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 first display is the orientation requested in Problem 2.8; the second is its reciprocal consequence.

Contents

• solution.pdf — the complete proof.
• solution.tex — its LaTeX source.
• certificates/p28_standalone_equations.py — mandatory, dependency-free expansion of the four cleared Ore identities, tail coefficient equations, terminating base/generic/top identities, ascension, and the hypergeometric differential equation. It uses rational coefficient dictionaries only: no division algorithm, simplifier, factorizer, special function library, root finder, or sample values.
• certificates/p28_dominant_product_algebra.py — mandatory, dependency-free verification of the balanced limit, characteristic polynomial, root-separation inequalities, left-eigenvector identity, and four positive-coordinate rewrites.
• certificates/STANDALONE_EQUATION_CERTIFICATES.md — the same Ore and terminating identities in a human-readable, denominator-cleared equation sheet.
• certificates/p28_full_closure_certificate.wl — optional independent symbolic cross-check of the differential gauge and closure.
• 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.
• ADVERSARIAL_AUDIT.md — defect ledger, repairs, replay evidence, and the exact trust boundary.

Reproduction

From this directory, run:

./run_checks.sh

The mandatory proof path is Python-standard-library only. The Wolfram cross-check can also be run directly:

wolframscript -file certificates/p28_full_closure_certificate.wl

The Python checks use only the standard library:

python3 certificates/p28_rank_ode_bound_verifier.py
python3 certificates/p28_convergence_constants.py
python3 certificates/p28_standalone_equations.py
python3 certificates/p28_dominant_product_algebra.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

Trust boundary

No numerical enclosure is used to infer equality. The recurrence proof is expanded into explicit equations and an elementary stable-graph contraction. The sole imported mathematical theorem is the classical Chudnovsky formula, identified precisely in solution.tex with a reference to a complete modular/CM derivation. Thus the package is self-contained relative to that published theorem; it does not claim to reconstruct the entire theory of the Chudnovsky formula from first principles.