1 min read
| Status | Active — read in full 2026-10-27 (NOTE-1223), from arXiv v6; worth reading as the theorem that black-boxing passive linear circuits is functorial: the Lagrangian relation a circuit imposes on its terminals' potentials and currents composes as the circuits do. It is also a clean account of eliminating an interior by minimum power, and of why that elimination alone has no identities. Its scope is narrow: passive, linear, two-terminal components, time only through the Laplace transform with zero initial conditions. |
| Version | 2 |
| Tags | compositionality, mathematics |
| Cited as “grounds” by | CLAIM-480 — Composing interactions can turn an interior into a constraint, and sometimes into a constraint of a new kind: in the chain E = ½k₁(x₁ − x₂)² + ½k₂(x₂ − x₃)², with x₂ internal and equilibrated, the boundary sees E_eff = ½·k₁k₂/(k₁ + k₂)·(x₁ − x₃)², a coupling between x₁ and x₃ that neither part has; if the interior relaxes slowly, eliminating it leaves a history-dependent constraint, a memory kernel, so interconnection fixes an effective constraint on the boundary's trajectories and separation of time scales decides only whether that constraint is approximately memoryless; and for cellular sheaves the reduced constraint can relate three boundary values where every relation among the parts is pairwise |
| Cited as “paper” by | NOTE-1223 — Baez & Fong — A compositional framework for passive linear networks |
| Cited as “source” by | THEORY-1063 — Black-boxing a passive linear circuit, which keeps only the Lagrangian relation its terminals' potentials and currents can jointly satisfy, is a hypergraph functor from open circuits to Lagrangian relations: the external behaviour of a circuit glued from parts is the relational composite of the parts' external behaviours, so circuits with the same behaviour are interchangeable in every composite, and the effective relation a composite presents at its boundary is fixed by its parts' black boxes alone, THEORY-1089 — In a passive linear network, eliminating interior nodes by the principle of minimum power, which over a general field with positive elements is stationarity, takes a Dirichlet form to a Dirichlet form on the remaining nodes, linearly in their values; eliminating one node s adds c_is·c_js/Σ_k c_ks to each coupling c_ij, so two resistors in series act at their ends as one of resistance r_AB + r_BC; composing Dirichlet forms by this elimination is associative but lacks identities, because an ideal wire of zero impedance has no power functional |
Baez, J. C., & Fong, B. (2018). A compositional framework for passive linear networks. Theory and Applications of Categories, 33(38), 1158–1222. arXiv:1504.05625.
Read in full on 2026-10-27 (NOTE-1223).
A23 §IV cited it as “a rigorous theory of precisely this phenomenon”: composed interactions yield an effective boundary behavior that constrains further interaction, the composition CLAIM-480 demonstrates in its simplest case. It bears on CLAIM-480 and CLAIM-477.