# 1. The author's certificates (written by verify_paper.py) for the counterexamples G62, min62 (the
#    mirror image of G62, Remark 3.2), H73, G62sub = G62^(2), G49 and G59 (Remark 6.3).
G62      |V|=62 cells=231 : X(G) collapse certificate VALID ; 21 sections, 21 pieces : all piece certificates VALID
min62    |V|=62 cells=231 : X(G) collapse certificate VALID ; 21 sections, 21 pieces : all piece certificates VALID
H73      |V|=73 cells=265 : X(G) collapse certificate VALID ; 27 sections, 27 pieces : all piece certificates VALID
G62sub   |V|=231 cells=917 : X(G) collapse certificate VALID ; 39 sections, 39 pieces : all piece certificates VALID
G49      |V|=49 cells=175 : X(G) collapse certificate VALID ; 21 sections, 21 pieces : all piece certificates VALID
G59      |V|=59 cells=219 : X(G) collapse certificate VALID ; 21 sections, 21 pieces : all piece certificates VALID
ALL CERTIFICATES VALID

# 2. The third referee's own certificates for G49 and G59 (../referee2/G49_*_collapse_ref2.txt,
#    ../referee2/G59_*_collapse_ref2.txt), replayed on the author's data/G49.json and data/G59.json.
G49      |V|=49 cells=175 : X(G) collapse certificate VALID ; 21 sections, 21 pieces : all piece certificates VALID
G59      |V|=59 cells=219 : X(G) collapse certificate VALID ; 21 sections, 21 pieces : all piece certificates VALID
ALL CERTIFICATES VALID

# 3. The trough T12 and its subdivision T12sub (paper, Section 6, 'Two directions'). X(G) is collapsible.
#    Their y-pieces for 0 <= y < 1 (resp. 0 <= y < 2) are hexagons, so these pieces are not collapsible;
#    their certificate entries say 'collapsible=False' and contain no collapse sequence. This is expected.
T12      |V|=12 cells=39 : X(G) collapse certificate VALID ; 11 sections, 11 pieces : all piece certificates VALID; not collapsible, as expected (hexagons): [('y', 0.0), ('y', 0.5)]
T12sub   |V|=39 cells=141 : X(G) collapse certificate VALID ; 19 sections, 19 pieces : all piece certificates VALID; not collapsible, as expected (hexagons): [('y', 0.0), ('y', 0.5), ('y', 1.0), ('y', 1.5)]
ALL CERTIFICATES VALID (the only pieces without a collapse sequence are the expected hexagon y-pieces of T12 and T12sub, which are not collapsible)
