A Log-Scale Operator Package for the Riemann Xi Function: A Foundation for APM-Linked Mathematics
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| Format: | Recurso digital |
| Sprache: | Englisch |
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2026
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| _version_ | 1866901595069874176 |
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| author | Thomson, David |
| author_facet | Thomson, David |
| contents | <p>This paper standardizes <em>Operator Package 1</em>: a reusable, ledger-compatible operator module on the Hilbert space $L^{2}(\mathbb{R},dq)$ intended to support future Aether Physics Model (APM) work that touches analytic number theory.</p> <ol> <li><strong>What the package contains (Operator Package 1).</strong><br><strong>Log-scale translation generator.</strong> A self-adjoint operator with explicit domain:<br>$$<br>P=-i\,\frac{d}{dq},\qquad \mathrm{Dom}(P)=H^{1}(\mathbb{R}).<br>$$<br>With the unitary Fourier transform $\mathcal{F}$ on $L^{2}(\mathbb{R})$, the operator is diagonalized as<br>$$<br>\mathcal{F}P\mathcal{F}^{-1}=M_{t},<br>$$<br>where $M_t$ is multiplication by $t$.</li> <li><strong>Xi-convolution operator.</strong> A bounded self-adjoint convolution operator<br>$$<br>(Wf)(q)=(\Phi*f)(q),<br>$$<br>whose kernel $\Phi$ is constructed explicitly from Jacobi's theta function (via a derived $\omega$-kernel and the change $x=e^{2q}$). The kernel is real, even, smooth, and rapidly decaying, and is chosen to satisfy the Fourier identity<br>$$<br>\Xi(t)=\int_{-\infty}^{\infty}\Phi(q)\,e^{itq}\,dq,<br>$$<br>where $\Xi(t)=\xi\!\left(\tfrac12+it\right)$ is the Riemann xi function restricted to the critical line.</li> </ol> <p><strong>Main closure results.</strong> A lemma--theorem--proof chain establishes the analytic closure required for reuse: kernel regularity/decay, $L^{1}\cap L^{2}$ membership, boundedness of $W$ on $L^{2}$, self-adjointness of both $P$ and $W$, and Fourier diagonalization:<br>$$<br>\mathcal{F}W\mathcal{F}^{-1}=M_{\Xi}.<br>$$<br>Equivalently, by the functional calculus for the self-adjoint operator $P$, the construction closes as the operator identity<br>$$<br>W=\Xi(P).<br>$$</p> <p><strong>Scope.</strong> The goal is not to solve the Riemann Hypothesis (RH), but to provide a mathematically explicit operator backbone suitable for subsequent RH-oriented strategies and for APM-linked spectral constructions. The paper makes explicit why bounded convolution on $L^{2}(\mathbb{R})$ does not by itself discretize the zero set into a nontrivial $L^{2}$ point spectrum, motivating the need for additional structure (boundary conditions, alternate Hilbert spaces, or Fredholm-determinant/trace-class frameworks).</p> <p><strong>APM/QMU integration.</strong> The package is embedded in QMU-normalized log-scale mechanics by interpreting the log-scale variable as a scale ratio:<br>$$<br>q=\log(\ell/\lambda_C).<br>$$<br>Using the Ledger-One propagation scale $\kappa_{0}=F_q\lambda_C$, the paper motivates a ledger-normalized generator<br>$$<br>P_L=\kappa_{0}P.<br>$$<br>In this interpretation, $\Xi(t)$ functions as a scale-spectral response multiplier on log-scale harmonics, and $W$ provides a benchmark self-adjoint ``scale filter'' for later APM operator development.</p> <p><strong>Planned continuation (Operator Package 2).</strong> The next module is outlined as a discretized, self-adjoint log-scale Hamiltonian on a bounded interval $[q_-,q_+]$, with boundary conditions motivated by Aether-unit packing/closure, enabling genuinely discrete spectra suitable for APM mode predictions and zero-ordinate discretization strategies:<br>$$<br>H_{\mathrm{APM}}=-\frac{d^{2}}{dq^{2}}+V_{\mathrm{APM}}(q),\qquad q\in[q_-,q_+].<br>$$</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_18158307 |
| institution | Zenodo |
| language | eng |
| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | A Log-Scale Operator Package for the Riemann Xi Function: A Foundation for APM-Linked Mathematics Thomson, David Riemann Hypothesis Riemann xi function $\Xi(t)$ analytic number theory Jacobi theta function Fourier transform Mellin transform convolution operator self-adjoint operator functional calculus Hilbert–Pólya program log-scale spectral mechanics Aether Physics Model Quantum Measurement Units QMU APM Ledger One scale harmonics <p>This paper standardizes <em>Operator Package 1</em>: a reusable, ledger-compatible operator module on the Hilbert space $L^{2}(\mathbb{R},dq)$ intended to support future Aether Physics Model (APM) work that touches analytic number theory.</p> <ol> <li><strong>What the package contains (Operator Package 1).</strong><br><strong>Log-scale translation generator.</strong> A self-adjoint operator with explicit domain:<br>$$<br>P=-i\,\frac{d}{dq},\qquad \mathrm{Dom}(P)=H^{1}(\mathbb{R}).<br>$$<br>With the unitary Fourier transform $\mathcal{F}$ on $L^{2}(\mathbb{R})$, the operator is diagonalized as<br>$$<br>\mathcal{F}P\mathcal{F}^{-1}=M_{t},<br>$$<br>where $M_t$ is multiplication by $t$.</li> <li><strong>Xi-convolution operator.</strong> A bounded self-adjoint convolution operator<br>$$<br>(Wf)(q)=(\Phi*f)(q),<br>$$<br>whose kernel $\Phi$ is constructed explicitly from Jacobi's theta function (via a derived $\omega$-kernel and the change $x=e^{2q}$). The kernel is real, even, smooth, and rapidly decaying, and is chosen to satisfy the Fourier identity<br>$$<br>\Xi(t)=\int_{-\infty}^{\infty}\Phi(q)\,e^{itq}\,dq,<br>$$<br>where $\Xi(t)=\xi\!\left(\tfrac12+it\right)$ is the Riemann xi function restricted to the critical line.</li> </ol> <p><strong>Main closure results.</strong> A lemma--theorem--proof chain establishes the analytic closure required for reuse: kernel regularity/decay, $L^{1}\cap L^{2}$ membership, boundedness of $W$ on $L^{2}$, self-adjointness of both $P$ and $W$, and Fourier diagonalization:<br>$$<br>\mathcal{F}W\mathcal{F}^{-1}=M_{\Xi}.<br>$$<br>Equivalently, by the functional calculus for the self-adjoint operator $P$, the construction closes as the operator identity<br>$$<br>W=\Xi(P).<br>$$</p> <p><strong>Scope.</strong> The goal is not to solve the Riemann Hypothesis (RH), but to provide a mathematically explicit operator backbone suitable for subsequent RH-oriented strategies and for APM-linked spectral constructions. The paper makes explicit why bounded convolution on $L^{2}(\mathbb{R})$ does not by itself discretize the zero set into a nontrivial $L^{2}$ point spectrum, motivating the need for additional structure (boundary conditions, alternate Hilbert spaces, or Fredholm-determinant/trace-class frameworks).</p> <p><strong>APM/QMU integration.</strong> The package is embedded in QMU-normalized log-scale mechanics by interpreting the log-scale variable as a scale ratio:<br>$$<br>q=\log(\ell/\lambda_C).<br>$$<br>Using the Ledger-One propagation scale $\kappa_{0}=F_q\lambda_C$, the paper motivates a ledger-normalized generator<br>$$<br>P_L=\kappa_{0}P.<br>$$<br>In this interpretation, $\Xi(t)$ functions as a scale-spectral response multiplier on log-scale harmonics, and $W$ provides a benchmark self-adjoint ``scale filter'' for later APM operator development.</p> <p><strong>Planned continuation (Operator Package 2).</strong> The next module is outlined as a discretized, self-adjoint log-scale Hamiltonian on a bounded interval $[q_-,q_+]$, with boundary conditions motivated by Aether-unit packing/closure, enabling genuinely discrete spectra suitable for APM mode predictions and zero-ordinate discretization strategies:<br>$$<br>H_{\mathrm{APM}}=-\frac{d^{2}}{dq^{2}}+V_{\mathrm{APM}}(q),\qquad q\in[q_-,q_+].<br>$$</p> |
| title | A Log-Scale Operator Package for the Riemann Xi Function: A Foundation for APM-Linked Mathematics |
| topic | Riemann Hypothesis Riemann xi function $\Xi(t)$ analytic number theory Jacobi theta function Fourier transform Mellin transform convolution operator self-adjoint operator functional calculus Hilbert–Pólya program log-scale spectral mechanics Aether Physics Model Quantum Measurement Units QMU APM Ledger One scale harmonics |
| url | https://doi.org/10.5281/zenodo.18158307 |