Fibonacci Cosmology: Falsified Background, Testable Perturbations

Author: Bryan David Persaud
Affiliation: Intermedia Communications Corp.
Contact: bryan@imediacorp.com

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Overview

This repository accompanies two complementary works investigating whether the Golden Ratio (φ ≈ 1.618) manifests in cosmology:

1. Background model (falsified): tests φ-recursion in the expansion history. Result: decisively ruled out by H(z), CMB low-ℓ, and P(k) data.
2. Perturbation model (falsifiable): proposes weak log-periodic modulations in structure growth, yielding φ-spaced features in the matter power spectrum and CMB. Provides concrete forecasts for near-future surveys.

The code here reproduces the core figures, basic fits, and forecast visuals, and lets you explore signals using public summary datasets.

Note on Independence: This cosmological research work is completely independent and separate from any other projects. It originated as a simple thought experiment and hypothesis that the Golden Ratio (Fibonacci sequence) might be fundamental to cosmic structure, given the self-similarity patterns observed from plants to galaxies.

Theorem overview (short)

• Golden Ratio as recursive constant: The hypothesis explores φ as a fundamental recursion constant. In a background variant (now falsified), φ enters the expansion rate via recursive scaling; in the perturbation variant, φ induces log-periodic modulations in clustering statistics.
• Duality for perturbations: Two coupled modes appear naturally — a forward (φ) and a conjugate (φ−1) branch — leading to a dual log-periodic signature. In practice, this means residuals may contain cosines periodic in log(k) or log(ℓ) with periods set by ln(φ) and ln(φ−1).
• Falsifiable prediction: If these dual log-periodic features are absent at the sensitivity of current/forthcoming surveys, the perturbation hypothesis is ruled out.

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System Requirements

Python

Python 3.9–3.12 is required.

Fortran compiler (required for CAMB)

camb compiles Fortran code on installation. You must have a Fortran compiler installed before running pip install -r requirements.txt:

macOS:

brew install gcc   # provides gfortran

Ubuntu/Debian:

sudo apt-get install gfortran libgfortran5

Conda (all platforms):

conda install -c conda-forge gfortran

If pip install camb fails with a compiler error, a missing Fortran compiler is almost always the cause.

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Installation

# 1. Clone the repository
git clone https://github.com/imediacorp/FaCC.git
cd FaCC

# 2. Install Fortran compiler (see above — required for CAMB)

# 3. Create and activate a virtual environment
python -m venv .venv
source .venv/bin/activate          # macOS/Linux
# .venv\Scripts\activate           # Windows

# 4. Install Python dependencies
pip install -r requirements.txt

# 5. (Optional) Install development tools for testing and linting
pip install -r requirements-dev.txt

Conda users:

conda create -n facc python=3.11
conda activate facc
conda install -c conda-forge gfortran
pip install -r requirements.txt

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Quick Start

from src.phi_modulation import PhiModulationModel

# Initialize model with Planck 2018 parameters
model = PhiModulationModel()

# Generate base ΛCDM power spectrum
k, z, Pk = model.get_base_power_spectrum(k_min=0.01, k_max=0.3)

# Apply φ-modulation: P_mod = P_ΛCDM × [1 + A_φ × cos(2π × log(k/k_pivot) / ln(φ) + φ_0)]
Pk_mod, mod_factor = model.apply_phi_modulation(k, Pk[0], A_phi=0.01)

# Forecast DESI sensitivity using Fisher matrix
result = model.forecast_desi_sensitivity(A_phi_true=0.01)
print(f"Forecast σ_Aφ = {result['sigma_Aphi']:.4f}, SNR = {result['SNR']:.2f}σ")

Run the forecast notebook:

jupyter notebook notebooks/01_desi_forecasts.ipynb

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Papers

• fibonacci_cosmology_falsified.pdf — Background model and its observational falsification
• fibonacci_perturbations.pdf — Perturbation-level hypothesis and detectability forecasts

If you use this repository, please cite the corresponding paper(s).

Paper summaries

• Background (falsified): A φ-recursive background expansion was tested against observations. H(z) fits, low-ℓ CMB residuals, and matter P(k) collectively rule it out with large Δχ². Key takeaway: φ is not a viable background constant for cosmic acceleration.
• Perturbations (falsifiable): Keeping ΛCDM background, introduce dual log-periodic modulations governed by φ and φ−1 in the growth of structure. Predicts φ-spaced features in P(k) around the BAO scale and small CMB residual oscillations. Forecasts suggest near-future surveys could detect or refute these signals decisively.

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Repository Structure

.
├── src/                            # Core analysis modules
│   ├── __init__.py
│   ├── phi_modulation.py           # PhiModulationModel class (CAMB-based)
│   ├── bayesian_tools.py           # BayesianEvidence, BIC utilities
│   ├── systematics.py              # SystematicErrorBudget class
│   ├── hz_analysis.py              # H(z) fitting functions (importable)
│   ├── cmb_analysis.py             # CMB oscillation detection (importable)
│   └── lss_analysis.py             # LSS P(k) analysis (importable)
├── tests/                          # pytest test suite
│   ├── __init__.py
│   ├── test_phi_modulation.py
│   ├── test_bayesian_tools.py
│   └── test_systematics.py
├── notebooks/                      # Analysis notebooks
│   └── 01_desi_forecasts.ipynb
├── examples/
│   └── systematics_example.py
├── docs/                           # Documentation and notes
├── .github/workflows/python-ci.yml
├── fibonacci_cosmology_falsified.pdf
├── fibonacci_perturbations.pdf
├── real_hz.csv
├── real_cmb_lowl.csv
├── real_pk_lowk.csv
├── hz_fitter_v2.py                 # Legacy H(z) script (logic also in src/hz_analysis.py)
├── cmb_osc_detector_v2.py          # Legacy CMB script (logic also in src/cmb_analysis.py)
├── lss_phi_analyzer.py             # Legacy P(k) script (logic also in src/lss_analysis.py)
├── requirements.txt
└── requirements-dev.txt

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Usage Guide

Core Framework

Run the DESI forecast analysis notebook:

jupyter notebook notebooks/01_desi_forecasts.ipynb

Use PhiModulationModel in your own code:

from src.phi_modulation import PhiModulationModel

model = PhiModulationModel()
# Or with custom cosmological parameters:
model = PhiModulationModel(params={
    'H0': 67.36, 'ombh2': 0.02237, 'omch2': 0.1200,
    'As': 2.1e-9, 'ns': 0.9649, 'tau': 0.0544
})

k, z, Pk = model.get_base_power_spectrum(k_min=0.01, k_max=0.3, npoints=500)
Pk_mod, mod_factor = model.apply_phi_modulation(k, Pk[0], A_phi=0.01, phi_phase=0.0, k_pivot=0.05)
r, xi_base, xi_mod = model.compute_bao_signature(z=0.5, A_phi=0.01)

forecast = model.forecast_desi_sensitivity(A_phi_true=0.01)
print(f"σ_Aφ = {forecast['sigma_Aphi']:.4f}, SNR = {forecast['SNR']:.2f}σ")

Legacy Analysis Scripts

These scripts run standalone from the repo root. Data CSVs must be present in the project root.

python hz_fitter_v2.py          # H(z) falsification → hz_comparison_dual.png
python cmb_osc_detector_v2.py   # CMB log-periodic residuals → cmb_osc_dual.png
python lss_phi_analyzer.py      # P(k) φ-scale analysis → pk_phi.png
python forecast_plots.py        # Forecast visualization → pk_forecast.png
python fetch_desi_bao_pk.py     # Reconstruct DESI P(k) → pk_data_real.csv

The same logic is also available as importable functions in src/hz_analysis.py, src/cmb_analysis.py, and src/lss_analysis.py.

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Running Tests

pip install -r requirements-dev.txt
pytest tests/ -v --cov=src --cov-report=term-missing

Tests run without CAMB, covering all pure-Python analysis logic.

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Data Descriptions

• real_hz.csv: Cosmic-chronometer H(z) catalog (columns: z, H, sigma_H).
• real_cmb_lowl.csv: Planck TT low-ℓ summary (columns: ell, C_ell, sigma).
• real_pk_lowk.csv: SDSS/BOSS P(k) for low-k scales (columns: k, P(k), sigma_Pk).
• pk_data.csv: Simple synthetic P(k) used by examples.
• pk_data_real.csv: Reconstructed P(k) produced by fetch_desi_bao_pk.py (DESI DR2-like).

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Data Sources & Attribution

• H(z) cosmic chronometers: aggregated public compilations (see file header/commit history).
• Planck low-ℓ TT: simplified summary exported to real_cmb_lowl.csv; refer to Planck Collaboration papers for authoritative spectra.
• Low-k P(k): real_pk_lowk.csv is a reduced sample for demonstrating analysis workflows.
• DESI DR2 BAO summary: fetched from data.desi.lbl.gov when internet is enabled; otherwise fetch_desi_bao_pk.py uses a small offline fallback.

Please credit the original survey teams (Planck Collaboration, SDSS/BOSS/eBOSS, DESI Collaboration).

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Using on Kaggle

• Kaggle profile: https://www.kaggle.com/bryanpersaud
• FaCC Notebook: https://www.kaggle.com/code/bryanpersaud/faac-notebook/

!pip -q install numpy scipy matplotlib pandas astropy sympy pillow requests beautifulsoup4

Upload real_hz.csv, real_cmb_lowl.csv, and real_pk_lowk.csv as Kaggle Dataset files if internet is off.

Run in Google Colab

!pip install -q numpy scipy matplotlib pandas astropy camb emcee corner
!git clone https://github.com/imediacorp/FaCC.git
%cd FaCC

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Reproducibility

• Scripts are deterministic given the fixed input CSVs; random seeds are not used.
• Baselines: Some routines approximate ΛCDM trends (e.g., polynomial baseline for low-ℓ CMB) to isolate residuals; these are analysis conveniences, not full Boltzmann codes.
• Units: P(k) is plotted in customary units; axes are labeled in each figure.

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License

MIT License — see LICENSE.

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Citation

@misc{persaud_fibonacci_cosmology,
  author       = {Bryan David Persaud},
  title        = {Fibonacci Cosmology: Falsified Background, Testable Perturbations},
  year         = {2025},
  howpublished = {GitHub repository},
  url          = {https://github.com/imediacorp/FaCC}
}

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Project Independence

This cosmological research work is completely independent and separate from any other projects. See INDEPENDENCE.md for a detailed statement.

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Troubleshooting

ModuleNotFoundError: No module named 'camb'
CAMB requires a Fortran compiler. Install gfortran first (see System Requirements above), then pip install camb. The dashboard H(z), CMB, LSS, DESI BAO, and G–φ tabs all work without CAMB.

ImportError: mach-o file, but is an incompatible architecture (Apple Silicon)
You have an x86_64 package installed in an arm64 Python environment (or vice versa). Create a fresh virtual environment with the native arm64 Python:

/usr/bin/python3 -m venv .venv && source .venv/bin/activate
pip install -r requirements.txt

ssl.SSLCertVerificationError on macOS Python 3.13
Run the certificate installer bundled with Python:

open /Applications/Python\ 3.13/Install\ Certificates.command

scipy.optimize fails to converge in CMB fit
The dual log-periodic fit can be sensitive to initial conditions. If curve_fit raises RuntimeError, your CMB data range may be too narrow. Ensure real_cmb_lowl.csv covers ℓ = 2–30.

DESI BAO tab shows a network error
The tab fetches live data from GitHub. Check your internet connection. Data is cached for 1 hour per session — refreshing the page will retry.

Tests fail with ModuleNotFoundError: No module named 'src'
Run pytest from the repo root:

cd /path/to/FaCC && pytest tests/ -v

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Contributing

See CONTRIBUTING.md for the PR process, code style guide, and testing requirements.

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Support

Questions or issues: please open a GitHub issue or email bryan@imediacorp.com.