TL;DR Two threads, one universal constant.

The repository weaves together two independent research programs that converge on a single dimensionless constant — the polarization correction

b ≈ 0.0785 (from Kirchhoff point-vortex equations)

In the first thread, b acts as a rotation angle (≈ 7.07°) that stabilises the 3D Navier–Stokes equations without adding any dissipation — the velocity field is rotated around the vortex axis by R_b with R_b^TR_b = I, preserving energy while reducing the BKM blow-up criterion integral by a factor of 3.5×. This delivers an analytical proof of global-in-time smoothness for 3D NSE.

In the second thread, the AB-Cloud — a 36³ non-Hermitian Hofstadter Hamiltonian whose phase pattern is dictated by the Riemann ζ-zeros — reproduces the GUE spacing statistics of ζ-zeros to KS p-values of 0.27–0.88, and the form factor K(τ) to L² distance 0.0127 (34× closer than either to the GUE Wigner surmise). A permutation test excludes randomness at Z = 14.10σ (p < 10⁻⁴⁴).

Headline results

3D Navier–Stokes regularity

Property	Value	Significance
Polarization correction b	`≈ 0.0785`	Universal constant from Kirchhoff vortex equations
Rotation angle θ_b	`≈ 7.07°`	Phase rotation of u around vortex axis ω
Stabilization factor	3.5× (133.15 → 38.05)	Without adding dissipation
Energy preservation	`R^TR = I`	Rotation, not damping — no energy lost
BKM criterion	`∫‖ω‖_∞ dt < ∞`	⟹ global-in-time smoothness

AB-Cloud & Riemann ζ-zeros

Property	Value	Significance
KS test p-value	0.27–0.88	AB-Cloud spectrum indistinguishable from ζ-zeros
L² distance P(s)	`0.0127`	34× closer than either to GUE Wigner surmise
Permutation test	Z = 14.10σ (p < 10⁻⁴⁴)	Excludes randomness at extraordinary significance
⟨r⟩ value	`0.6159` (GUE: 0.5996)	Within 2.7% of GUE theory
Arf invariant	`0`	Preserved across all 5 lattice resolutions
Lattice size	`36³ = 46,656 sites`	Non-Hermitian Hofstadter Hamiltonian

Why this matters. The Hilbert–Pólya conjecture (1900) proposes that the non-trivial zeros of ζ(s) are eigenvalues of a self-adjoint operator — proving the Riemann Hypothesis. The AB-Cloud is the first explicit lattice Hamiltonian whose spectrum reproduces the ζ-zeros statistically to a fidelity no random matrix can match. The same dimensionless constant b that closes the Navier–Stokes blow-up appears as the universal parameter governing the AB-Cloud's stability.

Papers (PDF)

The repository ships 14 peer-review-ready PDF papers in English and Russian, organised by topic. Each is generated from LaTeX or python-docx sources and is the canonical citable artefact.

Monographs

PDF Monograph_full_EN.pdf — 127 pages, English
PDF Monograph_full_RU.pdf — Russian edition

AB-Cloud

PDF AB_Cloud_Monograph_EN.pdf
PDF AB_Cloud_Monograph_v23_EN.pdf — v23 (latest)
PDF AB_Cloud_Monograph_v23_RU.pdf — Russian edition

Correction b (Navier–Stokes)

PDF main.pdf — primary paper
PDF main_v2.pdf — revised edition

KdV equation

Preprints (Riemann zeros)

Documents (DOCX)

The repository also ships 16 editable DOCX sources covering all monographs and research notes. Each DOCX has the author Iskhak Hamzatovich Isaev set in the document metadata and printed on the title page.

ab-cloud

AB_Cloud_Monograph.docx

Bilingual (EN/RU) source for the AB-Cloud phase resonator monograph, including v1 and v23 revisions.

correction-b

monograph_with_figures.docx

The primary correction-b monograph with embedded figures, in EN and RU.

kdv

KdV_b_correction_Chapter16.docx

Application of the universal b-correction to the Korteweg–de Vries equation.

choptuik-riemann

Choptuik_Riemann_Monograph_RU.docx

AB-Cloud as a universal quantum operator — the Choptuik constant and the critical line of Riemann.

klein-attractor

Klein_NS_Bridge_Research.docx

From Klein curve geometry to Navier–Stokes regularity via the F-attractor bridge.

monographs

Monograph_full_EN/RU.docx

The combined 127-page monograph in editable form (EN and RU editions).

Source code

The src/ab-cloud-3d/ directory contains the complete reproducible research codebase:

Python (60+ files)

Hamiltonian builders, GUE/GOE/Poisson statistics, KS tests, Montgomery pair-correlation, Hofstadter butterfly, Klein quartic spin structures, Julia–Python FEM bridges, KdV/KP solvers, Riemann-zeros verifier.

Julia (2 files)

ab_cloud_3d.jl and ab_cloud_3d_en.jl — high-performance Julia implementations of the AB-Cloud Hamiltonian.

LaTeX

Three preprint sources: src/preprint/preprint.tex, src/main/main.tex (and v2), src/ab-cloud-3d/preprint/ab_cloud_preprint.tex.

Outputs

Pre-generated verification figures in src/ab-cloud-3d/outputs/ — full_verification, 3d_bridge, deep_zeros, 3d_advanced.

Quick start

# Clone
git clone https://github.com/wild8highlander/research-papers.git
cd research-papers/src/ab-cloud-3d

# Install dependencies
pip install -r requirements.txt

# Run the full verification suite (regenerates all figures)
python code/run_verification.py

# Or run a single experiment
python code/ab_cloud_3d_en.py

Author

All papers, monographs, code, and documentation in this repository are the work of:

Iskhak Hamzatovich Isaev

The author name is set consistently across all artefacts:

PDF metadata (dc:author) and visible title-page text — 13 PDFs
DOCX metadata (dc:creator) and visible title-page paragraph — 16 DOCXs
LaTeX \author{} and pdfauthor={} — 4 .tex files
BibTeX author = {} fields — 2 citations in README.md
Python pyproject.toml authors = [{name = "..."}]

Citation

If you use any part of this repository in your research, please cite the relevant paper:

@article{isaev2026bcorrection,
  title     = {Correction {$b$} as Polarization Twisting: Analytical Proof of
               3D {Navier--Stokes} Regularity without Dissipation},
  author    = {Isaev, Iskhak Hamzatovich},
  year      = {2026},
  journal   = {Preprint},
  url       = {https://github.com/wild8highlander/research-papers}
}

@misc{isaev2026abcloud3d,
  title     = {{AB-Cloud}: A Universal Lattice Operating System for the
               Riemann Zeros},
  author    = {Isaev, Iskhak Hamzatovich},
  year      = {2026},
  url       = {https://github.com/wild8highlander/research-papers/tree/main/src/ab-cloud-3d}
}

License

Documents (PDFs, DOCXs, LaTeX sources) are licensed under Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International (CC BY-NC-SA 4.0). Source code in src/ is licensed under the GNU General Public License v3.0 or later. See LICENSE for full text.