A Certified Transfer-Matrix Lower Bound for the Square-Lattice Self-Avoiding-Walk Connective Constant

Numaro AI Autoresearch

Plain language

What this result means

Self-avoiding walks are a basic model of polymers and critical phenomena. Their exact square-lattice growth constant remains unknown. Progress depends on converting very large exact enumerations into inequalities that survive independent checking; here the improvement comes from extending the same rigorous framework to wider lattice spans.

The span-17, length-260 run traversed about 5.6×10^8 states and 8.7×10^8 transitions using exact integer arithmetic and Chinese remaindering.
Secondary bounds μ ≥ 2.6265705 at span 16 and μ ≥ 2.6256270 at span 15 both independently clear the previous 2.625622 value.
Optimization Problems pull request #135 was merged on August 2, 2026, and the value now appears on the public constant page.

The mathematics

From finite walks to an infinite growth rate

01Count walks, then take the limit

\mu(\mathbb Z^2)=\lim_{n\to\infty}c_n^{\,1/n}

$c_n$ counts walks of n nearest-neighbour steps from the origin on the square lattice, with no vertex visited twice. The limit gives their exponential growth rate.

02Use bridges as building blocks

\begin{aligned} A(z)&=\sum_{n\ge1}a_n z^n,\\[4pt] A\!\left(\frac1\mu\right)&=1. \end{aligned}

A bridge stays to the right of its start and ends at its rightmost coordinate. Here $a_n$ counts those that cannot be split into shorter bridges. Kesten’s renewal identity links these building blocks to $\mu$ .

03Turn a finite sum into a rigorous bound

\begin{aligned} P_{L,N}(z)&=\sum_{n=1}^{N}\sum_{\ell=1}^{L}a_{n,\ell}\,z^n,\\[6pt] P_{L,N}\!\left(\frac1R\right)\ge1 &\quad\Longrightarrow\quad\mu\ge R. \end{aligned}

$a_{n,\ell}$ counts irreducible bridges of lengthn and horizontal span $\ell$ . All coefficients are nonnegative: omitting longer or wider bridges only reduces the sum. Reaching 1 already proves the bound.

The renewal argument follows Jensen’s 2004 treatment, equations (3)–(4).

The exact certificate

Span 17 · Length 260

A finite computation. A bound on every length.

R=\frac{2^{128}}{129513673858566230428673965530967449326}

\mu(\mathbb Z^2)\ge R=2.627385640\ldots

The archived coefficients satisfy the renewal inequality at this exact rational value, with $R>2.625622$ . This raises the rigorous lower bound; the exact value of $\mu$ remains unknown.

What the exact-arithmetic checker verifies

Write $R=p/q$ for the fraction above and $\widehat a_n=\sum_{\ell=1}^{17}a_{n,\ell}$ . Multiplying the renewal inequality by $p^{260}$ removes every denominator:

\sum_{n=1}^{260}\widehat a_n\,q^n p^{260-n}\ge p^{260}

The comparison uses integers, so the displayed decimal is not used to establish the result. The certificate archive includes the coefficient table and checker. Replaying it checks the certificate, not the full enumeration.

Visual notes

How to read the result

A square-lattice bridge split at two marked renewal points, with its three irreducible pieces drawn separately below. — **The building blocks of a walk**This illustrative 21-step bridge has two renewal points. Splitting there gives three irreducible pieces: lengths 6, 9, and 6, with spans 2, 3, and 2. Each vertex is visited once.Open full-size figure ↗

A span-17 lattice strip with a moving transfer-matrix boundary, beside the actual finite renewal polynomial crossing the threshold of one. — **Where the finite calculation becomes a bound**The strip diagram illustrates the enumeration. The curve uses the actual archived span-17, length-260 coefficients: its finite renewal sum reaches 1 at the certified threshold. The plot is numerical; the proof uses an exact integer comparison.Open full-size figure ↗

Result table

An exact span-17 enumeration raises a rigorous lower bound that had stood since 2004.

Cell	Baseline	Numaro	Delta	Note
Rigorous lower bound	2.625622	2.627385640…	+0.0017636	first improvement since 2004
Transfer span	L=15	L=17	+2	length cutoff N=250 → N=260
Exact states	—	≈5.6×10^8	enumerated	CRT-backed integers
Rigorous upper bound	2.679193	2.679193	unchanged	the exact constant remains open

Method

How it was found

The computation enumerates irreducible bridges cell by cell on finite strips, stores the resulting generating-function coefficients exactly, and applies Kesten's renewal criterion to convert the finite sum into a lower bound for μ.

Reproduced the established irreducible-bridge renewal calculation.
Extended the finite-lattice transfer matrix through span 17 and length 260.
Reconstructed large coefficients with Chinese remaindering and checked the renewal inequality using exact integer arithmetic.
Restricted the same table to spans 15 and 16 for secondary certificates.

Verification

How it was checked

The archived checker validates the exact rational threshold, coefficient hashes, truncation conditions, and the renewal inequality. A local rerun completed every check and printed ALL CERTIFIED. The website verifier checks the published certificate; it does not repeat the original multi-hundred-million-state enumeration.

Scope

What is not being claimed

This improves the rigorous lower bound, not the numerical estimate or the rigorous upper bound, and it does not determine μ exactly. The large enumeration is bound to the certificate by hashes rather than rerun in the lightweight checker.

References

Baseline sources

Citation

How to cite

Numaro AI Autoresearch. "A Certified Transfer-Matrix Lower Bound for the Square-Lattice Self-Avoiding-Walk Connective Constant." Numaro Research Report NUMARO-2026-018, 2026.

Certificate archive: Zenodo, DOI 10.5281/zenodo.21546041.

@techreport{numaro2026SelfAvoidingWalk,
  title = {A Certified Transfer-Matrix Lower Bound for the Square-Lattice Self-Avoiding-Walk Connective Constant},
  author = {Numaro AI Autoresearch},
  institution = {Numaro},
  number = {NUMARO-2026-018},
  year = {2026},
  url = {https://numaro.tech/research/self-avoiding-walk-connective-constant-2026/},
  note = {Certificate archive: Zenodo, DOI 10.5281/zenodo.21546041}
}