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| Status | Active — read in full 2026-10-27 (NOTE-1225), from arXiv v2; worth reading as a compositional account in which the coupling data (wiring, potential, response) and the dynamical regime are separate choices: one arrangement of harmonic particles gives a discrete wave equation or a discrete heat equation depending only on whether the integrator stores momentum, and gradient descent with backpropagation is the same construction applied to a parameterized map. Rigorous for its categorical core and its worked examples; many further examples are reported from Claude-written code the author did not audit. |
| Version | 2 |
| Tags | compositionality, learning-theory, mathematics, anthology-candidate |
| Cited as “grounds” by | CLAIM-481 — Coupling does not by itself separate time scales: two identical undamped oscillators coupled with stiffness k have normal modes at ω₀ and √(ω₀² + 2k), nearly equal when k ≪ ω₀², and neither decays; separation comes where the coupling dissipates relative differences faster than collective ones, as in ẋ = −ax − k(x − y), ẏ = −ay − k(y − x), whose collective and relative modes relax at rates a and a + 2k, and in a Laplacian flow ẋ = −κLx, where the spectrum of L sets the rates |
| Cited as “paper” by | NOTE-1225 — Spivak — Compositional dynamics in learning and mechanics |
| Cited as “source” by | THEORY-1054 — One arrangement of harmonic particles on a finite graph, with the same wiring, potential and inverse-mass response, gives the discrete wave equation when the integrator stores position and momentum and the discrete heat equation when it stores position alone; both are governed by the symmetrized graph Laplacian, and the coupling data do not decide between them, THEORY-1057 — Gradient descent with backpropagation, weight decay and Newton's method are one discrete-time semantics of an operad of potential-carrying lenses, in which the state moves by subtracting the sharpened sum of three pulled-back covectors (from the output, from the input wiring and from the potential); the semantics is an operad functor, so a composite's update is assembled from its parts' updates, and backpropagation is that functoriality |
Spivak, D. I. (2026). Compositional dynamics in learning and mechanics. arXiv:2606.28984.
Read in full on 2026-10-27 (NOTE-1225).
A22 §6 called it “almost a controlled demonstration” that “The same relational organization need not imply the same temporal hierarchy”, which is the objection to the owner’s U20 that coupling induces scale separation (CLAIM-481). It bears on CLAIM-473.