Why it is a question
The owner, at U6: “it feels like temporal discounting and hawkes processes would be useful tools to integrate here.” A6 §2: “The more intriguing hypothesis is that the influence canvas dynamically regulates the branching structure, changing which events can recruit others and how far cascades can travel.” Also: “I would not assume psychological systems optimize criticality, though. Different subsystems might function productively at different distances from criticality.” A7’s H4: “Influence allocation modulates cascade sizes and recurrent activation regimes.”
For a stationary linear excitatory Hawkes process the branching ratio is the spectral radius of the integrated excitation matrix. Below one, cascades die out, which is the Hawkes–Oakes cluster representation (LIT-1028). The exchange’s kernels are signed and change with state. A6 and A7 granted that these “do not inherit the simple stability guarantees”, so Brémaud and Massoulié (LIT-1035) is where the analysis starts.
CLAIM-163 objects that “synchronization” among an agent’s interests is “an analogy without a mechanism”, with no coupling and no order parameter. A Hawkes model of cascades has the coupling, and a control parameter, not an order parameter. It does not rescue the synchronization reading of Ainslie, but it shows what an answer of that kind looks like. An answer fits a marked point process with influence-dependent kernels, to r/place repairs (CASE-079) or to a simulation, and reports whether the effective branching ratio moves with the allocation of influence.
Note of 2026-10-13: what Brémaud and Massoulié prove, and do not
Brémaud and Massoulié (LIT-1035) was read in NOTE-728, and its result is filed as THEORY-215. Three points correct this question’s framing. The status is unchanged.
- Their kernels are signed but fixed. The rate is a nonlinear function of a fixed filter of the past. Kernels that change with the allocation of influence, which is what this question asks about, are outside every theorem in the paper.
- The spectral radius is a branching ratio only in the linear excitatory case. For Lipschitz rates and signed kernels, a spectral radius below one of the matrix α_i∫|h_ji| is a sufficient condition for one stable regime. Its distance from one measures no proved property of cascades, and for bounded rates no size condition is needed at all, so a model with saturating influence has no criticality to be near.
- It is a control parameter, not an order parameter. The branching ratio is computed from the couplings before the dynamics run. The summary’s “the branching ratio is the coupling and order parameter” runs the two together. A Hawkes model has a coupling (the kernels) and this control parameter; an order parameter would be a collective variable of the state, such as the event rate, and the paper proves nothing about transitions.