步进递归：资源剖面与下降商
Step Recursion: Resource Profiles and Descent Quotients
AI总结：
该研究提出步进递归的资源表示，明确可变状态宽度与递归下降两个独立参数，推导广义逆下降的深度并关联标准有限分支计算，证明多项式宽度重参数化后下降仍是非冗余资源坐标。
AI中文摘要：
我们提出了一种用于步进递归的资源表示方法，其中可变状态宽度与递归下降是显式且独立的参数。宽度界$u$控制编码机器状态的大小，而有效下降量$ρ$决定可用的递归深度$δ_ρ(u)$。对于广义逆下降，我们直接从生成元增长推导深度，并刻画了可作为生成元轨道出现的递增序列。\n 随后我们将这种深度-宽度几何结构与标准有限分支计算建立关联。每一个确定性有界状态动力学都可通过固定有限数值基上的单个普通有界步进递归实现。对同一局部动力学使用确定性、存在性、全称性或交替性聚合，可得到对应的机器语义。在通过吸收固定局部开销所需的宽度重参数化完成闭包后，得到的语言类恰好是剖面$(δ_ρ(u),u)$上的机器时间-空间类。\n 最后，剖面支配通过可容许宽度重参数化对有效下降进行商化。部分深度曲线会坍缩，但多项式宽度可在规范的多项式深度与指数深度剖面之间支撑一个显式的无限严格分层。因此，在多项式宽度重参数化后，下降仍是一个非冗余的资源坐标；标准复杂性类是校准点。
英文摘要：
We develop a resource representation for step recursion in which mutable-state width and recursion descent are explicit and independent parameters. A width bound $u$ controls the size of the encoded machine state, while an effective descent $ρ$ determines the available recursion depth $δ_ρ(u)$. For generalized-inverse descents, we derive the depth directly from generator growth and characterize the increasing sequences that can occur as generator orbits. We then connect this depth--width geometry to standard finite-branching computation. Every deterministic bounded-state dynamics is realizable by a single ordinary bounded step recursion over a fixed finite numerical basis. Using deterministic, existential, universal, or alternating aggregation on the same local dynamics yields the corresponding machine semantics. After closure under the width reparameterizations needed to absorb fixed local cost, the resulting language classes are exactly the machine time--space classes on profiles $(δ_ρ(u),u)$. Finally, profile domination quotients effective descents by admissible width reparameterization. Some depth curves collapse, yet polynomial widths support an explicit infinite strict hierarchy between the canonical polynomial- and exponential-depth profiles. Thus descent remains a nonredundant resource coordinate after polynomial width reparameterization; standard complexity classes are calibration points.
