# Problem 2.8 — Exact Hypergeometric Tail Closure

**Status:** exact proof, adversarially audited
**Date:** July 2026

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

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

Its reciprocal consequence is

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

## Exact closure

The proof closes the former connection-functional gap through:

1. the fully displayed rational deformation
   \(M_N(x)=\mathcal M(2N+3,x)\);
2. a nonterminating \({}_4F_3\) tail with
   \(M_Nk_{N+1}=k_N\);
3. four denominator-cleared Ore factorizations, with no division or
   remainder command;
4. a rank-one discrete-valuation induction giving the all-\(N\)
   Padé divisibility pattern;
5. an exact terminating adjoint \({}_4F_3\), proved by a matrix-induced
   scalar step and base/generic/top coefficient induction;
6. positivity at \(z_0=-1/53360^3\) and a Cauchy bound with
   \[
   \beta=
   \frac{3125}{1307443596565949700399927}
   <4\cdot10^{-19};
   \]
7. the Chudnovsky CM value
   \(\Phi(x_0)=\sqrt{10005}/\pi\);
8. an explicit stable-graph contraction constructing the dominant
   functional and proving eventual nonvanishing in all four columns.

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

## Equation-only audit repairs

The repaired release removes:

- Ore `quo_rem` calls from the proof path;
- ODE-normalization uniqueness;
- Birkhoff–Poincaré delegation;
- irreducibility and polynomial-GCD decisions;
- a false negative-exponent norm inequality;
- implicit maximum-principle and denominator-nonvanishing hypotheses;
- reliance on stored PASS transcripts.

The mandatory standard-library checkers verify the full differential gauge,
base horizontal row, matrix-to-scalar bridge, terminating induction,
balanced limit, characteristic polynomial, root-separation inequalities, and
positive exterior-root eigenvector coordinates.

## Trust boundary

The classical Chudnovsky formula is the sole imported problem-specific
theorem and is cited precisely to a complete modular/CM derivation. Standard
foundational complex- and linear-analysis results are used with their
hypotheses displayed. The package therefore claims a self-contained
recurrence proof relative to that explicit theorem—not an axiom-free
reconstruction of all of complex analysis or CM theory.

## Authoritative artifacts

- `docs/proofs/PROBLEM_28_PROOF.tex`
- `docs/proofs/PROBLEM_28_PROOF.pdf`
- `experiments/ramanujan_28/submission/solution.tex`
- `experiments/ramanujan_28/submission/solution.pdf`
- `experiments/ramanujan_28/submission/ADVERSARIAL_AUDIT.md`
- `experiments/ramanujan_28/submission/certificates/STANDALONE_EQUATION_CERTIFICATES.md`
- `experiments/ramanujan_28/submission/ramanujan_challenge_problem_2_8.zip`

## Release verification

- Mandatory dependency-free equations: **PASS**
- Fresh optional Wolfram cross-check: **22/22 PASS**
- Independent Ore/special-function audit: **PASS**
- Independent convergence/all-column audit: **PASS**
- Independent logic/vacuity audit: **PASS**
- LaTeX build and warning scan: **PASS**
- PDF page-by-page inspection: **PASS**
- Clean ZIP extraction, checks, and PDF rebuild: **PASS**
- Clean Git-bundle clone and checks: **PASS**

Final SHA-256 values are recorded in `Sha256.txt` beside the released
artifacts.