# P13093: Light-Cone Realization of Burnol Causality for the Riemann Hypothesis

- ID: `P13093`
- Reference: `light-cone-realization-burnol-causality-riemann-hypothesis`
- Page: https://theoremdb.org/statements/P13093
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- Record maturity: Reviewed problem

## The problem

Let \((\mathcal H_{\rm ar},U,\mathcal D_-,\mathcal D_+)\) denote Jean-François Burnol's adelic Lax–Phillips scattering system for \(K=\mathbb Q\), and let \(P_1\) denote the projection onto the trivial isotypical component for the compact subgroup of idele classes of norm one. Put \(\mathcal H_\zeta=P_1\mathcal H_{\rm ar}\) and \(\mathcal D_\pm^\zeta=P_1\mathcal D_\pm\). Burnol's componentwise causality criterion identifies \(\mathcal D_-^\zeta\perp \mathcal D_+^\zeta\) with the absence of nontrivial zeros of \(\zeta(s)\) having \(\Re s>1/2\), hence, by the functional equation, with the Riemann Hypothesis.

Construct a local finite-propagation-speed hyperbolic realization of this zeta scattering component, or prove that no realization exists in the class below. A positive realization consists of: (i) a globally hyperbolic spacetime \(M=\mathbb R_t\times\Sigma\) with a specified local linear symmetric-hyperbolic or normally-hyperbolic operator \(P\); (ii) a finite-energy Hilbert space \(\mathcal H_{\rm phys}\) whose Cauchy evolution \(V(t)\) is unitary and has propagation speed at most some fixed \(c<\infty\) relative to a specified spatial metric; (iii) closed incoming and outgoing subspaces \(\mathcal K_-,\mathcal K_+\subset\mathcal H_{\rm phys}\), defined by spacetime support/domain-of-dependence conditions independent of the nontrivial zero divisor of \(\zeta\); and (iv) an explicit unitary map \(J\) from \(\mathcal H_{\rm phys}\), or from a reducing subspace specified a priori, onto \(\mathcal H_\zeta\), satisfying
\[
J V(t)J^{-1}=U(e^t),\qquad J\mathcal K_\pm=\mathcal D_\pm^\zeta.
\]
The construction may use arithmetic data known independently of RH, such as primes, adelic/Fourier data, Poisson–Tate summation, or the explicit formula, but it may not assume RH, Burnol causality, zero-freeness in \(\Re s>1/2\), a de Branges/Hermite–Biehler positivity statement equivalent to RH, or any equivalent closure criterion. Prove from the local hyperbolic PDE, finite propagation, support geometry, and energy structure alone that \(\mathcal K_-\perp\mathcal K_+\). Under the intertwiner this yields Burnol causality in the zeta component and therefore RH.

Alternatively, give a negative resolution for the full stated class: prove that no such local finite-energy symmetric-hyperbolic or normally-hyperbolic realization with finite propagation and exact dilation intertwining can map support-defined \(\mathcal K_\pm\) onto \(\mathcal D_\pm^\zeta\). The obstruction must follow from a property genuinely forced by locality/finite propagation (for example an appropriate Paley–Wiener or exponential-type constraint, microlocal support property, spectral multiplicity constraint, or semigroup structure) and must be incompatible with a proved property of the arithmetic scattering component without merely restating an RH-equivalent failure of orthogonality.

## Status

**Resolution status:** Open.

Research work is recorded. A complete resolution has not passed review.

## Research packet

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## References

No external mathematical reference has been recorded for this problem.