rlds-paired

Regime-Limited Dynamical Systems — Paired Dark-Sector Signature

A Python library for reproducing the numerical results of the RLDS cosmology series (Papers III–V). rlds-paired is the observational–cosmological branch of the RLDS SDK family.

The central result is the paired growth–expansion signature:

Δ(f·σ₈) / (f·σ₈)_ΛCDM  =  c · K + O(K²)
c  ≈  −0.19      (negative: growth suppression)

The coefficient is computed from the model code, not stored as a literal — recompute it with rlds_paired.reference.reference_results().

Theory references

This library implements the methods from:

1. Kubanski, A. (2026). Paired observational signature of the RLDS dark sector: growth suppression and phantom excursion. Zenodo. doi:10.5281/zenodo.19636371 [Paper III]

2. Kubanski, A. (2026). Cosmological consequences under RLDS closure. Zenodo. doi:10.5281/zenodo.19640061 [Paper IV]

3. Kubanski, A. (2026). Microphysical descent to RLDS. Zenodo. doi:10.5281/zenodo.19640630 [Paper V]

The mathematical core (Papers I–II) is implemented in rlds-mat (doi:10.5281/zenodo.20342965).

Features

• Algebraic IDE model: E²(z) with tanh-transition, self-consistent R(z), phantom w_eff(z).
• Full ODE trajectory: Coupled [R, u] two-phase system (Paper IV), global attractor R* ≈ 0.103.
• Growth factor: Linear growth f·σ₈(z) on the full coupled-ODE RLDS background (Paper III).
• K-scan: Paired-signature coefficient scan; linear scaling with a negative slope (growth suppression).
• Spectral gap: δ(N) = |dR/dN|/3 validation (Paper V, Appendix C). The Fenichel persistence bound (0.10) is reported pass/fail; at the canonical benchmark it is violated (δ_max ≈ 0.245), so the slow-manifold reduction is a marginal (~25%) correction in the transition window.
• CLI: rlds-paired verify | scan | spectral | fsigma8.
• Canonical inputs: Paper III–V benchmark parameters in one place; computed results live in rlds_paired.reference.

Installation

pip install rlds-paired

# Development install
pip install -e ".[dev]"

Quick start

From Python

import rlds_paired

# Canonical benchmark *inputs* (computed results live in rlds_paired.reference)
print(rlds_paired.CANONICAL)
# {'K': 0.0106, 'alpha': 3.0102, 'Rc': 0.565, 'Omega_m': 0.297, ...}

# ODE trajectory (Paper IV)
traj = rlds_paired.RLDSTrajectory()
print(f"R* = {traj.R_star:.4f}")        # 0.1027  (attractor, ≈ sqrt(K))
print(f"R(z=0) = {traj.R_today:.4f}")   # 0.2974  (present-day matter fraction)
print(f"E(z=0.5) = {traj.E(0.5):.4f}")

# Spectral gap validation (Paper V, Appendix C)
sg = rlds_paired.compute_spectral_gap()
print(sg.summary())
# delta_max ≈ 0.245 at z ≈ 0.42  →  Fenichel bound (0.10) VIOLATED (expected)

# Recomputed headline results (paired coefficient, delta_max)
print(rlds_paired.reference_results().summary())

# Growth factor (Paper III)
fs8 = rlds_paired.growth_factor(rlds_paired.OMEGA_M, K=rlds_paired.K_BEST, z_eval=0.5)
print(f"f·σ₈(z=0.5) = {fs8:.4f}")

# K-scan
result = rlds_paired.k_scan(n_points=20)
print(result.summary())

# f·σ₈ comparison figure
fs8_res = rlds_paired.fsigma8_comparison()
fs8_res.save_figure("fsigma8.png")

From the command line

# Verify installation
rlds-paired verify

# Spectral gap table (Paper V, Appendix C)
rlds-paired spectral --table

# K-scan (20 points)
rlds-paired scan --n 20

# f·σ₈ comparison figure
rlds-paired fsigma8 --figure fsigma8.png

Reproducing the figures

A full map from each paper's figures/results to the exact entry point is in REPRODUCE_FIGURES.md. In short: Paper III's paired coefficient and K-scan come from rlds_paired.analysis.k_scan(); Paper IV's R*, z_acc, and w_eff from rlds_paired.core.ode / rlds_paired.core.algebraic; and Paper V's spectral gap (δ_max, Figure 5) from rlds_paired.core.spectral.compute_spectral_gap().

Paper V microphysical reproduction

Paper V's microphysical validation figures (Figures 1–4) are regenerated by a dedicated script bundled in the source archive:

python scripts/paper_v/fig5_generate.py --out outputs/paper_v

It writes four figures (PDF + PNG), four CSVs, and a machine-readable fig5_summary.json. The recipe (λ=0.5, β_*=6e-4, Δφ=0.3, φ_c=0.85, V₀≈1.886e-5 by shooting) and the RLDS branch are evaluated on the DR2 canonical frame. See scripts/paper_v/README_PAPER_V_REPRODUCTION.md for outputs and expected key values. (Paper V's Figure 5 / δ_max is reproduced by rlds_paired.core.spectral, not by this script.)

Model-comparison reference values and external data policy

The DR2 model comparison reports the RLDS shooting form as disfavored relative to ΛCDM, with Δχ² ≈ +36 and ΔBIC ≈ +58. The sign convention is:

Δχ²  = χ²_RLDS  − χ²_LCDM
ΔBIC = BIC_RLDS − BIC_LCDM

so positive Δχ² / ΔBIC means the RLDS fit is disfavored relative to ΛCDM for the corresponding comparison.

The raw Pantheon+ / Pantheon+SH0ES dataset is not redistributed in this archive: it is an external third-party dataset. A full likelihood rerun requires the user to obtain it from the official source and place it under the documented local data path. What this archive does include from the original run is the set of derived artifacts: archived chains, summary arrays, chains/omega_m_derived.npy, and a reference model-comparison file:

data/reference/model_comparison_reference.json

These make the paper-level Δχ² / ΔBIC claims traceable, not fully re-derivable from the archive alone — the comparison is supernova-inclusive, so the likelihood-level values are stored reference numbers from the papers rather than quantities the SDK recomputes at run time. The model-level results (paired coefficient, spectral gap, attractor, present-day matter fraction) remain fully reproducible from the code, as described above.

Canonical benchmark values

Quantity	Value	Source
Coupling K	0.0106	Paper IV, DR2 shooting MCMC
Dilution rate α	3.0102	Paper IV (DR2; railed at prior floor 3.01)
Switching threshold R_c	0.565	Paper IV
Switching width ΔR	0.170	Paper IV, DR2 shooting best-fit
Derived Ω_m	0.297	Paper IV, self-consistency
Attractor R*	0.1027 (≈ √K)	Paper IV
Present-day matter fraction R(z=0)	0.2974	computed
Equation of state w_DE	−1.003	Paper IV
Acceleration onset z_acc	0.7144	Paper IV, q(z)=0 on DR2 background
Paired coefficient c	≈ −0.1936 ± 0.005 (computed; r² ≈ 0.97)	Paper III
Spectral gap δ_max	≈ 0.245 — Fenichel (0.10) violated	Paper V, Appendix C

Package structure

rlds_paired/
├── _version.py       ← single source of truth for version
├── constants.py      ← all canonical benchmark values
├── exceptions.py     ← custom exception hierarchy
├── analysis.py       ← k_scan(), fsigma8_comparison()
├── cli.py            ← rlds-paired command-line tool
└── core/
    ├── algebraic.py  ← algebraic IDE model (E², R, F, w_eff)
    ├── ode.py        ← two-phase coupled [R, u] ODE (Paper IV)
    ├── growth.py     ← linear growth factor (Paper III)
    └── spectral.py   ← spectral gap δ(N) = |dR/dN|/3 (Paper V)

SDK family

Library	Papers	Description
rlds-mat	I–II	Mathematical core: ODE analysis, attractor proofs
rlds-paired	III–V	Observational: paired signature, growth, spectral gap

Author

Aleksander Kubanski Independent Researcher, Poland ORCID: 0009-0006-7239-5074 Research site: research.kubanski.pro

License

MIT License. See LICENSE.