Distributed computational substrates rely on two elementary operations: bundling, the act of populating a shared physical medium with independently retrievable components, and binding, the act of composing components into outputs whose identity depends on their relations. We study these two primitives on the simplest closed substrate carrying a continuous symmetry, a cycle graph of N nodes, in two parameter regimes of a single master equation of motion. The linear regime sorts a temporal input across the substrate's U(1)-organised eigenmodes, providing a feature representation that matches a windowed-FFT baseline at high signal-to-noise ratio and modestly outperforms it for transient signals at low SNR. The Duffing regime activates a cubic mode-mixing operation constrained by the substrate's symmetry into a sparse selection rule on integer wavenumbers, generating shape-dependent harmonic content that the linear regime cannot produce. We identify a single-number observable, $φ_0$, that summarises the bound representation's response to input shape, and we analyse its symmetry structure: a $π$-periodicity in the shape parameter is exact, while a time-reversal symmetry that would render $φ_0$ degenerate is broken by the substrate's dissipation. The asymmetric status of these two symmetries is what licenses $φ_0$ as a meaningful single-number observable; its trajectory across the quotient domain encodes the joint response of binding and dissipation to the input shape. Numerical experiments confirm that $φ_0$ retains its information content under additive band-limited noise, with seed-averaged means staying clearly above the symmetric-attractor value down to 0 dB input SNR. The framework is developed on synthetic signals only; extensions to richer substrates, more elaborate drives, and real biological signals are open questions for the work that follows.

翻译：分布式计算基板依赖两种基本操作：捆绑——用独立可检索的组件填充共享物理介质的行为；以及绑定——将组件组合为依赖其关系决定身份的输出的行为。我们在具有连续对称性的最简单闭合基板（即由N个节点构成的环图）上，于单个运动主方程的两个参数体系中研究了这两种基本操作。线性体系将时间输入排序到基板U(1)对称性组织的本征模式上，在信噪比高时提供与窗函数FFT基线匹配的特征表示，在信噪比低时对瞬态信号的性能略优于该基线。达芬体系激活了由基板对称性约束的三次模式混合操作，在整数波数上形成稀疏选择规则，产生线性体系无法生成的形状相关谐波内容。我们识别出一个单数值可观测量$φ_0$，它总结了绑定表示对输入形状的响应，并分析了其对称性结构：形状参数中的$π$周期性是严格的，而会使$φ_0$退化的时间反演对称性则被基板耗散破缺。这两种对称性的非对称状态使得$φ_0$成为有意义的单数值可观测量；其在商域内的轨迹编码了绑定与耗散对输入形状的联合响应。数值实验证实，在加性带限噪声下$φ_0$能保留其信息内容，在输入信噪比低至0 dB时，种子平均均值仍明显高于对称吸引子值。该框架仅基于合成信号开发；扩展到更丰富的基板、更复杂的驱动及真实生物信号，是后续工作的待解决问题。