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S.T.A.R. Program — Overview

S.T.A.R. — Symbolic Topoloical Arithmetic Relativity

A unified mathematical and theoretical physics framework that proposes the large-scale structure of the cosmos is predetermined by the arithmetic invariants of elliptic curves over ℚ, dynamically conserved through entropy cohomology, and realized as an effective relativity-cosmology model.

Core Idea

The physical universe operates as the ultimate Symbolic Field: a self-consistent projection of deep number-theoretic structures. Fundamental physical constants and the geometry of the cosmic web emerge not from chance or anthropic selection, but from a number-theoretic necessity.

Trilogy Architecture

  • ACSC — Arithmetic–Cosmic Structure Conjecture (Geometry) A metric-preserving projection map $\Phi: \mathcal{E} \to M_{\rm cosmo}$ from elliptic curves $E/\mathbb{ℚ}$ to 3D cosmic geometry. Uses bounded normalizations of the discriminant $\Delta_E$, conductor $N_E$, rank $r_E$, regulator, and real period $\Omega_E$.

  • ECC — Entropy Cohomology Conjecture (Dynamics & Field Theory) Introduces an entropy field $\mathcal{M}(x)$ and cohomology framework that enforces conservation laws and provides the dynamical stability for the arithmetic projection.

  • S.T.A.R. — Combined Cosmology Synthesizes the above into an effective relativity theory with Scale-dependent effective Hubble parameter $H_{\rm eff}(z) = H_0 \cdot \langle \Omega_E \rangle_z$ where $\langle \Omega_E \rangle_z$ is a weighted average over the projected point cloud ${\Phi(E)}$, with weights determined by projection distortions, isogeny class density, and local-versus-global sampling. This naturally produces different effective expansion rates at late times (local distance ladder, $\sim 72{-}74$ km/s/Mpc) versus early times (CMB, $\sim 67.4$ km/s/Mpc), offering a potential resolution to the Hubble tension through arithmetic self-tuning.

Key Innovations

  • Global-to-Local Mapping Paradox Correction (GLMPCT): Dynamic topological corrections (sinusoidal, oblate flattening, force-directed repulsion) that preserve metric structure.

  • Symbolic Action Principle: Derived via symbolic regression on topological and arithmetic features.

  • Testable Predictions: Persistent homology of the arithmetic point cloud compared to CMB anomalies (low quadrupole suppression, hemispherical asymmetry, low-multipole alignment).

  • Inverse Mapping: Recover arithmetic rank from cosmological observables via $\rho_{\rm obs} \to r_E$.

Repository Structure

 src/ — Core modules (acsc, ecc, physics, tda, symbolic_regression, pipeline)
 data/ — Raw & processed LMFDB + Cremona datasets
 notebooks/ — Exploratory analysis and Hubble tension fits
 scripts/ — Reproducible pipelines
 manuscript/ — Full theoretical monograph (LaTeX + PDF)
 docker/ & environment.yml — Reproducible environments

How to Get Started

Open in GitHub Codespaces

git clone https://github.com/LcosmosS/S.T.A.R.-Program.git
cd S.T.A.R.-Program
conda env create -f environment.yml
conda activate star-env

See notebooks/hubble_tension_fit.ipynb and scripts/ for quick starts.

Status

The framework combines rigorous number theory, topological data analysis, and symbolic regression with explicit cosmological predictions. It is under active development with strong emphasis on computational reproducibility and falsifiability.

Citation

Patrick J. McNamara, The S.T.A.R. Program for Mathematical and Theoretical Physics, 2025–2026.
GitHub Repository