Locally 2-homogeneous block designs

Referee: [Main classification theorem (likely §3)] The central reduction from local 2-homogeneity to Kantor's 2-transitive cases is load-bearing but lacks an explicit statement of the main theorem (presumably in §3 or §4) that enumerates all surviving designs and rules out non-symmetric examples. Without this, it is impossible to verify that no infinite families with varying block sizes or designs whose automorphism group is only locally 2-homogeneous survive.

Authors: We agree that an explicit, standalone statement of the classification theorem is needed for clarity and verifiability. In the revised manuscript we will insert a dedicated theorem (new Theorem 3.1) at the start of Section 3 that enumerates all designs satisfying the local 2-homogeneity condition: the known 2-transitive symmetric families from Kantor’s 1985 classification together with the finitely many additional cases identified in our analysis. The statement will explicitly record that no infinite families with varying block sizes survive and that no designs exist whose automorphism group is merely locally 2-homogeneous. This formulation makes the reduction from local to global symmetry immediate and allows direct checking against the parameter constraints derived earlier in the paper. revision: yes

Referee: [Proof of the reduction (likely §2.3 or §4)] The argument that local 2-homogeneity forces global 2-transitivity or symmetry appears to rely on connectivity of the incidence graph and parameter constraints, but the manuscript does not supply a self-contained check (e.g., via the incidence matrix or block intersection numbers) that excludes designs whose local action is 2-homogeneous only on blocks through a fixed point.

Authors: The existing argument in Section 4 already uses connectivity of the incidence graph together with the numerical constraints implied by local 2-homogeneity to deduce global 2-transitivity. To meet the referee’s request for a self-contained verification, we will add a short auxiliary lemma (new Lemma 4.2) that works directly with the incidence matrix and the block intersection numbers. The lemma shows that if the stabilizer of a point acts 2-homogeneously on the blocks through that point, then the same action must extend to the full point set, thereby ruling out any design in which 2-homogeneity holds only locally on the blocks incident with a fixed point. This addition keeps the original logic intact while supplying the requested explicit check. revision: yes