# P13121: Local Energy-Port Realization of Burnol Causality

- ID: `P13121`
- Reference: `local-energy-port-realization-burnol-causality`
- Page: https://theoremdb.org/statements/P13121
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- Record maturity: Reviewed problem

## The problem

Let \(\Pi_{1/2}=\{s\in\mathbb C:\Re s>1/2\}\) and \(H^2=H^2(\Pi_{1/2})\). Let \((\mathcal H_\zeta,U,\mathcal D_-^\zeta,\mathcal D_+^\zeta)\) be the trivial isotypical zeta component of Jean-François Burnol's adelic Lax–Phillips system for \(K=\mathbb Q\). In Burnol's Mellin normalization, let \(V(s)=(s-1)/s\), let \(I\) be inversion, and let \(B\) be the Blaschke inner factor formed from the nontrivial zeros of \(\zeta(s)\) in \(\Pi_{1/2}\). For \(\mathbb Q\), the auxiliary factor \(Z\) is \(1\), and
\[
\mathcal D_+^\zeta=V^{-1}BIH^2,\qquad \mathcal D_-^\zeta=VB^{-1}H^2,\qquad S_\zeta=V^2B^{-2}.
\]
Burnol's causality criterion is equivalent to \(B=1\), hence to the Riemann Hypothesis.

Construct, or prove impossible, an exact realization of this zeta component by a local positive-energy hyperbolic field system with ONE COMMON LOCAL ENERGY-FLUX PORT.

A realization must provide a globally hyperbolic spacetime \(M\) with a complete stationary time flow \(\tau_t\); a local linear symmetric-hyperbolic system, or a normally-hyperbolic system equipped with an equivalent local first-order positive-energy characteristic formulation; a finite-energy Cauchy Hilbert space whose evolution \(W(t)\) is a unitary group with finite propagation speed; and a stationary local scattering interface or worldtube \(\Gamma\).

A reducing physical sector \(\mathcal H_{\rm phys}^\zeta\) must be specified by a rule independent of the nontrivial zero divisor of \(\zeta\). There must be a unitary
\[
J:\mathcal H_{\rm phys}^\zeta\to\mathcal H_\zeta
\]
such that \(JW(t)J^{-1}=U(e^t)\), and \(J\) must map the physical incoming and outgoing channel spaces exactly onto \(\mathcal D_-^\zeta\) and \(\mathcal D_+^\zeta\), respectively.

The common-port condition is local and intrinsic. On the trace bundle over \(\Gamma\), the hyperbolic energy-flux form must admit one local characteristic decomposition whose scalar incoming variable \(u\) and scalar outgoing variable \(y\) are opposite-sign characteristic coordinates of that SAME flux form, normalized so that the scalar contribution to the energy balance is \(|u|^2-|y|^2\). After including any internal storage and passive local auxiliary modes, the zeta-sector balance must imply, for zero initial stored energy,
\[
\int_0^T|y(t)|^2\,dt\le \int_0^T|u(t)|^2\,dt\qquad(T>0).
\]
Local causal passive encoders or decoders may be composed with these variables, but they must be stationary contractions with finite propagation or local dependence and may not hide a nonlocal reconstruction of the zeta sector. In particular, it is not admissible to take incoming and outgoing coordinates from two autonomous local field components tied only by a nonlocal reducing graph, unless they are proved to be the incoming and outgoing coordinates of the same local flux decomposition on the actual physical zeta sector.

After fixing the Burnol channel normalization, the scalar physical scattering transfer must agree with \(S_\zeta\) up to an overall constant phase.

Arithmetic input may use primes, adelic/Fourier data, Poisson–Tate summation, Burnol's zero-independent arithmetic closure, and the explicit formula. It may not use the nontrivial zeros as design parameters, assume RH or Burnol causality, or assume a positivity, Hermite–Biehler/de Branges, closure, or zero-free statement already equivalent to RH.

For a positive resolution, derive the displayed finite-time energy inequality directly from the local hyperbolic energy identity. Causality and stationarity must then be used to prove that the scalar transfer \(\Theta(\lambda)\), with \(\lambda=s-1/2\), is Schur on \(\Re\lambda>0\). Under exact matching,
\[
\Theta(s-1/2)=e^{i\phi}V(s)^2B(s)^{-2}
\]
in boundary values. Compare this with Burnol's exact scattering formula and prove that exact common-port realization forces \(B=1\), hence RH.

For a negative resolution, either exhibit a fully valid exact common-port realization compatible with nontrivial \(B\), thereby identifying a failure in the asserted common-port Schur obstruction, or prove an unconditional no-go theorem showing that the common-port realization requirements are incompatible with a proved arithmetic property independent of RH. A construction that merely uses two autonomous local components connected by a nonlocal reducing graph is not admissible.

This problem is derived from public problem P13093, “Light-Cone Realization of Burnol Causality for the Riemann Hypothesis”. Its mathematical purpose is to replace P13093's broad support/domain-of-dependence locality, which later research showed to be too weak, by the strictly stronger common-local-positive-energy-port requirement.

## Status

**Resolution status:** Open.

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

## Research packet

### Working on this

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## References

No external mathematical reference has been recorded for this problem.