First public release: April 2026

Author: Matti S. Kärki

Vortex Operator Theory (VOT)

A dynamical resonance framework for modeling the nontrivial zeros of the Riemann zeta function.

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Overview

Vortex Operator Theory (VOT) is a research program exploring whether the zeros of the Riemann zeta function can be understood as outcomes of a prime-driven dynamical system.

Instead of treating zeros as static spectral points, VOT models them as:

emergent structures of a constrained resonance field

The framework integrates:

• arithmetic input (primes)
• dynamical signal ( A_0(t) )
• local spectral geometry
• vortex-based structure (shell / 0-point / flow)
• event system (anchor / bridge / transition)
• hybrid modeling (global + local mechanisms)
• and now: a rigorous spectral operator core

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Two-Layer Structure

VOT is explicitly divided into two layers:

1. Operator Program (long-term goal)

• Prime input: [ K(t) = \sum_{p,k} \log p , \delta(t - k \log p) ]

• Dynamic field: [ \frac{dA_0}{dt} = -\alpha A_0 + \beta \sin(\omega t) K(t) ]

• Operator: [ H = -i \frac{d}{dt} + \frac12 + A_0(t) ]

Goal:

• derive zeta zeros from spectral properties of ( H )
• connect to ( \xi(1/2 + iE) )

⚠️ Not proven.

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2. Empirical Vortex Architecture (current working model)

• Gaussian-windowed local spectrum (STFT)
• purity functional ( \mathcal P(T,\omega) )
• dynamic programming → drifting paths
• multi-layer, multi-arc model
• zero-zone detection (interval-based)

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Vortex Ontology

Each local structure is interpreted via:

• Shell → local resonance layer
• 0-point → attractor center
• Flow → directional alignment

Event roles:

• shell_contact (anchor)
• shell_approach (bridge)
• shell_transition
• shell_failure

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Hybrid Model

The current best-performing structure:

final model = V1 backbone + local corrections

Components:

• V1 → global scaffold (scales better)
• precision → sharp local corrections
• rescue → flexible adjustment
• local_rescue → targeted fixes

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Zero Classification

Each zero is assigned:

• mode: V1 / precision / rescue / local_rescue
• role: shell_contact / shell_approach / etc.
• shell: S1–S5

Key empirical insight

Zeros are not homogeneous.

• majority → shell_contact
• minority → shell_approach
• rare → rescue-required

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Scaling Results (current)

zeros	@1.0	@1.5
20	17	18
100	62	70
300	151	178
1000	248	308

Interpretation:

• purely local models degrade with scale
• hybrid structure improves robustness
• global scaffold (V1) is essential

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NEW: Rigorous Spectral Core

We now have a formal operator-theoretic core:

Construction

From purity functional: [ \mathcal P(T,\omega) ]

define smoothed reduction: [ F_\beta(T)

\frac{1}{\beta}\log\int e^{\beta \mathcal P(T,\omega)},d\omega ]

then: [ V_\beta(T) = -F_\beta(T) ]

and operator: [ H_\beta = -\frac{d^2}{dT^2} + V_\beta(T) ]

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Proven result

If ( \mathcal P \in C^2 ) on a compact domain:

• ( V_\beta \in C^2 \subset L^\infty )
• ( H_\beta ) is self-adjoint
• spectrum is discrete

👉 This is the first rigorous mathematical core of VOT.

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Interpretation

• 0-points → maxima of (F_\beta)
• maxima → minima of (V_\beta)
• minima → spectral localization

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What is Proven vs Open

Proven (mathematically)

• construction of self-adjoint operator ( H_\beta )
• discrete spectrum
• well-defined potential from ( \mathcal P )

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Empirical (observed)

• vortex structure
• hybrid scaling behavior
• zero classification patterns

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Open (core problem)

[ \mathrm{spec}(H_\beta) \stackrel{?}{=} {\gamma_n} ]

This is the exact RH-critical step.

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Repository Structure

preprint/
  vot_preprint_arxiv_style.html
  vot_formal_core_spectral.html

data/
  zero_classification.json
  scaling_results.json

notes/
  state_summary.txt

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What This Is

• a structured research program
• a hybrid of empirical + theoretical work
• a candidate framework for RH-related structure

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What This Is NOT

• not a proof of RH
• not a finished theory
• not parameter-free

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Next Steps

• spectral approximation: [ \lambda_n(H_\beta) \approx \gamma_n ]

• shell invariants

• large-scale classification (1000+ zeros)

• operator reconstruction from ( A_0 )

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Philosophy

VOT suggests:

• zeros arise from structured arithmetic dynamics
• multiple mechanisms coexist
• global + local structure is necessary

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Status

⚠️ Research preprint ⚠️ Partial rigor (operator core only) ⚠️ Main theorem still open

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License

Recommended:

• CC BY 4.0 or
• MIT (if code-focused)