== A. Lemma 3.2 (counting), exhaustive
  n=1: 1 labelled graphs, 1 connected, 1 (F,K) cases with Z all cut vertices; max(|B|-(2r-2))=0; tight cases (r>=2)=0
  n=2: 2 labelled graphs, 1 connected, 1 (F,K) cases with Z all cut vertices; max(|B|-(2r-2))=-2; tight cases (r>=2)=0
  n=3: 8 labelled graphs, 4 connected, 7 (F,K) cases with Z all cut vertices; max(|B|-(2r-2))=-1; tight cases (r>=2)=0
  n=4: 64 labelled graphs, 38 connected, 90 (F,K) cases with Z all cut vertices; max(|B|-(2r-2))=0; tight cases (r>=2)=12
  n=5: 1024 labelled graphs, 728 connected, 1938 (F,K) cases with Z all cut vertices; max(|B|-(2r-2))=0; tight cases (r>=2)=60
  n=6: 32768 labelled graphs, 26704 connected, 67720 (F,K) cases with Z all cut vertices; max(|B|-(2r-2))=0; tight cases (r>=2)=360
  n=7: 2097152 labelled graphs, 1866256 connected, 3964184 (F,K) cases with Z all cut vertices; max(|B|-(2r-2))=0; tight cases (r>=2)=7560
  tightness path example: |B| = 2r-2 for r=1..8 checked
== B. Lemma 3.3 (core lemma; no density)
  exhaustive n=2: 0 feasible (G,S), 0 optimal Y*, 0 (Y*,H) pairs; bound attained by 0 Y*; Y* with non-D vertices: 0
  exhaustive n=3: 15 feasible (G,S), 15 optimal Y*, 27 (Y*,H) pairs; bound attained by 15 Y*; Y* with non-D vertices: 0
  exhaustive n=4: 456 feasible (G,S), 468 optimal Y*, 1292 (Y*,H) pairs; bound attained by 468 Y*; Y* with non-D vertices: 0
  exhaustive n=5: 20690 feasible (G,S), 22250 optimal Y*, 94910 (Y*,H) pairs; bound attained by 22250 Y*; Y* with non-D vertices: 60
  exhaustive n=6: 1625052 feasible (G,S), 1860152 optimal Y*, 12330516 (Y*,H) pairs; bound attained by 1859792 Y*; Y* with non-D vertices: 7560
  random sparse/tree-like n=7..14: 24000 instances, 29902 optimal Y*, 1016936 (Y*,H) pairs; attained 28304; with non-D: 5099
== D. Theorem 3.4 (algorithm) vs brute force
  401 dense instances (218 random, 183 cluster+long-path), opt up to 14, c up to 8: algorithm = brute force in all
== C. Lemma 3.1 (hubs)
  3000 dense instances (eps in 0.1..1, k<=60): all bounds hold; max greedy steps / bound = 1.000
== E. Lemma 4.1 (gadget)
  checked (q,d) in [(2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (3, 1), (3, 2), (3, 3), (5, 1), (5, 2), (7, 2), (3, 4), (2, 6)]: counts and tau = d
== F. Lemma 4.2 (product)
  3000 random pairs of set systems: tau(product) = min
== G. Lemma 4.3 (dense Steiner instances)
  200 instances: density, |S|, |V| and its bound, opt = min(tau,d)
== H. Lemma 4.4 (block encoding)
  4635 (formula,k) pairs (618 satisfiable formulas): tau<=k iff satisfiable
== I. full chain 3-SAT -> Set Cover -> dense Steiner Tree
  40 formulas (23 satisfiable): opt<=k iff satisfiable; density checked
== J. size bounds (Theorems 1.1 and 1.2)
  Theorem 1.1 exponent bound and M <= 8v^3 verified on ranges;
  max log2(N)/sqrt(v) over v <= 3000 and v = 1e4,1e5,1e6: eps=0.3,q=2: 6.36, eps=0.49,q=2: 8.00, eps=0.6,q=3: 7.78, eps=0.9,q=11: 12.02, eps=0.99,q=101: 19.09
== K. Remark 4.5 and the eps=1/|N| remark
  200 set systems: dense Set Cover density and eps=1/|N| reduction (opt = tau)
== L. repetition-free variant of the gadget (suggested fix for Remark 4.5)
  projective indexing: distinct sets, each point in a (q^d-q^(d-1))/(q^d-1) fraction, tau = d
TOTAL failures = 0; time 25 s
exit 0
