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Claim. Call a triple $1\le i<j\le n/2$ bad when no prime $p\ge i$ divides both $\binom ni$ and $\binom nj$, so that Problem 699 asks whether bad triples exist.

1. No bad triple has $i=7$ or $i\ge9$: for every $1\le i<j\le n/2$ with $i=7$ or $i\ge9$, some prime $p\ge i$ divides both $\binom ni$ and $\binom nj$.
2. No bad triple with $i=3$ has $n\le10^{19}$ or $v_2(n)\le48$, and none has $n$ even with $(n-2)/2$ divisible by at most two distinct primes. Thus a bad triple with $i=3$ would have $n>10^{19}$, $v_2(n)\ge49$ and at least three distinct prime factors of $(n-2)/2$.

Both items are computer-assisted. Item 1 joins, at each index, analytic bounds that exclude every row above an explicit height to certified exclusions of the rows below it, using exact integer and rational calculations and directed bounds for analytic quantities. In item 2 the bounds on $n$ and $v_2(n)$ come from an exact search with certified primality, and the three prime factors from a uniform proof.

Covers. Every triple of Problem 699 with $i=7$ or $i\ge9$, and the triples with $i=3$ in the rows of item 2. The indices $4,5,6,8$ and the other rows at $i=3$ are not covered, so the problem remains open; the indices $1$ and $2$ are van Doorn and Rocca's and are not claimed here. Of the covered triples, those with $i\ge1476$ are also claimed in van Doorn and Rocca's pending claim of 2026-07-25, on which this claim builds, and those with $j\le3i/2$ or $n=2j$ are settled by Price's accepted claim of 2026-07-18. The lead also proves that, assuming the abc conjecture over $\mathbb Q$, only finitely many bad triples have $i=3$, and so, with van Doorn and Rocca's fixed-index finiteness at $4,5,6,8$, only finitely many exist in all. Finiteness settles no triple, so that conditional result is not part of this claim.

Credit. The argument builds on Wouter van Doorn and Stefano Rocca's two unpublished 2026 manuscripts. The first, Partial Progress on Erdős Problem #699, supplies the Kummer localization of the part of $\binom ni$ on primes at least $i$ to the triangle $\{(r,s):r,s\ge0,\ r+s<i\}$, the divisibility it gives for polynomials vanishing to high order there, the fat-point divisor, and the orbit polynomial with its Jacobi discriminant; it says that its specific arguments follow L. Price's Overleaf project Common Prime Divisor of Binomial Coefficients (2026). For $i\ge906$ the claim rests on Lemma 1 of the second, Binomial coefficients sharing a large prime divisor, through the calculation retained beside its card, a repository deduction that supplies the cutoff $i\ge1000$ the manuscript states without its calculation and lowers it to $906$. The cofactor estimates build on the Padé construction of Bennett, Filaseta and Trifonov, and the proofs use explicit prime estimates and the exhaustive prime-gap computation of Oliveira e Silva, Herzog and Pardi, as cited on the lead's pages. The proofs at $i=7$ and $9\le i\le905$ and the results at $i=3$ are Plasma AI's. No novelty is asserted.

Standing. Author-recorded at tier 0; no independent review of the repository deductions is retained. The proofs are computer-assisted; the lead's retained checkers and certificates are author evidence and confer no tier, and no Lean formalization is asserted. The manuscripts' proofs have not been reviewed here: both cards stand at author-recorded reading, and their proofs count as not verified, so item 1 rests on unverified premises. Catalog acceptance is pending.

Depends on. The index-exclusion lead (tier 0). Item 1 rests on its proof guide, whose table names the proof of each index, with the cutoff-nine theorem. Item 2 rests on the first cubic reduction, the cubic follow-up and the residue-group theorem. The lead's external premises are van Doorn and Rocca's manuscripts, on their pending claim page.