The claim
Take operators 1, …, N with budgets b_k > 0, endorsements E_kj ≥ 0 with E_kk = 0 and each row summing to at most 1, and a damping 0 < η < 1. CLAIM-282’s damped recursive rule is g = (1 − η)(I − ηEᵀ)⁻¹b, and under it an operator’s authority depends on its own endorsements through every cycle through it.
The rule. Let E₍₋ᵢ₎ be E with row i, i’s own outgoing endorsements, set to zero. Operator i’s authority is g_i = [(1 − η)(I − ηE₍₋ᵢ₎ᵀ)⁻¹b]_i. Each operator’s authority is read from the recursive rule on a graph in which it endorses no one.
- What it counts. Expanded as a series, g_i is (1 − η)b_i plus, for every walk of endorsements that ends at i and does not visit i before its end, (1 − η)b_s η^ℓ times the product of the endorsements along it, s being the walk’s start and ℓ its length. Authority reaches i along paths of any length, and cycles that avoid i still feed it. What never reaches i is authority that has already passed through i, its own budget included. In CASE-077’s pair, g₁ = (1 − η)b₁ + η(1 − η)b₂, against (b₁ + ηb₂)/(1 + η) under the recursive rule. Where no cycle passes through i, g_i is its value under the recursive rule.
- Its own endorsements do not move it. E₍₋ᵢ₎ does not contain row i, so g_i is a function of the other rows and of b alone. This is exact independence, not a vanishing derivative: any change to i’s endorsements, of any size, leaves g_i where it was.
- Its endorsements carry beyond one hop. For k ≠ i, let M = (I − ηE₍₋ₖ₎ᵀ)⁻¹, whose entries are non-negative. Then ∂g_k/∂E_ij = (1 − η)η M_kj (Mb)_i, with (Mb)_i ≥ b_i > 0, and M_kj > 0 exactly when j reaches k by a path of positive endorsements. So raising E_ij raises the authority of j and of every operator j reaches, however many hops away. It lowers no one’s, and it leaves i’s own unchanged.
So an operator’s authority can be independent of its own endorsements while endorsement carries authority further than one hop.
The check
p19/claim282_loo.py (numpy) checks each step on random graphs, with controls:
- Step 2. Redrawing i’s whole row moved g_i by exactly 0 in 300 graphs. The recursive rule moved it by up to 0.58, and the one-hop rule by 0.
- Step 1. An absorbing-walk computation with no inverse agrees with the formula to 6e-16. The pair and the no-cycle cases match their closed forms.
- Step 3. Raising E_ij raised every reachable k ≠ i, 1,325 cases, and lowered none. The 213 unreachable operators stayed put. Two hops out, the one-hop rule gives zero.
The eighteenth review’s script (p18/cluster_da_checks.py, part B) found the same by finite differences. The derivative in step 3 was re-derived by hand at the nineteenth review’s ruling.
What it does not say
- That authority is conserved. Each operator’s authority is read off a different graph, so the vector is not one fixed point. In the check the total ran from 17% to 99.7% of the budgets, where the recursive rule keeps it equal to them. A2 said the same of its path-based rule with origin tags (CLAIM-282), whose open conventions this rule may be one way to fix.
- That it closes the social route. g_i still depends on what others endorse, so an operator can change its authority by changing what others endorse (CLAIM-282).
- That delegating to a nearer ally pays, as under one hop. CLAIM-375’s guarantee does not carry over. Raising an endorsement of an ally nearer the endorser’s policy than the authority-weighted decision moved the decision toward the endorser in 1,772 of 2,141 trials, against all 2,138 under one-hop allocation. The added authority also flows on from the ally to those it endorses. Whether the incentive still does work is CLAIM-285’s question.
- That it is a cycle ban. The graph keeps its cycles. Only each operator’s own accounting stops at its boundary, so CLAIM-282’s point about cycle bans stands.