CONNECTION MATRICES IN COMBINATORIAL TOPOLOGICAL DYNAMICS

The paper proves that for a gradient combinatorial vector field V on a regular Lefschetz complex X, the Morse decomposition by critical cells has exactly one connection matrix, and it coincides with the (X̄, X̄)-matrix of ∂κ restricted to Fix Φ (up to graded similarity) . The core steps—constructing Γ on matched pairs, defining Φ := id + ∂Γ + Γ∂, showing stabilization Φ∞ and that Fix Φ is spanned by x̄ := Φ∞x for critical x, establishing filtered chain equivalences between (V, C(X), ∂κ) and (X̄, C(X̄), ∂κ̄), and invoking uniqueness for natural filtrations—are explicitly in the paper . The model’s solution mirrors these arguments closely and reaches the same conclusions. Minor issues: the model asserts “E kills every regular vector,” which is not established (Φ∞ projects onto Fix Φ; images of noncritical cells need not be zero), and it states exact equality rather than “up to graded similarity.” These do not affect the main conclusions, especially once the canonical x̄-basis is fixed.