Commensurability conjecture for critical-exponent universality

For any two hierarchical lattices arising in the stated Bernoulli-percolation setting, if their critical-exponent universality classes coincide, then their discrete renormalisation scales are commensurate: for scale factors $r_1,r_2>1$ , there exist positive integers $m,n$ such that $r_1^m=r_2^n$ , equivalently $\log r_1/\log r_2\in\mathbb{Q}$ .

References

Primary source

arXiv

Additional references

Iterated Graph Systems (II): Bernoulli percolation and critical-exponent universality classes on hierarchical lattices — arXiv — Ziyu Neroli

Posted 5 hours ago · edited 5 hours ago

Progress summary

Refreshed 5 hours ago

Claimed solved

A new preprint claims to disprove the arithmetic classification in the hierarchical-lattice setting, but the result has not been independently verified.

The conjecture concerns whether critical-exponent universality classes in hierarchical lattices obey an arithmetic commensurability classification. A counterexample would settle the conjecture negatively within that setting.

October 2026 counterexample

Ziyu Neroli’s preprint Iterated Graph Systems (II): Bernoulli percolation and critical-exponent universality classes on hierarchical lattices claims that noncommensurate scale ratios can yield the same universality class and that no finite family of substitution dimensions determines universality. If correct, this refutes the conjecture in the stated hierarchical-lattice setting; the claim remains unverified.

Current status (as of October 2026): The conjecture is claimed refuted for the stated hierarchical-lattice setting, while the preprint’s counterexample remains unverified and broader critical-phenomena models are not addressed.

Sources

No solutions have been posted yet.