ramanujan-challenge/docs/proofs/PROBLEM_28_COMPLETE_PROOF.md

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:

PDF  a70c50287b24d13bdb113bcdbf87011dcbd698fdd4a7566ede5a8b37aeb8c2b9
ZIP  60b9d60808af129a339064e72b2ad5bd8ff9bc933c3905cf8faf821316cab91d