The Topological Nesting Structure of Quantum Wells ——A Special-Topic Analysis Based on the "Dual-Channel Quantum-Well Annihilation-Field Model" (the nesting, arrangement and topological self-similarity of the three types of quantum wells in fields of different density)

The Topological Nesting Structure of Quantum Wells ——A Special-Topic Analysis Based on the “Dual-Channel Quantum-Well Annihilation-Field Model” (the nesting, arrangement and topological self-similarity of the three types of quantum wells in fields of different density) Author: Lu Weihui This paper is a special-topic sequel to the “Dual-Channel Quantum-Well Annihilation-Field Model”; the overall conventions of the model are given in the parent paper, “The Dual-Channel Quantum-Well Annihilation-Field Model: A Self-Consistent Derivation from Underlying Postulates to Macroscopic Phenomena.” Contents Abstract Chapter 1 The Problem and the Conventions Adopted 1.1 The Question This Paper Answers 1.2 The Model Conventions and Formula Numbering Adopted 1.3 The Derivation Discipline of This Paper Chapter 2 All Three Types of Quantum Wells Can Nest 2.1 Self-Nesting of the Neutral Quantum Well: A Well Within a Well 2.2 Nesting of the Positive Quantum Well: Layering Under Self-Catalytic Aggregation 2.3 Nesting of the Negative Quantum Well: Film Formation Under Self-Catalytic Repulsion 2.4 Common Features and Differences of the Three Types Chapter 3 Nesting Depth Varies with Local Density 3.1 Density Determines the Number of Nesting Levels 3.2 High-Density Regions: Many Levels, Convergent Apparent Size 3.3 Low-Density Regions: Few Levels, Ample Room to Turn 3.4 Transitional Regions: Nesting and Arrangement Together Chapter 4 Arrangement, Nesting and the Transitional State: Three Basic Forms 4.1 Sparse Regions: Arrangement Predominates 4.2 High-Density Regions: Nesting Predominates 4.3 The Transitional Mixed State 4.4 Single Quantum Well in Appearance, Many in the Interior: the External-Equivalence Criterion Chapter 5 The High-Density Limit: Self-Similar Nesting in the Near Field of the Black-Hole Singularity 5.1 Outwardly a Single “Hole” 5.2 Self-Similar Levels Inside 5.3 The Model Meaning of “Inexhaustible, Endlessly Available” 5.4 Link with the Parent Paper’s “Nesting Is the Structure of the Universe” Chapter 6 Low-Density Regions: Sparse Arrangement Filling Space 6.1 Sparse Is Not Empty 6.2 Arrangement Constitutes Continuous Spacetime 6.3 Room to Turn and “Few Levels” Chapter 7 Testable Predictions and Falsifiability Chapter 8 Open Problems and Statement of Boundaries Appendix A Comparison of Formula Numbering Between This Paper and the Parent Paper Appendix B Key Terms in Chinese and English Appendix C List of Testable Predictions Abstract Within the postulate system of the “Dual-Channel Quantum-Well Annihilation-Field Model,” this paper examines the topological structure of quantum wells themselves. Postulate III of the parent paper already states that the infinite nesting through which light travels is the very gigantic quantum well to which light itself belongs, and that this nesting necessarily consists jointly of positive, negative and neutral quantum wells; Postulate V further states that this infinitely nested (matryoshka-style) structure is itself the structure of the universe. Following these two statements, this paper presses four questions. First: since the interior of a neutral quantum well may contain “a well within a well,” can this self-nesting continue indefinitely, so that an arbitrarily large number of levels is accommodated within a finite outward appearance? Second: positive quantum wells draw closer to one another through self-catalytic aggregation, and negative quantum wells are squeezed into thin layers through self-catalytic repulsion—can these two types likewise form nested structures? Third: is whether a quantum well occupies space by “nesting” or by “arrangement” determined by the local density at its location—high density giving many nesting levels, low density giving a single quantum well ample room to turn, rather than “a small region meaning few wells”? Fourth: within a structure that looks outwardly like a “single quantum well,” can there be inexhaustible, endlessly available levels, while for external fields it still remains equivalent to a single quantum well? The conclusion of this paper is as follows: all three types of quantum wells can nest—nesting is not the privilege of the neutral quantum well; how many levels a quantum well nests inside is determined by the local density at its location—high-density regions (including the near field of the black-hole singularity) have many nesting levels while their outward size converges to a finite value, and low-density regions have few nesting levels while a single quantum well has ample room to turn; space is neither “empty,” nor monopolized by any one type, but is jointly filled by these nested and arranged quantum wells. The derivations in this paper rest only on the postulates of the parent paper (Postulates I–VI, including Postulate VI on the neutral quantum well), the five postulate-level laws, and the simplest algebraic expressions, and introduce no external ready-made formula; wherever this model cannot yet give a numerical value, it is uniformly and explicitly marked “to be calibrated.” Keywords: quantum well; topological nesting; a well within a well; self-similarity; three types of quantum wells; local density; arrangement; transitional mixed state; external equivalence; near field of the black-hole singularity; vacuum; the Dual-Channel Quantum-Well Annihilation-Field Model Chapter 1 The Problem and the Conventions Adopted 1.1 The Question This Paper Answers Postulate III of the parent paper gives a key statement: the infinite nesting through which light travels is the very gigantic quantum well to which light itself belongs, and it necessarily consists jointly of positive, negative and neutral quantum wells; Postulate V further states that this infinitely nested (matryoshka-style) structure is itself the structure of the universe. Together, the two statements amount to saying that the “nesting” of quantum wells is not an isolated phenomenon but a universal form of the structure of the universe. What this paper sets out to do is to make the matter of “nesting” itself clear. The question is broken down into four successive sub-questions: • First layer (can it nest at all): may a neutral quantum well contain another neutral quantum well, and can this go on indefinitely? Can all three types of quantum wells nest?—This is Chapter 2. • Second layer (how deep): what determines how many levels a quantum well nests inside? Is it determined by the local density at its location?—This is Chapter 3. • Third layer (how it occupies space): does a quantum well occupy space by “nesting” or by “arrangement”? Is there a transitional form between the two?—This is Chapter 4. • Fourth layer (appearance and interior): within a structure that looks like a “single quantum well,” can there be infinitely many levels? Why, for external fields, does it still remain equivalent to a single quantum well?—This is Chapters 5 and 6. It should be stated at the outset that the “topological structure” referred to in this paper means how the boundaries of quantum wells are partitioned, how they close, and how they nest layer upon layer; it is a matter of geometry and connectivity, and does not involve the concrete form of the reactions inside quantum wells. This paper does not pursue what is actually going on “inside” a quantum well; it pursues only how its boundaries are organized. 1.2 The Model Conventions and Formula Numbering Adopted This paper introduces no new assumption and adopts the postulate system of the parent paper throughout. So that the reader need not consult the parent paper, the conventions used here are restated together. To avoid confusion of numbering, the formulas of this paper use the numbering of this paper, and wherever a conclusion of the parent paper is cited, it is marked “parent paper (x-y).” • Three types of quantum wells: the positive quantum well ρ⁺ (carrying positive annihilation pairs, self-catalytically aggregating), the negative quantum well ρ⁻ (carrying negative annihilation pairs, self-catalytically repelling), and the neutral quantum well (the “vacuum” redefined in the parent paper, Postulate VI). • Neutral quantum well (vacuum): the state in which, within one quantum well, two independent annihilation pairs of equal content, positive and negative, are isolated from each other and mutually neutralize outwardly; its essence is “being neutralized,” not “having no reaction” (parent paper 1.2(6), Postulate VI). • A well within a well: within a neutral quantum well the positive and negative annihilation pairs are partitioned and isolated, and a single well may have a nested structure—that is, within one well there may be another well, and it may be “half positive, half negative” (parent paper 1.2(6), Postulate VI). • Positive-quantum-well equation: ∂ρ⁺/∂t = α⁺(ρ⁺)² − β⁺ρ⁺ + D⁺∇²ρ⁺ − ηρ⁺ρ⁻ (parent paper (2-1)); equilibrium density ρ⁺* = β⁺/α⁺ (parent paper (3-3)). • Negative-quantum-well equation: ∂ρ⁻/∂t = −α⁻(ρ⁻)² − β⁻ρ⁻ + D⁻∇²ρ⁻ − ηρ⁺ρ⁻ (parent paper (2-2)); the negative quantum well is self-catalytically repulsive and cannot spontaneously aggregate into a bound region (parent paper 1.1). • Characteristic radius: a particle is a sphere of conserved total positive energy N, r_eq ≈ [ 3N/(4π) · α⁺/β⁺ ]^(1/3) (parent paper (3-4)). • The three types of regions coexist: the local form of spacetime is divided into positive, negative and neutral regions, and the three coexist at the same place; “compression” compresses the three together, only the attribute manifested differing from place to place (parent paper 1.3, 4.1). • Neutralization energy ε_N and neutralization energy density u_N = ε_N·n_N (parent paper 3.2(b)); n_N is the number of neutral quantum wells per unit volume. • Infinite nesting is the structure of the universe: this infinitely nested (matryoshka-style) structure is itself the structure of the universe, and it necessarily consists jointly of positive, nega