The claim
A2 §5 set up the recursive rule g = (1 − η)b + ηEᵀg, with 0 ≤ η < 1, base budgets b, a row-stochastic endorsement matrix E and E_ii = 0. Its solution is g = (1 − η)(I − ηEᵀ)⁻¹b. A2: “Suppose D₁ endorses D₂, and D₂ endorses D₁. Even though neither endorses itself directly, their reciprocal relationship can recycle delegated authority between them.” And: “The constraint E_ii = 0 therefore prevents direct self-endorsement, but not indirect amplification.”
This is right, and it is a fact about the fixed point, not about any psychology. In the two-operator case each endorses only the other, so g₁ = (1 − η)b₁ + ηg₂ and g₂ = (1 − η)b₂ + ηg₁. Then g₁ = (b₁ + ηb₂)/(1 + η). Operator 1’s authority depends on its partner’s endorsement, which returns operator 1’s own contribution after two hops. With more operators, any cycle through i makes g_i depend on E_ij. So the condition CLAIM-285 needs, |∂g_i/∂E_ij| ≈ 0, fails.
A3 §4 gave the case where it holds. Under one-hop allocation, I_j = I_j⁰ + Σ_{i≠j} b_i E_ij, where the budgets b and baseline I⁰ are fixed independently of outgoing endorsements. Then I_i depends on what others endorse and not on E_i·. A3: “For the one-hop aggregation above, this condition holds exactly if budgets and baseline authority are fixed independently of outgoing endorsements. Other agents’ responses can still change I_i later.” Its proposal: “one-hop authority allocation within a governance epoch, with endorsements updated between epochs.”
A2 also named the route no accounting rule closes: “an agent could influence its own authority indirectly by causing other agents to change their endorsements. That is a genuine social feedback effect, not an accounting loop. It cannot be eliminated by cycle prevention alone.”
A6 §4 restated the principle for nested organizations, where the constituents of a higher-level operator legitimately endorse it: “Authority may circulate, but its accounting cannot multiply through circulation alone.” That calls for tracking where delegated authority came from, not for banning every return to its source. It is held here as A6’s slogan, not as a result. A2’s own path-based rule with origin tags admits it “needs an explicit convention for the length-zero path and for authority terminated at cycle boundaries; the total amount of authority actually arriving need not remain constant under this rule.”
Why it is granted and Active
The record states the limit of the design it files as CLAIM-285.
The algebra is elementary and was checked above. It holds whatever any line
concludes, so it carries formal-apparatus as a second line (ADR-040).
What it does not say
- It does not say multi-hop delegation is useless. It says multi-hop delegation and insensitivity to one’s own endorsements cannot both be had from cycle bans alone.
- It does not say the slower social route is a defect. A5 and the owner hold that durable trust is how authority should arise (CLAIM-277). The same route is also how authority is captured.
- It does not say one-hop allocation is the only rule that makes an operator’s authority independent of its own endorsements. Leave-one-out delegation, which counts the authority reaching an operator only along paths that have not passed through it, does so while carrying authority beyond one hop (CLAIM-502). That rule is not a cycle ban: cycles that avoid the operator still feed it.
Note of 2026-10-29: evidence-based subjective logic, an instance with a ceiling
Škorić, de Hoogh and Zannone’s evidence-based subjective logic, LIT-1130, is now read (NOTE-1354). Its fixed point is a damped recursive delegation with self-rating forbidden (A_xx = U), the design this claim is about, in an evidence algebra. The paper’s loop case shows the claim’s point. User 1’s trust in user 2, R_12, depends on user 3’s opinion of 2 weighted by 1’s trust in 3, which depends on R_12 itself, and a near-certain A_32 can make R_12 near-certain whatever A_12 says.
What it adds is a ceiling: no user’s evidence about j can exceed the evidence about j that others hold directly, so recycling cannot create evidence, only re-weight it. It does not make an operator’s standing independent of its own opinions, as one-hop allocation does. The claim is unchanged.