Collective Pride Coupled Gating, Belief Fields and Group Folds under Mean, Sum and Extremal Aggregation

Abstract The two central conclusions of this paper are: collective pride does not require proud individuals—each person alone can be far below the fold while the group as a block crosses it; and the aggregation rule matters more than the shape of the network—whether others’ commitments enter a person’s cost of admission as a mean, a sum, a maximum or a minimum decides whether, where and how a group slides into collective overestimation. Let N agents each update their belief bias x_i about their own ability by the asymmetric rule of [1], with the booked share of failures a_i = a₀·exp(−χ_s·x_i − χ_o·S_i(x)) + a_ext, where χ_s is the elasticity of self-gating, S_i is the operator that aggregates others’ commitments into agent i’s cost of admission, and χ_o is the elasticity of social gating. Four operators are compared: mean (share gating), sum (absolute gating), maximum and minimum. First, a general criterion (Theorem 1, A). With a₀ = 1 and a_ext = 0, the fully calibrated state x = 0 is a fixed point for every operator; for a linear operator M it loses stability if and only if χ_o·λ_max(M) > √(2π)/σ − χ_s. At this point the loss of stability is a symmetric pitchfork; for a₀ ≠ 1 it unfolds into a pair of saddle-nodes with hysteresis. Second, mean coupling: collective pride without proud individuals (Proposition 2, A). On any connected undirected network, with gating by the mean commitment of neighbours, the synchronous manifold is invariant, and the group dynamics on it are exactly the single-agent model with gating elasticity χ_s + χ_o; moreover, every branch that is stable in the synchronous direction is transversely stable. So when χ_s < √(2π)/σ < χ_s + χ_o, the group as a block shows a saddle-node with hysteresis that no member would show alone. The threshold does not depend on network topology. Third, sum coupling: the threshold is set by the largest eigenvalue of the network, and pride nucleates on hubs (Proposition 3, A; nonlinear part B). With gating by the sum of neighbours’ commitments, the collective threshold is c_c = (√(2π)/σ − χ_s)/λ_max(A). Since λ_max ≥ max(⟨k⟩, √k_max), the more uneven the degree distribution, the lower the threshold; numerically, for 200 nodes with mean degree about 4, the thresholds of a ring, a random network, a scale-free network and a star are 0.377, 0.309, 0.190 and 0.107, matched by nonlinear simulation. On the scale-free network pride nucleates first on hubs and then spreads in islands; on the star, over a wide interval only the centre enters the pride branch. Fourth, the maximum operator: leader reduction and a size effect (Proposition 4, A; size formula B). When one agent holds the highest commitment, the group’s fold is that agent’s fold; the others follow passively and have no fold of their own. In a homogeneous group, noise makes the maximum exceed the mean by about s·e_N (e_N the expected maximum of N standard normals), so that below the collective threshold the systematic overestimation of the group, x̄ ≈ χ_o·s·e_N/(√(2π) − χ_s − χ_o), grows with size; in simulation it is 0.17, 0.27 and 0.34 for N = 5, 20 and 100, against linear-response predictions of 0.19, 0.26 and 0.37. Fifth, what one ungated member can do. Under the mean operator the member’s effect is 1/N, and in simulation the group stays on the pride branch; under the maximum operator the effect is zero; only under the minimum operator does the member bring the whole group back—to the calibrated point under one-sided gating, while under symmetric gating the group overcorrects and falls as a whole onto the self-abasement branch. This gives the role of “multiple centres and the option to exit” [4] a precise place, and states its cost. Sixth, the stabilising effect of group size, and shared readings. Under mean coupling the noise of the group mean shrinks as 1/√N and noise-induced transitions are suppressed: the low well from which a single agent is pushed out by noise inside the window is fully retained in groups of 20 or more. This makes large groups more dependent on history. But if members’ feedback shares one reading, then by Many Lakes, One Watchtower [6] the effective size is no greater than 1/ρ, and the stabilising effect fails: at ρ = 0.4 the low-well retention of groups of 20 and 100 is only 0.25 and 0.53. Seventh, two-layer coupling (Proposition 5, A). A joint fixed point of two layers (such as a superior and subordinates) loses stability when the product of the cross-gating elasticities exceeds the product of the layers’ own margins, the same form as K_c = Δ_a·Δ_b in [5]; when the product of cross elasticities is negative there is no fold, only damped oscillation. Seven falsifiable predictions are registered. Number of executions on external data: zero.