Exact-Mean Decompositions for Cost-Preserving Single-Source Unsplittable Flows: Demand Scales and Local Interaction Structure

This is Part III of a series on cost-preserving single-source unsplittable flow. Parts I and II proved lower bounds (records $1.1397\ldots$, $1.28249\ldots$, refined later to $1.28260069\ldots$); this paper proves the first unconditional positive results. Main contributions: a mixture characterization: on every class closed under arc deletion, the optimal cost-preserving additive constant equals the least radius at which every fractional flow is an exact-mean convex combination of support routings; the route-difference matrix of every exact-two-path instance is a network matrix, hence totally unimodular: pairwise interactions have exact constant 1 at any cycle rank, and sign-mixed or odd-hole patterns are unrealizable; the exact local envelope ladder: $E(2) = 1$, $E(3) = 9/8$, and $E(4) = (299 - 41\sqrt{41})/32 = 1.13974707\ldots$, a solver-free closure of all 2,015 boundary cells identifying the record lower bound constants as exact envelopes of the general theory; a tree-path four-coloring theorem giving radius 2D for every arity-three system and explicit bounds for every arity, the first finite constants for unbounded classes; exact class constants: out-trees 0, two-layer hubs 1, outerplanar two-exit interval spines 1 (sharp), series-parallel at most 1, and a new planar lower bound $$\frac{58676765987259}{50000000000000} = 1.17353531974518$$ against the known ceiling 2; scale-sum, chain-cover and shifted-chain theorems for structured demand sets, including the Morell-Skutella constant 2 on all two-valued instances, and a reduction of the universal question to a single factor-two merger statement with certified wall $K >= 2.565201380250176725...$. Every numerical claim is certified in exact rational arithmetic by at least two independently written programs that rebuild each instance from its raw arc list; the final proofs are solver-free. The artifacts are published in the companion repository: https://github.com/snikolenko/unsplittable-flows.