Conductance-based deterministic approximation of oriented cycle-rooted forests and long-subdivision sink-free orientations

This paper gives a deterministic relative-accuracy algorithm for the positive-weight oriented cycle-rooted spanning-forest partition function. Running time is polynomial in graph size, inverse accuracy, input bit length, and the numerical largest/smallest positive weight ratio, not its logarithm. In particular, the established unweighted quantity has a deterministic FPTAS on arbitrary explicitly represented graphs. A conductance subsolution, rational rounding, adaptive recursion, and weighted series reduction supply the algorithm. The paper also proves exact subdivision-kernel zero geometry and, for each fixed alpha > 0, a deterministic FPTAS for sink-free orientations of long subdivisions of simple cubic kernels with minimum path length at least n^alpha, where n counts kernel vertices. Compressed unequal lengths retain numerical weight-ratio dependence. General minimum-degree-two sink-free orientation counting is not resolved. Complete proofs, precise input models, prior-work credit, a standalone reference algorithm, and verification scripts are included. AI assistance in research development, proof checking, computational verification, and manuscript preparation is disclosed; no formal verification or external peer review is asserted.