AI 中文总结
该研究定位自指裁决中可判定性的边界，提出带自指门的撤销结构，证明有限层级可判定、极限处收敛失效，裁决器最小图灵度为神谕的跳转度，未用哥德尔不完备定理。
AI 中文摘要
撤销结构由带编号的区分域和一个Σ¹₁显现谓词构成；裁决器是一个部分映射，为区分指派“已撤销”或“豁免”的裁决。我们探究哪些域存在一个在使命上是完全、正确且完备的裁决器——即撤销所有非显现成员，并精确定位该边界。正面结果：具有可判定显现性的域存在带证书的规范裁决器，适用于Presburger算术和有限状态重入系统。负面结果：添加一个零元门（区分可查询自身裁决）会统一破坏可判定性：扩展域存在三分法，其中每个裁决器在预先固定的一个区分处会丧失完全性、穷尽性或可靠性。限制自指深度可细化结论：每个有限层级仍可判定，迭代裁决的层级是查询图上布尔网络的同步更新，对显式呈现的网络，判定稳定性是PSpace完全的，且在闭包大小为n时会出现高达2ⁿ−1的周期。极限情况下失效的是收敛性而非可判定性；对角区分以周期2振荡，其1/2的平均频率是反思神谕在该处必须返回的值。在神谕X对应的标准域上，具备全部三个性质的裁决器的最小图灵度是X'的度：裁决每层级需付出一次跳转的代价。该边界是确定性、正确性与裁决层级间的三方权衡。不动点核心在Ershov意义下的每个预完全编号上成立。该框架属于Kleene两个递归定理划分中的内涵侧。未使用哥德尔不完备定理。
英文摘要
An annulment structure consists of a numbered domain of distinctions with a $Σ^0_1$ manifestation predicate; an adjudicator is a partial map assigning to distinctions the verdicts annulled or exempt. We ask which domains admit an adjudicator that is total, correct, and complete in its mission -- annulling every non-manifesting member -- and locate the boundary exactly. On the positive side, domains with decidable manifestation admit canonical adjudicators with certificates, instantiated for Presburger arithmetic and finite-state re-entry systems. On the negative side, adjoining a single nullary gate, by which a distinction may query the verdict passed on itself, destroys decidability uniformly: the extended domain carries a trichotomy in which every adjudicator fails totality, exhaustiveness, or soundness at one distinction fixed in advance. Bounding the depth of self-address refines this: each finite level remains decidable, the hierarchy of iterated verdicts is the synchronous update of a Boolean network on the query graph, deciding stabilisation is PSpace-complete for explicitly presented networks, and periods as large as $2^n-1$ occur at closure size $n$. In the limit what fails is convergence, not decidability; the diagonal distinctions oscillate with period two, and their mean frequency of $1/2$ is the value a reflective oracle is forced to return there. Over the standard domain relative to an oracle $X$, the least Turing degree of an adjudicator with all three properties is the degree of $X'$: adjudication costs one jump per level. The boundary is a three-way trade among determinacy, correctness, and residence at the level adjudicated. The fixed-point core holds over every precomplete numbering in the sense of Ershov. The framework falls on the intensional side of the divide between Kleene's two recursion theorems. Gödel's incompleteness theorems are nowhere used.
