# P2538: Eventual monotonicity in a signed subset-sum local limit

- ID: `P2538`
- Reference: `signed-subset-sum-local-clt-monotone`
- Page: https://theoremdb.org/statements/P2538
- Record maturity: Reviewed problem with recorded work

## Problem

Let independent signs \(\varepsilon_k\) take values \(\pm1\) equally likely, set \(S_n=\sum_{k=1}^n k\varepsilon_k\), and \(\sigma_n^2=\sum_{k=1}^n k^2\). Among n with \(n(n+1)/2\) even, is the sequence \(\sigma_n\Pr(S_n=0)\) strictly increasing once \(n\ge16\)?

### Remarks

- **Remark.** The parity condition is necessary and sufficient for zero to lie on the support lattice.
- **Remark.** The admissible n are those congruent to 0 or 3 modulo 4, ordered in the usual way.

### What counts as a solution

- Prove every consecutive admissible comparison is strict after n=16, or exhibit an admissible counterexample.

## Status

Exact computation proves the claim through n=1000; the located asymptotic theorem gives convergence without a termwise inequality or an effective threshold. [1](#reference-1) [3](#reference-3) [2](#reference-2)

## Work

### Evidence for the current status

**Claim 1 (The all-n monotonicity claim remains unresolved in this audit).** Exact computation proves the claim through n=1000; the located asymptotic theorem gives convergence without a termwise inequality or an effective threshold.

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.

### Background and intake notes

Exact coefficient tables make any proposed threshold checkable. Analytic work should record a remainder bound strong enough to dominate the next-term difference.

- Original intake status: Status not established. No literature search was performed. Local central limit expansions for weighted Bernoulli sums may decide it.
- The attractive route is a local central limit expansion whose leading term tends to \(\sqrt{2/\pi}\).
- The obstruction is the interleaving of the two parity subsequences and the sign of the lattice correction. The normalized sequence decreases at several small transitions, so a leading asymptotic term is insufficient.

- Recorded example: At n=16 the normalized value is approximately 0.775498980775.

### Other known results

- **Theorem 1** (established): A central coefficient recurrence gives the exact probability, and squaring clears every square root and power-of-two denominator. [2](#reference-2)
- **Computation 1** (reproduced): Exact coefficient extraction and integer comparison find no counterexample between admissible indices 16 and 1000. [1](#reference-1)

### Prior approaches

- **Route 1** (inconclusive): The located papers identify the coefficient and its limit; neither supplies strict normalized monotonicity from n=16. [1](#reference-1)

### Runnable artifacts

- **Artifact 1** (reproduced): One Python integer stores the complete coefficient table in base 2^1001, allowing exact shift-and-add polynomial multiplication. [1](#reference-1)

### Computational notes

- Arbitrary-precision integer subset-sum dynamic programming checked every admissible n through 500. The only decreases in the combined admissible sequence occurred at 3 to 4, 8 to 11, and 15 to 16. Values at n=100, 200, and 500 were approximately 0.794303753371, 0.796091731844, and 0.797166849838. The exact zero-sum count at n=200 was 780463610226751719065842218999070243255558586796769387244.

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `signed-subset-sum-local-clt-monotone`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome.

## References

1. <a id="reference-1"></a>Blair D. Sullivan, On a Conjecture of Andrica and Tomescu, Journal of Integer Sequences 16 (2013), Article 13.3.1. Theorem 4 and its proof https://cs.uwaterloo.ca/journals/JIS/VOL16/Sullivan/sullivan8.html
   - Also cited at Sullivan 2013, Theorem 4: first-order asymptotic for the central coefficient
   - Also cited at Independent exact computation in ssclt-artifact-bitpacked-dp, executed 2026-07-24
   - Also cited at Inline CPython standard-library computation executed on 2026-07-24
   - journal_article; primary source; version of record; checked 2026-07-24
   - Open copy: https://cs.uwaterloo.ca/journals/JIS/VOL16/Sullivan/sullivan8.pdf
   - Source use: citation_only
   - Proves the first-order asymptotic for the zero-sum sign count, without an effective adjacent-monotonicity threshold.
   - Source named by the research packet.
2. <a id="reference-2"></a>OEIS Foundation Inc., A063865, number of solutions to ±1 ±2 ... ± n = 0 (checked 27 July 2026). Definition, coefficient formula, references, and term table https://oeis.org/A063865
   - Also cited at A063865, zero-sum sign counts, coefficient formula, references, and term table; accessed 2026-07-24
   - reference_database; reference source; web version checked 2026-08-01; checked 2026-07-24
   - Source use: citation_only
   - Records the exact zero-sum sign counts and cross-references the two analytic papers used in the audit.
3. <a id="reference-3"></a>Dorin Andrica and Ioan Tomescu, On an Integer Sequence Related to a Product of Trigonometric Functions, and Its Combinatorial Relevance, Journal of Integer Sequences 5 (2002), Article 02.2.4. Abstract, coefficient identity, and main conjecture https://cs.uwaterloo.ca/journals/JIS/VOL5/Tomescu/tomescu4.html
   - journal_article; primary source; version of record; checked 2026-07-24
   - Source use: citation_only
   - Gives the central-coefficient identity, the four-step growth bound, and the original asymptotic conjecture.