The Contagion of Selfishness Under Frame Uncertainty: A Repaired Network Bateson Game with Comparison, Propagation, Mean-Field Limits, and Belief–Behavior Non-Identification

This paper formalizes and repairs a network-theoretic model of frame-based belief contagion, extending the two-player Bateson game to a finite connected graph of receivers subject to exploitative sender exposure. The central object is a projected Rescorla–Wagner belief recursion in which a non-negative effective leverage term, derived from sender-induced payoff prediction errors, creates a systematic downward drift in each receiver's credence that a cooperative frame is operative. Five results are established as conditional theorems, each with explicit hypotheses. First, a componentwise domination theorem shows that the Bateson belief process lies weakly below the frame-free benchmark at every date, with strict separation whenever leverage is active and the update is interior. The proof requires an explicit order-compatibility condition on the leverage family; this missing hypothesis is identified and repaired from the earlier draft. Second, a pathwise contagion theorem establishes that frame infection, defined cumulatively rather than as a one-step event, propagates along simple graph paths in O(d/α) rounds under stated direct- and indirect-exposure drift conditions. The indirect-exposure step is grounded in an explicit no-upward-input buffer hypothesis rather than an invalid monotonicity inference. Third, the network-to-mean-field reduction is formulated as a projected scalar ODE with a rigorous Euler/fluid proof under an explicit finite-horizon concentration assumption; the tipping threshold is characterized as an implicit fixed point of the scalar field with a first-order expansion in sender-exposure share, and no exact global affine identity is claimed. Fourth, the Kullback–Leibler representation of leverage is made dimensionally consistent by imposing a weak-signal scaling under which the KL divergence between manipulated and cooperative observation laws is O(α), yielding a slow-time leverage profile of the form A(p) = p(1−p)K. Fifth, a belief–behavior non-identification theorem shows, via explicit connected-graph construction, that matched aggregate cooperation does not identify the cross-sectional belief distribution; a no-go theorem against uniformly Lipschitz frame-free model classes is stated for matched local input histories, which is the correct level of observation for the result. Irreversibility after sender removal is also established under a lower certainty-trap condition.