Write \[ A_n=\sigma_n\Pr(S_n=0),\qquad \sigma_n^2=\sum_{k=1}^n k^2. \] The targeted search located the exact coefficient sequence, a constructive growth bound, and the proved first-order asymptotic. It found no paper proving that \(A_n\) increases at every consecutive admissible index after 16.

Sullivan proves \[ C_n\sim \sqrt{\frac6\pi}\,2^n n^{-3/2} \] for \(n\equiv0,3\pmod4\), where \(C_n\) is the zero-sum sign count. Since \(\sigma_n\sim n^{3/2}/\sqrt3\), this yields \(A_n\to\sqrt{2/\pi}\). Convergence to the limit permits occasional decreases and supplies no finite threshold. A formal local Edgeworth calculation gives the first correction \[ A_n=\sqrt{\frac2\pi}\left(1-\frac9{20n}+O(n^{-2})\right), \] which predicts eventual increase. The displayed error order is still the same size as a consecutive difference. A proof for all \(n\ge16\) needs a sharper effective remainder, or a direct coefficient inequality, joined to the finite computation.

The companion artifact establishes all 492 requested comparisons whose larger endpoint is at most 1000. The infinite tail remains open in this record.

Let \[ F_n(x)=\prod_{k=1}^n(1+x^k)=\sum_s c_{n,s}x^s, \qquad T_n=\frac{n(n+1)}2. \] Choosing the indices assigned sign \(+1\) shows \[ S_n=0\quad\Longleftrightarrow\quad \sum_{k:\varepsilon_k=1}k=\frac{T_n}{2}. \] Hence, for admissible \(n\), \[ C_n=c_{n,T_n/2}=[x^{n(n+1)/4}]F_n(x), \qquad \Pr(S_n=0)=\frac{C_n}{2^n}. \] The coefficients obey the exact subset-sum recurrence \[ c_{n,s}=c_{n-1,s}+c_{n-1,s-n}, \] with \(c_{0,0}=1\). Also \[ Q_n=\sigma_n^2=\frac{n(n+1)(2n+1)}6. \] For consecutive admissible indices \(a<b\), all quantities are positive. Squaring and clearing denominators gives \[ A_b>A_a \quad\Longleftrightarrow\quad C_b^2Q_b-C_a^2Q_a4^{b-a}>0. \] This equivalence is the comparison used by the executable artifact. Every decision is an integer sign test.

The exhaustive replay constructs every coefficient polynomial through \(F_{1000}\) and extracts \(C_n\) at each of the 500 positive admissible indices up to 1000. It checks the sign of the cleared integer margin for every adjacent pair.

All 492 comparisons beginning at \(n=16\) are positive. The normalized endpoints include \[ A_{16}=0.775498980775123\ldots, \] \[ A_{999}=0.797525249765113\ldots, \qquad A_{1000}=0.797525608979770\ldots. \] As a check on the earlier behavior, the complete list of decreases through 1000 is \[ 3\to4,\qquad 8\to11,\qquad 15\to16. \] The count at \(n=200\) is `780463610226751719065842218999070243255558586796769387244`, matching the candidate's prior computation.

Andrica and Tomescu identify \(C_n\) as the middle coefficient of \(\prod_{k=1}^n(1+x^k)\), derive an integral representation, and prove the constructive bound \(C_n\geq6C_{n-4}\) for \(n\geq8\). Their bound compares indices four apart and concerns the unnormalized count.

Sullivan proves the Andrica-Tomescu asymptotic by Laplace's method. The proof separates a neighborhood of zero in the cosine-product integral and shows the remaining integral is lower order. Its conclusion is a first-order equivalence as \(n\to\infty\). The paper states no effective error bound that decides each adjacent admissible comparison.

OEIS A063865 records the exact sequence and points to both papers. Searches for the sequence identifier, normalized central coefficients, weighted Rademacher local limits, and monotonicity found no direct theorem for the present claim. This is a targeted audit rather than a proof of novelty.

Set \(B=2^{1001}\) and encode \(F_n(x)=\sum_sc_{n,s}x^s\) as the integer \(P_n=\sum_sc_{n,s}B^s\). Every coefficient is below \(2^n<B\), so base-\(B\) digits never carry into each other. Multiplication by \(1+x^n\) is therefore the single exact operation ``` P_n = P_{n-1} + (P_{n-1} << (1001*n)). ``` A mask extracts the desired base-\(B\) digit. The program asserts known counts, constructs all cleared comparison margins, and requires every post-16 margin to be positive.

The canonical `n:count:variance` record has SHA-256 digest `c42ba94e917ea48f4c8761938c3e52989f5a01143bb0bc34dedd19425eb417dc`. The canonical `a,b:margin` record has digest `d28f8f888dedb6d9fb9f27858685d315a242c7276e9e73c40a0a47f0a15ac8b5`. The ten-line output digest is `1c970286ddadc5532e0766c4e36bd3acdb5c898d358435e9999f0108909831e2`.