Source
Hansen and Ghrist (2020; SIAM J. Appl. Math. 81(5), 2021), LIT-1646, §§8–9 (pp. 13–17 of arXiv 2005.12798v1); read in NOTE-1222. The setting is THEORY-1062’s.
What was actually shown
Expression diffusion, opinions fixed (§8). Each agent sees its neighbours’ expressions and changes only its own maps, by dF_{v⊴e}/dt = −β(F_{v⊴e}x_v − F_{u⊴e}x_u)x_vᵀ (eq. 8.1), so as to “remove the part of the image that contributes to disagreement with neighbors” (p. 13). Theorem 8.1: F(t) converges to “the nearest sheaf such that x is a global section”, in the distance ‖δ − δ′‖²_F (eq. 8.4). Each edge’s flow is a linear positive-semidefinite flow, which projects δ_e onto {δ_e : δ_e x_e = 0}. In Example 8.2 (Fig. 6), the constant sheaf on one edge with opinions −4 and +1, one map changes sign. That agent “learns to lie about their opinion”, flipping its sign and downplaying its magnitude. The reading computes the limit as −3/17 and 12/17, where the figure labels −0.2 and 0.8.
Joint diffusion (§9). With dx/dt = −αδᵀδx as well (eq. 9.1), the flow is gradient descent on Ψ = ½‖δx‖² (Lemma 9.1, p. 15). Trajectories converge to a sheaf together with a global section of it, and smooth equilibria are Lyapunov stable (Theorem 9.2). There is “no global asymptotic stability” because the equilibria are not isolated.
The trivial solution, named and excluded for x. Most of the equilibrium variety has ker δ = 0, “This would seem to indicate that we should expect most trajectories to converge to points with x = 0. However, this is not the case” (p. 16). Theorem 9.3: the diagonal blocks of αδᵀδ − βxxᵀ are constant in time, so if one is indefinite at the start, x does not go to zero. Hence some rescaling κx₀ of any start converges to a sheaf with a nonzero section. The printed proof writes M = δᵀδ − xxᵀ and tr(M) = ‖δ‖²_F − ‖x‖², which are correct only for α = β. The reading’s numerical check with α = 2, β = 0.5 found the weighted blocks conserved and the unweighted ones not. Theorem 9.4: ‖δ‖²_F, ‖x‖², ‖δx‖² and the Rayleigh quotient ‖δx‖²/‖x‖² are nonincreasing, so x “becomes a global section of F(t) at least as quickly as it approaches zero” (p. 17).
What drives the flows. Both flows depend on the state only through the edge disagreement δx (eqs. 8.1–8.3, 9.1). Neither has a term relating an expression F_{v⊴e}x_v to the opinion x_v it expresses, to the agent’s initial maps, or to anything outside the network. This is a property of the equations as written. Example 9.5: when opinions are far apart, “the sheaf is able to adapt more easily than the opinions themselves” (p. 17), and an agent again ends up lying.
What this does not say
- That the learned maps cannot collapse. Theorem 9.3 bounds x away from zero, not δ. ‖δ‖_F only decreases (Theorem 9.4). In §8 the nearest-sheaf property keeps each change minimal, a rank-one change along x_v at each vertex, which is not a bar to zero maps. (The reading derives, beyond the paper, that a vertex’s maps cannot all vanish when its weighted block starts with a positive eigenvalue.)
- That the guards are evidence. Minimal change and a conserved quantity keep the outcome non-degenerate. They do not make the agreement reached any more faithful to the opinions held. The consensus is coherence by construction.
- That people learn to communicate this way, or that “lying” here is more than a sign change in a linear map. The authors set the sociological question aside (p. 13).
- Which equilibrium is selected. The equilibria form a singular variety, and the paper characterizes stability only at its smooth points.
- Anything about the knowledge-sheaf case directly. The zero section is common to both (THEORY-299), but there the maps are trained against negatives, and here there are none.