a₀(z): Epoch-Dependent MOND Acceleration Scale from Bounded Topology

Code and data for the paper:

B. Shatto, "Epoch-Dependent Acceleration Scale from Bounded Topology: Predictions for High-Redshift Galactic Dynamics" (2026).

This repository contains the analysis pipeline, prediction tables, and figure-generation scripts needed to reproduce all numerical results in the paper.

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Overview

The paper derives an epoch-dependent MOND acceleration scale

$$a_0(z) = a_0(0) \cdot E(z)$$

within a bounded-topology measurement framework, where $E(z) = H(z)/H_0$ is the Friedmann dimensionless Hubble parameter. The local Milgrom ratio $a_0/(cH_0)$ is predicted as the structural ratio of two phase-operator values at Fibonacci wells of the framework's 120-domain, $C(13/120)/C(34/120) = 0.1845$, agreeing with observation at the 0.7% level.

Five distinct observable channels follow from the single relation $a_0(z) \propto E(z)$ as different powers of $E(z)$:

• BTFR normalization shifts as $E(z)^{-1}$ (§3.1)
• MOND transition radius contracts as $E(z)^{-1/2}$ (§3.2)
• Asymptotic flat velocity at fixed baryonic mass rises as $E(z)^{+1/4}$ (§3.2)
• Lensing-inferred $M_\text{dyn}/M_b$ enhancement scales as $E(z)^{+1/2}$ (§3.3)
• Free-fall collapse-time speedup as $E(z)^{-1/4}$ (§3.4, supporting evidence)

Reproducible results in this repository:

• a₀(z) tabulation across $z = 0$ to $15$ (Table 1, §3)
• BTFR evolution at six reference redshifts (Table 2, §3.1)
• Rotation-curve archetype predictions for four baryonic-mass anchors (Table 3, §3.2)
• Lensing $M_\text{dyn}/M_b$ at three apertures across five redshifts (Table 4, §3.3)
• Five-prediction summary at $z = 2$ (Table 5, §3.5)
• Constraint-status summary across five regimes (Table 6, §4.5)
• Falsification criteria for each predicted exponent (Table 7, §5)
• Near-term test schedule (Table 8, §5)
• ΛCDM lensing discriminator with pess/repr/opt bracketing (Table B.1, App B.3)
• Übler 2017 KMOS3D σ-tension under three uncertainty budgets (§4.1)
• Übler 2017 forward-model bias sweep (four-bias Monte Carlo: RPU, BS, NA, SI) showing no literature-plausible combination reproduces the observed BTFR pattern (§4.1)
• CMB leakage-coupling bound $\varepsilon \leq 1.2 \times 10^{-5}$ at 0.5% Planck tolerance (§4.3)
• Combinatorial baseline for the Milgrom-ratio match: 24 of 7,021 pairs match within 1%; 6 unique value-pairs after reflection collapse; 1 of 6 within the Fibonacci subset ${13, 21, 34, 55}/120$ (§2)
• Labbé candidate speedup factors $E(z)^{1/4} \in [1.92, 2.06]$ (§3.4)

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Citation

If you use this code or data, please cite the paper and the Zenodo archive of this repository.

The repository is archived on Zenodo:

• Concept DOI (resolves to the latest version): 10.5281/zenodo.19980665
• Version DOI (this release, v1.2.0): 10.5281/zenodo.20045115

For paper citations and machine-readable use the concept DOI; it auto-resolves to the latest version. For citing the exact code state used in a specific analysis, use the version DOI.

@software{ShattoA0ZCode2026,
  author = {Shatto, B.},
  title  = {a0z: Code and Data for "Epoch-Dependent Acceleration
            Scale from Bounded Topology"},
  year   = {2026},
  doi    = {10.5281/zenodo.19980665},
  url    = {https://doi.org/10.5281/zenodo.19980665},
  note   = {Concept DOI; resolves to the latest version}
}

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Repository contents

.
├── README.md                            This file
├── LICENSE                              MIT
├── requirements.txt                     Python dependencies
├── data/
│   ├── README.md                        Provenance and reference for observational inputs
│   └── labbe_2023_candidates.csv        Labbé et al. 2023 candidate IDs and photometric redshifts
├── scripts/
│   ├── README.md                        Module overview and verification target list
│   ├── cosmology.py                     E(z), H(z), t_age(z) under flat ΛCDM (Planck 2018)
│   ├── framework.py                     C(Θ), N_H, scaling-law evaluation, well assignments
│   ├── combinatorial_baseline.py        §2 sparsity check on the 120-domain
│   ├── btfr.py                          §3.1 BTFR normalization and velocity scaling
│   ├── rotation_curves.py               §3.2 r_M, v_flat at archetype masses
│   ├── lensing.py                       §3.3 + App B M_dyn/M_b, ΛCDM NFW comparison
│   ├── jwst_speedup.py                  §3.4 free-fall collapse, Labbé candidates
│   ├── ubler_sigma_tension.py           §4.1 analytic σ-tension table
│   ├── ubler_forward_model.py           §4.1 four-bias Monte Carlo forward analysis
│   ├── cmb_leakage.py                   §4.3 ε bound from Planck first-peak amplitude
│   ├── verify.py                        Single-command pass/fail harness across all modules
│   ├── make_tables.py                   Generate Tables 1–8 + Table B.1 from the analysis modules
│   └── make_figures.py                  Generate the four paper figures from the analysis modules
├── figures/
│   ├── figure1.{pdf,png}                §2 Milgrom-ratio sparsity on the 120-domain
│   ├── figure2.{pdf,png}                §3.3 + App B Euclid DR1 lensing discriminator
│   ├── figure3.{pdf,png}                §3.5 five exponents from one input
│   └── figure4.{pdf,png}                §4.1 Übler trend-shape tension
└── tables/
    ├── table{1..8}.csv                  One CSV per paper table (§3 / §3.1 / §3.2 / §3.3 / §3.5 / §4.5 / §5 / §5)
    └── tableB1.csv                      App B.3 ΛCDM bracketing at L*, R = 100 kpc

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Installation

Python 3.10 or later recommended.

git clone https://github.com/dmobius3/a0z.git
cd a0z
python -m venv venv
source venv/bin/activate
pip install -r requirements.txt

Dependencies (requirements.txt):

numpy>=1.24
scipy>=1.10
matplotlib>=3.7
pandas>=2.0

Total install footprint about 100 MB. The full pipeline (all tables + all figures) runs in under a minute on a recent laptop. No MCMC; the predictions are deterministic given the framework's well assignments and the standard ΛCDM cosmology.

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Verification

A single command checks every numerical claim in the paper against the corresponding script output:

python scripts/verify.py

Exit code 0 plus All modules report green against the paper. means every assertion the paper depends on is reproducible from the scripts. Exit code 1 prints the failure list module-by-module with the relevant stdout tail.

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Quickstart: reproducing all paper numbers

# Generate Tables 1–8 (and Table B.1)
python scripts/make_tables.py

# Generate paper figures
python scripts/make_figures.py

Each analysis module also runs standalone and prints its own per-claim verification. Examples:

python scripts/cosmology.py            # §3 cosmology + Table 1 row spot-checks
python scripts/framework.py            # §2 phase-operator ratios
python scripts/combinatorial_baseline.py  # §2 sparsity (accepts --target, --tol, --list-matches)
python scripts/btfr.py                 # §3.1 Table 2 + L* worked example
python scripts/rotation_curves.py      # §3.2 archetype rotation curves
python scripts/lensing.py              # §3.3 + App B M_dyn/M_b
python scripts/jwst_speedup.py         # §3.4 Labbé speedup factors
python scripts/ubler_sigma_tension.py  # §4.1 σ-tension table
python scripts/ubler_forward_model.py  # §4.1 four-bias forward-model sweep
python scripts/cmb_leakage.py          # §4.3 ε bound

Pipeline runtime is well under one minute on a recent laptop. No fitting, no MCMC; predictions are deterministic given the Planck 2018 cosmology and the framework's $z = 0$ well assignments ($\Theta_{a_0} = 13/120$, $\Theta_H = 34/120$).

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Data sources and provenance

Dataset	Source	Reference
SPARC rotation curves	astroweb.cwru.edu/SPARC	Lelli, McGaugh, Schombert, Astron. J. 152, 157 (2016)
Übler 2017 KMOS3D BTFR	Tabulated from publication	Übler et al., Astrophys. J. 842, 121 (2017)
Labbé 2023 candidate sample	Public Nature catalog	Labbé et al., Nature 616, 266 (2023)
Wisnioski 2015 σ_0 medians	KMOS3D survey paper	Wisnioski et al., Astrophys. J. 799, 209 (2015)
Planck 2018 cosmology	pla.esac.esa.int	Planck Collaboration VI, Astron. Astrophys. 641, A6 (2020)

The Labbé candidate file under data/ is a formatted derivative of the public Nature supplement above, repackaged for direct loading by jwst_speedup.py. No proprietary data is redistributed; the SPARC, Übler, Wisnioski, and Planck values used elsewhere in the pipeline are taken directly from the cited publications and hardcoded in the relevant scripts.

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Reproducibility notes

• No fitting: the predictions in this paper are deterministic given the framework's well assignments at $z = 0$ ($\Theta_{a_0} = 13/120$, $\Theta_H = 34/120$) and the standard ΛCDM cosmology (Planck 2018). There are no MCMC chains to converge.
• No Monte Carlo: every prediction in this pipeline is closed-form. The §4.1 σ-tension table is analytic; the §3.3 / Appendix B ΛCDM comparison uses closed-form NFW enclosed mass; the §3.4 speedup factors are direct evaluations of $E(z)^{1/4}$.
• Numerical accuracy: the cosmic-age integral uses SciPy's quad with epsrel=1e-8; NFW enclosed-mass evaluation is closed-form.
• Random seeds: where Monte Carlo is added in subsequent modules (e.g., a forward-model variant of the §4.1 Übler analysis), every script accepts a --seed argument; default is 42 for deterministic outputs.
• Platform: tested on macOS 14.x and Ubuntu 22.04 with Python 3.11. No GPU required.

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License

This repository is released under the MIT License. See LICENSE for the full text.

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Contact

• Author: B. Shatto, bshatto.pe@gmail.com
• Issues and questions: please open a GitHub issue.