Coupling defects of large-prime fibres in optimal path partitions of the divisor graph

Let G_N be the divisor graph on {1, ..., N}, and let P(X) be the minimum number of vertex-disjoint paths covering V(X). For a prime q > √N, put s = ⌊N/q⌋ and H_N = G_N − {1}. The multiples of q in H_N induce a copy of G_s; if R_{N,q} is the induced complement, define the coupling defect D_N(q) = P(R_{N,q}) + P(G_s) − P(H_N). Following the factorizing framework of Melotti–Saias, call a partition of H_N q-factorizing if every path meeting this copy is contained in it. Then D_N(q) is exactly the excess number of paths forced by q-factorization; in particular, it vanishes precisely when some optimal partition is q-factorizing. We prove, uniformly for integers r ≥ 1, #{√N < q ≤ N : q prime, D_N(q) ≥ r} ≪ √N / (r max{r, log N}). Hence fibres admitting no q-factorizing optimum have vanishing relative density. Estimates of Guth–Maynard also imply D_N(q) = 0 whenever s ≤ N^(13/43−ε) and N is sufficiently large; an almost-all estimate gives density-one vanishing for quotient pairs (N, s) throughout every range s ≤ N^γ, γ < 1/3. For an individual fibre we establish the exact variational formula D_N(q) = max_X (|X| − ρ(X)), where ρ(X) is the least total loss in linear-forest rank needed to realise X. Component contraction and a degree-two relaxation give splicing and matching–cycle descriptions. For q ≥ 29, a maximising interface may be chosen without prime-ended crossings, so for N ≥ 529 only composite–composite crossings need be considered. Exact primal–dual certificates give D_507(23) = 2, so imposing 23-factorization costs exactly two paths. Natural local interface invariants do not control the extremal problem: the remaining obstruction is terminal realisability on maximum or near-maximum optimiser faces. Whether D_N(q) is uniformly bounded remains open.