Introduction and Problem Formulation

The Hilbert–Pólya program seeks a self-adjoint operator whose spectrum encodes the imaginary parts of the non-trivial zeros of the Riemann zeta function. Let $$\begin{align} \Lambda(s) &\coloneqq \pi^{-s/2}\Gamma(s/2)\zeta(s),\\ \xi(s) &\coloneqq \frac12 s(s-1)\Lambda(s),\\ \Xi(E) &\coloneqq \xi\left(\frac12+iE\right). \end{align}$$ A spectral realization would have the schematic form $$\begin{equation} \Xi(E) = C\det\nolimits_{\mathrm{reg}}(E-H), \qquad H=H^*, \end{equation}$$ with zeros of $\Xi$ corresponding to spectral values of $H$.

If such a realization exists with the determinant identity established independently of the zeros, then $$\begin{equation} H=H^* \quad\Longrightarrow\quad \operatorname{Spec}(H)\subset\mathbb{R}, \end{equation}$$ and hence every non-trivial zero would have the form $$\begin{equation} \rho=\frac12+i\gamma, \qquad \gamma\in\mathbb{R}. \end{equation}$$ Thus self-adjointness supplies the final spectral implication required for the Riemann Hypothesis, but does not by itself construct the operator or establish the determinant identity.

In the mathematical-physics literature, related approaches have included local Schrödinger operators $$\begin{equation} H=-\frac{d^2}{dx^2}+V(x), \end{equation}$$ the semiclassical dilation Hamiltonian $$\begin{equation} H=\frac12(xp+px), \end{equation}$$ and hypothetical decompositions $$\begin{equation} H\stackrel{?}{=}\bigoplus_{p\in\mathbb P}H_p. \end{equation}$$ The purpose of this paper is not to rule out every conceivable Hilbert–Pólya construction, but to identify precise obstructions to simple local or finite reductions and to separate established harmonic architecture from the genuinely open dynamical step.

Local and Finite Obstructions

The Physical-Sheet Barrier

For a prime $p$, write $$\begin{equation} D_p(E) \coloneqq 1-p^{-1/2-iE}. \end{equation}$$ Its zeros satisfy $p^{-1/2-iE}=1$, hence $$\begin{equation} E_k = \frac{2\pi k}{\log p} +\frac{i}{2}, \qquad k\in\mathbb{Z}. \end{equation}$$ Thus all these zeros lie strictly in the upper half-plane.

The conclusion is deliberately limited to the physical perturbation determinant. It does not prohibit the meromorphic continuation of scattering quantities. Resonances are poles or zeros of analytically continued objects on non-physical sheets and therefore are not counterexamples to the proposition.

The Divisor Obstruction for Finite Euler Products

Let $S=\{p_1,\ldots,p_N\}\subset\mathbb P$ be a non-empty finite set of primes, and define $$\begin{equation} P_S(s) \coloneqq \prod_{p\in S}(1-p^{-s})^{-1}. \end{equation}$$ Its reciprocal $P_S(s)^{-1}=\prod_{p\in S}(1-p^{-s})$ is entire, so $P_S(s)$ has no zeros in $\mathbb{C}$. Its poles are located at $$\begin{equation} s_{p,k} = \frac{2\pi i k}{\log p}, \qquad p\in S,\quad k\in\mathbb{Z}, \end{equation}$$ all lying strictly on the imaginary axis $\Re(s)=0$.

Primitive Arithmetic Lengths and Their Iterates

The Euler product imposes a distinction essential for any periodic-orbit interpretation. The primitive arithmetic datum is a prime $p$, while the integer $k\geq1$ indexes repeated traversal of that primitive datum.

For $\Re(s)>1$, $$\begin{equation} \log\zeta(s) = \sum_{p\in\mathbb P} \sum_{k\geq1} \frac{p^{-ks}}{k}. \end{equation}$$ Differentiation gives $$\begin{equation} -\frac{\zeta'(s)}{\zeta(s)} = \sum_{p\in\mathbb P} \sum_{k\geq1} (\log p)p^{-ks}. \end{equation}$$ This formula has the formal structure of a primitive-orbit expansion with primitive arithmetic length $L_p=\log p$ and iterated lengths $L_{p,k}=kL_p=k\log p$.

Obstructions to Classical Geometric Prime-Orbit Realizations

Before considering infinite-dimensional or non-commutative frameworks, we investigate whether the primitive arithmetic lengths $L_p = \log p$ can be realized as the closed-orbit spectrum of an autonomous flow on a classical smooth geometric space.

Cross-Section and Suspension Flows

Hyperbolic Geodesic Flows

For geodesic flows on the unit tangent bundle of a hyperbolic surface, periods are not homological and the preceding proposition does not apply. Nevertheless, an algebraic number-theoretic obstruction rules out finite-area hyperbolic surfaces.

The Kinematic Foundation: Adelic Harmonic Analysis

Tate’s Thesis

The adelic ring is $\mathbb{A}_\mathbb{Q}= \mathbb{R}\times\prod_p'\mathbb{Q}_p$. For the standard factorized Schwartz–Bruhat function $f=f_\infty\otimes\bigotimes_p f_p$ with $f_\infty(x)=e^{-\pi x^2}$ and $f_p=\mathbf 1_{\mathbb{Z}_p}$, the local Tate zeta integrals are $$\begin{align} Z_\infty(f_\infty,s) &= \int_{\mathbb{R}^\times} e^{-\pi x^2}|x|^s\,d^\times x = \pi^{-s/2}\Gamma(s/2), \\ Z_p(f_p,s) &= \int_{\mathbb{Q}_p^\times} \mathbf 1_{\mathbb{Z}_p}(x)|x|_p^s\,d^\times x = \frac{1}{1-p^{-s}}. \end{align}$$ Consequently, $Z(f,s) = \pi^{-s/2}\Gamma(s/2)\zeta(s) = \Lambda(s)$.

Poisson summation and Fourier self-duality yield the functional equation $\Lambda(s)=\Lambda(1-s)$. The entire function $\xi(s)$ is obtained by removing the simple poles at $s=0$ and $s=1$ via $\frac12s(s-1)$. This is an established harmonic-analytic realization of the functional equation; it is not, by itself, a construction of a Hilbert–Pólya Hamiltonian.

The Weil Representation and Theta Correspondence

A second established structure is supplied by the Weil representation. Let $V$ be a non-degenerate rational quadratic space. The associated metaplectic representation provides theta kernels through the reductive dual-pair framework $(SL_2,O(V))$.

The Motivic Layer: Periods and Cyclotomic Expansions

Odd zeta values and special formulas such as BBP belong to a motivic and period-theoretic layer. In the theory of mixed Tate motives over $\mathbb{Z}$, the motivic Galois structure organizes odd zeta values through the weight grading. The period map relates motivic objects to numerical periods $\zeta(3),\zeta(5),\ldots$

BBP as a Coordinate Expansion of a Period

The Bailey–Borwein–Plouffe formula $$\begin{equation} \pi = \sum_{n=0}^\infty \frac{1}{16^n} \left( \frac{4}{8n+1} - \frac{2}{8n+4} - \frac{1}{8n+5} - \frac{1}{8n+6} \right) \end{equation}$$ is an efficient base-$16$ expansion of a period expressible through polylogarithmic integrals at cyclotomic points. It serves as an arithmetic coordinate projection of a period, but there is no theorem identifying BBP with a spectral determinant or with the $D_4/E_8$ theta correspondence.

The Explicit Formula as the Dynamical Target

Normalization of the Dynamical Generator

To compare Deninger’s skew-adjoint formulation with Hilbert–Pólya’s self-adjoint formulation, introduce a formal generator $\Theta$ satisfying $\Theta\psi_\rho = \left(\rho-\frac12\right)\psi_\rho$. If $\rho=\frac12+i\gamma$, then $\Theta\psi_\rho=i\gamma\psi_\rho$. Thus the desired operator relation is $\Theta^*=-\Theta$. Equivalently, $H\coloneqq-i\Theta$ satisfies $H^*=H$.

The Weil Explicit Formula

Let $h$ be a regular test function. The Weil explicit formula has the schematic form $$\begin{equation} \begin{aligned} \sum_\rho h\!\left(\frac{\rho-\frac12}{i}\right) ={}& \text{Archimedean contribution} \\ &- \sum_{p\in\mathbb P} \sum_{k\geq1} \frac{\log p}{p^{k/2}} \widehat h \left( \frac{k\log p}{2\pi} \right) + \text{pole/trivial-zero terms}. \end{aligned} \end{equation}$$ This formula constitutes the design specification for any proposed autonomous arithmetic dynamics: primitive arithmetic lengths $L_p=\log p$, iterates $k\log p$, and a relative minus sign on the arithmetic side.

Automorphic Scattering and the Unphysical-Sheet Complement

The physical-sheet obstruction of Proposition 2.1 is unconditional. Scattering theory does not invalidate it; rather, scattering theory provides a distinct analytic mechanism in which a self-adjoint operator has a real physical spectrum while its meromorphically continued scattering data possess complex poles and zeros.

For the modular surface $X=SL(2,\mathbb{Z})\backslash\mathbb{H}$, the Eisenstein scattering coefficient is $$\begin{equation} \phi(s) = \sqrt{\pi}\, \frac{c(s)}{c(1-s)} = \sqrt{\pi}\, \frac{\Gamma(s-\frac12)\zeta(2s-1)} {\Gamma(s)\zeta(2s)}. \end{equation}$$ Its logarithmic derivative contains $$\begin{equation} \frac{\phi'(s)}{\phi(s)} = \frac{\Gamma'}{\Gamma}\left(s-\frac12\right) - \frac{\Gamma'}{\Gamma}(s) + 2\frac{\zeta'}{\zeta}(2s-1) - 2\frac{\zeta'}{\zeta}(2s). \end{equation}$$

The minus sign in the second zeta logarithmic derivative is already present in a genuine relative/scattering quantity. However, the poles and zeros of the meromorphically continued scattering coefficient are not physical-sheet eigenvalues; they arise after analytic continuation across the continuous-spectrum boundary. Consequently: $$\begin{equation} \boxed{ \text{physical-sheet self-adjointness} \neq \text{meromorphic scattering continuation}. } \end{equation}$$

Candidate Sign Mechanisms and the Non-Necessity of Super-Trace

The negative prime contribution in the explicit formula is incompatible with the simplest interpretation as an ordinary positive scalar periodic-orbit trace. However, the explicit formula does not prove that a super-trace is mathematically necessary.

Automorphic Scattering as a Non-Supertrace Sign Mechanism

In the Selberg trace formula for $SL(2,\mathbb{Z})\backslash\mathbb{H}$, the continuous spectrum contribution to the relative trace is governed by the scattering phase shift: $$\begin{equation} \operatorname{Tr}_{\mathrm{rel}}(h(\Delta) - h(\Delta_0)) = \frac{1}{4\pi} \int_{-\infty}^\infty h(r) \left( -\frac{\phi'}{\phi}\left(\frac12+ir\right) \right) dr. \end{equation}$$ Evaluating $-\frac{\phi'}{\phi}(s)$ yields the term $+2\frac{\zeta'}{\zeta}(2s)$. In established automorphic trace theory (e.g., Venkov , Iwaniec ), moving the integration contour into the half-plane of absolute convergence $\Re(2s)>1$ and inserting the Dirichlet series $$\begin{equation} \frac{\zeta'(w)}{\zeta(w)} = -\sum_{p\in\mathbb P}\sum_{k\ge 1}(\log p)p^{-kw} \end{equation}$$ produces prime-power contributions carrying an overall negative sign.

This demonstrates an essential structural principle:

1. The underlying Hilbert space $L^2(SL(2,\mathbb{Z})\backslash\mathbb{H})$ is purely scalar, with no $\mathbb{Z}_2$-grading.

2. The operator $\Delta$ is standard non-negative self-adjoint.

3. The negative arithmetic sign arises from the fact that $\zeta(2s)$ resides in the denominator of the scattering coefficient $\phi(s)$, which combines with the orientation sign of the scattering phase shift.

Comparison of Candidate Mechanisms

Several mechanisms can in principle generate the required sign:

1. Relative Trace / Scattering Phase: $\operatorname{Tr}_{\mathrm{rel}}(e^{t\Theta} - e^{t\Theta_0})$ where arithmetic $L$-factors reside in the denominator of a scattering matrix.

2. Super-Trace: A $\mathbb{Z}_2$-graded space $\mathcal H = \mathcal H^0 \oplus \mathcal H^1$ with $\operatorname{STr}(e^{t\Theta}) = \operatorname{Tr}_{\mathcal H^0}(e^{t\Theta}) - \operatorname{Tr}_{\mathcal H^1}(e^{t\Theta})$.

3. Alternating Determinants: An alternating determinant $\det\nolimits_{\mathrm{reg}}(s-\Theta\mid H^0)^{-1}\det\nolimits_{\mathrm{reg}}(s-\Theta\mid H^1)$, as in the finite-field Lefschetz formula.

The Analytic Domain Problem

A formal vector field or derivation is not enough for a Hilbert–Pólya construction. A genuine generator must be defined on a Hilbert space $\mathcal H$ with a specified dense domain $\mathcal D(\Theta)\subset\mathcal H$ and must satisfy essential skew-adjointness: $\overline{\Theta}=\Theta^*$. One must specify: (1) the underlying space; (2) its measure structure; (3) the domain $\mathcal D(\Theta)$; (4) leaf boundary behavior; (5) self-adjointness/skew-adjointness proof; and (6) trace and determinant regularization schemes.

The Non-Circularity Criterion

The required logical direction is: $$\begin{equation} \boxed{ \text{geometry / representation / dynamics} \Longrightarrow \text{trace identity} \Longrightarrow \xi(s). } \end{equation}$$

The Function-Field Model

The strongest rigorous test of the architectural requirements occurs over finite fields. Let $X/\mathbb{F}_q$ be a smooth, projective, geometrically connected curve of genus $g$. Let $V_\ell = H^1_{\acute{e}t}(X_{\overline{\mathbb{F}_q}},\mathbb{Q}_\ell)$ and let $F=\operatorname{Frob}_q^*$ denote the chosen Frobenius action. The Grothendieck–Lefschetz trace formula yields $$\begin{equation} Z_X(T) = \frac{\det(1-TF\mid V_\ell)}{(1-T)(1-qT)}. \end{equation}$$

Normalized Frobenius

The Weil bounds give $|\alpha_j|=q^{1/2}$ for each eigenvalue of $F$ on $V_\ell$. Define the normalized Frobenius $$\begin{equation} U \coloneqq q^{-1/2}F. \end{equation}$$ Its eigenvalues satisfy $|\lambda_j|=1$. Equipped with an appropriate Hermitian inner product, $U$ is unitary: $$\begin{equation} U^*U=I, \qquad \operatorname{Spec}(U)\subset S^1. \end{equation}$$

Alternating Sign and the Odd Cohomological Sector

The middle cohomological degree carries the negative sign: $$\begin{equation} \#X(\mathbb{F}_{q^n}) = \operatorname{Tr}(F^n\mid H^0) - \operatorname{Tr}(F^n\mid H^1) + \operatorname{Tr}(F^n\mid H^2). \end{equation}$$ This is the Euler–Poincaré sign $(-1)^i$ built into the Lefschetz trace formula. Equivalently, the zeta function is the alternating determinant: $$\begin{equation} Z_X(T) = \prod_{i=0}^2 \det(1-TF\mid H^i)^{(-1)^{i+1}}. \end{equation}$$

The Missing Arrow

The established structures and the open Dynamical Arrow are summarized: $$\begin{equation} \boxed{ \begin{array}{ccccc} \text{Adelic Fourier Duality} & \longrightarrow & \text{Tate }L\text{-function} & \longrightarrow & \Lambda(s) \\[2mm] \downarrow && && \downarrow \\[2mm] \text{Weil Representation} & \longrightarrow & \text{Theta Correspondence} & \longrightarrow & \Theta_L \\[3mm] && \Downarrow\ \text{\bf OPEN} \\[-1mm] && \text{Autonomous Arithmetic Flow }\Theta \\[2mm] && \Downarrow\ \text{\bf OPEN} \\[-1mm] && \text{Relative / Index / Scattering Trace} \\[2mm] && \Downarrow\ \text{\bf OPEN} \\[-1mm] && \xi(s). \end{array}} \end{equation}$$

The Five Conditions Recast Precisely

A candidate autonomous arithmetic system should satisfy: $$\begin{align} &\Theta^*=-\Theta, \tag{A}\\ &\operatorname{Spec}(\Theta) \text{ encodes } \rho-\frac12, \tag{B}\\ &\text{primitive arithmetic periods are } L_p=\log p, \text{ with iterates }L_{p,k}=k\log p, \tag{C$_{\rm arith}$}\\ &\operatorname{Tr}_{\mathrm{rel}}, \text{ an index, super-trace, or scattering determinant recovers the explicit formula}, \tag{D}\\ &\det\nolimits_{\mathrm{reg}} \left(s-\frac12-\Theta\right) \propto \xi(s). \tag{E} \end{align}$$

Historical Paradigms

1. Berry–Keating and Local Hamiltonians: The semiclassical Hamiltonian $H=xp$ reproduces the smooth average counting law $\overline{N}(E) \sim \frac{E}{2\pi}\left(\log\frac{E}{2\pi}-1\right)$, but lacks an autonomous mechanism for arithmetic prime fluctuations. While standard one-dimensional Schrödinger operators impose rigid asymptotic spectral constraints, no accepted independent differential realization is known that non-circularly produces the zeros together with their arithmetic structure.

2. Connes and Meyer (Adèle Class Space): Connes’ noncommutative geometry on $\mathbb{A}_\mathbb{Q}/\mathbb{Q}^\times$ recovers the Weil explicit formula as an absorption-spectrum trace formula. Meyer realized this within a virtual representation of the idèle class group whose distribution character recovers the explicit formula. This provides a rigorous representation-theoretic framework, but it remains a virtual representation rather than a conventional discrete self-adjoint Hilbert–Pólya Hamiltonian.

3. Deninger’s Cohomological Program: Deninger proposes a foliated dynamical space where primes correspond to closed orbits and an arithmetic Lefschetz trace formula yields the explicit formula. In the formulation of this program, the prospective cohomological spaces carrying the infinitesimal generator $\Theta$ are intrinsically infinite-dimensional real pre-Hilbert spaces. The construction of the underlying space, its foliated measure, and its closed operator domain remain open.

4. de Branges’ Positivity Route and Conrey–Li: De Branges proposed an entire-function Hilbert-space framework $\mathcal{H}(E)$ wherein specific positivity conditions would force the zeros of $L$-functions to lie on the critical line. Conrey and Li proved that the specific positivity condition required by this version of the de Branges program fails for the Riemann zeta function and Dirichlet $L$-functions.

5. Automorphic Scattering: Scalar automorphic scattering ($SL(2,\mathbb{Z})\backslash\mathbb{H}$) demonstrates that relative traces and scattering phase shifts can rigorously generate negative prime-power contributions without requiring a $\mathbb{Z}_2$-graded super-trace. Non-real scattering poles occur as resonances on analytically continued unphysical sheets, preserving self-adjointness of the physical Laplacian while leaving the Hilbert–Pólya operator problem open.

Scope and Limitations

1. No new proof of the Riemann Hypothesis is claimed.

2. No explicit Hilbert–Pólya operator is constructed.

3. Tate’s thesis provides the functional equation, not a discrete spectrum.

4. Weil representation does not identify $\xi(s)$ with lattice theta series.

5. Euler products supply arithmetic lengths $k\log p$, but an autonomous flow realizing them as geometric periods remains unconstructed.

6. Geometric obstructions rule out cross-section flows with $\omega(X)\equiv 1$ and finite-area hyperbolic surfaces, but do not rule out all smooth flows.

7. 1D local Schrödinger models face strong structural spectral constraints; no accepted independent realization reproducing the zeros is known.

8. Automorphic scattering zeros are resonances, not physical eigenvalues.

9. A super-trace is not logically necessary; scalar scattering yields negative signs.

10. Characteristic-$p$ Frobenius verifies consistency, not characteristic-zero dynamics.

11. Any proposed generator requires a specified dense Hilbert domain.

Falsification Boundaries and Residual Open Problems

The obstruction results permit a sharper classification of the architectural questions surrounding a Hilbert–Pólya realization. It is vital to distinguish:

1. a proposed construction is impossible under specified hypotheses;

2. a particular mechanism cannot establish the desired spectral interpretation;

3. the underlying Hilbert–Pólya problem itself is impossible.

Only the first two conclusions are established by the obstruction results.

Classical Flows: Precise Obstruction Classes

The prime Euler product determines the primitive arithmetic lengths $L_p = \log p$, with iterates $L_{p,k}=k\log p$. The homological obstruction of Section 4 proves that a compact cross-section flow with $\omega(X)\equiv 1$, $d\omega=0$, cannot have all $\log p$ as primitive periods because all periods lie in a subgroup of finite $\mathbb{Q}$-rank, whereas $\{\log p\}_{p\in\mathbb P}$ has infinite $\mathbb{Q}$-rank.

Likewise, the algebraic obstruction rules out finite-area hyperbolic surfaces. These are class-specific no-go theorems, not universal theorems about all smooth flows.

Locally Symmetric Spaces

Scattering Does Not Evade the Physical-Sheet Obstruction

Automorphic scattering on $L^2(SL(2,\mathbb{Z})\backslash\mathbb{H})$ operates on the physical Hilbert space with real spectrum $\operatorname{Spec}(\Delta)\subset\mathbb{R}_{\ge 0}$. The resonances (zeros and poles of $\phi(s)$) occur on the unphysical sheet via meromorphic continuation. Scattering demonstrates that a negative arithmetic contribution in a relative trace does not require a $\mathbb{Z}_2$-graded super-trace, but it does not construct a physical-sheet Hilbert–Pólya operator.

The Determinant Problem

An identity $\det\nolimits_{\mathrm{reg}}\left(s-\frac12-\Theta\right) = C\xi(s)$ is meaningful only after the regularization genus, operator domain, and multiplicities are specified. While regularized product formulas for the non-trivial zero set are established (Illies ), finding an autonomous operator $\Theta$ remains the central open hurdle.

Conditions (A)–(E) and the Status of RH

If an operator $\Theta$ non-circularly satisfies (A) $\Theta^*=-\Theta$ and (B) $\operatorname{Spec}(\Theta)$ encodes $\rho-\frac12$, it would imply $\Re(\rho)=\frac12$ (RH). Conversely, the discovery of a single off-line zero $\rho_0$ with $\Re(\rho_0)\neq 1/2$ would definitively rule out the simultaneous realization of (A) and (B) in this form.

Authoritative Epistemic Classification

Table 1 summarizes the controlling epistemic status of candidate mechanisms and open problems.

Conclusion

The architectural boundary of the Hilbert–Pólya program can now be stated with precision:

First, a substantial portion of the harmonic-analytic kinematics is established. Adelic Fourier duality gives the Tate zeta integral and the functional equation, while the Weil representation and theta correspondence organize a broad family of lattice theta series including those of $D_4$, $E_8$, and the Leech lattice. These structures represent parallel manifestations of self-dual harmonic analysis rather than an approximation sequence terminating in $\zeta(s)$.

Second, simple local and finite reductions encounter genuine analytic obstructions. Individual Euler factors cannot be physical perturbation determinants of self-adjoint pairs on the physical sheet. Furthermore, under the horizontal-divisor hypothesis (which includes all standard finite-degree Archimedean Gamma factors), finite Euler products cannot reproduce the reflection symmetry of the completed zeta function due to the polar mismatch between $\Re(s)=0$ and $\Re(s)=1$.

Third, classical geometric realizations of the prime lengths $\log p$ are obstructed in specific natural settings. A compact flow admitting a closed transverse 1-form $\omega(X)\equiv 1$ has a period group of finite $\mathbb{Q}$-rank, which cannot contain the $\mathbb{Q}$-linearly independent prime logarithms. Similarly, closed geodesics on finite-area hyperbolic surfaces cannot have lengths $\log p$ because their traces would force a finitely generated field to contain multiquadratic extensions of arbitrarily large degree. These results rigorously motivate the necessity of looking beyond simple suspension geometries or modular geodesics, while leaving open the question for general smooth flows.

Fourth, automorphic scattering clarifies the sign problem. The relative trace of the self-adjoint Laplacian on $L^2(SL(2,\mathbb{Z})\backslash\mathbb{H})$ generates arithmetic logarithmic derivatives with a negative sign because $\zeta(2s)$ sits in the denominator of the scattering matrix. This disproves the claim that a super-trace is logically necessary, while respecting the physical-sheet barrier: scattering zeros appear as resonances on the unphysical sheet, rather than as physical-sheet eigenvalues.

Fifth, the finite-field model confirms that the overall spectral/determinantal/sign architecture is internally consistent. For a curve over $\mathbb{F}_q$, normalized Frobenius has unit-modulus spectrum, the Lefschetz trace formula supplies the alternating sign, and the cohomological determinant reproduces the zeta function.

The unresolved problem in characteristic zero is therefore isolated to Conditions (A)–(E): constructing an autonomous dynamics whose operator, domain, trace formula, and regularized determinant are defined independently of the non-trivial zeros and simultaneously recover the spectral and prime-power sides of the explicit formula. Until such an object is constructed, the Hilbert–Pólya program remains open in precisely this dynamical sense.