Discoveries / combinatorics

1.17353 planar lower bound and exact local envelopes for cost-preserving single-source unsplittable flow

Part III of a series, and the first unconditional positive results in it. Settled exactly: the local envelope ladder $E(2)=1$, $E(3)=9/8$ and $E(4)=(299-41\sqrt{41})/32=1.13974707\ldots$, which recasts the earlier record constants as exact envelopes of the general theory rather than isolated instances, plus exact constants for four classes - out-trees 0, two-layer hubs 1, outerplanar two-exit interval spines 1 (sharp), series-parallel at most 1. Improved but not settled: the planar lower bound rises to $1.17353531974518$ against the known ceiling 2, and every exact-two-path instance whose rows touch at most three terminals satisfies $C\le2$, the first unconditional constant for an unbounded class. The universal question is untouched - it reduces here to a single factor-two merger statement with certified wall $K^*\ge2.5652\ldots$, twice the refined general lower bound $1.28260069\ldots$.