99 lines
3.4 KiB
Markdown
99 lines
3.4 KiB
Markdown
# Problem 2.8 — Exact Hypergeometric Tail Closure
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**Status:** exact proof, adversarially audited
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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 the official orientation
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\[
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\boxed{\displaystyle
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\lim_{N\to\infty}\frac{P_{N,j}}{Q_{N,j}}
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=\frac{\sqrt{10005}}{\pi}}.
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\]
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Its reciprocal consequence is
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\[
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\lim_{N\to\infty}\frac{Q_{N,j}}{P_{N,j}}
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=\frac{\pi}{\sqrt{10005}}.
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\]
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## Exact closure
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The proof closes the former connection-functional gap through:
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1. the fully displayed rational deformation
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\(M_N(x)=\mathcal M(2N+3,x)\);
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2. a nonterminating \({}_4F_3\) tail with
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\(M_Nk_{N+1}=k_N\);
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3. four denominator-cleared Ore factorizations, with no division or
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remainder command;
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4. a rank-one discrete-valuation induction giving the all-\(N\)
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Padé divisibility pattern;
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5. an exact terminating adjoint \({}_4F_3\), proved by a matrix-induced
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scalar step and base/generic/top coefficient induction;
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6. positivity at \(z_0=-1/53360^3\) and a 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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7. the Chudnovsky CM value
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\(\Phi(x_0)=\sqrt{10005}/\pi\);
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8. an explicit stable-graph contraction constructing the dominant
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functional and proving eventual nonvanishing in 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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## Equation-only audit repairs
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The repaired release removes:
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- Ore `quo_rem` calls from the proof path;
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- ODE-normalization uniqueness;
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- Birkhoff–Poincaré delegation;
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- irreducibility and polynomial-GCD decisions;
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- a false negative-exponent norm inequality;
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- implicit maximum-principle and denominator-nonvanishing hypotheses;
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- reliance on stored PASS transcripts.
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The mandatory standard-library checkers verify the full differential gauge,
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base horizontal row, matrix-to-scalar bridge, terminating induction,
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balanced limit, characteristic polynomial, root-separation inequalities, and
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positive exterior-root eigenvector coordinates.
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## Trust boundary
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The classical Chudnovsky formula is the sole imported problem-specific
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theorem and is cited precisely to a complete modular/CM derivation. Standard
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foundational complex- and linear-analysis results are used with their
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hypotheses displayed. The package therefore claims a self-contained
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recurrence proof relative to that explicit theorem—not an axiom-free
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reconstruction of all of complex analysis or CM theory.
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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/solution.tex`
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- `experiments/ramanujan_28/submission/solution.pdf`
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- `experiments/ramanujan_28/submission/ADVERSARIAL_AUDIT.md`
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- `experiments/ramanujan_28/submission/certificates/STANDALONE_EQUATION_CERTIFICATES.md`
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- `experiments/ramanujan_28/submission/ramanujan_challenge_problem_2_8.zip`
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## Release verification
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- Mandatory dependency-free equations: **PASS**
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- Fresh optional Wolfram cross-check: **22/22 PASS**
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- Independent Ore/special-function audit: **PASS**
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- Independent convergence/all-column audit: **PASS**
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- Independent logic/vacuity audit: **PASS**
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- LaTeX build and warning scan: **PASS**
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- PDF page-by-page inspection: **PASS**
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- Clean ZIP extraction, checks, and PDF rebuild: **PASS**
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- Clean Git-bundle clone and checks: **PASS**
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Final SHA-256 values are recorded in `Sha256.txt` beside the released
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artifacts.
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