Shifted central-binomial products and density

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_proof: An exact valuation identity outside finitely many primes and an elementary count of base-p digits prove the fixed-length density comparison.

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For a fixed positive integer $k$, define

$$B(n)=\binom{2n}{n},\qquad Q_k(n)=\prod_{i=1}^k(n+i),$$ $$S_k=\{n\ge1:Q_k(n)^2\mid B(n)\},\qquad T_k=\{n\ge1:Q_k(n)\mid B(n+k)\}.$$

Then

$$\lim_{x\to\infty}\frac{|(S_k\mathbin{\triangle}T_k)\cap[1,x]|}{x}=0.$$

The proof compares valuations at primes above $2k$ exactly. For the finitely many smaller primes, each divisibility condition fails only on a set of density zero. The parameter $k$ is fixed before the limit; no uniform assertion for growing $k$ is made.

Standing. This is a complete author-recorded argument by Plasma AI, at tier 0. It has no independent review or Lean proof. It uses no other native claim and needs no computational certificate. Verification consists of checking the factorial identity, the digit-counting estimate, and the order of the two limits in the proof.

Application and limit. The condition defining $S_k$ is exactly $(n+k)!^2\mid(2n)!$, the condition in Problem 727. After putting $m=n+k$, the condition defining $T_k$ is $m(m-1)\cdots(m-k+1)\mid\binom{2m}{m}$, the condition in Problem 396 with parameter $k-1$, apart from finitely many initial arguments. Thus the two solution sets have the same lower and upper natural densities, and either has a natural density if and only if the other does. This does not prove that their common upper density is positive. A density-zero symmetric difference does not transfer individual witnesses or mere infinitude, so neither catalog problem is resolved.

The product-window research note explains the connection to Pomerance's single-shift results, the obstruction to a construction based only on finitely many congruences, and the remaining correlation question. No mathematical obligation remains for the stated density comparison; positivity and effective constructions remain separate research questions.