A Complete Proof of the Riemann Hypothesis within the Constraint Network Framework: Densification-Sparseification Contest and Five-Point Full Coverage

Menggang Yu
Independent Researcher
Honggutan District, Nanchang, Jiangxi Province, China, 330100
Email: ymg198702@163.com
ORCID: 0009-0004-1943-6776


Abstract

The Riemann Hypothesis asserts that all non-trivial zeros of the Riemann zeta function lie on the critical line σ = 1/2 in the complex plane, and that all trivial zeros are strictly the negative even integers. This conjecture, proposed over 160 years ago, is one of the most important unsolved problems in the history of mathematics. This paper presents a complete proof of the Riemann Hypothesis within the axiom system of Constraint Network Dynamics. The Constraint Network Model is defined by five axioms: the ontology of the universe is energy, energy always moves at the speed of light, encounters result in collision, the constraint network limits the free path of energy, and the total energy of the entire system is strictly conserved. Through the rigorous proof of five sub-propositions, this paper fully covers the entire content of the Riemann Hypothesis: Sub-proposition 1 provides an explicit construction function from the constraint network density to the complex variable of the zeta function, rigorously proving a one-to-one correspondence between the non-trivial zeros of the zeta function and the eigenvalues of the Hilbert-Pólya Hermitian operator H; Sub-proposition 2 proves that all non-trivial zeros satisfy Re(s) = 1/2; Sub-proposition 3, starting from the densification-sparseification contest equation, rigorously proves that the density set is continuously complete and fully covers the critical strip, ruling out any stray zeros; Sub-proposition 4 proves that all trivial zeros of the zeta function are strictly the negative even integers; Sub-proposition 5 proves that the zeta function is self-consistently analytically continued to the entire complex plane and that its functional equation holds globally. Together with the physical symmetry proof in [1] and the rigorous operator construction in [2], this paper forms a trilogy for the proof of the Riemann Hypothesis within the Constraint Network framework.

Keywords: Riemann Hypothesis, constraint network, densification-sparseification contest, Hilbert-Pólya operator, construction function


1. Introduction

The Riemann Hypothesis [3] asserts that all non-trivial zeros of the Riemann zeta function lie on the critical line σ = 1/2 in the complex plane, and that all trivial zeros are strictly the negative even integers. This conjecture, proposed over 160 years ago, is one of the most important unsolved problems in the history of mathematics.

This paper presents a complete proof of the Riemann Hypothesis within the axiom system of Constraint Network Dynamics. The Constraint Network Model is defined by five axioms, has been formalized within ZF set theory, and its emergent constant 1836 has been precisely solved [4]; the existence of the Yang-Mills mass gap has been rigorously proven [5]; and the boundary for the breakdown of the continuum hypothesis in the Navier-Stokes equations has been determined [6].

Reference [1] starts from the specular reflection symmetry of collision-rebound and proves that the fact that all non-trivial zeros of the zeta function lie on σ = 1/2 is an inevitable corollary of the axioms. Reference [2] rigorously constructs the Hilbert-Pólya Hermitian operator H within the Constraint Network framework and provides large-scale numerical verification. Building on these, this paper, through the rigorous proof of five sub-propositions, fully covers the entire content of the Riemann Hypothesis—from the non-trivial zeros to the trivial zeros, from the covering of the critical strip to the global analytic continuation. The three papers together form a trilogy for the proof of the Riemann Hypothesis within the Constraint Network framework.

The core innovation of this paper lies in introducing the densification-sparseification contest equation dρ/dt = η − γ as a core tool for global dynamics, and in providing an explicit construction function from the constraint network density to the complex variable of the zeta function. These two tools make possible the rigorous proofs of Sub-propositions 1 and 3, and constitute the key distinction of this paper from references [1] and [2].


2. Axiom System and the Densification-Sparseification Contest

2.1 Fundamental Axioms

Axiom One. The ontology of the universe is energy.

Axiom Two. Energy always moves at the speed of light.

Axiom Three. Encounter is collision.

Axiom Four. The constraint network limits the free path of energy; the free path determines the macroscopic displacement state.

Axiom Five. The total energy of the entire system is strictly conserved.

2.2 Global Dynamics and the Densification-Sparseification Contest

The global dynamics iteration formula is G_{t+1} = ℳ ∘ ℬ ∘ 𝒞 (G_t), where 𝒞 is the motion operator, ℬ is the collision operator, and ℳ is the constraint network update operator.

The core density evolution equation is:

dρ/dt = η − γ

η is the densification rate, defined as the rate of increase of the constraint network density caused by accretion events. γ is the sparseification rate, defined as the rate of decrease of the constraint network density caused by node vibrations and defects, γ(T) = ν_D · exp(−E_bind/(k_B T)).

When η > γ, the local density continuously increases. When η < γ, the local density falls back. When η = γ, the constraint network is in dynamic equilibrium.

2.3 Zero-Point Auxiliary Reference Frame and the Hilbert-Pólya Operator

Relying on the zero-point auxiliary reference frame, the energy of the constraint system is partitioned into the positive-direction return-to-zero subspace S₊ and the negative-direction return-to-zero subspace S₋. By the specular reflection symmetry of collision, S₊ and S₋ form a conjugate dual space. On this dual conjugate space, the Hilbert-Pólya operator H is rigorously constructed according to the rules of energy collision, accretion, and free-path evolution. The conjugate property of the space directly guarantees that H satisfies the definition of a Hermitian operator [2].


3. Complete Proof of the Five Sub-Propositions

3.1 Sub-Proposition 1: One-to-One Correspondence between the Non-Trivial Zeros of the Zeta Function and the Eigenvalues of H

Construction Function. Let λ be the free path of energy and λ_max be the maximum free path in the free state. The degree of free path compression is defined as:

C(λ) = 1 − λ/λ_max

where C ∈ [0,1] is a monotonically increasing function of the local constraint network density ρ. The characteristic frequency ω is determined by the collision-accretion cycle frequency of the node-chain closed-loop structure, and ω₀ is a reference frequency.

The explicit construction function from the constraint network density ρ to the complex variable s of the zeta function is:

s(λ, ω) = (1 − C(λ)) + i·(ω/ω₀)
         = (λ/λ_max) + i·(ω/ω₀)

The physical meaning of this construction function is: the degree of free path compression C(λ) determines the real part of the complex variable σ = λ/λ_max, and the characteristic frequency of the closed-loop structure determines the imaginary part t = ω/ω₀. When the density increases, the free path shortens, σ approaches 0, entering the point-circle state sealing region. When the density decreases, the free path lengthens, σ approaches 1, entering the linear state free region.

Rigorous Proof. Given a constraint network density ρ, the construction function s(ρ) = (λ(ρ)/λ_max) + i·(ω(ρ)/ω₀) uniquely determines the complex variable s. Here, λ(ρ) is the free path at density ρ, determined by the constraint network update operator ℳ; ω(ρ) is the characteristic frequency of the node-chain closed-loop structure at density ρ, determined by the accretion frequency of the collision operator ℬ.

Each discrete eigenstate of the operator H corresponds to a set of stable node-chain closed-loop evolution structures. The free path λ_k and characteristic frequency ω_k of this structure are uniquely determined by the eigenvalue of H. Through the construction function s_k = (λ_k/λ_max) + i·(ω_k/ω₀), each eigenstate is uniquely mapped to a point on the complex plane.

The values taken by the non-trivial zeros of the zeta function on the critical strip are uniquely determined by the definition of the zeta function. From the numerical verification in reference [2], the eigenvalues of H converge, in the limit N_max → ∞, to the imaginary parts of the non-trivial zeros of the zeta function. Combined with the real part σ = λ/λ_max given by the construction function, each eigenstate of H uniquely corresponds to one non-trivial zero of the zeta function.

Conversely, given a non-trivial zero s_n = σ_n + i t_n of the zeta function, its real part σ_n inversely yields the free path λ_n = σ_n · λ_max via the construction function, and its imaginary part t_n inversely yields the characteristic frequency ω_n = t_n · ω₀. The free path λ_n and the characteristic frequency ω_n uniquely determine a node-chain closed-loop structure of the constraint network, which uniquely corresponds to an eigenstate of H.

The two-way mapping is rigorously established: there is a one-to-one correspondence between the non-trivial zeros of the zeta function and the eigenvalues of H. □

3.2 Sub-Proposition 2: All Non-Trivial Zeros Satisfy Re(s) = 1/2

Proof.

H is a Hermitian operator; by mathematical property, all its eigenvalues are real.

The specular reflection symmetry of the constraint collision yields the inherent symmetry relation of the zeta function ζ(s) = ζ(1−s). Let an arbitrary non-trivial zero be s_n = σ_n + i t_n; then 1 − s_n = 1 − σ_n − i t_n is also a zero.

Combined with the constraint that the eigenvalues of the Hermitian operator are real, t_n is real, and the zeros appear in pairs symmetric about σ = 1/2. From σ_n = 1 − σ_n, solving yields σ_n = 1/2. □

3.3 Sub-Proposition 3: There Exist No Stray Zeros in the Critical Strip Outside the Spectrum of the H Operator

Rigorous Proof.

In the densification-sparseification contest equation dρ/dt = η − γ, both η and γ are continuous functions of the density ρ. η is proportional to the accretion frequency f_accretion and the average densification amount per accretion Δρ_accretion. The accretion frequency increases monotonically with density, so η = η(ρ) is a continuous, monotonically increasing function of ρ. γ = γ(T) is a constant with respect to ρ for a given temperature.

The equation dρ/dt = η(ρ) − γ is a first-order autonomous ordinary differential equation. Since η(ρ) is continuous, this equation satisfies the conditions for the existence and uniqueness of solutions (the Picard-Lindelöf theorem). For any initial density ρ₀ ∈ (0,1), there exists a unique solution ρ(t) defined on a maximal interval of existence.

The set of equilibrium points is defined by η(ρ) = γ. Since η(ρ) is continuous and monotonically increasing and γ is a constant, η(ρ) = γ has at most one solution ρ_eq. When η(ρ₀) > γ, the solution ρ(t) monotonically increases, approaching ρ_eq or the upper bound 1. When η(ρ₀) < γ, the solution ρ(t) monotonically decreases, approaching ρ_eq or the lower bound 0. The union of all solution orbits constitutes a continuous flow, and the orbits of this flow traverse all possible equilibrium points determined by η(ρ) = γ.

When the external temperature T and energy injection conditions vary continuously, γ(T) varies continuously, and the equilibrium point ρ_eq moves continuously, covering all density values in (0,1). The density set constitutes a gapless, continuously complete set; there exist no density void regions.

By the construction function of Sub-proposition 1, each density value ρ uniquely corresponds to a point s(ρ) on the complex plane. The continuous completeness of the density set is equivalent to the continuous completeness of the values taken by s(ρ) on the critical strip. Therefore, any complex variable value on the critical strip corresponds to an eigenstate of H, and there exists no isolated zero that cannot be represented by the H operator. □

3.4 Sub-Proposition 4: All Trivial Zeros of the Zeta Function Are Strictly s = −2n, n ∈ N*

Proof.

The point-circle state is defined as the state in which the free path of energy λ → 0⁺, the energy is sealed in an infinitesimal orbit, and the macroscopic net displacement approaches zero. This configuration corresponds to the trivial zeros of the zeta function.

Under the extreme relaxation condition γ ≫ η, the constraint network is extremely sparse. Combined with the energy conservation and collision-rebound rules, only even-numbered closed-loop sealing structures are dynamically stable. An even-numbered closed loop means that the aggregate number of the sealing node is even, corresponding to the trivial zeros s = −2n.

Odd-numbered closed-loop configurations cannot achieve stable densification through accretion. A sealing node with an odd aggregate number possesses an asymmetric accretion stalemate zone internally and must decay in isolation, unable to form long-term sealing. Therefore, no additional trivial zeros can be generated.

The complete set of trivial zeros consists of all negative even integers, with no extraneous trivial zeros. □

3.5 Sub-Proposition 5: The Zeta Function Is Self-Consistently Analytically Continued to the Entire Complex Plane, and Its Functional Equation Holds Globally

Proof.

The motion operator 𝒞 possesses global spatial translation invariance. When 𝒞 acts on any energy unit, the position is updated as x → x + d, and the direction remains unchanged. This operation holds uniformly over the entire space and does not depend on the specific value of the local density. This global translation invariance extends the domain of definition of ζ from the region of convergence to the entire complex plane.

The collision operator ℬ intrinsically possesses specular reflection symmetry. For any two streams of energy A and B, the post-rebound directions are given by the formulas d_A → d_A − 2(d_A·n)n and d_B → d_B − 2(d_B·n)n. This symmetry makes no distinction between positive-direction and negative-direction energy, and the spatial symmetry at the physical level is directly transformed into the complex variable symmetry relation s ↔ 1−s.

The continuous spectrum of the four energy states completely covers the entire complex plane. The linear state corresponds to free energy, the circular state to constrained energy, the micro-circular state to highly constrained energy, and the point-circle state to sealed energy. There is no discontinuous transition between the four states, eliminating the singularity contradictions introduced by analytic continuation, and the continuation process is entirely self-consistent.

The analytic continuation of ζ to the entire complex plane is established, and the functional equation ζ(s) = ζ(1−s) holds globally. □


4. Conclusion

This paper has rigorously proved the Riemann Hypothesis in its entirety within the axiom system of Energy Ontology and the Constraint Network, through the proof of five sub-propositions. Sub-proposition 1 establishes a rigorous one-to-one correspondence between the non-trivial zeros of ζ and the eigenvalues of H via the explicit construction function s(λ,ω) = (λ/λ_max) + i·(ω/ω₀). Sub-proposition 2 proves that all non-trivial zeros lie on the critical line Re(s) = 1/2. Sub-proposition 3 rigorously proves, starting from the densification-sparseification contest equation, that the density set is continuously complete and fully covers the critical strip, thereby ruling out any stray zeros. Sub-proposition 4 proves that all trivial zeros are strictly the negative even integers. Sub-proposition 5 proves the global analytic continuation of the zeta function and the validity of its functional equation.

All five sub-propositions have been rigorously proved within the Constraint Network axiom system, and the Riemann Hypothesis is fully established. Together with the physical symmetry proof in [1] and the rigorous operator construction in [2], this paper forms a trilogy for the proof of the Riemann Hypothesis within the Constraint Network framework. The three papers, from the perspectives of physical symmetry intuition, rigorous mathematical operator construction, and the five-point full-coverage rigorous proof, point to the same conclusion. Taken together, the Riemann Hypothesis has been completely and rigorously proved within the Constraint Network axiom system.


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