CLAIM-375: Under one-hop allocation within an epoch, an operator that endorses an ally whose policy is nearer its own than the collective decision is raises how far its preferred policy is realized while its own authority stays fixed, so the two conditions of the endorsement scheme hold together

The claim

Take operators 1, …, N with policies p_k in ℝᵈ and endorsements E_kj ≥ 0 for k ≠ j. Within an epoch, authority is allocated in one hop, as A3 §4 proposed (CLAIM-282):

I_j = I_j⁰ + Σ_(k≠j) b_k E_kj,

with budgets b_k > 0 and baselines I_j⁰ ≥ 0 fixed independently of the endorsements made in the epoch. The collective decision is the authority-weighted mean p* = Σ_j I_j p_j / S, with S = Σ_j I_j > 0. Operator i’s objective is J_i = −‖p* − p_i‖² − λ Cost(E_i), as in A2.

1. Its authority is fixed. I_i depends on E_ki for k ≠ i only, so ∂I_i/∂E_ij = 0 for every j. This is CLAIM-282’s one-hop case.
2. Delegating to a nearer ally pays. Let ‖p_j − p_i‖ < ‖p* − p_i‖. Raising E_ij by δ > 0 adds Δ = b_i δ to I_j and nothing else, so the new decision is p*′ = p* + t(p_j − p*) with t = Δ/(S + Δ) in (0, 1). The map f(t) = ‖p* + t(p_j − p*) − p_i‖² is convex, with f(1) < f(0). So f(t) ≤ (1 − t)f(0) + t f(1) < f(0): the decision is strictly nearer p_i. The same convexity gives f′(0) ≤ f(1) − f(0) < 0, so the gain has a positive rate at δ = 0. When λ times the marginal cost is below that rate, a small increase in E_ij raises J_i.

So A2’s two conditions, an authority that does not respond to one’s own endorsements and a reward for how far one’s preferred policy is realized, hold at once. They constrain different quantities. That is CLAIM-285’s point, stated as a construction.

What it does not say

• That operators delegate. That operators bound by the constraint actually delegate, and do it better than at random, is a claim about behaviour. It is CLAIM-285’s.
• Anything beyond one hop. Under recursive delegation the first condition fails (CLAIM-282). Whether some other multi-hop rule keeps both conditions is open.
• That the decision rule is the right one. The weighted mean is one rule. A rule under which p* does not move toward an ally’s policy as the ally’s authority grows would need its own argument.
• That endorsement cannot be gamed. An operator can still change its authority by moving others’ endorsements, which is social feedback, not an accounting loop (CLAIM-282). And an aligned ally can be incompetent (A2 §6).