# Problem 2.8 — Exact Hypergeometric Tail Closure

**Status:** PROVED  
**Date:** July 2026

For every official column \(j=1,2,3,4\), the authoritative recurrence
satisfies

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

Equivalently, in the orientation requested by Ramanujan Challenge
Problem 2.8,

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

## Exact closure

The proof closes the former connection-functional gap through:

1. an exact nonterminating \({}_4F_3\) tail with
   \(M_Nk_{N+1}=k_N\);
2. a rank-one discrete-valuation argument giving the all-\(N\)
   Padé divisibility pattern;
3. an exact terminating adjoint \({}_4F_3\) formula for the denominator;
4. positivity at \(z_0=-1/53360^3\) and a fixed-point Cauchy bound with
   \[
   \beta=
   \frac{3125}{1307443596565949700399927}
   <4\cdot10^{-19};
   \]
5. the Chudnovsky CM value
   \(\Phi(x_0)=\sqrt{10005}/\pi\);
6. an exact Rouché separation of the characteristic quartic, a positive
   denominator lower bound, and the cyclic-frame argument transferring the
   first-column result to all four columns.

The proof is structural and does not infer equality from the earlier
\(10^{-1052}\) numerical enclosure.

## Authoritative artifacts

- `docs/proofs/PROBLEM_28_PROOF.tex`
- `docs/proofs/PROBLEM_28_PROOF.pdf`
- `experiments/ramanujan_28/submission/`
- `experiments/ramanujan_28/submission/ramanujan_challenge_problem_2_8.zip`

The primary Wolfram Language certificate contains 22 exact symbolic checks
plus a consolidated PASS conclusion. Dependency-free Python checks verify
the rank-one algebra, the Rouché inequality, and the convergence constants.
Independent SageMath certificates provide secondary exact cross-checks.

## Release verification

- Independent adversarial proof audit: **PASS**
- Wolfram exact checks: **22/22 PASS**
- Python exact checks: **PASS**
- LaTeX build: **PASS**, zero warnings
- PDF visual inspection: **PASS**, all 10 pages
- Clean ZIP extraction and PDF rebuild: **PASS**

SHA-256:

```text
PDF  a70c50287b24d13bdb113bcdbf87011dcbd698fdd4a7566ede5a8b37aeb8c2b9
ZIP  60b9d60808af129a339064e72b2ad5bd8ff9bc933c3905cf8faf821316cab91d
```