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arkheion

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# SPDX-FileCopyrightText: 2026 Jung Soo Kim (Arkhenomia Institute) <https://orcid.org/0009-0006-5140-0604>
# SPDX-License-Identifier: CC-BY-NC-4.0 AND LicenseRef-Arkhenomia-Governance-v2.2.0
# ==============================================================================
# Author & Institute: Jung Soo Kim (ORCID: 0009-0006-5140-0604) / Arkhenomia Institute
# License : CC BY-NC 4.0 (Commercial licensing administered by Arkhenetics; see LICENSE.txt)
# Canonical Repo : https://github.com/arkhenomia/arkheion
# Dataset Archive : https://huggingface.co/datasets/arkhenomia/arkheion
# Official Websites : https://arkhenomia.org \| https://arkhenetics.com
# ------------------------------------------------------------------------------
# Commercial use, corporate R&D, in-house enterprise deployment, AI knowledge
# distillation, surrogate model training, automated scraping/corpus ingestion,
# OS kernel/hardware/firmware porting (ZINN / Ark-ZPF), network state-index
# synchronization/congestion-control protocol implementation, and high-risk or
# non-peaceful applications are strictly prohibited without a separate written
# commercial license agreement executed with the Author or Arkhenetics. All
# patent rights are expressly reserved; no patent or commercial rights are
# granted by implication. Governed by the latest LICENSE.txt (Policy v2.2.0+).
# ==============================================================================
#
# H.U.G.G.E.R + TZT Framework Python Dependencies
# Usage: pip install -r requirements.txt

numpy>=1.21.0
scipy>=1.7.0
pandas>=1.3.0
matplotlib>=3.4.0
sympy>=1.9
# SPDX-FileCopyrightText: 2026 Jung Soo Kim (Arkhenomia Institute) <https://orcid.org/0009-0006-5140-0604>
# SPDX-License-Identifier: CC-BY-NC-4.0 AND LicenseRef-Arkhenomia-Governance-v2.2.0
# ==============================================================================
# Author & Institute: Jung Soo Kim (ORCID: 0009-0006-5140-0604) / Arkhenomia Institute
# License : CC BY-NC 4.0 (Commercial licensing administered by Arkhenetics; see LICENSE.txt)
# Canonical Repo : https://github.com/arkhenomia/arkheion
# Dataset Archive : https://huggingface.co/datasets/arkhenomia/arkheion
# Official Websites : https://arkhenomia.org \| https://arkhenetics.com
# ------------------------------------------------------------------------------
# Commercial use, corporate R&D, in-house enterprise deployment, AI knowledge
# distillation, surrogate model training, automated scraping/corpus ingestion,
# OS kernel/hardware/firmware porting (ZINN / Ark-ZPF), network state-index
# synchronization/congestion-control protocol implementation, and high-risk or
# non-peaceful applications are strictly prohibited without a separate written
# commercial license agreement executed with the Author or Arkhenetics. All
# patent rights are expressly reserved; no patent or commercial rights are
# granted by implication. Governed by the latest LICENSE.txt (Policy v2.2.2+).
# ==============================================================================

# DNS Benchmark Environment and Ablation Counter-Example Notes
# 2026 Grand Unified Framework - Part 1 (v1.5 Canonical Release):
# Macroscopic Gravity-Fluid Dynamics (H.U.G.G.E.R) & Topological Zero Tensor (TZT) Validation

## 1. Note on Ablation Counter-Example Data and Figures
The enclosed ablation figures, `CounterExample_1_Ill_Posed_Ablation_Without_Solenoidal_Condition.png` and `CounterExample_2_Energy_Blowup_Without_Mantle_Dissipation.png`, visualize the rapid numerical and energetic blow-up that occurs when the solenoidal Helmholtz-Leray projection condition (\nabla \cdot \mathcal{X} = ...

This ablation test demonstrates that violating the coupled canonical constraints (solenoidality and localized counter-strain dissipation) causes the system to deteriorate into non-physical instability on an under-resolved grid. In contrast, the canonical pseudo-spectral solver (`Source_Code_1_TZT_Dynamics_Solver.py`), ...

## 2. Direct Numerical Simulation (DNS) Parameters
The numerical validations in the inviscid Euler limit and the high-Reynolds-number viscous regime (Re = 10,000) were conducted under the following benchmark specifications:

* Domain: 3D Periodic Flat Torus T^3 = [0, 2\pi)^3
* Spatial Discretization: Pseudo-spectral Fourier-Galerkin method with the 2/3 dealiasing rule
* Time Integration: Classical 4th-order explicit Runge-Kutta (RK4)
* Grid Resolutions (N^3): 32^3, 48^3, 64^3, and 96^3
* Time Step (dt): 0.025 (halved to 0.0125 for temporal convergence and robustness checks)
* Baseline Regimes: Inviscid Euler limit (\nu = 0) and Viscous Navier-Stokes at Re = 10,000 (\nu = 10^{-4})

## 3. TZT-Augmented Configuration and Localized Valve Execution
In `Source_Code_1_TZT_Dynamics_Solver.py`, the TZT counter-strain and rotational regularization operators are configured with the following baseline parameters:

* C_X (Solenoidal Intake Scaling): \pm 0.1
* C_O (Rotational Spin Tensor Scaling): 1.0
* \delta_m (Denominator Regularizer in \mathcal{X}): 1.0
* \varepsilon (Spin Saturation Regularizer in \mathcal{O}^{turb}): 10^{-2}
* \nu_{turb} (Subgrid Eddy Viscosity Scale): Coupled with C_s = 0.17 and grid filter scale \Delta = 2\pi / N
* --valve (Localization Mode, synchronized with Lean 4 `CriticalRegion`):
- `--valve omega` (Default in v1.5): Activates TZT forcing strictly on the supercritical vorticity region K = {x \in T^3 : \|\omega(x)\|^2 > \Lambda^2}, preserving unmodified Navier-Stokes/Euler kinematics outside K.
- `--valve global`: Applies the TZT operator across the entire domain K = T^3 (retained for comparison with baseline global-limit data).

---
## 4. License, Attribution, and Governance (Policy v2.2.2+)

* Author & Institute: Jung Soo Kim (ORCID: 0009-0006-5140-0604) / Arkhenomia Institute (https://arkhenomia.org)
* License: CC BY-NC 4.0 (Commercial licensing administered by Arkhenetics; see LICENSE.txt)
* Canonical Repository: https://github.com/arkhenomia/arkheion
* Dataset Archive: https://huggingface.co/datasets/arkhenomia/arkheion
* Commercial Licensing & Enterprise Inquiries: Arkhenetics (https://arkhenetics.com)

All manuscripts, formal verification proofs, symbolic CAS scripts, numerical solvers, and benchmark datasets in this repository are licensed under the Creative Commons Attribution-NonCommercial 4.0 International License (CC BY-NC 4.0, https://creativecommons.org/licenses/by-nc/4.0/) and governed by the latest version o...
# SPDX-FileCopyrightText: 2026 Jung Soo Kim (Arkhenomia Institute) <https://orcid.org/0009-0006-5140-0604>
# SPDX-License-Identifier: CC-BY-NC-4.0 AND LicenseRef-Arkhenomia-Governance-v2.2.0
# ==============================================================================
# Author & Institute: Jung Soo Kim (ORCID: 0009-0006-5140-0604) / Arkhenomia Institute
# License : CC BY-NC 4.0 (Commercial licensing administered by Arkhenetics; see LICENSE.txt)
# Canonical Repo : https://github.com/arkhenomia/arkheion
# Dataset Archive : https://huggingface.co/datasets/arkhenomia/arkheion
# Official Websites : https://arkhenomia.org \| https://arkhenetics.com
# ------------------------------------------------------------------------------
# Commercial use, corporate R&D, in-house enterprise deployment, AI knowledge
# distillation, surrogate model training, automated scraping/corpus ingestion,
# OS kernel/hardware/firmware porting (ZINN / Ark-ZPF), network state-index
# synchronization/congestion-control protocol implementation, and high-risk or

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Arkheion - Grand Unified Framework (v1.5 Canonical Release)

Official canonical archive for the Grand Unified Framework research series, mechanized Lean 4 (v4.34.1, 0 sorry, zero custom axioms) formal verification suites, SymPy symbolic audits, and 3D pseudo-spectral numerical validation benchmarks.

Author & Principal Investigator: Jung Soo Kim (ORCID: 0009-0006-5140-0604)
Research Institute: Arkhenomia Institute
Commercial Licensing & Applied R&D: Arkhenetics
Canonical Repository: github.com/arkhenomia/arkheion
Dataset & Telemetry Archive: huggingface.co/datasets/arkhenomia/arkheion
Interactive Simulation Spaces: huggingface.co/spaces/arkhenomia/sim-2026-v1.5.0

1. Canonical Research Series (v1.5)

Part 1 - Macroscopic Gravito-Fluid Dynamics & Topological Zero Tensor (TZT)
A Grand Unified Framework for Macroscopic Gravity-Fluid Dynamics and 0-Point Equilibrium Operation: Regularization of 3D Navier-Stokes Singularities and Proof of Global Phase Transitions via Topological Trivialization
Part 2 - Golden M0 Tensor Mechanics & Quasi-Periodic 3-Torus Stability
The Golden M0 Tensor Mechanics and Quasi-Periodic 3-Torus Stability: Resolution of Resonance Collapse, Homological Information Preservation, and Proof of Lyapunov 0-Point Convergence via Golden Integer Lattices
Part 3 - Microscopic Vorticity-Selective PDE Regularity (NS_TZT)
A Vorticity-Selective Dissipative Model (NS_TZT) for Incompressible Flows on the 3-Torus: Global Strong Well-Posedness, Calderon-Zygmund Cubic Regularity, and Subgrid-Scale Laminar Shielding
Foundational Monographs (Pre-Series I-III)
H.U.G.G.E.R v1.0-v1.2 Continuum Stress Closures and Covariant TZT Foundations

Scope Note: All mathematical regularity proofs in this series concern topologically closed, scale-regularized (l > 0) systems on 3-Torus manifolds and do not assert claims regarding the unforced, unconstrained Euclidean (R3, l = 0) Navier-Stokes Millennium problem.

2. Citation

@article{kim2026arkheion,
  author  = {Kim, Jung Soo},
  title   = {The Grand Unified Framework Series (Parts 1--3): Macroscopic Gravito-Fluid Dynamics, Golden M0 Tensor Mechanics on Quasi-Periodic 3-Torus Manifolds, and the Vorticity-Selective NS_TZT Model},
  journal = {Arkhenomia Institute Research Monographs},
  year    = {2026},
  note    = {v1.5 Canonical Release, ORCID: 0009-0006-5140-0604},
  url     = {[https://github.com/arkhenomia/arkheion](https://github.com/arkhenomia/arkheion)}
}

3. Licensing & Governance

SPDX-License-Identifier: CC-BY-NC-4.0 AND LicenseRef-Arkhenomia-Governance

Non-Commercial Academic Use (CC BY-NC 4.0): Non-commercial academic research, educational use, and independent verification are freely permitted with proper attribution and primary reference citation. Prior Art & Patent Reservation: All disclosed mathematical equations and system architectures constitute published Prior Art. All domestic and international patent rights are expressly reserved by the original author, Arkhenomia Institute, and Arkhenetics. Commercial Licensing (Arkhenetics): Any commercial use, corporate R&D, enterprise deployment, AI distillation/surrogate training, or kernel/hardware/firmware implementation requires a separate written commercial license agreement with the Author or Arkhenetics (arkhenetics.com), governed by the latest LICENSE.txt (Policy v2.2.2+).

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