How Much Infinity Does Physics Need?

Code, data, figures and manuscript for

R. Chen, How Much Infinity Does Physics Need? Reverse Mathematics of Physical Limits and the Empirical Boundary (2026). DOI: 10.5281/zenodo.23212848

Summary

• Four equivalences over RCA₀. Each is equivalent to arithmetical comprehension ACA₀:
  • Fekete's lemma for subadditive sequences (existence of thermodynamic limits);
  • the existence of the infrared limit of every bounded monotone c-function;
  • the existence of the scaling exponent of every Cantor construction with nondecreasing dimension parameters;
  • the existence of the set of indices whose computable fractal limit sets are connected.
• Individual quantities. A limit with a computable modulus that RCA₀ verifies is provably existent in RCA₀; a limit that is not a computable real is not, because the ω-model REC contains no real equal to it.
• Moduli in the series. The Ising free energy on an L×L torus has error 0.639913/L² at K_c, matching the Ising CFT torus amplitude ln(θ₂+θ₃+θ₄)/2η = 0.639912; the diamond-lattice critical-cluster mass converges geometrically with ratio Λ₂/Λ₁ = 0.3553; the embezzlement asymptotics has error O(1/(n ln n)). The Ω approximations have no computable modulus.
• The empirical boundary. The limits that entered explicit certification budgets are provable in RCA₀; those shown to be empirically inaccessible (Ω-dimensions, limit connectivity, finite-dimensional game values) require ACA₀. Since ACA₀ is conservative over Peano arithmetic, the infinity physical limits require is real but logically mild.

Repository structure

Path	Contents
paper/	LaTeX source and compiled PDF of the manuscript
code/reverse_math_moduli.py	Script producing every number and the figure in the paper
figures/	Figure as vector PDF (used by the paper) and PNG
data/	Numerical data as CSV (metadata in # header lines)
results/	Console output of the script (the numbers quoted in the paper)

Reproducing the results

Requirements: Python ≥ 3.10 and the packages in requirements.txt (tested with Python 3.12.4, NumPy 1.26.4, SciPy 1.13.1, Matplotlib 3.8.4). Runtime is about 15 seconds.

pip install -r requirements.txt
python code/reverse_math_moduli.py      # add --show to display the figure

Run from the repository root; the figure goes to figures/ and data to data/ (override with RM_FIG_DIR, RM_DATA_DIR).

Data file	Content	Paper
ising_free_energy_error.csv	free-energy error on the L×L torus at K_c and 0.8 K_c	§4, Fig. 1
dhl_mass_convergence.csv	convergence of the normalised critical-cluster mass	§4, Fig. 1
embezzlement_error.csv	error of the embezzlement asymptotics	§4, Fig. 1
omega_certified_interval.csv	certified lower and upper bounds for Ω of binary lambda calculus	§4, Fig. 1

To rebuild the paper (pdfLaTeX, two passes):

cd paper
pdflatex Chen_2026_Reverse_Math_Physics.tex
pdflatex Chen_2026_Reverse_Math_Physics.tex

Citation

@misc{Chen2026ReverseMathPhysics,
  author = {Chen, Ruqing},
  title  = {How Much Infinity Does Physics Need? Reverse Mathematics of Physical Limits and the Empirical Boundary},
  year   = {2026},
  doi    = {10.5281/zenodo.23212848},
  url    = {https://doi.org/10.5281/zenodo.23212848}
}

License

• Code (code/): MIT License
• Manuscript, figures, data and results (paper/, figures/, data/, results/): CC BY 4.0

Contact

Ruqing Chen — GUT Geoservice Inc., Montreal — ruqing@hotmail.com