Spectral-Dimension Obstructions for Operators with Superlinear Counting Laws

1. Problem Statement

The paper addresses a fundamental problem in the search for a Hilbert–Pólya operator for the Riemann Hypothesis. The Hilbert–Pólya conjecture posits that the nontrivial zeros of the Riemann zeta function, $\zeta(s)$, correspond to the eigenvalues of a self-adjoint operator.

The authors investigate whether such an operator can be constructed within two broad, natural universality classes:

1. Arithmetic Single-Valuation Class: Operators derived from entropy-maximizing exponential kernels on a logarithmic prime axis (a "single-valuation" framework).
2. Geometric Elliptic Class: Classical local elliptic differential operators defined on finite-dimensional compact manifolds.

The core question is whether an operator can simultaneously satisfy the spectral growth laws required by the Riemann zeros (superlinear counting) and the structural constraints imposed by these two classes. The paper proves that no such operator exists, establishing a "spectral-dimension obstruction."

2. Methodology

The authors employ a combination of variational analysis, spectral theory, Tauberian theorems, and Mosco convergence to analyze the small-time heat-trace asymptotics ($\Theta(t) = \text{Tr}(e^{-tH})$) of different operator classes.

• Variational Principle (Entropy Maximization): They formulate a variational problem for transition kernels on a countable label set (primes) subject to a linear coherence constraint. They prove that maximizing Shannon entropy forces the kernel to take a specific exponential (Gibbs) form.
• Continuum Limit via Mosco Convergence: They analyze the discrete arithmetic Laplacians generated by these kernels. By embedding the discrete system into a continuum space ($L^2(\mathbb{R}, du)$) and applying Mosco convergence, they derive the limiting differential operator.
• Heat-Kernel Asymptotics: They calculate the short-time behavior of the heat trace for the resulting operators.
• Tauberian Analysis: For the Riemann zeros, they use the Riemann–von Mangoldt formula and the Hardy–Littlewood–Karamata Tauberian theorem to invert the counting function $N(\lambda)$ into the heat trace $\Theta(t)$.
• Invariance Arguments: They utilize the stability of spectral dimension and leading heat-trace exponents under unitary equivalence and compact perturbations (Weyl's law) to show that these spectral signatures are structural invariants.

3. Key Contributions and Results

A. The Exponential Kernel Principle and the Arithmetic Obstruction

• Theorem 1.1 (Exponential Kernel Principle): Under mild regularity and integrability assumptions, the unique solution to the entropy-maximization problem for transition kernels on a label set is necessarily of the exponential form $T_{ij} = e^{-\beta \Delta_{ij}} / Z(\beta)$.
• Arithmetic Specialization: When applied to the logarithmic separation of prime numbers ($\Delta_{ij} = |\log p_i - \log p_j|$), the authors construct discrete arithmetic Laplacians.
• Continuum Limit (Theorem 1.2): Using Mosco convergence, they show that the continuum limit of these discrete operators is a biharmonic operator ($H_A = c_4 \partial_u^4$) on the real line.
• Spectral Dimension Result: The heat trace of this operator behaves as $\Theta_{H_A}(t) \sim C t^{-1/4}$ as $t \to 0$. This yields a spectral dimension of:
  $$d_s(H_A) = 2 \times (1/4) = 1/2$$
  This is a strictly fractional spectral dimension arising from a fourth-order operator.

B. The Geometric Obstruction (Seeley–DeWitt Class)

• Classical Behavior: For any elliptic differential operator of order $m$ on a compact $d$-dimensional manifold, the heat trace follows the Seeley–DeWitt expansion:
  $$\text{Tr}(e^{-tH}) \sim t^{-d/m} \sum a_k t^{k/m}$$
• Incompatibility: While the numerical value $d_s = 1/2$ is theoretically possible for high-order elliptic operators (e.g., $m=4d$), the authors argue that the arithmetic operator $H_A$ acts on a non-compact space with a continuous spectrum, whereas geometric elliptic operators on compact manifolds have discrete spectra. Thus, they cannot be unitarily equivalent.

C. The Analytic Obstruction (Riemann Zeros)

• Riemann–von Mangoldt Law: The counting function for the ordinates of the nontrivial zeros of $\zeta(s)$ is $N_\zeta(T) \sim \frac{T}{2\pi} \log T$.
• Tauberian Inversion (Proposition 6.1): Applying Tauberian theorems to this counting law forces the heat trace of any operator realizing these zeros to behave as:
  $$\Theta_{\zeta}(t) \sim \frac{\log(1/t)}{2\pi t} \quad \text{as } t \to 0$$
• Spectral Dimension: This corresponds to a spectral dimension of $d_s = 2$, but crucially, it includes an unavoidable logarithmic correction ($\log(1/t)$).
• Exclusion: This asymptotic form (power law with a slowly varying logarithmic factor) is incompatible with the pure power-law expansions of the Seeley–DeWitt class (geometric operators) and the pure power-law $t^{-1/4}$ of the arithmetic class.

D. The Main No-Go Theorem (Double Hilbert–Pólya Obstruction)

• Theorem 1.5 / Theorem 7.3: No self-adjoint operator $H$ can simultaneously satisfy:
  1. Being a limit of single-valuation exponential kernel constructions (implying $d_s = 1/2$ and pure power-law decay).
  2. Having a spectrum coinciding with the Riemann zeros (implying $d_s = 2$ with a logarithmic correction).
• Conclusion: The arithmetic scaling ($t^{-1/4}$) and the zeta scaling ($t^{-1}\log(1/t)$) are mutually exclusive. Therefore, a Hilbert–Pólya operator cannot belong to the single-valuation arithmetic class nor the standard geometric elliptic class.

4. Significance

• Ruling Out Natural Candidates: The paper rigorously eliminates two of the most promising avenues for constructing a Hilbert–Pólya operator: simple arithmetic models based on prime logarithms and standard geometric models on compact manifolds.
• Structural Rigidity: It demonstrates that spectral dimension and the specific form of heat-trace asymptotics (presence or absence of logarithmic corrections) are robust invariants that cannot be bridged by unitary equivalence or compact perturbations.
• Guidance for Future Research: The authors suggest that a viable Hilbert–Pólya operator must lie outside these established universality classes. They propose that future candidates must involve:
  • Multiple valuation directions (beyond single-valuation).
  • Genuinely nonlocal analytic structures (e.g., operators with symbols like $f(\Delta) \sim \Delta / \log \Delta$).
  • Quantum graph models with specific edge length distributions (e.g., $\log p$) that naturally organize prime contributions.

In summary, the paper establishes a "spectral obstruction" proving that the spectral geometry required to encode the Riemann zeros is fundamentally distinct from both the geometry of compact manifolds and the arithmetic geometry of single-valuation prime kernels.