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 pushing the same rigorous framework one lattice span farther.

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.

Visual notes

How to read the result

Timeline and magnified comparison of rigorous lower bounds for the square-lattice self-avoiding-walk connective constant. — **A 22-year step**The magnified panel isolates the new rigorous step. The much larger gap to the rigorous upper bound remains open.

Nested span 15, 16, and 17 transfer-matrix certificates and their lower bounds. — **Three independent spans**All three exact finite-lattice certificates clear the former frontier; span 17 produces the reported bound.

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=14	L=17	+3	length cutoff 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 evaluated the renewal inequality with directed bounds.
Ran smaller span-15 and span-16 certificates as independent checks.

Verification

How it was checked

The archived checker validates the exact dyadic 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}
}