Limit theory for the sensitive continuous-type regime

Develop the critical and geometric limit theory for the interacting diffusion cascade outside the saturated regime, where failure strength is a continuous type and the mean offspring operator acts on the failure-strength state space; in particular, establish the total-progeny \(3/2\) law with its correct constant and characterize the boundary dimension using the continuous-type Perron projection and spatial central-limit theory.

Background

Outside saturation, the failure strength of a vertex is inherited by its children and induces correlations among sibling transmissions. The natural branching object is therefore a continuous-type process on hazard domain × failure strength, governed by a positive integral mean operator rather than the finite matrix used in the saturated theory.

The paper provides finite-matrix upper and lower certificates for the operator's criticality, but does not establish the corresponding limit theorems. In particular, the authors identify the total-progeny exponent, its constant, and the Hausdorff dimension of the failed boundary as unresolved continuous-type questions requiring general-state-space branching methods, Perron projections, and spatial fluctuation results.