# 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:

```sh
./run_checks.sh
```

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

```sh
wolframscript -file certificates/p28_full_closure_certificate.wl
```

The Python checks use only the standard library:

```sh
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:

```sh
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:

```sh
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.