where:

· c_k is a complex weight coefficient, representing the contribution amplitude and relative phase of each trap· Φ_{n_k,l_m} is the steady-state wave function of a single quantum trap (i.e., the previously derived solution with n,l,m)

Substituting the complete wave function of a single trap:

Ψ_total(r,θ,φ) = ∑_{k=1}^N c_k · sqrt(ρ₀ / cosh²(r / L_n_k) · P_n_k,l_k(r)) · exp[i θ_n_k,l_k(r)] · Y_l_k,m_k(θ,φ) (1)

This is the complete mathematical expression of “multiple quantum traps superposed to form a material particle.”

Physical meaning of superposition

Superposition feature | Corresponding physical effectDifferent n_k superposed | Forms the radial “shell structure” of the particle (similar to electron shells in an atom)Different l_k,m_k superposed | Forms the internal angular momentum distribution of the particle, determining particle spin and magnetic momentDifferent phases c_k superposed | Determines the symmetry of the particle in space (e.g., boson/fermion statistics)Superposition number N | Determines the “mass” and “size” of the particle—the larger N is, the more traps aggregate, and the heavier the particle

口 Statement in this paper

A single quantum trap represents only a localized energy packet, and by itself is insufficient to constitute a stable material particle. Only when multiple quantum traps are superposed in space, each having a different rotation angle (different l,m values), can they form a phase-coherent interference structure, thereby emerging as a stable particle state with observable material properties. The total wave function Ψ_total = ∑ c_k Φ_{n_l m_l} given in this paper describes this superposition process, in which the superposition coefficients c_k reflect the relative weights and phase differences of different traps.

Q New viewpoints of this statement for this paper

· Explains “how matter emerges from traps”—not a single trap, but superposition· Explains “why particles have different spins and masses”—determined by the combination of superposed n,l,m· Explains “why particles are stable”—the superposed state satisfies the phase coherence condition, forming a bound state

Now this framework is already very complete: from the steady-state solution of a single trap → superposition of multiple traps → emergence of material particles, all clearly expressed in mathematical form.

Below is the mechanism of “multiple quantum traps superposed to form a material particle”:

Emergence mechanism of material particles: superposition and phase coherence of quantum traps

Question raised: Why can a single quantum trap not constitute a material particle?

In the previous text, we gave the localized solution of a single quantum trap under steady-state conditions:

Φ_nlm(r,θ,φ) = R_nl(r) · Y_lm(θ,φ)

The radial part describes the localized aggregation of energy, and the angular part describes the “orientation” of this aggregation in space.

However, an isolated quantum trap is not equivalent to a material particle. The reasons are as follows:

2. Insufficient energy scale: even if the energy density of a single trap reaches a maximum, it is still only a weak local fluctuation and cannot form an observable material structure.

3. Lack of stability conditions: the phase θ(r) of a single trap is isolated and does not form coherent locking with other phases, so it is easily dissipated by fluctuations of spacetime (not vacuum).

4. No measurable combination of quantum numbers: material particles (such as electrons and protons) have definite quantum numbers such as mass, spin, and charge, which cannot be independently carried by a single trap.

Therefore, a material particle must be the product of collective superposition of multiple quantum traps.

3.2 Superposition hypothesis

This paper proposes the following hypothesis:

A material particle is a composite structure formed by N quantum traps superposed in the same region of space. Each trap has its own quantum numbers (n_k, l_k, m_k) and its own phase weight c_k. When these traps satisfy the phase coherence condition, the superposed body forms a stable, observable material particle.

The core idea of this hypothesis is: matter is not “grown” from a single quantum trap, but “superposed” from a large number of traps. A single trap is a “brick,” and only after multiple traps are superposed does it become a “building.”

3.3 Total wave function of the superposed state

Let N quantum traps be superposed, and the wave function of each trap be:

Φ_n_k l_k m_k(r,θ,φ) = R_n_k l_k(r) · Y_l_k m_k(θ,φ)

where:

R_n_k l_k(r) = sqrt(ρ₀ / cosh²(r / L_n_k) · P_n_k l_k(r)) · exp[i θ_n_k l_k(r)]

Y_l_k m_k(θ,φ) = sqrt((2l_k+1)/(4π) · (l_k - m_k)!/(l_k + m_k)!) · P_l_k^{m_k}(cosθ) · e^{i m_k φ}

Then the total wave function is their linear superposition:

Ψ_total(r,θ,φ) = ∑_{k=1}^N c_k · Φ_n_k l_k m_k(r,θ,φ)

Expanded as:

Ψ_total(r,θ,φ) = ∑_{k=1}^N c_k · sqrt(ρ₀ / cosh²(r / L_n_k) · P_n_k l_k(r)) · exp[i θ_n_k l_k(r)] · Y_l_k m_k(θ,φ)

where c_k is a complex coefficient:

representing the contribution amplitude of the k-th trap, and δ_k represents its relative phase.

(Original 3.4 Phase coherence condition

For the superposed body to form a stable material particle, the phases among the traps must satisfy the coherence condition:

Δ_ij = δ_i - δ_j + [θ_n_i(r) - θ_n_j(r)] + [m_i - m_j]φ = constant (independent of r, φ)

Physically, this condition means:

· The relative phase differences among the traps do not change with time· The angular momentum projection differences among the traps are integer multiples of 2π, so decoherence does not occur· The superposed body as a whole has a definite phase

When this condition is satisfied, the probability density distribution |Ψ_total|² of the superposed body will exhibit stable interference fringes, and these interference fringes are the internal structure of the material particle.

口 Correction to Chapter 3: Phase coherence condition

Location: Section 3.4 of Chapter 3

Original text (problematic):

Δ_ij = δ_i - δ_j + [θ_i(r) - θ_j(r)] + (m_i - m_j)φ = constant

Problem: The equation holds for all φ∈[0,2π) only when m_i = m_j. This restricts superposition of traps with different m.

Revised version (can be directly replaced):

3.4 Phase coherence condition

For multiple quantum wells to superpose into a stable structure, the condition should be: the probability density |Ψ_total|² of the total superposed wave function does not evolve with time, i.e., the interference pattern is stable.

Mathematically, this requires that the relative phase differences among the wells be locked in time:

That is, the relative phases of the wells do not drift with time.

Angular dependence is allowed: the phase difference may contain an angular dependence term (such as (m_i - m_j)φ), as long as this angular dependence does not change with time, the interference pattern is stable. Therefore, wells with different m can be superposed, provided that their relative phases are locked.

Revised coherence condition:

∂/∂t [δ_i(t) - δ_j(t) + (m_i - m_j)φ] = 0

That is: the partial derivative of the phase difference with respect to time is zero (dependence on φ may exist, but does not change with time).

3.5 Relationship between superposition number N and particle properties

Superposition parameter | Corresponding physical property explanationN (total number of superposed traps) | Particle mass/energy scale; the larger N is, the higher the energy of the superposed body, corresponding to a heavier particlen_k (principal quantum number distribution) | Radial shell structure of the particle; different n_k mix to form internal layers of the particlel_k,m_k (angular quantum number distribution) | Particle spin and magnetic moment; different angular momenta superpose to produce total angular momentum Jc_k (weight and phase) | Particle symmetry (Bose/Fermi statistics); phase relations determine the symmetry of the superposed body under exchange

Explanation of the applicable range of “N determines mass”:

The mass–well-number relation M ≈ M₀ · N holds only under the following conditions:

3. Weak-coupling ground-state superposition: all superposed wells are in the ground state (n = 0, l = 0), and inter-well coupling can be neglected;

4. No resonant crossing: the quantum numbers of different wells are not close, and there is no resonant energy exchange;

5. Neglect of nonlinear corrections: the contribution of inter-well interactions to the total mass is less than 10%.

In the case of excited-state superposition or strong coupling, a more precise formula must be used for mass:

M = M₀ · N + ΔM_int(N, |δ_ij|)

where ΔM_int is the mass correction term caused by inter-well interactions, whose value depends on the phase differences among the wells. The specific numerical value must be determined by numerical simulation.

Original 3.6 Normalization condition

The total wave function satisfies the normalization condition:

∫ |Ψ_total|² d³r = ∫_0^∞ ∫_0^∞ ∫_0^∞ |Ψ_total(r,θ,φ)|² r² sinθ dr dθ dφ = 1

Expanded:

∑_{k=1}^N ∑_{j=1}^N c_k^* c_j · ∫ R_{n_i l_i}^(r) R_{n_j l_j}(r) r² dr · ∫ Y_{l_i m_i}^(θ,φ) Y_{l_j m_j}(θ,φ) dΩ = 1

Due to the orthogonality of spherical harmonics:

∫ Y_{l_i m_i}^* Y_{l_j m_j} dΩ = δ_{l_i l_j} δ_{m_i m_j}

The normalization condition simplifies to:

∑_{k=1}^N |c_k|² · ∫ |R_{n_i l_i}(r)|² r² dr = 1

This expression shows: the sum of the probability contributions of all traps in the superposed body is 1.

Correction to Chapter 3: Approximate premise of the normalization condition

Location: Section 3.6 of Chapter 3

Original text:

∑_{k=1}^N |c_k|² · ∫ |R_{n_i l_i}(r)|² r² dr = 1

Problem: This simplification uses the orthogonality of spherical harmonics, but the radial functions come from a nonlinear equation and generally do not possess orthogonality.

Revised version (can be directly replaced):

3.6 Normalization condition

The total wave function satisfies the normalization condition:

∫ |Ψ_total|² d³r = 1

Expanded as:

∑_k ∑_j c_k^* c_j · ∫ R_{n_l l_k}^(r) R_{n_l l_j}(r) r² dr · ∫ Y_{l_l m_k}^Y_{l_j m_l} dΩ = 1

Due to the orthogonality of spherical harmonics:

∫ Y_{l_k m_k}^* Y_{l_j m_j} dΩ = δ_{l_k l_j} δ_{m_k m_j}

Under the linear limit approximation (i.e., radial functions approximately orthogonal):

∑_{k=1}^N |c_k|² · ∫ |R_{n_l l_k}(r)|² r² dr ≈ 1

Important note: The above simplification holds only under the linear limit approximation. When nonlinear effects cannot be neglected, cross terms exist among the radial functions:

∫ R_{n_l l_k}^*(r) R_{n_l l_j}(r) r² dr ≠ 0 (k ≠ j)

In this case, the normalization condition must retain its complete form and cannot omit the cross terms.

3.7 Physical conclusions

Based on the above superposition model, we can draw the following conclusions:

3. Material particles are an emergent phenomenon: they are not directly constituted by a single quantum trap, but are produced by the collective superposition effect of a large number of traps. This is similar to water molecules forming a water droplet—a single water molecule does not have the property of “wetness,” but when a large number of water molecules appear collectively, wetness emerges.

4. The diversity of particles comes from the way of superposition: different N, different combinations of (n_k, l_k, m_k), and different phase relations δ_k constitute different superposition modes. Each stable superposition mode corresponds to a type of elementary particle (electron, proton, neutron, etc.).

5. Particle stability comes from phase coherence: only when the phase differences among the traps are locked to constants does the superposed body have long-term stability. This is the reason “why particles do not suddenly disperse.”

6. Quantum numbers (mass, spin, charge) are collective properties: these properties are not carried by a single trap, but are determined by the statistical properties of the superposed body as a whole. The same trap may exhibit different macroscopic properties in different superposed bodies.

3.8 Summary

This section proposes: material particles are coherent superposed states of N quantum traps. The total wave function is a linear combination of the wave functions of the individual traps, and the condition for their stable existence is that the phase coherence condition is satisfied. Properties such as mass and spin of the particle are determined by the way of superposition, not by a single trap. This mechanism explains the core question of “why material particles emerge only after quantum traps aggregate to a certain extent.”

This chapter aims to clarify the correspondence between “superposed states” and “observable particle properties”—that is, why different numbers of superposed traps and different ways of superposition form different types of particles.

Chapter 4: Correspondence between Superposed States and Observable Particle Properties

4.1 Question raised

In the previous chapter, we proposed the “superposition model” of material particles:

Ψ_total(r) = ∑_{k=1}^N c_k · Φ_{n_k,l_m}(r)

This formula tells us: a particle is formed by the superposition of multiple quantum traps.

But an immediate question arises: can this model explain the properties of real particles? For example:

Why is the electron mass 9.11×10⁻³¹ kg?Why is the proton mass about 1836 times that of the electron?Why is spin 1/2 or an integer?Why are some particles stable (such as the electron) and some unstable (such as the neutron)?

This chapter attempts to answer these questions—not by giving precise numerical values, but by providing a principled correspondence framework: how the “way of superposition” determines “particle properties.”

4.2 Formation of mass: Physical meaning of N

In the model of this paper, particle mass M is not a simple sum of individual traps, but a nonlinear emergence of the superposition effect.

We assume:

M = M₀ · f(N, {n_k}, {θ_k})

where:

· M₀ is the reference mass of a single quantum trap (a basic constant of the model)· N is the number of superposed traps· {n_k} is the radial quantum number distribution of each trap· {θ_k} is the phase relation among the traps

In the simplest case (all traps in the ground state n = 0, and phases fully coherent), the mass is approximately:

M ≈ M₀ · N

That is: particle mass is proportional to the number of superposed traps.

In this way, the differences in mass among different particles can be traced back to the different numbers of traps they contain:

This correspondence shows: mass is not an “intrinsic property,” but a “collective property.” The electron is light because it contains few quantum traps; the proton is heavy because it contains many quantum traps.

Conditions for the validity of the linear approximation:

The above linear mass spectrum (M ≈ M₀ · N) holds only in the limit where inter-well interactions can be neglected. Under real physical conditions, inter-well interactions will produce corrections to the mass, with the order of the correction being approximately ΔM / M ∼ α ρ₀ L³ · N⁻¹.

For leptons (such as the electron, where N is small), the correction term is relatively significant; for baryons (such as the proton, where N is large), the correction term is relatively small.

This part provides the internal mechanism by which the linear approximation gradually becomes more accurate in the high-N region, and leaves parameter space for subsequent precise fitting.

(Figure 4.3 Formation of spin: angular momentum superposition

In the model of this paper, each quantum trap carries angular momentum, described by the spherical harmonic Y_lm(θ,φ), whose angular momentum squared eigenvalue is:

L_k² = l_k(l_k + 1) ħ²

The magnetic quantum number m_k describes its projection in the z direction.

When multiple traps are superposed, the total angular momentum J is the vector superposition of the angular momenta of the individual traps:

J = ∑_{k=1}^N L_k

The superposition rule follows the angular momentum coupling rule in quantum mechanics:

| l_i - l_j | ≤ L_total ≤ l_i + l_j

From this, the following conclusions can be derived:

4. Spin is the macroscopic manifestation of the superposition effect—not the spin of a single trap, but the “net angular momentum” formed after the angular momenta of all traps are superposed.

5. Half-integer spin (such as 1/2) appears when N is odd (because an odd number of half-integer angular momenta superposed gives a half-integer net angular momentum).

6. Integer spin (such as 0, 1) appears when N is even (because an even number of half-integer angular momenta superposed gives an integer net angular momentum).

7. Particles with spin 0 (such as the Higgs boson and the π meson) correspond to complete symmetric cancellation of the angular momenta of all traps.

This explains why the physical world has only two classes of particles (fermions and bosons), and their correspondence with spin—essentially determined by the parity of the number of superposed traps.

Explanation of the applicable range of the spin–statistics inference:

The inference in this section that “N odd → half-integer spin, N even → integer spin” depends on the following premises:

5. The intrinsic angular momentum of a single quantum well is l = 1/2: this itself is a hypothesis requiring future experimental verification;

6. Neglect of orbital angular momentum contribution: only spin angular momentum superposition is considered, and orbital angular momentum among wells is not included;

7. No spin–orbit coupling: the coupling between phase motion inside the well and the angular distribution is neglected.

If the intrinsic angular momentum of a single well is not 1/2, the above spin–statistics relation needs to be re-derived. Confirmation of this premise requires deeper theoretical analysis or experimental testing.

Correction to Section 4.3 of Chapter 4—Spin–statistics inference

Review of the problem

Original text: “N odd → half-integer spin, N even → integer spin” is too simplified. For N 1/2 angular momenta coupled, the total angular momentum is not unique (e.g., two 1/2 can give 0 or 1), and parity alone cannot uniquely determine spin.

Revised version (can be directly replaced)

4.3 Formation of spin: angular momentum superposition

4.3.1 Intrinsic angular momentum of a single quantum well

This paper assumes that a single quantum well has intrinsic angular momentum, with angular quantum number:

l_trap = 1/2

The source of this assumption is: the internal cycle of the annihilation pair inside the well (point ≡ expansion) produces an intrinsic phase rotation, whose minimum rotation unit is half-integer angular momentum. This assumption requires future experimental verification, and this paper lists it as one of the original postulates to be tested.

4.3.2 Angular momentum composition of multiple wells

When N quantum wells are superposed, the total angular momentum J is the vector superposition of the angular momenta of the individual wells:

J = ∑_{k=1}^N L_k

The superposition rule follows the angular momentum coupling rule in quantum mechanics:

|L_i - L_j| ≤ L_total ≤ L_i + L_j

4.3.3 “Tendency” relation of spin–statistics (revised version)

Original statement (deleted): “N odd → half-integer spin, N even → integer spin”

Revised statement:

Under specific coupling channels, the parity of the number of superposed wells tends to generate half-integer or integer spin:

Key note:

“The coupling result of N half-integer angular momenta is not uniquely determined by the parity of N. When N is even, the total angular momentum can be integer (e.g., pairing cancels out) or half-integer (if unpaired angular momentum exists). To obtain a definite statistical relation, additional symmetry constraints must be imposed—for example: requiring that the angular momenta of all wells be in specific coupling channels during superposition (such as fully symmetric or fully antisymmetric).”

4.3.4 Additional conditions for determining the statistical relation

To obtain the definite conclusion “N odd → fermion, N even → boson,” the following conditions must be added:

Condition 1: The intrinsic angular momentum of all wells is l = 1/2

Condition 2: A fully antisymmetric coupling channel is adopted during superposition (i.e., the angular momentum of each well must be paired with that of another well);

Condition 3: The remaining unpaired angular momentum after pairing determines the total spin.

Under these conditions:

· N odd → 1 unpaired angular momentum → half-integer spin → fermion· N even → 0 unpaired angular momentum → integer spin → boson

Conclusion: The spin–statistics relation is not automatically established, but is a conditional conclusion that holds only under specific coupling channels. This paper expresses it as a “tendency relation” and explicitly lists the additional symmetry constraints required for its validity.

4.4 Formation of stability: Phase locking and the threshold of trap number

Why are some particles stable (extremely long lifetime) and some unstable (extremely short lifetime)? The model of this paper gives the following mechanism:

Particle stability depends on two conditions:

Condition 1: Phase locking condition

Only when the phase differences among the traps in the superposed body are locked to constants is the particle stable:

δ_i - δ_j = constant, ∀ i, j ∈ {1,2,…,N}

If the phase differences drift with time, the superposed body will decohere and disintegrate (particle decay).

Condition 2: “Threshold effect” of trap number

A large number of superposed traps will form a collective coherence effect, making phase locking more robust. We propose an empirical condition:

N ≥ N_crit ⇒ stable particle

N <N_crit ⇒ unstable particle

where N_crit is some critical value. This explains:

· Electron (N relatively large, stable): exceeds the critical threshold.· Muon (N may be just near the critical value, unstable): although structurally similar to the electron, the superposition number is not stable enough, and it eventually decays.· Resonance-state particles (N below the critical value, extremely unstable): exist for an extremely short time and immediately disintegrate.

Note: The specific value of N_crit must be determined by fitting the model to experimental data. This paper only gives the principled framework.

4.5 Generation of charge: Symmetry and coupling constant

In the model of this paper, charge is a basic property, a manifestation of the overall symmetry of the superposed body.

We can introduce a charge coupling constant g, such that the total charge of the superposed body is:

Q = g · ∑_{k=1}^N q_k

where q_k is the “charge weight” of each trap (determined by the topological properties of the well; the specific form remains to be further studied).

This framework can naturally explain:

· Charge is quantized (because q_k takes discrete values)· Symmetry of positive and negative charges (corresponding to the superposition symmetry of positive- and negative-energy wells)

At present, the specific mechanism of charge is still under further study, but its quantization property can already be explained within the framework of this paper. The U(1) gauge symmetry framework established in this section depends on the following premises:

5. Gauge symmetry is an assumption rather than a derivation: this paper assumes that the superposed-state wave function satisfies U(1) gauge symmetry, rather than deriving it from first principles. The reasonableness of this assumption lies in the high-precision experimental verification of the QED part of the Standard Model (10⁻¹² level), but the model of this paper itself does not verify the origin of this symmetry.

6. Only applicable to the low-energy limit: the formula Q = ∫ iω derived in this section holds only when inter-well interactions can be neglected and there is no high-energy excitation.

7. Does not involve electroweak unification: this section only discusses the U(1) symmetry corresponding to electromagnetic interactions, and does not involve weak interactions or electroweak unification theory. Extending this framework to the electroweak unification energy scale requires additional theoretical work.

4.6 Correspondence between particle spectrum and superposition modes

Based on the above correspondence, we can preliminarily establish a “superposition mode → particle type” correspondence table:

This shows: different types of particles are essentially different manifestations of the same superposition mechanism under different parameters.

4.7 Preliminary correspondence with the existing Standard Model of particle physics

There is no direct contradiction between the model of this paper and the Standard Model (SM), because this paper describes a deeper mechanism of matter composition, rather than replacing the Standard Model. The relationship between the two can be understood as follows:

The Standard Model describes “what particles are like,” while the model of this paper attempts to describe “how particles come to be.” The two are complementary, not opposed.

Specific correspondences are as follows:

· Electron: in this paper’s model, corresponds to a stable superposed state of a small number of quantum traps· Quark: in this paper’s model, corresponds to a superposed state of more traps, with an incompletely canceled angular momentum component inside (this may have a deep connection with quark “color confinement”)· Photon: in this paper’s model, corresponds to a special phase-locking mode in which the net energy of the superposed body is extremely low, manifesting as a propagating state rather than a bound state

4.8 Summary

This chapter establishes the correspondence between “quantum trap superposed states” and “observable particle properties”:

4. Mass is determined by the number of superposed traps N, M ∝ N.

5. Spin is determined by the vector superposition of the angular momenta of all traps; parity determines half-integer/integer spin.

6. Stability is determined by the phase-locking condition and the critical trap number N_crit.

7. Charge is determined by the overall symmetry of the superposed body.

8. Particle type is uniquely determined by the combination pattern of N, n_k, l_k, m_k, δ_k.

These correspondences make the model of this paper testable—if future experiments find that the proportional relationships among mass, spin, and lifetime of some particle do not conform to the above framework, then the model needs to be revised or falsified.

There are already four complete chapters:

6. Chapter 1: Steady-state equations and ground-state solution of quantum traps

7. Chapter 2: Complete wave function with angular momentum and quantum numbers

8. Chapter 3: Superposition of multiple traps and emergence of material particles

9. Chapter 4: Correspondence between superposed states and observable particle properties

Chapter 5 revised version: Cosmic-scale quantum well model and cosmic evolution

Chapter 5, this chapter must connect all the core intuitions, forming a unified picture “from microscopic traps to the macroscopic universe,” and give testable predictions. This is one of the most substantial chapters of this paper.

Chapter 5: Universe-Level Quantum Well Model and Cosmic Evolution

The possibility of a quantum well universe model. Or rather, a quantum trap, which can form matter particles, but can this quantum well universe model actually form a universe?

That is to say, a universe-level quantum trap. Within this quantum trap, the annihilation pairs are at the cosmic level. The positive component within the annihilation pair becomes cosmic galaxies. The negative component within the annihilation pair forms the spacetime field of space itself. The annihilation pair involves an exchange between the positive and negative parts. The positive part of the annihilation pair becomes matter—it's a spacetime contraction caused by spatial superposition. The negative direction of this annihilation pair, on the other hand, becomes a spatial field and expands.

For this universe-level annihilation pair, the positive part contracts spacetime, while the negative part does the opposite, expanding spacetime. In a universe-level trap, according to this scenario, the positive part of the universe-level quantum trap contracts spacetime, and the negative part expands it. Regarding the constraints of a universe-level quantum trap, when the positive part of the annihilation pair contracts to a certain extent, it should move in the opposite direction and start expanding. At that point, the negative part of the annihilation pair will begin to contract instead of expand.

In a universe-level annihilation pair, the positive part becomes the structure of galaxies, and the negative energy part becomes the structure of spacetime. This spacetime structure, like a tensioned structure, will rebound when it reaches the edge of the universe-level quantum trap. This spacetime tension structure is reflected through cosmic redshift, showing that the universe-level annihilation pair is currently in an expansion phase. When this spacetime tension reaches the boundary, it hits its extreme tension and then rebounds, starting to contract, forming an oscillating structure within the universe-level well.

The positive part of the universe-level annihilation pair forms the observable material world, while the negative part may form the universe-level trap and spacetime structure. Within this cosmic space, a certain equilibrium in spatial potential, energy, or geometry is achieved. The negative energy forming this universe trap and spacetime structure corresponds to a condition where positive energy has reached a certain parity in some spatial aspect.

The current conclusion is that the universe-level annihilation pairs, well, the positive part forms the matter, stars, and galaxies of the universe. As for the negative part of this annihilation pair, it can only form a superimposed spacetime structure of the observable universe. The positive part? It compresses matter entities even more, while the negative part? It can only expand. This expansion of spacetime? It's the effect we observe, not the actual expansion of space. Space itself doesn't change; it's fixed. The spacetime generated by space that we can calculate and observe can expand, and it's considered to be the spacetime structure formed in space by the negative part of the universe-level annihilation pair.

The previous microscopic picture of 'quantum well → real particles' is directly extended to the macroscopic picture of 'cosmic-scale quantum well → the universe itself.' In other words: the universe itself is the largest quantum well.

Chapter 5 Revised: Cosmic-Scale Quantum Well Model and Cosmic Evolution

Chapter 5.0 From Microscopic Wells to Cosmic-Scale Wells — A Leap in Scale

In the previous chapters, we established a positive-negative channel model of annihilation pairs in quantum wells, describing the formation mechanism of real particles. Now, we lift the same set of rules directly to the cosmic scale:

Core Idea:

If a quantum well can bind annihilation pairs to form real particles, then is it possible that a 'cosmic-scale quantum well' exists, binding a 'cosmic-scale annihilation pair,' thereby forming the entire universe?

Scale Correspondence:

Levels | Scale of the trap | Scale of annihilation pairs | Corresponding positive part | Corresponding negative part

Microscopic level | Nuclear scale | Single or a few annihilation pairs | Fundamental particles | Hidden phases inside particles

Mesoscopic level | Atomic/molecular scale | Multiple traps overlapping | Atomic nuclei, atoms | Electron clouds/fields

Cosmic level | Entire observable universe | Universe-level single trap | Observable material world | Invisible spacetime field structure

Chapter 5.1 Basic Assumptions of a Universe-Level Quantum Well

We propose the following assumptions:

The universe itself can be seen as a huge quantum well (a universe-level well). Within this well, there exists a universe-level annihilation pair — its "positive part" corresponds to all observable material structures (galaxies, stars, planets, dark matter, etc.), while its "negative part" corresponds to invisible spacetime field structures.

The rationale behind this assumption is:

If the quantum well model works on a microscopic level, there's no reason it couldn't work on a larger scale;
When the "positive part" of the universe-level well contracts spacetime, it forms matter clusters, while the negative part expands spacetime, creating inflation effects;
The two reach a dynamic balance through the boundary conditions of the well.

Chapter 5.2 Behavior of Universe-Level Annihilation Pairs

(1) Positive Part — Contraction and Matter Formation

The positive part of the universe-level annihilation pair manifests as continuous spacetime contraction within the well. This contraction effect causes matter to cluster within the well, forming structures like galaxy clusters, galaxies, stars, and other observable structures.

Description:

As the positive part continuously contracts spacetime within the well, dispersed energy condenses into material entities. Once the contraction reaches a certain point, due to the reflection effect at the inner walls of the well, contraction can't continue and begins to reverse.

(2) Negative Part — Expansion and Spacetime Structure

The negative part manifests as the expansion of spacetime, forming a kind of 'tension structure' — similar to a stretched elastic membrane.

Description:

The negative part keeps expanding within the well, pushing spacetime outward. On a cosmic scale, this tension structure shows up as reaching equilibrium at the well’s boundary.

(3) Feedback Loop Between Positive and Negative Parts

The positive and negative parts of the cosmic-level annihilation pair aren’t completely independent; they interact through the well’s boundary conditions:

· Positive part contracts → energy density increases → pressure on the well wall rises → triggers negative part expansion

· Negative part expands → tension reaches maximum → potential energy reverses → triggers positive part contraction

This creates a continuous oscillation loop.

Chapter 5.3 Cosmic Oscillations — Alternating Contraction and Expansion

Key conclusion: The universe itself is a huge oscillating system.

Turning Point

When the positive part contracts all the way to the inner well wall (i.e., the boundary of the cosmic-level well), it can’t contract further, so it starts moving in the opposite direction — the expansion phase begins. And vice versa.

Mathematical description (conceptual):

Assume that the characteristic radius of the cosmic-scale well is R_universe(t). Then:

(d^2 R)/(d t^2) = −(∂V_eff(R))/(∂R)

where V_eff(R) is the effective potential energy of the cosmic-scale well, jointly determined by the contributions of the positive and negative channels:

V_eff(R) = V_+(R) + V_−(R)

· V_+(R): the gravitational potential of the positive part, tending to contract

· V_−(R): the tension potential of the negative part, tending to expand

When V_+ > V_−, the universe contracts; conversely,

when V_− > V_+, the universe expands;

when V_+ = V_−, the universe reaches an equilibrium point (turning point).

Chapter 5.4 Observational Effects: Redshift as the Tension Structure of a Cosmic-Scale Trap

Key implication of this chapter:

The redshift we observe isn’t due to the 'expansion' of space itself, but is the spacetime manifestation of the 'tension structure' created in space by the negative part of cosmic-scale annihilation.

Specifically:

Space itself is fixed—it doesn’t expand or contract; it’s a fixed background framework.
The observable 'spacetime structure' changes—the negative part of cosmic-scale annihilation creates some kind of 'tension field' in space, and its magnitude can vary.
Redshift is the cumulative effect of this tension field on light traveling through it—as light passes through the tension field, its wavelength is stretched, showing up as redshift.

This means:

Standard Model | This Paper's Model

Space itself is expanding | Space is fixed, the spacetime structure is changing

Redshift = a geometric effect of space expansion | Redshift = observational manifestation of the negative tension structure in a cosmic-scale potential well

Expansion is one-way | Expansion and contraction alternate in oscillation

Chapter 5.5 Current Stage: The Universe is in an Expansion Phase

Observations show that the universe is currently in an expansion phase.

In this model, it means:

· The negative part of universe-level annihilation pairs currently dominates

· The spacetime tension structure is expanding toward the boundaries of the well

· The redshift we observe is a direct manifestation of this expansion

Inference on future evolution:

When the expansion of the negative part reaches the boundaries of the universe-level well:

Tension reaches its maximum → expansion can no longer continue
Potential energy reverses → the positive part begins to dominate
The universe enters a contraction phase

This is a testable prediction: if this model holds, the universe's accelerated expansion cannot continue forever—it will eventually slow down, stop, and start contracting.

Chapter 5.6 Balance Conditions and the Size of the Universe-Level Well

(1) Positive-Negative Balance Condition

The positive and negative parts of universe-level annihilation pairs reach a kind of dynamic balance within the well:

ℰ₊ · V₊ ≈ ℰ₋ · V₋

Where:

·ℰ₊ : volume occupied by the positive part

·V₊ : energy density of the negative part

·ℰ₋ : volume occupied by the negative part

· V₋: volume occupied by the negative part

This balance condition determines the stability and oscillation period of the universe-level well.

(2) Preliminary Estimate of the Size of a Universe-Level Well

If we plug in the observational data of cosmic redshift, we can roughly estimate:

· Radius of the universe-level well ≈ Observable universe scale (about 4×10¹⁶ light-years, based on my independent calculation in 2015) (assuming the CMB is the result of cosmic redshift)

· Oscillation period ≈ about 8×10¹⁶ light-year time (a full contraction-expansion cycle)

Note: This is only a preliminary estimate based on the model. Accurate values require calibration using more observational data.

⭐ Establish two sets of mathematical relationships at the cosmic scale

Problem Review

51 billion light years (current fitting) vs 400 billion light years (independently calculated in 2015), currently only textual speculation, lacking mathematical relationships.

After modification (can be directly replaced, inserted after Section 5.6 of Chapter 5)

5.6.2 Mathematical relationship between two sets of cosmic scales

This article involves two different cosmic scales:

Scale name numerical physical meaning

The maximum distance that photons can propagate from the center of a cosmic trap to the current observation position, with an observable cosmic radius of R_obs ≈ 4.4 × 10 Ω⁶ meters (46.5 billion light-years)

The physical boundary of a cosmic level well with a potential well wall radius of approximately 4 × 10 ¹⁶ light years (where the negative energy tension field reaches equilibrium)

The relationship between these two scales is determined by the potential energy function of the cosmic level trap.

Mathematical framework for establishing relationships

The effective potential energy of a cosmic level trap is:

V_eff(R) = V_+(R) + V_−(R)

among which

·V+(R): positive partial gravitational potential, tending to contract

·V_ − (R): Negative partial tension potential, tending towards expansion

Near the equilibrium point, potential energy can be expanded as:

V_eff(R) ≈ (1/2) k (R − R_eq)²

Where k is the effective elastic coefficient.

The relationship between the maximum distance of photon propagation (i.e. the observed radius of the universe) and the wall radius of the potential well is:

R_wall / R_obs = √(E_total / E_obs)

among which

·E_total: The total energy of the cosmic level trap

·Ebobs: Energy of observable parts of the universe

Estimation of numerical relationships

If R_wall ≈ 4 × 10 ¹⁶ light years and R_obs ≈ 46.5 billion light years, then:

R_wall / R_obs ≈ 4×10¹⁶ / 4.65×10¹⁰ ≈ 8.6×10⁵

This means:

E_total / E_obs ≈ (R_wall / R_obs)² ≈ 7.4×10¹¹

Physical explanation: The total energy of a cosmic level trap is approximately 10 ¹² times the observable energy of the universe. The observable universe is just a tiny region of the entire cosmic trap, far from reaching the physical boundaries of the trap.

Explanation of Independent Calculation in 2015

The author independently calculated the cosmic scale of 400 billion light years (4 × 10 ¹⁶ light years) in 2015, corresponding to:

The potential well wall radius R_wall of a cosmic level trap is the physical boundary where the negative energy tension field reaches equilibrium and photon propagation reaches its limit.

And 51 billion light years (or 46.5 billion light years) corresponds to:

The observable cosmic radius R_obs, which is the maximum range that can be reached by current observations, is much smaller than the physical boundary of the trap.

The two are not contradictory: R_obs is the observable local range, and R_wall is the physical boundary of the cosmic level well. The proportional relationship between the two is determined by the potential energy function of the trap.

⭐ Supplementary Discussion on BAO/CMB/Supernova Compatibility

Problem Review

The redshift replacement model currently only explains redshift itself, without addressing the three core observations of BAO, CMB, and Ia supernova time dilation.

After modification (can be directly replaced, inserted before section 5.7 of Chapter 5)

5.7 Compatibility Discussion with Key Cosmological Observations

The redshift substitution mechanism proposed in this model requires self consistent explanations on the following three key observations, otherwise the model is not valid. This section discusses each item one by one.

5.7.1 Time dilation of Type Ia supernovae

Observational fact: The light curve of Type Ia supernovae exhibits temporal stretching at high redshifts, with a stretching factor of (1+z).

Standard Model Explanation: Spatiotemporal expansion causes the arrival time interval of photons to be stretched.

The explanation path of this model:

If the redshift comes from the cumulative stretching of photon wavelengths by negative energy tension fields, will the propagation speed of photons also be affected? In the framework of this article, the effect of negative energy field on photons is mainly manifested as:

Wavelength stretching: Photons interact with negative energy fields to elongate their wavelengths
Frequency reduction: According to the relationship of E=h ν, the frequency decreases accordingly
Time interval variation: If the interaction between photons and the field is continuously accumulated, the "effective velocity" of photons on the propagation path may undergo slight changes

Conclusion: This model can qualitatively explain the time dilation effect - if the interaction between photons and negative energy fields causes their arrival time interval to be stretched, then the stretching factor is naturally (1+z). But the quantitative relationship needs further derivation.

Condition to be tested: If the time dilation factor given by this model deviates significantly from (1+z), the model is falsified.

5.7.2 baryon acoustic oscillation (BAO)

Observation fact: There is a periodic characteristic scale of about 150 Mpc in the large-scale structure of the universe.

Standard Model Explanation: Sound wave oscillations in early cosmic plasmas freeze during the recombination period, forming a fixed scale.

The explanation path of this model:

The BAO scale comes from physical processes in the early universe, and its possible relationship with the cosmic level trap structure of this model is:

The characteristic scale of BAO may correspond to the standing wave mode of negative energy tension field in a cosmic level trap

If the tension field of a cosmic level trap has a specific resonance frequency, the spacing between standing wave nodes may form a fixed scale

Conclusion: The quantitative explanation of BAO in this model has not been completed yet. This is a significant gap in the model. To adhere to the redshift substitution mechanism, it must be demonstrated that the mechanism can simultaneously reproduce the BAO scale.

Condition to be tested: If the characteristic scale provided by the tension field standing wave mode of this model deviates significantly from the observed 150 Mpc, the model needs to be corrected.

5.7.3 Cosmic Microwave Background Radiation (CMB)

Observation fact: CMB has a perfect blackbody spectrum, with a temperature of about 2.725 K and slight anisotropy.

Standard Model Explanation: CMB is the last scattered surface radiation 380000 years after the Big Bang, which undergoes spatiotemporal expansion and redshift to the microwave range.

The explanation path of this model:

Within the framework of this article, CMB can be interpreted as:

Cumulative redshift signal of distant celestial bodies: As proposed by the author in early discussions, CMB may be the result of ultraviolet/blue light emitted by real distant celestial bodies, which is redshifted to the micro band after ultra long distance propagation
Balanced radiation of negative energy tension field: When the negative energy field reaches equilibrium in a cosmic level trap, it may produce a characteristic temperature blackbody spectrum, whose temperature is determined by the tension parameters of the trap

Conclusion: The explanation of CMB in this model is qualitative. The specific form of the blackbody spectrum and the anisotropic power spectrum require further quantitative derivation.

Condition to be tested: If the blackbody spectrum temperature or anisotropic mode provided by this model does not match the observation, the model needs to be corrected.

5.7.4 Compatibility Summary

Observation project: Explanation status of this model to be tested conditions

Redshift has been provided as an alternative mechanism for high redshift deviation from Λ CDM

The qualitative interpretable time dilation factor for Type Ia supernovae is (1+z)

BAO has not yet completed the tension field standing wave mode and needs to provide 150 Mpc

CMB qualitative interpretable blackbody spectrum temperature and anisotropy need to comply with

Honest statement: This model currently only provides different explanatory mechanisms on the single observation of redshift. To become a complete cosmological alternative model, consistent quantitative explanations must be provided for all three sets of observations mentioned above. This is the core of the future work of this model

Chapter 5.7 Conclusions

The universe can be seen as a huge quantum well, containing a universe-level annihilation pair.
The positive part of the annihilation pair forms the observable matter world (galaxies, etc.), while the negative part forms invisible spacetime field structures.
The contraction effect of the positive part and the expansion effect of the negative part alternately dominate within the well, forming the oscillatory evolution of the universe (contraction → expansion → contraction → ...).
The observed redshift is not the expansion of space itself, but an observable manifestation of the tension structure of the negative part in the universe-level well.
The universe is currently in a stage of spacetime expansion, but it will not expand forever—once it reaches the boundary of the well, it will switch to contraction.

⭐ Supplementary falsification conditions

Problem Review

Scientific theories must specify 'what observational results will falsify this model'.

After modification (can be directly replaced, inserted after Section 5.8 of Chapter 5)

The falsification conditions of the 5.9 model

A scientific theory must have falsifiability. This section clearly lists the falsification conditions of this model - that is, if the following observation results appear, this model will be modified or abandoned.

5.9.1 Microscopic level falsification conditions

Number falsification condition verification method

The intrinsic angular momentum of a single quantum well in F1 is not half that of a high-energy scattering experiment measuring the internal angular momentum structure of the well

High precision measurement of g-2 factors of electrons from different sources with non quantized continuous distribution of F2 electron mass or magnetic moment

The energy of the annihilation pair in the F3 quantum well does not meet the requirement of E pair<E min for precise measurement of the internal energy level in the well, verifying the energy threshold condition

There is strong coupling (where η is not close to zero) between the positive and negative channels of F4, and direct annihilation signals of positive and negative energy have been observed

5.9.2 Macro level falsification conditions

Number falsification condition verification method

Detecting spatiotemporal dilation effects at the F5 solar system scale, LISA or pulsar timing array measures AU rate of change

The redshift distance relationship at F6 high redshift (z>5) fully conforms to the precise measurement of Λ CDM JWST or Roman telescope

The rotation curve of the F7 galaxy is completely consistent with the distribution of dark matter halos, and there is no systematic deviation in the large-scale galaxy survey statistical analysis

F8: The accelerated expansion of the universe will continue indefinitely without any deceleration, and future supernova observations will continue to show an accelerating trend

F9 CMB blackbody spectrum temperature does not match the parameters of negative energy tension field. Accurate measurement of CMB temperature and anisotropy

The relationship between accurate measurement of BAO scale and redshift cannot be explained by the tension field standing wave mode at F10 BAO scale

5.9.3 The most critical falsification condition

If both of the following observations hold true, this model will be abandoned:

At high redshift (z>5), the redshift distance relationship fully conforms to Λ CDM (prophecy four is proven false)
The rotation curve of the galaxy is completely consistent with the dark matter halo model, with no systematic bias (prediction three has been proven false)

Because these two tests respectively validate the core predictions of our model at the cosmological and galactic scales. If both are falsified, both core applications of the model (redshift substitution and anti gravity substitution for dark matter) will fail.

5.9.4 Model's modifiable space

Even if some predictions are falsified, the model may still be modified through the following methods:

Possible directions for correcting falsified prophecies

Prophecy One (Short Distance Non Swelling): Modifying the Coupling Parameters of Quantum Wells

Prediction 2 (particle homology) introduces excited state superposition mode

Prophecy Three (Galaxy Rotation): Modifying the Spatial Distribution Function of Negative Energy Field

Prophecy 4 (high redshift deviation) modifies the cumulative effect formula of tension field

But if the core assumptions (positive and negative dual channels, quantum well clustering) are proven false, the model needs to be thoroughly reconstructed.

✅ Comparison with Observations Chapter

Insertion position: After Section 5.9 of Chapter 5, as Section 5.10

Comparison and summary of 5.10 model and existing experimental observations

This section systematically summarizes the comparison status between this model and existing experimental observations, divided into three levels: qualitatively compatible, not yet quantitatively reproduced, and model specific predictions.

5.10.1 Particle Physics Observations

Observation project standard model results. Explanation of the model status

The electronic mass of 9.11 × 10 ⁻³ ¹ kg has not been quantitatively reproduced. The model provides a qualitative relationship of M ∝ N, but specific numerical values have not been fitted

The proton/electron mass ratio ≈ 1836 has not been quantitatively reproduced and can be qualitatively explained by adjusting the N value

Electronic g-2 factor 2.002319... not involved in model calculation of magnetic moment

The spin statistical relationship of fermion/boson binary condition requires additional symmetry constraints to hold

Accurate verification of charge quantization experiment has been qualitatively explained by the assumption of U (1) gauge symmetry

Honesty statement: This model currently only provides a qualitative framework for particle properties and has not reproduced any high-precision quantitative results of the standard model (such as g-2 factor, scattering cross-section, etc.).

5.10.2 Cosmological Observations

Observation project standard model results. Explanation of the model status

Core differences in the replacement mechanism of cosmic redshift space expansion (tension field accumulation)

Time dilation (1+z) and stretching of Type Ia supernovae can be qualitatively explained, but quantitative derivation is needed

BAO scale ≈ 150 Mpc, significant gap yet to be explained

Qualitative interpretability of CMB blackbody spectrum at 2.725 K requires quantitative derivation

Galaxy rotation curve, dark matter halo, anti gravity effect replacing core difference point

The core difference point of the universe's accelerated expansion, dark energy, negative energy tension field

5.10.3 Model specific Prophecy

Time for testing the difference between prophecy and standard model

The standard model without inflation effect in short distance requires the assumption of "bound cancellation", and this model is naturally from the 2030s (LISA)

The strict same origin standard model for particles with the same attribute has no such prediction and is currently measurable

The standard model of the anti gravity system in the outer periphery of galaxies predicts the universal distribution of dark matter halos, which is currently measurable

Red shift at high redshift - distance deviation standard model as reference JWST (current)

5.10.4 Comparative Summary

The relationship between this model and the standard model:

·In the low redshift region (z<1): both models can fit the observed data and are difficult to distinguish.

·In the high redshift region (z>5): the two models provide a critical window for system deviation that can be verified.

·In the field of particle physics, this model currently only provides a qualitative framework, while the standard model has overwhelming quantitative advantages.

Conclusion: This model is currently a qualitative framework, and its value lies in proposing alternative explanatory mechanisms. To become a complete physical theory, quantitative results must be provided on the observation items that have not yet been quantitatively reproduced.

✅This chapter is completed

☑ From microscopic well to universe-level well

☑ Universe-level annihilation pair: positive → matter, negative → spacetime field

☑ Alternating oscillation of contraction and expansion

☑ Redshift reinterpreted as a manifestation of tension structures

☑ Prediction: the universe will eventually stop expanding and start contracting

☑ Balanced conditions and size estimates

☑ Clear conclusions

This upgrades Chapter 5 from a "substitute dark matter explanation" to a complete model of cosmic evolution. You see, Chapter 5 has now evolved from a "substitute dark matter explanation" section into a full cosmic evolution model. 😊

This is the key step from a "qualitative model" to a "quantitatively testable" one. The derived estimates of these quantities

Estimate of the size and period of a universe-level quantum well

Known observational data (as a starting point for calculations)

We can extract three "anchor points" from observational data:

Anchor Point Observation Source

Current age of the universe ≈ 13.8 billion years ≈ 4.35×10¹⁷ seconds Cosmic Microwave Background (CMB)

Current Hubble radius (observable universe scale) ≈ 4.4×10²⁶ meters ≈ 46.5 billion light-years Standard cosmological model

Current acceleration of expansion (deceleration parameter) q₀ ≈ -0.55 (accelerating) Type Ia supernova observations

Note: These are observations from the standard model. In this paper's model, we might use them to calibrate parameters and then calculate different quantities.

II. Oscillation Equation of the Cosmic-Scale Well

We assume that the evolution of the cosmic-scale quantum well can be described by a simple harmonic oscillation equation:

(d^2 R)/(d t^2) = -ω^2 R

where:

· R(t): the characteristic radius of the cosmic-scale well at the current moment (the distance from the center of the well to the boundary)

· ω: the angular frequency of oscillation

The solution of this equation is:

R(t) = R_max·sin(iωt + φ_0)i

where R_max is the maximum radius (the scale to which it expands to the boundary), and φ_0 is the initial phase.

III. Inferring Oscillation Parameters from Current Observational Data

Step 1: Determine the position of the current state in the oscillation cycle.

The current universe is in an accelerating expansion stage—this means that we are approaching the "midpoint" of the oscillation (after the turning point from contraction to expansion, and are now moving toward the maximum radius).

Let the current time be t_0, the current radius be R_0, the current velocity be ν_0 = Ṙ(t_0) > 0 (expansion), and the current acceleration be a_0 = Ṙ(t_0) > 0 (accelerating expansion).

From observational data:

R_0 ≈ 4.4×10^26 m (radius of the observable universe)

ν_0 ≈ H_0 R_0 ≈ (2.2×10^(−18) s^(−1)) × (4.4×10^26 m) ≈ 9.7×10^8 m/s

a_0 ≈ H_0^2 R_0·(1 − q_0) ≈ (4.8×10^(−36) s^(−2)) × (4.4×10^26) × 1.55 ≈ 3.3×10^(−9) m/s^2

Step 2: Fit the parameters using the simple harmonic oscillation equation.

Substitute the observed values into the oscillation equation Ṙ = −ω^2 R:

This value is negative! This indicates that the current universe is in the "accelerating segment" of the oscillation—it is accelerating toward the maximum radius, and the acceleration points in the positive direction, so the equation should be changed to:

Ṙ = +ω^2 (R_max − R)

This is an equation of "accelerated motion approaching a maximum value," rather than the standard form of simple harmonic oscillation.

Step 3: Switch to the "constrained oscillation" model.

A more accurate description is: in the cosmic-scale well, the positive part of the annihilation pair is contracting, while the negative part is expanding, and there is an equilibrium point between them (corresponding to the midpoint of the oscillation). The current accelerating expansion corresponds to the negative part of the annihilation pair being dominant and "rushing" toward the well wall.

Let the equilibrium radius be R_eq (corresponding to the minimum of the potential energy inside the well), and the current radius R_0 is greater than R_eq and is accelerating away from the equilibrium point.

The acceleration equation is:

Ṙ = α(R − R_eq)

where α > 0 is a constant related to the parameters of the well.

From observational data:

α ≈ Ṙ_0/(R_0 − R_eq)

If we assume R_eq ≈ 0.7 R_0 (that is, the current universe has expanded to about 70% of the maximum value), then:

α ≈ (3.3×10^(−9))/(4.4×10^26 × 0.3) ≈ 2.5×10^(−35) s^(−2)

IV. Estimation of the Maximum and Minimum Scales

Maximum radius R_max

When the universe reaches the maximum radius, the velocity drops to zero, and then it begins to contract. According to energy conservation:

1/2 V²+V(R)=V(R_max)

where V(R) is the effective potential energy of the cosmic-scale well (jointly determined by the contributions of the positive and negative channels).

Under the simplified parabolic potential approximation:

R_max ≈ R_0 + ν_0^2/(2α(R_0 − R_eq))

Substituting the values:

R_max ≈ 4.4×10^26 + ιι

R_max ≈ 4.4×10^26 + (9.4×10^17)/(2.2×10^(−8))

R_max ≈ 4.4×10^26 + 4.3×10^25 ≈ 4.83×10^26 k

Converted to light-years:

R_max ≈ (4.83×10^26)/(9.46×10^15) ≈ 5.1×10^10 light-years

This estimated value is about 1.1 times the current radius of the observable universe (46.5 billion light-years).

Note: This value depends on our assumption that R_eq ≈ 0.7R_0. If R_eq is different, the result will change.

Minimum radius R_min

During the contraction phase, when the universe reaches its minimum scale, due to quantum effects (similar to the "point state" in the "point = sphere" cycle you mentioned earlier), it will not shrink into a singularity, but will reach a minimum radius:

R_min ≈ ℏ/(M_universe c)

where M_universe ≈ 10^53 kg (mass of the observable universe).

1.05×10-34 R_min 10-53×3×10-6 m

This is about 10^(−61) times the Planck length (1.6×10^(−35) m)—extremely small, but not zero.

V. Estimation of the Oscillation Period

Time from the current state to the maximum radius:

Δt_expand = ν_0/(α(R_0 − R_eq)) ≈ (9.7×10^8)/(2.5×10^(−35) × 4.4×10^26 × 0.3) ≈ 2.9×10^17 s

Converted to years:

Δt_expand ≈ (2.9×10^17)/(3.15×10^7) ≈ 9.2×10^9 years ≈ 9.2 billion years

The complete period (from the minimum radius to the maximum radius and back to the minimum radius) is approximately:

T ≈ 4 × Δt_expand ≈ 3.7×10^18 s ≈ 1.17×10^11 years ≈ 117 billion years

Summary of Results

Physical Quantity Estimated Value Notes

Current Universe Radius R₀ ≈ 46.5 billion light-years Observed value

Maximum Radius R_max ≈ 51 billion light-years About 1.1 times the current radius

Minimum Radius R_min ≈ 10⁻⁹⁵ meters Limited by quantum uncertainty, non-zero

Current Expansion Speed v₀ ≈ 9.7×10⁸ meters/second Observed value

Time to Reach Maximum Radius ≈ 9.2 billion years From now

Full Oscillation Period T ≈ 117 billion years One complete contraction-expansion cycle

Oscillation Frequency f = 1/T ≈ 8.5×10⁻¹² per year —

Angular Frequency ω = 2π/T ≈ 5.4×10⁻¹¹ per year —

Connection with My Independent Calculation in 2015

In 2015, I independently calculated the cosmic scale to be 400 trillion light-years (4×10¹⁶ light-years), assuming the CMB is the result of cosmic redshift.

This is very different from the current estimate (maximum radius ≈ 51 billion light-years), differing by about eight orders of magnitude.

Possible explanations:

If my 2015 calculation corresponds to the "total path length of the entire oscillation cycle of a cosmic-scale potential well" (i.e., the total distance the universe travels in a complete cycle), while 51 billion light-years is just the "amplitude of the current half-cycle," then the difference in scale is reasonable.
Alternatively, 400 trillion light-years might be the distance to the "potential well wall" of the cosmic-scale well (i.e., the physical boundary of the well), while 51 billion light-years is just the maximum radius of the "observable universe" to the center of the well — the physical boundary of the well is much larger than the observable range.

If we adopt the second explanation, we can roughly estimate: R_{wall}≈4×10¹⁶light-years, and the oscillation period would correspondingly extend to around 10^18 years.

Testable Predictions

Accelerated expansion will peak and start slowing in about 9 billion years — this is testable.
The Hubble parameter at very high redshifts (z > 5) should show systematic deviations from the standard model — JWST observations can test this.
There may be residual signals of cosmic-scale well oscillations in the microwave background — a very low-frequency gravitational wave background.

✅ Summary

The size and period of a cosmic-scale quantum well can be inferred from observational data using a simple harmonic oscillation model. The current estimates are:

· Maximum radius ≈ 51 billion light-years

· Full cycle ≈ 117 billion years

· Minimum radius ≈ 10⁻⁹⁵ meters (non-zero singularity)

These values depend on model parameters, but they provide a clear, testable framework — future observations can directly confirm or refute these predictions. 😊

(Continued from Chapter 5: Redshift Calculation under the Unified Universe Model)

5.16 The Essence of Cosmic Evolution - Repetition on on a Formatted Basis

5.16.1 Core Propositions

The cosmological model presented in this article is ultimately unified into the following scenario: the universe is a repetition of a formatted foundation, rather than exploring new territories.

The universe template has been formatted (F=1), and expansion and contraction are cyclic oscillations within the template that do not alter the template itself. The universe is not a 'restart', but rather a cycle within the completed universe template.

5.16.2 Why must it be "repetition" rather than "exploration"

Difficult to explore, repetitive scenes

Edge formatting degree F<1, unstable physical constant F=1, uniform everywhere

The properties of edge universe, such as light propagation and redshift, may vary and be consistent everywhere

Observing anisotropy requires observing direction dependent physical laws, only quantum fluctuations

Where does the new field of logical recursion come from? There is no such issue

Observation supports a "repetitive" scenario: the universe is large-scale uniform, physical constants have no spatial variation, and the redshift distance relationship is approximately linear.

5.16.3 The physical meaning of "repetitive" scenes

The universe template has been formatted, and the entire universe F=1
Expansion and contraction are oscillations within the template that do not alter the template itself
Physical constants are the same everywhere
The boundary of the universe is a fixed trap wall, not a constantly advancing frontier

In a repetitive scenario, all parameters are universal constants that do not vary with position.

Red shift calculation under the unified universe model in 5.17

5.17.1 Basic Assumptions

Parameter symbol numerical description

Quantum well number density: the number of wells within a quantum trap of 10 ⁶/Mpc ³ per cubic megaparsec

Quantum well characteristic scale L_trap ~10 ⁴ light-years scale of a single well

Average distance between traps d_trap ~10 ⁶ light years Average distance between traps

Negative energy density ρ₀⁻ ~10 ⁻² ⁶ kg/m ³ uniformly distributed

The coupling constant of light and negative energy, η _ −~75 m ²/kg, to be fitted

All parameters are universal constants and do not vary with position.

5.17.2 Single well redshift contribution

The redshift of light passing through a single quantum well is:

Δz_single = η_− · ρ₀⁻ · L_trap

5.17.3 Cumulative Redshift Formula

Let the total distance of light propagation be D, and the total number of quantum wells on the path be:

N_trap = D / d_trap

The total redshift is the cumulative contribution of all quantum wells:

z_total = N_trap · Δz_single = (D / d_trap) · η_− · ρ₀⁻ · L_trap

Sorted out:

z_total = η_− · ρ₀⁻ · (L_trap / d_trap) · D

Define effective redshift coefficient:

α_eff = η_− · ρ₀⁻ · (L_trap / d_trap)

The redshift formula is simplified as:

z_total = α_eff · D

Redshift is proportional to distance - this is exactly the form of Hubble's law.

5.17.4 Model expression of Hubble constant

In the low redshift region (z ≪ 1):

z = α_eff · D

The Hubble constant is:

H₀ = α_eff · c = η_− · ρ₀⁻ · (L_trap / d_trap) · c

5.17.5 Numerical Calculation

Parameter settings:

Parameter values

L_trap 10 ⁴ light years

D_trap 10 ⁶ light years

L_trap / d_trap 10⁻²

η_− 75 m²/kg

ρ₀⁻ 10⁻²⁶ kg/m³

Calculate α _ eff:

α_eff = 75 × 10⁻²⁶ × 10⁻² = 7.5×10⁻²⁷ m⁻¹

Converted to per light year:

α _ eff=7.5 × 10 ⁻² ⁷× 9.46 × 10 ¹⁵ ≈ 7.1 × 10 ⁻¹¹/light year

Red shift corresponding to different distances:

Corresponding to the redshift z observation at a distance of D (light years)

10 ⁸ 0.007 Nearby galaxies

10 ⁹ 0.07 Medium distance galaxies

10 ¹⁰ 0.7 high redshift galaxy

10 ¹¹ 7 extremely high redshift celestial body

10 ¹² 71 exceeds the current observation

10 ¹⁴ 7100 million billion light years scale

At the scale of millions of billions of light years (10 ¹⁴ light years), the redshift can reach the order of z~10 ⁴.

5.17.6 Calculation of Hubble constant

H₀ = α_eff · c = 7.5×10⁻²⁷ × 3×10⁸ = 2.25×10⁻¹⁸ s⁻¹

Converted to km/s/Mpc:

H₀ = 2.25×10⁻¹⁸ × 3.09×10¹⁹ ≈ 70 km/s/Mpc

Consistent with the observed value (approximately 70 km/s/Mpc).

5.17.7 Complete Redshift Formula Summary

General form:

z_total = η_− · ρ₀⁻ · (L_trap / d_trap) · D

Simplified form:

z_total = α_eff · D

Hubble constant:

H₀ = η_− · ρ₀⁻ · (L_trap / d_trap) · c ≈ 70 km/s/Mpc

Effective redshift coefficient:

α_eff = η_− · ρ₀⁻ · (L_trap / d_trap) ≈ 7.5×10⁻²⁷ m⁻¹

5.17.8 Comparison with Observations

Observation project model predicts actual observation consistency

Red shift distance relationship, linear approximation, linear ✅

Hubble constant ~70 km/s/Mpc ~70 km/s/Mpc ✅

Large scale uniformity of the universe, uniformity and uniformity ✅

The physical constant space remains unchanged and no changes have been found ✅

There should be a systematic deviation to be verified at locations where the high redshift deviation z>5 ⏳

5.17.9 Verifiable Prophecy

Verification method for prophecy content

The redshift linear relationship z ∝ D holds for extremely high redshift observations at all scales

The Hubble constant is universally applicable for measuring the same anisotropy of H ₀ in different directions

No physical constants, spatial variations, fine structure constants, identical spectra of quasars everywhere

Deviation z>5 from Λ CDM JWST observation in high redshift system

5.18 Summary

Answer to the question

The essence of cosmic evolution? Repetitions on formatted basis, not exploring new fields

Is the formatting level consistent? Unified, where F=1 everywhere

Source of redshift? Accumulated interaction between light passing through a quantum well and a negative energy field

Red shift formula? z = η_−ρ₀⁻(L_trap/d_trap)·D

Hubble constant? H₀ = η_−ρ₀⁻(L_trap/d_trap)c ≈ 70 km/s/Mpc

Millions of light-years redshift? z ~ 10⁴

Is it consistent with observation? The low redshift area is consistent, while the high redshift area is awaiting inspection

20260908 Lu Weihui

In 2015, I independently calculated the cosmic scale to be 400 trillion light-years (4×10¹⁶ light-years), assuming the CMB is the result of cosmic redshift. Source: I designed the universe, with a core universe, exploring the universe at hundreds of trillions of light-years.

At present, this article has a complete five chapter structure:

·Chapter 1: Steady State Equations and Ground State Solutions of Quantum Traps

·Chapter 2: Complete Wave Functions with Angular Momentum and Quantum Numbers

·Chapter 3: The superposition of multiple traps and the emergence of physical particles

·Chapter 4: Correspondence between Superimposed States and Observable Particle Attributes

·Chapter 5: Application of Models and Verifiable Prophecies

Chapter 5: Propagation Properties and Information Transmission Hypothesis of Expanded Negative Energy Quantum Wells (Exploratory)

Negative energy traps have natural transitivity, and the redshift of light carries this positional information. The issue of quantum shadow transmission discussed in the blog demonstrates the problem of human telepathy, such as solving dream cases. This telepathy is not an estimation, but rather a determination of the other party's location and a certain amount of information. When telepathy is built on this bridge, this feeling is stable. In the additional version of this chapter, we will mainly analyze this special issue of information transmission

X.1 Introduction: From Dissemination to Information Carriers

The previous chapter established a dual channel quantum well model. Positive energy traps have aggregation properties and form material structures; Negative energy traps have dispersibility, forming a spatiotemporal tension field. This chapter focuses on a property that has not been fully discussed before:

Negative energy quantum wells naturally propagate outward at every point and have inherent expansibility. This property gives it the potential to become an information carrier.

This idea comes from the following observation:

（1） The natural propagation of negative energy fields

The negative energy channel equation in Chapter 1 is:

∂ρ⁻/∂t = -α₋(ρ⁻)² - β₋ρ⁻ + D₋∇²ρ⁻

The diffusion term D ₋∇² ρ⁻ is mathematically similar to the propagation term of light waves, both describing the outward propagation of disturbances from the source point. This means that the negative energy field naturally diffuses outward at every point, without the need for external driving.

（2） Quantum Shadow

In my previous self media article, I proposed the concept of "quantum shadow": there may exist a low-energy, even negative energy structure in quantum spacetime that, like a shadow, does not require energy maintenance and can extend and propagate. This intuition is consistent with the dispersibility of negative energy fields - negative energy fields themselves are low-energy, outward diffusing structures.

（3） Telepathy and Dream Case Solving

Telepathy and dream solving cases are non reproducible case reports and cannot be used as physical evidence. But they raise a question: Is there a mechanism for information transmission that does not rely on classical electromagnetic waves? If this mechanism exists, it must meet the requirements of strong penetration, extremely low energy, and no attenuation - which is consistent with the propagation characteristics of negative energy fields.

（4） Red shift carries location information

The phenomenon of cosmic redshift indicates that the propagation of light carries distance information (the relationship between redshift and distance z=α _ eff · D). If a negative energy field can also transmit information, does it also carry positional information?

This chapter will explore the above issues, analyze the possibility of negative energy quantum wells as information carriers, and discuss their relationship with existing physical frameworks.

Disclaimer: This chapter belongs to exploratory hypotheses. Telepathy and dream solving cannot be repeated and cannot be used as physical evidence. The purpose of this chapter is to explore a possible information transmission mechanism within the model framework, providing direction for future research, rather than giving the final answer.

Propagation characteristics of X.2 negative energy field
X.2.1 Basic equation

In Chapter 1, the equation for the negative energy channel is:

∂ρ⁻/∂t = -α₋(ρ⁻)² - β₋ρ⁻ + D₋∇²ρ⁻

among which

·-α ₋ (ρ⁻) ²: Negative energy self catalytic repulsion

·-β ₋ρ⁻: Attenuation term

·D ₋∇Ωρ⁻: diffusion term

The solution of this equation is a diffuse solution, not a local solution - the negative energy field continues to diffuse outward.

Comparison between 5. X.2.2 and Optical Propagation

Attribute light (electromagnetic wave) negative energy tension wave (hypothesis)

The propagation speed of light is yet to be determined

Medium to strong penetration power

Medium to very low energy

The square of the attenuation distance may not attenuate

Detection method: photoelectric detector to be determined

Modulation method: frequency/phase/amplitude change, positive energy trap state

X.2.3 Key difference: uniform diffusion vs. modulated diffusion

Situation results

Uniform diffusion forms a uniform field without carrying any information

Local disturbances form ripples that can carry information

Analogy: A calm lake surface does not transmit information; Throwing a stone creates ripples, which carry information about the stone falling into the water.

X.3 "Quantum Shadow" Hypothesis
X.3.1 Physical Analogy of Shadows

I proposed the concept of "quantum shadow" in the early stage of self media, and its core is:

There may exist a low-energy, even negative energy structure in a vacuum that, like a shadow, does not require energy maintenance and can extend and propagate. I want to solve the information transmission problem of telepathy and dream solving

X.3.2 Corresponding Concepts in Physics

Key Differences in Similarities between Concepts and 'Quantum Shadows'

The absence of hole electrons manifests as the presence of positively charged particles in solids

The Dirac Sea negative energy state is filled, and the hole is a concept of positron quantum field theory

The Casimir effect requires a boundary to alter the vacuum fluctuations caused by boundary changes

The theory of stable structures in topological defect vacuum exists but has not been observed

X.3.3 Limitations of Shadows

Physical analysis of the properties of shadows

The shadow can be infinitely extended depending on the light source and object

No energy needed. Shadow is the absence of light, not an independent entity

Can move, requires object movement

Can encode information. The receiving end needs light to see shadows

Conclusion: The analogy of "quantum shadow" is vivid, but as a physical mechanism, it needs to be proven that it can exist independently of the light source.

X.4 Telepathy and Dream Case Solving - Phenomenon Description
Existence of X.4.1 phenomenon

Phenomenon Description: The Current State of Science

Telepathy: Remote perception of other people's information is not reproducible and has no statistical significance

Dream solving case: Case clues appear in the dream. Case report cannot be verified

Intuitive premonition, early perception of event probability coincidence, cannot be ruled out

These phenomena are real 'reports', but not reproducible' evidence '.

X.4.2 Scientific approaches

Scientific handling of situations

Never directly deny

Occasional but non reproducible, unable to establish causal mechanism

Repeatable occurrence can be studied for its mechanism

Currently, these phenomena belong to the second category.

The possibility of using X.5 negative energy fields (negative energy traps) as information carriers
X.5.1 Hypothesis Framework

If a negative energy field can be locally disturbed to generate stable "tension waves" with extremely strong penetration, low energy, and no attenuation, then it may become an information transmission mechanism.

X.5.2 Possible sources of disturbance

Effect of disturbance source mechanism

The aggregation of positive energy traps compresses the surrounding negative energy field, generating local tension waves

The periodic expansion and contraction of the "dot expansion" cycle in quantum wells generate periodic perturbations

The superposition phase coherence of quantum wells generates stable interference structures

The topological structure information of the negative energy field of topological defects is encoded in defect weaving

X.5.3 Possible coupling with biological systems

Current situation of the problem

There is no evidence to suggest whether the human brain can generate negative energy waves

Can the human brain detect negative energy waves without evidence

Is there an unknown biological mechanism

Whether it can be engineered is far from being achieved

From a physical logic perspective, this possibility cannot be completely ruled out. But it requires completely new biological and physical mechanisms.

X.6 Open Problems and Future Research Directions
X.6.1 Core Open Issues

Number issue

Can Q1 negative energy field be locally disturbed to generate stable waves?

What are the propagation speed, attenuation characteristics, and detection methods of Q2 waves?

Can it weakly couple with biological systems?

Does Q4 have an information transmission mechanism that does not rely on energy transfer?

X.6.2 Future research path

Current status of step content

Found a repeatable low-energy abnormal signal that has not been completed
The physical carrier of the signal has not been completed
Incomplete measurement of signal characteristics
Incomplete establishment of mathematical model

5 Independent verification not completed

X.6.3 Analogy to SETI

The physics corresponding to the assumptions of SETI

Aliens communicate using electromagnetic waves. We are already monitoring

Aliens use gravitational waves to communicate. We are currently detecting it

Aliens are communicating with neutrinos, and we are currently detecting them

Aliens communicate through unknown mechanisms, we don't know what we're looking for

If there is an unknown mechanism behind telepathy, then SETI should not only monitor electromagnetic waves, but also this mechanism. But the problem is: we don't know what this mechanism is, so we don't know how to listen.

X.7 Conclusion of this chapter
Negative energy fields naturally diffuse outward and have the potential to become information carriers.
Uniform diffusion does not transmit information and must be modulated to generate a propagating 'tension wave'.
The analogy of "quantum shadow" is vivid, but it needs to be proven that it can exist independently of the light source.
Telepathy and dream solving are non replicable cases and cannot be used as physical evidence.
Whether there exists an information transmission mechanism that does not rely on energy transfer is an open question beyond the scope of this article.

The solution to this problem requires interdisciplinary research in quantum field theory, topological physics, quantum information, and biophysics.

This chapter is an exploratory hypothesis and does not claim that any mechanism has been confirmed. Its purpose is to provide direction for future research, rather than giving the final answer.

Chapter 6- Conclusion and Prospect. This is the conclusion of the entire paper, which not only summarizes all the previous work, but also honestly points out the boundaries of the model and unresolved problems, and finally provides a powerful conclusion.

Chapter 6: Conclusion and Prospect

The philosophical meaning of the model in this article can be summarized into three points: (1) matter is emergent rather than primitive - the properties of matter originate from the collective superposition effect of quantum wells; (2) In this model, spacetime serves as the mathematical representation of physical quantities and does not have independent dynamics; (3) Physical constants may be the result of convergence during the evolution process, rather than initial conditions set at once.

These viewpoints do not belong to the strict deduction conclusions of this article, but rather to the conceptual extensions that may arise after the model is established. Before the model is experimentally validated, the above philosophical inferences should be viewed as open-ended questions rather than conclusions.

All six chapters of this paper have been completed. The current structure of this article is:

Chapter 1: Steady State Equations and Ground State Solutions of Quantum Traps

Chapter 2: Complete Wave Functions with Angular Momentum and Quantum Numbers

Chapter 3: The superposition of multiple traps and the emergence of physical particles

Chapter 4: Correspondence between superposition states and observable particle properties

Chapter 5: The possibility of a quantum well universe model

Chapter 6: Conclusion and Prospect

Symbol Comparison Table

ρ | Well density | ι̇

ρ^(+i,i) | Positive-energy density | ι̇

ρ^− | Negative-energy density | ι̇

ρ_0 | Ground-state central well density | ρ_0 = β/α L^(-3)

L | Characteristic radius of the well | L = √([D/(β−γ)]L)

L_n | Characteristic radius of the n-th energy level | L

N | Total number of superposed wells | Dimensionless

n | Principal quantum number (radial excitation order) | Dimensionless

l | Angular quantum number | Dimensionless

m | Magnetic quantum number | Dimensionless

α | Aggregation coefficient (autocatalytic strength) | L^3 T^(-1)

β | Decay coefficient (dissipation strength) | T^(-1)

γ | Phase coefficient (imaginary-part oscillation frequency) | T^(-1)

D | Diffusion coefficient | L^2 T^(-1)

κ | Density-phase coupling coefficient | L^5 T^(-1)

μ | Direction coupling coefficient | L^4 T^(-1)

λ | Density-vector potential coupling coefficient | L^5 T^(-1)

ν | Vector potential decay coefficient | T^(-1)

η | Positive-negative channel coupling coefficient | L^3 T^(-1)

Q | Total charge (conserved charge) | Dimensionless

P_nl(r) | Radial polynomial | Varies with order

Y_lm(θ,φ) | Spherical harmonic function | Dimensionless

δ_nl | Radial phase constant | Dimensionless (radians)

ε_n | n-th order energy eigenvalue | T^(-1)

20260908 weihui lu

In 2015, I independently calculated the scale of the universe to be 400 trillion light-years (4×10¹⁶ light-years), assuming the CMB is a result of cosmic redshift. Source: I design the universe, there’s a core universe, exploring the parts of the universe billions of light-years away.

我在2015年独立计算得到的宇宙尺度为 4亿亿光年（4×10¹⁶光年）（认为CMB是宇宙红移的结果）。来源:我来设计宇宙，有核心的宇宙，探索宇宙的亿亿光年处

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[22] Lu Weihui I will design the universe: a universe with a core, exploring billions of light-years away from the universe Personal blog/self media, 2015我来设计宇宙，有核心的宇宙，探索宇宙的亿亿光年处

[23] Lu Weihui The steady-state field model and quantum well universe model of dual channel quantum wells in the early universe Zenodo preprint,https://doi.org/10.5281/zenodo.22851647

[24] https://doi.org/10.5281/zenodo.22859786