We prove that Hilbert's Tenth Problem over $\mathbb{N}$ remains undecidable when restricted to cubic equations (degree $\leq 3$), resolving the open case $\delta = 3$ identified by Jones (1982) and establishing sharpness against the decidability barrier at $\delta = 2$ (Lagrange's four-square theorem). For any consistent, recursively axiomatizable theory $T$ with G\"odel sentence $G_T$, we effectively construct a single polynomial $P(x_1, \ldots, x_m) \in \mathbb{Z}[\mathbf{x}]$ of degree $\leq 3$ such that $T \vdash G_T$ if and only if $\exists \mathbf{x} \in \mathbb{N}^m : P(\mathbf{x}) = 0$. Our reduction proceeds through four stages with explicit degree and variable accounting. First, proof-sequence encoding via Diophantine $\beta$-function and Zeckendorf representation yields $O(KN)$ quadratic constraints, where $K = O(\log(\max_i f_i))$ and $N$ is the proof length. Second, axiom--modus ponens verification is implemented via guard-gadgets wrapping each base constraint $E(\mathbf{x}) = 0$ into the system $u \cdot E(\mathbf{x}) = 0$, $u - 1 - v^2 = 0$, maintaining degree $\leq 3$ while introducing $O(KN^3)$ variables and equations. Third, system aggregation via sum-of-squares merger $P_{\text{merged}} = \sum_{i} P_i^2$ produces a single polynomial of degree $\leq 6$ with $O(KN^3)$ monomials. Fourth, recursive monomial shielding factors each monomial of degree exceeding $3$ in $O(\log d)$ rounds via auxiliary variables and degree-$\leq 3$ equations, adding $O(K^3 N^3)$ variables and restoring degree $\leq 3$. We provide bookkeeping for every guard-gadget and merging operation, plus a unified stage-by-stage variable-count table. Our construction is effective and non-uniform in the uncomputable proof length $N$, avoiding any universal cubic equation. This completes the proof that the class of cubic Diophantine equations over $\mathbb{N}$ is undecidable.

翻译：我们证明了希尔伯特第十问题在 $\mathbb{N}$ 上限制于三次方程（次数 $\leq 3$）时仍然是不可判定的，从而解决了 Jones (1982) 提出的 $\delta = 3$ 这一公开情形，并确立了相对于 $\delta = 2$（拉格朗日四平方定理）可判定性边界的锐利性。对于任何具有哥德尔语句 $G_T$ 的一致、递归可公理化理论 $T$，我们有效地构造了一个次数 $\leq 3$ 的单个多项式 $P(x_1, \ldots, x_m) \in \mathbb{Z}[\mathbf{x}]$，使得 $T \vdash G_T$ 当且仅当 $\exists \mathbf{x} \in \mathbb{N}^m : P(\mathbf{x}) = 0$。我们的归约过程通过四个阶段进行，并给出了明确的次数和变量计数。首先，通过丢番图 $\beta$ 函数和 Zeckendorf 表示进行证明序列编码，产生 $O(KN)$ 个二次约束，其中 $K = O(\log(\max_i f_i))$，$N$ 为证明长度。其次，公理-肯定前件验证通过保护装置实现，将每个基本约束 $E(\mathbf{x}) = 0$ 包装进系统 $u \cdot E(\mathbf{x}) = 0$, $u - 1 - v^2 = 0$ 中，在保持次数 $\leq 3$ 的同时，引入 $O(KN^3)$ 个变量和方程。第三，通过平方和合并 $P_{\text{merged}} = \sum_{i} P_i^2$ 进行系统聚合，产生一个次数 $\leq 6$、具有 $O(KN^3)$ 个单项式的单个多项式。第四，递归单项式屏蔽通过辅助变量和次数 $\leq 3$ 的方程，在 $O(\log d)$ 轮中对次数超过 $3$ 的每个单项式进行因子分解，添加 $O(K^3 N^3)$ 个变量并恢复次数 $\leq 3$。我们为每个保护装置和合并操作提供了详细的记录，以及一个统一的分阶段变量计数表。我们的构造是有效的，并且在不可计算的证明长度 $N$ 上是非均匀的，避免了任何通用的三次方程。这完成了 $\mathbb{N}$ 上三次丢番图方程类不可判定的证明。