# P2718: Nearest hard-square partition-function zero for the sixteen grid

- ID: `P2718`
- Reference: `hard-square-sixteen-zero-radius`
- Page: https://theoremdb.org/statements/P2718
- Record maturity: Reviewed problem with recorded work

## Problem

Let \(Z_{16}(z)=\sum_I z^{|I|}\), where \(I\) ranges over independent vertex sets of \(P_{16}\square P_{16}\). Determine the zero of \(Z_{16}\) having least modulus, in a rigorous complex rectangle of diameter at most \(10^{-12}\).

### Remarks

- **Remark.** The polynomial has integer coefficients and degree 128.
- **Remark.** If several zeros have equal least modulus, isolate and report all of them.

### What counts as a solution

- Provide the exact coefficient vector, certified root isolating rectangles, and a proof that every other zero has strictly larger modulus or belongs to a reported tie.

## Status

The zero radius lies between 27/256 and 0.122005371541038. A simple real zero is isolated near -0.122005371541037445, while its global least-modulus status remains open. [3](#reference-3)

## Work

### Evidence for the current status

**Computation 1 (A certified radius bracket and an isolated real zero for the sixteen grid).** The zero radius lies between 27/256 and 0.122005371541038. A simple real zero is isolated near -0.122005371541037445, while its global least-modulus status remains open.

Let \(\rho_{16}=\min\{|z|:Z_{16}(z)=0\}\). Exact row-mask transfer gives all 129 integer coefficients of \(Z_{16}\). An exact rational Rouché calculation isolates a simple real zero in
\[
[-0.122005371541038,-0.122005371541037]+i[-10^{-15},10^{-15}].
\]
The rectangle has diameter below \(3\mathbin{\cdot}10^{-15}\).

The grid has maximum degree four. The Shearer disk bound, in the independent-polynomial form developed by Scott and Sokal, gives
\[
Z_{16}(z)\ne0\qquad\text{for }|z|<\frac{(4-1)^{4-1}}{4^4}=\frac{27}{256}.
\]
Consequently,
\[
\frac{27}{256}\leq\rho_{16}\leq0.122005371541038.
\]

The local Rouché disk proves that the displayed zero is simple. It does not count the other 127 zeros or exclude a nonreal zero in the remaining annulus. The requested least-modulus identification remains open in this fixture.

### Background and intake notes

A complete smaller-grid computation supplies a pipeline check and a numerical target near -0.12425.

- Original intake status: Novelty remains unverified. No primary-source status audit was completed for this finite-volume zero.
- Row-mask transfer can compute all coefficients exactly; arbitrary-precision floating roots alone do not certify the least modulus.
- Coefficient growth makes standard companion-matrix roots ill-conditioned near clusters.
- A solution must isolate every root whose modulus interval overlaps the proposed minimum.

- Recorded example: At z=1, the polynomial counts all independent sets.

### Prior approaches

- **Route 1** (supported): The located sources supply the hard-square setting, compatible-mask transfer, and universal disk bound. None reports the finite 16 by 16 nearest zero. [1](#reference-1) [2](#reference-2) [3](#reference-3)

### Runnable artifacts

- **Artifact 1** (reproduced): A short C++17 program computes all coefficients with unsigned 256-bit addition and a subset zeta transform. [3](#reference-3)
- **Artifact 2** (reproduced): Fraction arithmetic proves one zero in a radius 5e-14 disk, then places its real representative in a rectangle of diameter below 3e-15. [3](#reference-3)

### Computational notes

- Exact row-mask transfer for the 12 by 12 grid produced degree 72 and Z_12(1)=162481813349792588536582997. The serialized coefficient vector has SHA-256 c36cdf2efa005e88c21469a24059de8c03f63010de218a42ac3e990411d1e820. Forty-digit numerical roots placed the least-modulus candidate at -0.1242498124533312648219610478237016098572.

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `hard-square-sixteen-zero-radius`, 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>R. J. Baxter, I. G. Enting, and S. K. Tsang, “Hard-square lattice gas”. Journal of Statistical Physics 22(4) (1980), 465-489. DOI 10.1007/BF01012867. R. J. Baxter, I. G. Enting, and S. K. Tsang, Hard-square lattice gas, Journal of Statistical Physics 22(4) (1980), 465-489; Neil J. Calkin and Herbert S. Wilf, The Number of Independent Sets in a Grid Graph, SIAM Journal on Discrete Mathematics 11(1) (1998), 54-60; Alexander D. Scott and Alan D. Sokal, The repulsive lattice gas, the independent-set polynomial, and the Lovasz local lemma, Journal of Statistical Physics 118 (2005), 1151-1261 https://doi.org/10.1007/BF01012867
   - Also cited at Journal of Statistical Physics 22(4) (1980), 465-489
   - scholarly_publication; reference source; version of record; checked 2026-08-01
   - Source use: citation_only
   - For Nearest hard-square partition-function zero for the sixteen grid: The located sources supply the hard-square setting, compatible-mask transfer, and universal disk bound. None reports the finite 16 by 16 nearest zero.
2. <a id="reference-2"></a>Neil J. Calkin and Herbert S. Wilf, “The Number of Independent Sets in a Grid Graph”. SIAM Journal on Discrete Mathematics 11(1) (1998), 54-60. DOI 10.1137/S089548019528993X. SIAM Journal on Discrete Mathematics 11(1) (1998), 54-60, Section 1 https://doi.org/10.1137/S089548019528993X
   - scholarly_publication; reference source; version of record; checked 2026-08-01
   - Source use: citation_only
   - For Nearest hard-square partition-function zero for the sixteen grid: Primary literature supports the model, transfer, and zero-free disk. The located sources supply the hard-square setting, compatible-mask transfer, and universal disk bound. None reports the finite 16 by 16 nearest zero.
3. <a id="reference-3"></a>Alexander D. Scott and Alan D. Sokal, The repulsive lattice gas, the independent-set polynomial, and the Lovasz local lemma Journal of Statistical Physics 118 (2005), 1151-1261 https://doi.org/10.1007/s10955-004-2055-4
   - Also cited at Exact transfer in hs16-artifact-exact-polynomial and rational Rouché certificate in hs16-artifact-root-enclosure
   - Also cited at Inline C++17 exact computation executed on 2026-07-25
   - Also cited at Inline Python 3 rational certificate executed on 2026-07-25
   - scholarly_publication; reference source
   - Source use: citation_only
   - Source named by the research packet.