🧬 Symbolic Algebraization of Rational Hodge Classes

A computational and epistemological approach to the Hodge Conjecture via symbolic convergence.

Symbolic Algebraization of Rational Hodge Classes

Author: Daniel Iván Campos Espinoza

Author: Daniel Iván Campos Espinoza
Preprint: [arXiv submission pending]
Repository: All simulations, code, and visualizations for the symbolic operator ( \widehat{\mathcal{S}} ) Date of public release: April 22, 2025
📄 License - 📘 Preprint PDF: CC BY-NC-ND 4.0 — Attribution-NonCommercial-NoDerivatives. - 💻 Code: MIT License

📄 Preprint PDF → main.pdf
📦 Live Repo → GitHub
📘 Extended Docs → docs/README_extended.md

🧠 Overview

Symbolic Algebraization of Rational Hodge Classes via Iterative Lefschetz Operators
A novel approach to the Hodge Conjecture using symbolic contraction dynamics. We introduce: [ \widehat{\mathcal{S}} := \Lambda \circ \Pi{\text{prim}} \circ Pk ] which acts on rational classes ( \omega \in H^{2p}(X, \mathbb{Q}) \cap H^{p,p}(X) ) to contract the transcendental part ( \phi ) and converge to ( \text{Im}(cl) ).

📁 Repository Structure

. ├── paper/ # PDF of the paper │ └── main.pdf ├── code/ # Core symbolic operators │ ├── operator_S_hat.py │ ├── algebraic_filters.py │ ├── primitive_projectors.py │ └── lefschetz_tools.py ├── notebooks/ # Symbolic iteration examples │ ├── K3_example.ipynb │ ├── CY3_example.ipynb │ └── symbolic_iteration_engine.ipynb ├── data/ # Base class definitions │ ├── base_classes_CY3.npy │ ├── base_classes_K3.npy ├── figures/ # Visualizations │ └── *.png ├── docs/ │ └── README_extended.md ├── requirements.txt ├── LICENSE ├── README.md └── CITATION.cff

🚀 How to Use

1. Install dependencies: bash pip install numpy sympy matplotlib scipy

2. Run the symbolic iteration: Use notebooks/symbolic_iteration_engine.ipynb to load a class ( \omega_0 ) and apply ( \widehat{\mathcal{S}} ) iteratively.

3. Visual outputs include:

4. Decay plots of ( |\phi_k| )

5. 3D symbolic projections

6. Heatmaps on K3 and CY3 bases

🔬 Mathematical Core

The operator acts symbolically as: [ \omegak := \widehat{\mathcal{S}}^k(\omega0) = \alpha + \phik, \quad \text{with } \phik \to 0 ] Implemented modularly as: - ( \Pi{\text{prim}} ) (primitive projector) - ( Pk ) (symbolic algebraizing filter) - ( \Lambda ) (Lefschetz inverse contraction)

All components are built over algebraic bases with tunable polarization metrics.

📊 Reproducibility

Includes open-source code, symbolic notebooks, and visualizations for:

• K3 surfaces
• CY3 varieties
• Convergence dynamics

📘 License

This work is open for research and educational use under the MIT License. Attribution required.

🌐 Contact

For questions, collaborations, or symbolic resonance discussions:
📧 danielcampos.cl@gmail.com