A semantic framework that interprets proof reduction (cut-elimination) as a dynamic flow or interaction of information, often modeled by operators, traces, or token machines that track the movement of information through a proof network.

Principle

Replace static syntactic reduction by a semantic dynamical process where interaction paths, operator composition, or token movement encode the computational behavior of proofs and elucidate invariants preserved by cut-elimination.

Demonstration

In linear logic proofs, represent cuts as connections in a proof net and model cut-elimination by tokens traversing the net or by composing linear operators whose trace records information flow, thereby characterizing normalization as matrix-style dynamics.

Misapplication

Using an interaction model that ignores structural constraints (for example failing to account for resource sensitivity in linear logic) can produce models that do not reflect proof-theoretic normalization and give misleading invariants.

Consequence

Goes beyond static proof identities to provide operational accounts of normalization, yields new invariants (execution formulas, traces), connects to operator theory and machine models, and suggests semantic implementations of computation extracted from proofs.

Reversal

The reversal is viewing cut-elimination purely as syntactic, local rewrite rules without emphasizing the global dynamic flow; such a view misses global invariants and connections to operator semantics.

Boundary

Applies best to systems with circuit-like or net representations of proofs (proof nets, interaction graphs) and where dynamics can be given operator or token semantics; it is less directly applicable to systems lacking a suitable compositional representation.

Semantic Tension

Tension appears between algebraic/operator presentations that emphasize spectral or trace properties and combinatorial/token presentations that emphasize stepwise movement; both capture interaction but highlight different invariants.

Synthesis

The Geometry of Interaction reconceives cut-elimination as information flow: proofs become networks through which tokens or operators move and interact, producing an operational semantics that uncovers dynamic invariants and unifies syntactic reduction with operator-theoretic behavior.