ramanujan-challenge/docs/proofs/PROBLEM_28_COMPLETE_PROOF.md

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.