This is an undirected matching-diagram model, retained for comparison with the typed interaction in the main essay. Pick a morphism f : A → B and g : B → C. Composition is “glue along B” (a cut), then execute by following paths; hiding the internal interface yields the composite wiring on A and C, plus any closed feedback loops (a scalar/traced residue).
Treat every drawn wire as an undirected “conductor.” A token launched at an exterior port follows its unique incident wire. At each internal B port it continues along the other incident wire, so paths concatenate across the cut. A diagnostic launch at an internal B port selects one of its two directions first. Each exterior port lies on a path to another exterior port. Closed cycles are separate components: clicking an internal B port can inspect one, but no exterior request can enter it.
This is a deliberately minimal “matching-diagram” picture: morphisms are wiring constraints, composition is gluing along a shared interface, and execution is path-following (the classic “token chasing” intuition behind Girard’s execution formula). The reduced view is the observable composite after hiding the internal interface B — categorically, think “trace”/feedback; diagrammatically, think “erase the middle column and reconnect endpoints by chasing paths,” counting any closed cycles as a scalar residue.
A possible connection to linear logic, requiring additional correctness conditions: A,B,C are interfaces/types, gluing is analogous to proof composition, and the token is a caricature of the GoI dynamics that computes cut-elimination without rewriting the whole net — it “measures” the composite by running through it.
Mathematical setting: a small fragment of compact-closed matching diagrams with formal loop counts. It is not a proof-net correctness checker or a game-strategy model. Related sources: Girard’s GoI I–III (operator-algebraic semantics), Danos–Regnier proof nets, and the traced/symmetric-monoidal account (Joyal–Street–Verity). The picture here is closest in spirit to compact-closed/string-diagram “interaction = trace” folklore.
Joining components can change their exterior connections and produce detached loops. That is what this model computes. A formal loop count belongs to this matching-diagram model; it does not measure learning, amplification, coordination cost or feasibility. The main reading-room essay instead uses typed response rules, so you can inspect how one coordinator’s answer changes another’s action. Those are different models with different meanings for feedback.