Zero-divisor algebras of graph functions: quantum caging, entangling routing and stochastic first-passage exclusion

We construct Lie-bialgebraic differential structures on vertex functions of a finite graph using coefficients in commutative algebras with zero divisors. We derive the exact Jacobi criterion for the graph bracket and classify its solutions. Over an integral domain, each connected component of the nonzero support is a uniformly weighted clique; over $\mathbb C^q$, the general solution is a superposition of such clique layers. A four-vertex diamond built from overlapping triangle layers shows that Jacobi compatibility is strictly broader than the matching geometry generated by proper edge colouring.
For the canonical cobracket, we prove a rigidity theorem over commutative $2$-torsion-free rings: Lie-bialgebra compatibility is equivalent to the local annihilation condition $w_{ij}w_{ik}=0$ for distinct incident edges. Hence the canonical bialgebra selects matching layers from the wider Jacobi-compatible class. The same structure makes the weighted graph Laplacian an inner derivation and yields an incidence-type vertex--edge calculus with a positive squared-Laplacian factorization.
Representing the idempotent channels by internal-state projectors gives exact quantum caging and channel-controlled transfer that creates path--channel entanglement. The ordered real realization gives an intrinsic graph Fokker--Planck equation, exact first-passage exclusion, and a solvable crossover to escape under weak channel switching. These models are deliberately reducible; their role is to exhibit the quantum and stochastic consequences of the matching geometry selected by canonical bialgebra compatibility.