ramanujan-challenge/experiments/ramanujan_28/submission

Ramanujan Challenge, Problem 2.8

This package proves, for each of the four official columns,

[ \lim_{N\to\infty}\frac{P_{N,j}}{Q_{N,j}} =\frac{\sqrt{10005}}{\pi}, \qquad \lim_{N\to\infty}\frac{Q_{N,j}}{P_{N,j}} =\frac{\pi}{\sqrt{10005}}. ]

The second display is the orientation requested in Problem 2.8.

Contents

• solution.pdf — the complete proof.
• solution.tex — its LaTeX source.
• certificates/p28_full_closure_certificate.wl — the primary, self-contained exact symbolic certificate. It proves the authoritative differential gauge, both hypergeometric contiguity identities, the terminating denominator formula, CM-seed annihilation, and the singular lattice step. No numerical sampling is used.
• certificates/p28_full_closure_certificate.PASS.txt — transcript of a stateless Wolfram Language run (22 exact checks plus the consolidated conclusion).
• certificates/p28_convergence_constants.py and certificates/p28_rank_ode_bound_verifier.py — dependency-free exact rational checks for the fixed-point convergence bound.
• certificates/p28_kernel_contiguity_certificate.sage, certificates/p28_lattice_hypotheses_certificate.sage, and certificates/all_four_columns_certificate.sage — independent exact SageMath cross-checks.
• certificates/p28_parametric_pade_probe.py — finite exact regression, included as a diagnostic only and not used as proof.

Reproduction

From this directory, run:

./run_checks.sh

The primary symbolic check can also be run directly:

wolframscript -file certificates/p28_full_closure_certificate.wl

It should print 22 exact-check lines beginning with PASS:, followed by the consolidated certificate conclusion. The Python checks use only the standard library:

python3 certificates/p28_rank_ode_bound_verifier.py
python3 certificates/p28_convergence_constants.py

For the independent SageMath checks:

sage certificates/p28_kernel_contiguity_certificate.sage
sage certificates/p28_lattice_hypotheses_certificate.sage
sage certificates/all_four_columns_certificate.sage

To rebuild the manuscript:

latexmk -pdf solution.tex