We develop a discipline-agnostic emergence calculus that treats theories as fixed points of idempotent operators acting on descriptions. We show that, once processes are composable but access to the underlying system is mediated by a bounded observational interface, a canonical toolkit of six closure-changing primitives (P1--P6) is unavoidable. The framework unifies order-theoretic closure operators with dynamics-induced endomaps $E_{τ,f}$ built from a Markov kernel, a coarse-graining lens, and a time scale $τ$. We introduce a computable total-variation idempotence defect for $E_{τ,f}$; small retention error implies approximate idempotence and yields stable "objects" packaged at the chosen $τ$ within a fixed lens. For directionality, we define an arrow-of-time functional as the path-space KL divergence between forward and time-reversed trajectories and prove it is monotone under coarse-graining (data processing); we also formalize a protocol-trap audit showing that protocol holonomy alone cannot sustain asymmetry without a genuine affinity in the lifted dynamics. Finally, we prove a finite forcing-style counting lemma: relative to a partition-based theory, definable predicate extensions are exponentially rare, giving a clean anti-saturation mechanism for strict ladder climbing.

翻译：我们发展了一种与学科无关的涌现演算，将理论视为作用于描述上的幂等算子的不动点。我们证明，一旦过程可组合但对底层系统的访问受限于一个有界的观测接口，六个改变闭包的原语（P1–P6）构成的正则工具集便是不可避免的。该框架将序论闭包算子与由马尔可夫核、粗粒化透镜和时间尺度 $τ$ 构建的动力学诱导自映射 $E_{τ,f}$ 统一起来。我们为 $E_{τ,f}$ 引入了一种可计算的全变差幂等性缺陷；小的保留误差意味着近似幂等性，并在固定透镜内于所选 $τ$ 处产生稳定的“对象”。对于方向性，我们将时间箭头泛函定义为正向与时间反转轨迹之间路径空间的KL散度，并证明其在粗粒化（数据处理）下是单调的；我们还形式化了一个协议陷阱审计，表明若无提升动力学中真正的亲和性，仅凭协议和乐无法维持不对称性。最后，我们证明了一个有限力迫风格的计数引理：相对于基于划分的理论，可定义的谓词扩展是指数级稀少的，这为严格的阶梯式攀升提供了一个清晰的反饱和机制。