Empirical Structure of the Gilbreath Decay Constants

Companion repository for the note Empirical Structure of the Gilbreath Decay Constants (M. M. Ross, July 2026, zenodo.21326025), a computational study of open questions raised by Chase, Hunter, and Tao (CHT), Gilbreath's conjecture: a Cramér random model and a deterministic analysis (arXiv:2607.08712). OEIS: A397880 and A395556.

For the CHT stationary model (iid Exp(1) top row, c_i = E a(i,j)):

1. Digit-sum law. c_i ≈ C·λ^{s₂(i)}/i, where s₂(i) is the binary digit sum. The 1/i envelope holds within fixed digit-sum classes; the effective λ drifts through ≈1.14–1.20 at accessible depths, so no closed-form constant is claimed. At extreme digit sums the modulation saturates below its geometric extrapolation.
2. New exact values. c₄ = 778959731701/1447295850000, plus exact c₅ and c₆ (see data/exact_values.json), extending the exactly computed values of CHT (which ends at c₃ = 227/288). Certified by an exact partition-of-unity identity and independent Monte Carlo.
3. Growth threshold. For a_j ~ Unif[0, R(j)], every tested polynomial rate is subcritical at accessible depths while every tested exponential rate (down to 2^{j/64}) is supercritical — probing the linear-vs-exponential gap CHT describe as difficult to narrow. This is a family-specific finding, not a worst-case one.
4. Transient laws. Full-row grind-down time τ(G) ≍ G^{0.63–0.66} (not logarithmic); a spike of amplitude G in a diverse background decays at ≈1 unit per column and survives to distance d*(G) ≈ G.

plots/    generated figures
src/      all generating code
data/     raw Monte Carlo data, figure data, exact-value certificates

pip install -r requirements.txt

# smoke tests (exact pipeline must reproduce CHT's values)
python3 src/exact_ci.py 2        # -> 7/9
python3 src/exact_ci.py 3        # -> 227/288

# new exact constants
python3 src/exact_ci.py 4
python3 src/exact_ci.py 5
python3 src/exact_ci.py 6 --workers 8    # ~2M sign patterns

# analyses and figures from the shipped data
python3 src/lambda_analysis.py           # run from data/ or adjust paths
python3 src/make_figures.py

# regenerate raw experiments (stochastic; seeds fixed in-script)
python3 src/cht_experiments.py
python3 src/grind_down.py

Note on exact_ci.py: it requires pycddlib built with GMP (import cdd.gmp must succeed). Official Windows wheels lack GMP; use WSL/Linux (apt install libcdd-dev libgmp-dev && pip install pycddlib) or a conda environment providing it. Without GMP the script refuses to run unless --allow-float is passed, and results are then approximate.

The exact c₄–c₆ runs print a volume check: 1 line — an exact partition-of-unity identity over all sign cones that serves as the correctness certificate for each value.

• Companion note on the parity mechanism: Is Gilbreath's conjecture garden-variety numerology?
• Z. Chase, A random analogue of Gilbreath's conjecture, Math. Ann. 388 (2024), 2611–2625.

MIT (see LICENSE).