Abstract models of computation often treat the successor function $S$ on $\mathbb{N}$ as a primitive operation, even though its low-level implementations correspond to non-trivial programs operating on specific numerical representations. This behaviour can be analyzed without referring to notations by replacing the standard interpretation $(\mathbb{N}, S)$ with an isomorphic copy ${\mathcal A} = (\mathbb{N}, S^{\mathcal A})$, in which $S^{\mathcal A}$ is no longer computable by a single instruction. While the class of computable functions on $\mathcal{A}$ is standard if $S^{\mathcal{A}}$ is computable, existing results indicate that this invariance fails at the level of primitive recursion. We investigate which sets of operations have the property that if they are primitive recursive on $\mathcal A$ then the class of primitive recursive functions on $\mathcal A$ remains standard. We call such sets of operations \emph{bases for punctual standardness}. We exhibit a series of non-basis results which show how the induced class of primitive recursive functions on $\mathcal A$ can deviate substantially from the standard one. In particular, we demonstrate that a wide range of natural operations, including large subclasses of primitive recursive functions studied by Skolem and Levitz, fail to form such bases. On the positive side, we exhibit natural finite bases for punctual standardness. Our results answer a question recently posed by Grabmayr and establish punctual categoricity for certain natural finitely generated structures.

翻译：计算抽象模型通常将自然数集 $\mathbb{N}$ 上的后继函数 $S$ 视为原始操作，尽管其底层实现对应于操作特定数值表示的非平凡程序。这种行为可以在不涉及记法的情况下进行分析：将标准解释 $(\mathbb{N}, S)$ 替换为同构副本 ${\mathcal A} = (\mathbb{N}, S^{\mathcal A})$，其中 $S^{\mathcal A}$ 不再可由单条指令计算。当 $S^{\mathcal{A}}$ 可计算时，$\mathcal{A}$ 上可计算函数的类是标准的，但已有结果表明，这种不变性在原始递归层面失效。我们研究哪些操作集具有如下性质：若它们在 $\mathcal A$ 上是原始递归的，则 $\mathcal A$ 上原始递归函数的类保持为标准。我们将此类操作集称为**准时标准性基**。我们展示了一系列非基结果，表明 $\mathcal A$ 上诱导出的原始递归函数类如何能显著偏离标准类。特别地，我们证明了一类广泛的自然操作（包括由 Skolem 和 Levitz 研究的原始递归函数的大子类）无法构成此类基。在积极方面，我们展示了准时标准性的自然有限基。我们的结果回答了 Grabmayr 近期提出的问题，并确立了某些自然有限生成结构的准时范畴性。