437b439c6c
Portable proof handoff intended for verified mirror base 1229ab9e61bee936cb1a29c0693ee56922d2d908.
73 lines
2.2 KiB
Markdown
73 lines
2.2 KiB
Markdown
# Problem 2.8 — Exact Hypergeometric Tail Closure
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**Status:** PROVED
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**Date:** July 2026
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For every official column \(j=1,2,3,4\), the authoritative recurrence
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satisfies
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\[
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\lim_{N\to\infty}\frac{P_{N,j}}{Q_{N,j}}
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=\frac{\sqrt{10005}}{\pi}.
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\]
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Equivalently, in the orientation requested by Ramanujan Challenge
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Problem 2.8,
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\[
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\boxed{\displaystyle
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\lim_{N\to\infty}\frac{Q_{N,j}}{P_{N,j}}
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=\frac{\pi}{\sqrt{10005}}}.
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\]
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## Exact closure
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The proof closes the former connection-functional gap through:
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1. an exact nonterminating \({}_4F_3\) tail with
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\(M_Nk_{N+1}=k_N\);
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2. a rank-one discrete-valuation argument giving the all-\(N\)
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Padé divisibility pattern;
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3. an exact terminating adjoint \({}_4F_3\) formula for the denominator;
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4. positivity at \(z_0=-1/53360^3\) and a fixed-point Cauchy bound with
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\[
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\beta=
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\frac{3125}{1307443596565949700399927}
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<4\cdot10^{-19};
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\]
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5. the Chudnovsky CM value
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\(\Phi(x_0)=\sqrt{10005}/\pi\);
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6. an exact Rouché separation of the characteristic quartic, a positive
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denominator lower bound, and the cyclic-frame argument transferring the
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first-column result to all four columns.
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The proof is structural and does not infer equality from the earlier
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\(10^{-1052}\) numerical enclosure.
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## Authoritative artifacts
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- `docs/proofs/PROBLEM_28_PROOF.tex`
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- `docs/proofs/PROBLEM_28_PROOF.pdf`
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- `experiments/ramanujan_28/submission/`
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- `experiments/ramanujan_28/submission/ramanujan_challenge_problem_2_8.zip`
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The primary Wolfram Language certificate contains 22 exact symbolic checks
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plus a consolidated PASS conclusion. Dependency-free Python checks verify
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the rank-one algebra, the Rouché inequality, and the convergence constants.
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Independent SageMath certificates provide secondary exact cross-checks.
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## Release verification
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- Independent adversarial proof audit: **PASS**
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- Wolfram exact checks: **22/22 PASS**
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- Python exact checks: **PASS**
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- LaTeX build: **PASS**, zero warnings
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- PDF visual inspection: **PASS**, all 10 pages
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- Clean ZIP extraction and PDF rebuild: **PASS**
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SHA-256:
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```text
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PDF a70c50287b24d13bdb113bcdbf87011dcbd698fdd4a7566ede5a8b37aeb8c2b9
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ZIP 60b9d60808af129a339064e72b2ad5bd8ff9bc933c3905cf8faf821316cab91d
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```
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