From Eternal Actual Existence to PR: A Self-Contained Derivation of the First-Beat Structure. A Conditional Ontological Theorem for Self-Contained, Continuing, Non-Fixed Actuality.

English description **From Eternal Actual Existence to PR: A Self-Contained Derivation of the First-Beat Structure** *A Conditional Ontological Theorem for Self-Contained, Continuing, Non-Fixed Actuality* Can an actual existence that is eternal or beginningless nevertheless possess a first-beat structure? This paper separates a globally earliest event from a locally minimal structure of occurrence and asks the latter question within an explicitly selected ontological domain. The target is actual occurrence rather than bare formal satisfiability: actuality is self-contained, continuing and non-fixed, and `ActualStep` is treated as a primitive actuality predicate rather than as membership in a completed static graph. At the event-structure level, an actuality model is defined as \[ \mathfrak A=(X,\mathcal E,s,t,\prec), \] with role types, particular event tokens, source and target assignments, and actual succession. Directed non-coincidence, continuation and successor compatibility establish that at least two roles are required. When the reachable role quotient is minimal, it has exactly two roles and every successor reverses the ordered role pair. Up to relabelling, the unique minimum transition skeleton is therefore \[ \alpha\rightleftarrows\beta. \] If identical event tokens cannot recur in a finite succession cycle, continuation also yields infinitely many event tokens. Weak eternity permits a one-sided infinite alternating realization; strong eternity permits a bi-infinite realization with no globally first event. Local first-beat structure is therefore compatible with global beginninglessness. At the normal-form level, the reachable quotient is written \(D=\{Y,N\}\). Among the four total maps \(D\to D\), label symmetry and non-fixation leave only the exchange map \[ \sigma(Y)=N,\qquad \sigma(N)=Y. \] When this map is actually enacted, is self-contained, and returns its result as the role of the next event, the paper identifies the resulting minimum actuality core as PR (Paradox–Reference). This is a conditional classification theorem: it does not derive the actuality commitments from neutral logic, does not claim that every possible dynamics is binary, and does not identify a static swap diagram with occurrence. The paper then places PR within the PR–ER–LE architecture. PR supplies the minimum non-fixed self-referential actuality core; ER carries relational configurations and structured histories; LE supplies local, delayed or finite evaluation; a concrete Rule determines the realized dynamics. PR alone is a period-two skeleton and does not imply chaos. The path from PR to chaos, persistent motifs, reproducible interactions, proto-chemistry and cellular life is therefore presented as a Rule-dependent research ladder with explicit additional conditions and failure modes, not as an automatic deduction. The paper includes explicit definitions, ontological commitments, event axioms, normal-form restrictions, dependency tables, twelve adversarial countermodels, a comparison with earlier first-beat arguments, open rival model classes and a full Chinese parallel text. Version 1.0 passed four rounds of internal adversarial review assisted by AI; this is not external peer review. ## 中文说明 **从永恒实际存在到PR：第一拍结构的自足推导** *关于自足、持续、非固定实际性的条件本体论定理* 本文研究：一个永恒或无始的实际存在，是否仍然具有“第一拍结构”？论文严格区分“全局最早事件”与“发生得以成立的局部最小结构”，并明确选择实际发生而非静态形式可满足性作为研究对象。实际性被限定为自足、持续、非固定；`ActualStep` 是实际性原语，而不是已经完成的静态图中的成员关系。 在事件结构层，模型 \(\mathfrak A=(X,\mathcal E,s,t,\prec)\) 区分角色类型、具体事件实例、源角色、目标角色和实际接续。定向不重合、持续性与后继相容性共同给出二角色下界；当可达角色商集达到最小时，每个后继事件必须反转有序角色对，因此唯一最小类型骨