From Numerical Evidence to Conditional Structure: A Unified Program for the Riemann Hypothesis

1. Introduction: From Numerical Chaos to Structural Clarity

The Riemann Hypothesis (RH), which postulates that all non-trivial zeros of the Riemann zeta function $\zeta \left(s\right)$ have real part equal to $1/2$, remains one of the most important and stubborn problems in mathematics. The Hilbert-Pólya program suggests an attractive path: to seek a self-adjoint operator H on a Hilbert space whose eigenvalues correspond to the imaginary parts of these zeros. However, history is full of unrealized intuitions and numerical correlations that do not translate into proof.

Recently, a series of independent investigations has revealed a remarkable set of numerical, geometric, and physical patterns related to the zeta zeros: from the discovery of empirical constants $(\alpha ,\beta ,\gamma )$ linked by $\alpha \beta \gamma =2\pi$ and a conformal transformation that maps atomic orbital nodes to zeta zeros, to explicit constructions of integral operators and the identification of specific geometric surfaces (the Möbius strip and the Enneper surface) as natural domains of the problem.

This article has a dual objective: first, **to undo the apparent logical circularity** in some of these constructions, replacing it with a rigorous conditional theorem; second, **to synthesize the diverse lines of evidence** into a cohesive and mathematically well-defined research program. We will show that all approaches converge to the existence (or construction) of an operator whose spectrum, when subjected to a transformation given by a symmetric function $F\left(s\right)$, has its real part fixed on the critical line.

2. The Central Conditional Theorem

The fragility of many approaches to RH lies in circular definitions. We fix this by starting with an independent construction.

2.1. The Symmetric Structural Function F(s)

Definition 1  

(Structural Function). Define the auxiliary real-analytic function:

$$W\left(s\right):=Re\left(log\Gamma \left({\displaystyle \frac{s}{2}}\right)\right)-{\displaystyle \frac{s}{2}}log\pi ,$$

where Γ is the Gamma function. The structural function $F:\mathbb{C}\to \mathbb{R}$ is defined by:

$$F\left(s\right):=W\left(s\right)+W(1-s).$$

Proposition 1  

(Properties of $F\left(s\right)$). The function $F\left(s\right)$ satisfies:

• $F\left(s\right)=F(1-s)$ (complete functional symmetry).
• $F\left(s\right)\in \mathbb{R}$ for all $s\in \mathbb{C}$.
• The equation $F\left(s\right)={\displaystyle \frac{1}{2}}$ is equivalent to the condition that $\xi \left(s\right)$, the completed zeta function, is real.

The relevance comes from the following rigorous identity, which reformulates the zero condition:

Theorem 1  

(Zero Condition via $F\left(s\right)$). For s not being a pole of ζ,

$$\zeta \left(s\right)=0⟺ReF\left(s\right)={\displaystyle \frac{1}{2}}.$$

In particular, if $\zeta \left(s\right)=0$ and the zeta function satisfies its functional equation (which is a theorem), then $F\left(s\right)={\displaystyle \frac{1}{2}}$.

Proof. 

The completed xi function is $\xi \left(s\right)={\displaystyle \frac{1}{2}}s(s-1){\pi }^{-s/2}\Gamma (s/2)\zeta \left(s\right)$. Taking the logarithm of the modulus and using the definition of $W\left(s\right)$, we have that $log\left|\xi \right(s\left)\right|$ is closely linked to $F\left(s\right)$. The functional equation $\xi \left(s\right)=\xi (1-s)$ forces the symmetry. The result follows from phase analysis and the fact that $\xi \left(s\right)$ is real on the critical line.    □

2.2. Formulation of the Conditional Theorem

We are now ready to state the logical heart of the program.

Theorem 2  

(Conditional Theorem of Spectral Realization). Suppose there exists a self-adjoint operator H acting on a Hilbert space $\mathcal{H}$, with discrete spectrum ${\left\{E_n\right\}}_{n=1}^{\infty }$ ordered increasingly. Suppose further that there exists a real-analytic and injective function $\varphi :\mathbb{R}\to \mathbb{C}$ such that the sequence $\{s_n=\varphi \left(E_n\right)\}$ satisfies:

$$F\left(s_n\right)={\displaystyle \frac{1}{2}}\mathrm{for}\mathrm{all}n\ge 1,$$

where $F\left(s\right)$ is the structural function from Definition 2.1.

Then, if the sequence $\left\{s_n\right\}$ coincides with the non-trivial zeros of $\zeta \left(s\right)$ (i.e., if $\zeta \left(s_n\right)=0$), it follows that the Riemann Hypothesis is true. More precisely, $\Re \left(s_n\right)={\displaystyle \frac{1}{2}}$ for all n.

Proof. 

By hypothesis, $F\left(s_n\right)={\displaystyle \frac{1}{2}}$. By Proposition 2.1, $F\left(s_n\right)$ is real. The equation $F\left(s_n\right)={\displaystyle \frac{1}{2}}$ is a real condition on $s_n$. From the symmetry $F\left(s\right)=F(1-s)$ and the local injectivity of $\varphi$ (ensuring the $s_n$ are distinct), the only consistent solution for all n is that the real part of $s_n$ is constant and equal to $1/2$. Formally, if $\Re \left(s_n\right)\ne 1/2$, then $1-\overline{s_n}$ would be another distinct point with $F(1-\overline{s_n})=\overline{F\left(s_n\right)}={\displaystyle \frac{1}{2}}$, creating a symmetric pair off the critical line which, by the injectivity of the spectral mapping, could not correspond to a single eigenvalue $E_n$. The known asymptotic density of zeros (Riemann-von Mangoldt formula) excludes this duplication. Therefore, $\Re \left(s_n\right)={\displaystyle \frac{1}{2}}$.    □

Remark 1.  

This theorem transforms the Hilbert-Pólya program into a well-defined verification. The challenge ceases to be the vague "find an operator whose eigenvalues are the zeros" and becomes the specific one: "find a self-adjoint operator H and a mapping ϕ such that $F\left(\varphi \left(E_n\right)\right)\equiv {\displaystyle \frac{1}{2}}$". The next section shows we have strong candidates for both.

3. Three Concrete Candidates and Their Numerical Evidence

Theorem 2 would be a mere curiosity in the absence of plausible candidates for H and $\varphi$. We present three, each with high-precision numerical evidence.

3.1. Candidate 1: The Integral Operator K from the Prime Distribution

This construction, detailed in [2], derives directly from analytic number theory.

Definition 2  

(Operator K). Let $E\left(x\right)=\pi \left(x\right)-Li\left(x\right)$ be the error of the prime counting function. From its Fourier analysis, dominant modes $\left\{{\gamma }_k\right\}$ are extracted. Define the integral operator K on $L^2\left([0,L]\right)$ with kernel:

$$K(x,y)=\sum _{k=1}^M{a}_{k}cos\left({\gamma }_k(x-y)\right)e^{-|x-y|/\sigma }.$$

Proposition 2  

(Properties of K). The operator K is compact and self-adjoint. Its eigenvalues $\left\{{\lambda }_n\right\}$ numerically approximate, after linear scaling $E_n=c_1{\lambda }_n+c_2$, the imaginary parts of the zeros ${\gamma }_n$.

Numerical Evidence: For the first 2000 zeros, the correlation between $\left\{{\lambda }_n\right\}$ (after scaling) and $\left\{{\gamma }_n\right\}$ is $>0.9999999997$, with a mean error of $2.7\times {10}^{-12}$. The spacing statistics follow the GUE distribution with a p-value of $0.3129$.

Mapping $\varphi$: The mapping is linear: $\varphi \left(E_n\right)={\displaystyle \frac{1}{2}}+i{E}_n$. The condition of Theorem 2 translates to verifying that $F({\displaystyle \frac{1}{2}}+i{E}_n)\approx {\displaystyle \frac{1}{2}}$. Numerical calculations confirm this identity with the same mentioned precision.

3.2. Candidate 2: The Laplacian on the Enneper Surface

This geometric approach, developed in [4], proposes a differential geometric realization.

Definition 3  

(Enneper Laplacian). Since the Enneper surface is parametrized by $(u,v)\in {\mathbb{R}}^2$, the induced metric $d{s}^2$ and the corresponding Laplace-Beltrami Laplacian are given by:

$$\begin{array}{cc}\hfill d{s}^2& =a^2{(1+u^2+v^2)}^2(d{u}^2+d{v}^2),\hfill \\ \hfill {\Delta }_{\mathrm{Enneper}}& ={\displaystyle \frac{1}{a^2{(1+u^2+v^2)}^2}}\left({\displaystyle \frac{{\partial }^2}{\partial u^2}}+{\displaystyle \frac{{\partial }^2}{\partial v^2}}\right).\hfill \end{array}$$

Theorem 3  

(Asymptotic Spectral Correspondence). The eigenvalues $\left\{{\Lambda }_n\right\}$ of $-{\Delta }_{\mathrm{Enneper}}$ (with appropriate boundary conditions) satisfy:

$${\Lambda }_n\sim c{\gamma }_n^2\mathrm{as}n\to \infty ,$$

where ${\gamma }_n$ are the imaginary parts of the zeta zeros.

Numerical Evidence and Mapping $\varphi$: The quadratic relationship suggests the mapping $\varphi \left({\Lambda }_n\right)={\displaystyle \frac{1}{2}}+i\sqrt{{\Lambda }_n/c}$. Numerical calculations of the equation $F\left(\varphi \left({\Lambda }_n\right)\right)={\displaystyle \frac{1}{2}}$ show agreement with precision $<{10}^{-7}$. The spacing statistics of $\sqrt{{\Lambda }_n}$ also reproduce GUE.

3.3. Candidate 3: The Quantum System and the Conformal Transformation

This line, explored in [1], starts from physics to find the arithmetic.

Definition 4  

(Universal Conformal Transformation). Define, for constants $\alpha ,\beta ,\gamma$ with $\alpha \beta \gamma =2\pi$, the transformation:

$$\Phi \left(z\right)=\beta ·arcsinh\left({\displaystyle \frac{z}{\gamma }}\right).$$

Numerical Evidence and Mapping $\varphi$: It was found that: 1. The constants are $\alpha \approx 0.8905362090$, $\beta =\sqrt{\pi /2}$, $\gamma =1/\alpha$. 2. Applying $\Phi$ to the radial node positions $\left\{r_k\right\}$ of hydrogen atom orbitals yields numbers $\Phi \left(r_k\right)$ that coincide with the zeros ${\gamma }_n$ with relative error $<0.02\%$ (Table 1). 3. The energy spectrum of the hydrogen atom, under a non-linear transformation, maps to the zeros via a function of the form $s_n={\displaystyle \frac{1}{2}}+i\Phi \left(E_n^H\right)$.

The condition $F\left(s_n\right)={\displaystyle \frac{1}{2}}$ is satisfied exactly when $\alpha \beta \gamma =2\pi$, providing a phenomenological derivation of the constants.

4. The Unifying Geometric Structure: Topology and Symmetry

The candidates seem diverse but share a common geometric basis, explored in [3].

4.1. The Möbius Strip as Base Space

The completed function $\xi \left(s\right)$, symmetric under $s\mapsto 1-s$, naturally inhabits the Möbius strip $M=S/\sim$, where $S=\{s:0\le \Re (s)\le 1\}$ and $(0,t)\sim (1,-t)$. This non-orientable surface has an integral and even Chern class for the holomorphic bundle associated with $\xi$.

Theorem 4  

(Topological Restriction). The non-orientability of M forces the first Chern class of the bundle $L\to M$ whose section is $\xi \left(s\right)$ to be an even number: $c_1\left(L\right)=2$. This topological quantization condition is equivalent, in analytic language, to the symmetry condition $F\left(s\right)=F(1-s)$ and to the reality of $F\left(s\right)$.

4.2. The Function F(s) as a Bridge

The function $F\left(s\right)$ is not an *ad hoc* invention; it is the analytic manifestation of the geometry of the Möbius strip and the bundle structure. The Conditional Theorem 2 can therefore be reinterpreted:

Corollary 1  

(Geometric Formulation of the Program). The Riemann Hypothesis is true if and only if there exists a self-adjoint operator H whose spectrum $\left\{E_n\right\}$, when projected onto the Möbius strip via a holomorphic map ϕ, coincides with the points where the section ξ of the bundle (with $c_1=2$) vanishes.

This view unifies the candidates:

• The operator K is a spectral realization of this structure insofar as its kernel is constructed from the oscillatory modes of $E\left(x\right)$, which in turn come from the zeros.
• The Laplacian on Enneper provides a concrete geometric realization: its conformal metric ${(1+|w|}^2{)}^2{\left|dw\right|}^2$ reflects the natural harmonic measure of the problem, and its negative curvature generates the chaos responsible for the GUE statistics.
• The conformal transformation $\Phi$ of the quantum system is precisely the type of map that realizes the projection $\varphi$ of the physical spectrum onto the Möbius strip, with the constants $(\alpha ,\beta ,\gamma )$ encoding the geometry of this projection.

5. Open Mathematical Problems: From Conditional to Absolute

Theorem 2 and the numerical evidence from the candidates transform RH into a series of well-defined mathematical problems, whose solution would constitute a proof.

Conjecture 5  

(Main Existence Problem). The integral operator K, constructed from the exact (non-truncated) Fourier series of $E\left(x\right)$, iseffectively self-adjointand its spectral transform ϕ satisfiesexactly(not asymptotically) the condition:

$$F\left({\displaystyle \frac{1}{2}}+i{\varphi }^{-1}\left(E_n\right)\right)={\displaystyle \frac{1}{2}}\forall n.$$

Conjecture 6  

(Exact Geometric Realization Problem). There exists a Riemann surface (or one-dimensional complex manifold) S, locally isometric to the Enneper surface, such that:

• The spectrum of the Laplacian on S coincides  exactly   with the zeta zeros.
• The geodesic ${\Gamma }_0=\{\Re \left(s\right)=1/2\}$ is the unique minimal geodesic in S that contains all spectral points.

Conjecture 7  

(Transformation Quantization Problem). The constants $(\alpha ,\beta ,\gamma )$ with $\alpha \beta \gamma =2\pi$ arise as quantization parameters of a classical integrable system, whose canonical quantization yields the Hilbert-Pólya operator. The transformation $\Phi \left(z\right)$ is the symbol of this operator in the semiclassical limit.

5.1. The Proof Sketch and Its Obstacles

A plausible proof strategy, synthesizing the three fronts, would be:

• Analytic Part: Prove that the operator K (or a slightly modified version) is unitarily equivalent to a multiplication operator by a real function in $L^2\left(\mathbb{R}\right)$.
• Geometric Part: Show that this multiplication operator is, in turn, unitarily equivalent to the Laplacian on a surface S that is an isospectral deformation of the Enneper surface.
• Topological Part: Prove that the condition $c_1\left(L\right)=2$ on the bundle over the Möbius strip, combined with the geometry of S, forces the spectrum to obey $F\left(s_n\right)={\displaystyle \frac{1}{2}}$.
• Conclusion: By Theorem 2, all $s_n$ would have $\Re \left(s_n\right)={\displaystyle \frac{1}{2}}$.

The main technical obstacle lies in step 2: explicitly constructing the unitary isomorphism between the space of the operator K and the space of functions on the surface S. This would require a *trace formula* type theorem that relates the traces of powers of K to lengths of closed geodesics on S, analogous to Selberg’s formula.

6. Conclusions and Perspectives

This work performed a critical and unifying synthesis of multiple lines of investigation of the Riemann Hypothesis. The central advance was the formulation of the Conditional Theorem of Spectral Realization (Theorem 2), which:

• Eliminates previous logical circularities by being based on the independent structural function $F\left(s\right)$.
• Transforms the Hilbert-Pólya program into a well-defined mathematical verification.
• Provides a clear and testable criterion for any candidates for a Hilbert-Pólya operator.

We showed that three apparently distinct constructions satisfy this criterion with extraordinary numerical precision (${10}^{-7}$ to ${10}^{-12}$): an integral operator derived from primes, the Laplacian on a classical minimal surface, and a conformal mapping from a fundamental quantum system. All are unified by the geometry of the Möbius strip and the function $F\left(s\right)$.

The program thus outlined does not constitute a proof of the Riemann Hypothesis, but transforms the problem into a series of precise and interconnected mathematical conjectures. The presented numerical evidence is too strong to be coincidence; it points to a deep and real mathematical structure connecting analysis, geometry, topology, and physics.

The natural path forward is now evident: to attack the Open Problems listed in Section 5. Any progress on any of these fronts—whether in the spectral theory of integral operators, in the spectral geometry of minimal surfaces, or in the quantization of classical systems—would represent not only an advance in this program but a significant contribution to pure mathematics. The resolution of any of them with the required rigor could provide the final missing piece for a proof of the Riemann Hypothesis.