Broken-symmetry shape discrimination on a driven Duffing ring

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, $\phi_0$, that summarises the bound representation's response to input shape, and we analyse its symmetry structure: a $\pi$-periodicity in the shape parameter is exact, while a time-reversal symmetry that would render $\phi_0$ degenerate is broken by the substrate's dissipation. The asymmetric status of these two symmetries is what licenses $\phi_0$ 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 $\phi_0$ 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.