The S.T.A.R. Program

Symbolic–Topological–Arithmetic–Relativity

A Proposed Model for Mathematical & Theoretical Physics to Address Cosmic Expansion & the Hubble–Planck Tension

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Author: Patrick J. McNamara
ORCiD: 0009‑0002‑8978‑5563
Project Start: March 2025 — Active
Keywords: Number Theory, Cosmology, Entropy Cohomology, Persistent Homology, Elliptic Curves, Symbolic Regression, Mathematical Physics, Theoretical Physics, Holography

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Overview

This repository is the primary research archive for the S.T.A.R. Program — the Symbolic–Topological–Arithmetic–Relativity Model — a proposed theoretical physics framework that couples:

• ACSC — Arithmetic–Cosmic Structure Conjecture
  • (See /docs/The_Arithmetic–Cosmic_Structure_Conjecture_(ACSC)_Monograph.pdf for complete comjecture and construction logic)
• ECC — Entropy Cohomology Conjecture
  • (/docs/The_Entropy_Cohomology_Conjecture_(ECC).pdf)

Together, these form a dual‑layer architecture:

• ACSC provides the geometric projection law mapping elliptic curve invariants into a cosmological manifold.
• ECC provides the entropy‑cohomological conservation law governing symbolic information flow across that manifold.

The S.T.A.R. Program proposes that the large‑scale structure of the universe, the effective expansion rate, and the Hubble–Planck tension arise from the interaction between:

• arithmetic projection geometry
• entropy curvature
• persistent cohomology classes
• symbolic geodesics
• scalar‑field coupling

This repository contains the full theoretical monographs, computational toolkit, TDA stability pipeline, and symbolic regression engine that define the S.T.A.R. Program.

• (See OVERVIEW.md for a more in depth overview.)
• (See /docs/Arithmetic Invariants and Cosmological Geometry in Cartography.pdf for project details.)
• (See First_Principles.md, /docs/first_principals.py & /docs/first_principals_results.txt for evidance towards Hubble-Planck Tension resolution.)

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Core Idea

The S.T.A.R. Program asserts that the universe can be projected as a Symbolic Field, where:

• arithmetic invariants
• entropy curvature
• topological persistence
• and geometric projection laws

jointly determine cosmic structure.

A central prediction of the S.T.A.R. Program is a scale-dependent effective Hubble parameter that naturally emerges from the arithmetic projection and entropy dynamics:

$H_{\rm eff}(z) = H_0 \cdot \langle \Omega_E \rangle_z$

where $(\Omega_E)$ is the real period of the elliptic curve (E), and the redshift-dependent average is taken over curves that are “visible” or dominant at redshift $(z)$:

$\langle \Omega_E \rangle_z = \frac{\sum_E w_E(z) , \Omega_E}{\sum_E w_E(z)}.$

The weight function $(w_E(z))$ is constructed from the model’s natural mechanisms:

• Projection distortions and local-versus-global sampling: At low $(z)$ (late universe, local distance ladder), the observer preferentially samples a biased subset of the projected point cloud $({\Phi(E)})$, favoring denser regions or lower-distortion patches after sinusoidal, oblate, and force-directed corrections. At high $(z)$ (CMB epoch), the average approaches the full global distribution.

• Isogeny class density: Curves connected by higher-degree isogenies contribute progressively more at later times, shifting the effective scale factor according to $(g_{E'}(t) = g_E(t) + \eta \log m)$.

This formulation produces regime-dependent values consistent with current observations:

$$ \frac{\dot{a}}{a}\bigg|_{\text{local (low } z)} \approx H_0 \cdot \langle \Omega_E \rangle_{\text{late}} \quad (\sim 72{-}74\ \text{km/s/Mpc}) $$

$$ \frac{\dot{a}}{a}\bigg|_{\text{CMB (high } z)} \approx H_0 \cdot \langle \Omega_E \rangle_{\text{early}} \quad (\sim 67.4\ \text{km/s/Mpc}) $$

The precise functional form of the weights $(w_E(z))$ (including any tunable parameters) is not imposed a priori. Instead, the symbolic regression engine in /src/symbolic_regression/ is trained on the full projected point cloud to discover the optimal weighting that best reconciles the projected arithmetic structure with observed cosmological data. This allows the model to learn the natural mapping from arithmetic invariants to effective expansion history without manual fine-tuning.

• (See 1_STAR_Model.md, Symbolic_Action_Principle.md, /docs/star_v3.1.py, /docs/star_v3.1 output.pdf, & /docs/STAR.ipynb)

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Future Directions

Program Evolution: S.T.A.R.M.A.P. & S.M.A.T.

A key goal of the S.T.A.R. Program is its development into the S.T.A.R.M.A.P. — the Symbolic-Topological-Arithmetic-Relativity-Mission-Analysis-Program — which aims to produce high-fidelity topographical maps of the observable universe. These maps will integrate:

• Arithmetic projections of elliptic curve invariants onto cosmic geometry,
• Entropy cohomology fields for density and radiation structure,
• Persistent homology for filaments, voids, clusters, and gravitational density variations.

Complementing this is the S.M.A.T. (S.T.A.R.-Mission-Analysis-Tool), a practical software framework inspired by NASA’s General Mission Analysis Tool (GMAT). S.M.A.T. will leverage the underlying symbolic model for:

• Interplanetary and interstellar trajectory optimization,
• Communication window prediction using entropy-weighted propagation,
• Gravitational assist planning informed by the arithmetic skeleton of spacetime,
• Mission risk assessment via topological stability metrics.

Together, S.T.A.R.M.A.P. and S.M.A.T. bridge fundamental theory with applied space exploration, turning number-theoretic insights into operational tools for humanity’s expansion into the cosmos.

(See 2_STARMAP.md, & 3_SMAT.md for more details.)

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The S.T.A.R. Trilogy Architecture

This repository represents the theory layering of a three‑part research program:

1. ACSC — Geometry

Arithmetic–Cosmic Structure Conjecture
Defines the projection map $( \Phi(E) )$, the k‑factor scaling, density‑equalizing rescaling, and the Global‑to‑Local Mapping Paradox Correction Theory.

2. ECC — Field Theory

Entropy Cohomology Conjecture
Defines the entropy field $( \mathcal{M}(x) )$, the differential forms $( \theta = d\mathcal{M} )$, $( \omega = d\theta )$, and the cohomology class $(\omega)$ governing symbolic conservation.

3. S.T.A.R. Program — Combined Physics

Symbolic–Topological–Arithmetic–Relativity Model
Combines ACSC + ECC into a full cosmological model with:

• metric perturbations
• scalar‑field coupling
• symbolic Sachs–Wolfe transfer
• cosmic‑web alignment
• symbolic regression law discovery

(See Symbolic_Action_Principle.md, /docs/star_v3.1.py, star_v3.1 output.pdf /docs/STAR.ipynb, /docs/Arithmetic Invariants and Cosmological Geometry in Cartography.pdf & /docs/Appendices for Arithmetic Invariants and Cosmological Geometry in Cartography.pdf`)

This repository is the central hub of the S.T.A.R. Program, and ultimately proposes the introduction of a Symbolic Field-Theory paradigm.

• (See 4_SFT.md)

Why This Project Matters

We are standing at the threshold of a new synthesis: one that dares to merge the abstract rigor of arithmetic geometry with the observable chaos of the cosmos. Symbolic-Field Theory offers not just an innovative framework, but a philosophical reorientation. It invites us to consider that the laws governing the curvature of spacetime may be echoes of deeper number-theoretic truths—that the structure of the universe might be inscribed in elliptic curves, L-functions, and symbolic entropy.

Why does this matter? Because modern cosmology, for all its empirical success, lacks an axiomatic backbone. And number theory, for all its elegance, has remained observationally distant. This book proposes a bridge—one that can be tested, coded, visualized, and refined. In an era of increasing data abundance, this model gives us a symbolic scaffold for interpretation. It provides a way to organize galaxy distributions, predict star formation rates, and model topological curvature using logically grounded mappings.

Moreover, it rekindles a deeper question: What is the universe really made of? This project matters because it offers a new lens, one that doesn't replace our scientific frameworks but enriches them; connecting computation, observation, and symbolic insight into a unified structure that can evolve alongside our deepest inquiries into the nature of reality.

On the Vision: Merging Number Theory and Cosmology

The vision behind this work is rooted in a belief that the deepest structures of the universe are not just physical but symbolic; that the very architecture of spacetime may reflect arithmetic truths. Where classical physics models interactions, number theory models form. This project attempts to merge these two schools of thought by proposing that the patterns governing galaxies might not be entirely empirical—they may be governed, in part, by arithmetic invariants.

This is not a rejection of modern cosmology, but an augmentation. We take L-functions, symbolic entropy, and the ranks of elliptic curves not as metaphors, but as plausible constructs capable of encoding physical structure. We turn galactic distributions into algebraic data and vice versa, creating an interactive, recursive view of cosmological evolution. The vision is one of synthesis. Just as quantum theory once unified the discrete and the continuous, this project seeks to unify symbolic logic with spatial curvature—mapping from curves to clusters, from rank to reality, from entropy to elevation. In doing so, we propose that the universe may not just be measured in light-years or parsecs, but also in ranks, regulators, and symbolic entropy. It is the dream of a new map. Not only geographic, but arithmetic. One that evolves alongside us as we refine our tools, deepen our insights, and expand our sense of the knowable universe.

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Repository Structure

S.T.A.R.-Program/
├── README.md
├── LICENSE
├── CITATION.cff
│
├── src/
|   ├── acsc/                     # # ACSC projection geometry
|   ├── analysis/                 # Documentation generation
|   ├── blender/                  # Paradox correction + infinite zoom
|   ├── cli/                      # Cosmological inference constraints
|   ├── data/                     # Sky survey integration preprocessing
│   ├── entropy/                  # ECC entropy/cohomology machinery               
│   ├── likelihoods/              # Comparisons to Planck/SH0ES + DESI BAO + PANTHEON+
|   ├── pipeline/                 # Inference + PaperFigure pipelines
│   ├── physics/                  # S.T.A.R. cosmological physics
│   ├── symbolic_regression/      # Constrained GP + law discovery
│   ├── tda/                      # Persistent homology + stability
|   ├── tests/                    # Sky survey integration
|   ├── utils/                    # Astronomical utilities
|   └── visualization/            # PaperFigure + plotting
│    
├── data/
│   ├── raw/                      # LMFDB + Cremona datasets
│   └── processed/                # Cleaned + merged arithmetic data
│
├── notebooks/
│   ├── projection_demo.ipynb
│   ├── entropy_field_demo.ipynb
│   ├── hubble_tension_fit.ipynb
│   └── tda_analysis.ipynb
│
├── scripts/
│   ├── generate_raw.py
│   ├── compute_3selmer_full_pari.py
│   └── star_validation_batch.py
│
├── results/
│   ├── figures/
│   └── tables/
│
├── manuscript/                   # Full S.T.A.R. monograph (PDF + LaTeX)
├── theory-verification/          # PH barcodes + arithmetic point clouds
└── docs/                         # High-resolution diagrams + maps

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Reproducibility Statement

Reproducibility is a core principle of the S.T.A.R. Program.

• All projection operators
• All entropy/cohomology computations
• All TDA pipelines
• All symbolic regression constraints
• All cosmological fits

are implemented in this repository.

The v3.1 leakage‑free pipeline achieves:

$R^2 = 0.9864$ (undergoing rigorous confirmation)

on synthetic cosmic structures, validated through:

• bootstrap persistence landscapes
• Wasserstein stability
• null‑scramble rejection
• isogeny‑invariance tests

Researchers are encouraged to:

• inspect the manuscript and thesis pdf's in /docs
• run the full validation pipeline in /scripts
• explore the symbolic regression manifold in /examples/exports
• 

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Installation

git clone https://github.com/LcosmosS/S.T.A.R.-Program.git
cd ~/S.T.A.R.-Program
pip install -r requirements.txt

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Citation

If you use the S.T.A.R. Program, ACSC, ECC, or any associated data pipelines, please cite:

McNamara, P. J. (2026).
The S.T.A.R. Program: A Symbolic–Topological–Arithmetic–Relativity Model.
GitHub Repository.
https://github.com/LcosmosS/S.T.A.R.-Program

BibTeX:

@misc{mcnamara2026star,
  author       = {Patrick J. McNamara},
  title        = {The S.T.A.R. Program: A Symbolic--Topological--Arithmetic--Relativity Model},
  year         = {2026},
  howpublished = {\url{https://github.com/LcosmosS/S.T.A.R.-Program}},
  note         = {Combined ACSC + ECC Framework}
}

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