Why it is a question
The owner, at U33: “also, I’m wondering if the modular duality program might be useful to us here.” A33 took it up through Bernstein and Newhouse’s modular duality (ANTH-LIT-438, read in the anthology, where ANTH-THEORY-024 holds that orthogonalizing an update is dualizing it). A modular norm says which parameter changes are large, so it fixes the budgeted residual Φ_r and which modules are bottlenecks. A33 kept the dualities apart: “a gradient is not an update, and an update is not an endorsement”, and “the mathematical duality map is not itself an endorsement relation”. A34 repeated it for the coupled update rule it proposed, d = argmin [Σ w_i⟨g_i, d⟩ + (λ/2)‖d‖²_𝓜]: “the modular duality map is not itself an endorsement mechanism”.
A33’s next result has two halves:
- Frustration in a modular geometry. The static half is largely in hand. CLAIM-365 gives Φ_r and its dual for any norm, a modular one included, and CLAIM-366 says which parts survive a change of chart. What is open is A33 §3B’s localization: whether frustration can be assigned to a layer or module, “whether a particular layer or subnetwork becomes a bottleneck through which several competing learning demands must pass”.
- Persistence through endorsement. A33 §8: “a system may become trapped in a self-reinforcing frustration regime even though better alternatives remain available.” In CLAIM-357’s terms, Γ stays positive while Φ ≤ 0, and stays so because the allocation of influence w_t is updated on outcomes the frustration itself produces.
The record holds the pieces of the second half but not the result. CLAIM-297 says authority updated on the interpretation it helped make dominant can amplify without fidelity. CLAIM-330, in CASE-099’s linearization, gives the threshold βγ > (1 − α)(1 − δ) beyond which the symmetric state is unstable, with no demands in the model. TERM-020 names the regime that generates the conditions for its own continuation. QUESTION-117 asks whether such coupled models do anything that opinion dynamics with evolving influence cannot. A39’s case study 2 proposed adding demands and a feasible-update oracle to CASE-099.
What would count as an answer
A model with demands g_i, an update geometry, and influence weights w_t updated by an endorsement rule, and a condition, in its parameters, under which a conflict present at t₀ and absent later (Φ ≤ 0 from some t₁) leaves Γ bounded away from zero. Better still if the condition shows a dependence on the norm: that a modular geometry and a Euclidean one give different persistence for the same demands and the same endorsement rule. A negative answer would also count: that persistence is fixed by the endorsement rule alone and the update geometry never enters, which would make modular duality irrelevant to this half.