Self-test of the arithmetic
PASS  pi and sqrt(2) agree with their first 50 decimals; pi in [3.141592653589793238462643383279, 3.141592653589793238462643383280]
PASS  atan(1) = pi/4, acos(1/2) = pi/3, acos(-1/2) = 2pi/3, atan2(-1, -1) = -3pi/4
PASS  atan(1/2) + atan(1/3) = atan(1), atan(2) + atan(1/2) = pi/2, atan(-3/7) = -atan(3/7)

(i) The heptagon Q of Table 1
PASS  the vertices computed from the stereographic data equal the coordinates printed in Table 1
      v0 = (32000/251073, 7000/251073, 248927/251073)
      v1 = (610000/1226981, 732000/1226981, 773019/1226981)
      v2 = (86500/206577, 182500/206577, 43423/206577)
      v3 = (63000/511369, 137000/511369, 488631/511369)
      v4 = (-158000/1011141, -140000/1011141, 988859/1011141)
      v5 = (86200/125181, -51400/125181, 74819/125181)
      v6 = (802000/1167857, -168000/1167857, 832143/1167857)

(ii) Simplicity and star-shapedness of Q (exact)
PASS  Q: all 7 vertices satisfy x^2 + y^2 + z^2 = 1 exactly
PASS  Q: all vertices have z > 0 (open northern hemisphere)
      min z = 0.210202..., so every polar distance is < pi/2
PASS  Q: x_i y_(i+1) - y_i x_(i+1) > 0 for all i (longitude gaps in (0, pi))
PASS  Q: winding number of the longitudes = 1 (must be 1)
PASS  Q: gnomonic image is a simple polygon (all 14 non-adjacent edge pairs disjoint, adjacent edges not overlapping; exact)
PASS  Q: gnomonic image is counterclockwise (twice its signed area = 2.328357...)
PASS  Q: the origin is strictly to the left of every directed edge of the gnomonic image
PASS  Q is a simple spherical heptagon in the open hemisphere about N, star-shaped with respect to N, N in the interior of the region
PASS  no turning angle is 0 or +-pi (all sine numerators s_i nonzero, exact): adjacent developed segments meet only at their common vertex
      signs of sin(tau_i): ['-', '+', '+', '-', '+', '+', '+'] (exact); v0 and v3 are reflex
      edge lengths l_i = acos(v_i . v_(i+1)), turning angles tau_i, interior angles pi - tau_i:
      l0 = [0.788896063149188012, 0.788896063149188013]   tau0 = [-1.917573753423170418, -1.917573753423170417]   interior angle [289.8688, 289.8689] deg
      l1 = [0.520324652473502158, 0.520324652473502159]   tau1 = [0.399668806273697256, 0.399668806273697257]   interior angle [157.1006, 157.1007] deg
      l2 = [1.059709072629907850, 1.059709072629907851]   tau2 = [2.737397626101988088, 2.737397626101988089]   interior angle [23.1586, 23.1587] deg
      l3 = [0.498849449922444674, 0.498849449922444675]   tau3 = [-0.163758055996438226, -0.163758055996438225]   interior angle [189.3826, 189.3827] deg
      l4 = [1.007747884058091704, 1.007747884058091705]   tau4 = [1.762853468189857035, 1.762853468189857036]   interior angle [78.9959, 78.9960] deg
      l5 = [0.291463381858529937, 0.291463381858529938]   tau5 = [1.986779351881962574, 1.986779351881962575]   interior angle [66.1659, 66.1660] deg
      l6 = [0.660047704558384196, 0.660047704558384197]   tau6 = [1.049457433073317079, 1.049457433073317080]   interior angle [119.8705, 119.8706] deg

(iii) The development of Q cut at v0
      w0 = ([0.000000000000000, 0.000000000000000], [0.000000000000000, 0.000000000000000])
      w1 = ([0.788896063149188, 0.788896063149189], [0.000000000000000, 0.000000000000000])
      w2 = ([1.268213886260627, 1.268213886260628], [0.202465227679671, 0.202465227679672])
      w3 = ([0.208515668571714, 0.208515668571715], [0.207261688987692, 0.207261688987693])
      w4 = ([-0.283286827569349, -0.283286827569348], [0.290814537397217, 0.290814537397218])
      w5 = ([-0.259332030476840, -0.259332030476839], [-0.716648596181286, -0.716648596181285])
      w6 = ([0.004400246481246, 0.004400246481247], [-0.592567215406752, -0.592567215406751])
      w7 = ([0.058189206960420, 0.058189206960421], [0.065285138866881, 0.065285138866882])
PASS  cut at v0: segments 0 and 6 (the images of e0 and e6) cross; the other 14 pairs of non-adjacent segments are certified disjoint; no pair undecided
PASS  the four orientation determinants have the certain signs (-, +, +, -)
      orient(w0, w1, w6) = [-0.46747394338566341173, -0.46747394338566341172]   (width < 1e-94)
      orient(w0, w1, w7) = [0.05150318903423074704, 0.05150318903423074705]   (width < 1e-93)
      orient(w6, w7, w0) = [0.03476828703784041768, 0.03476828703784041769]   (width < 1e-93)
      orient(w6, w7, w1) = [-0.48420884538205374109, -0.48420884538205374108]   (width < 1e-93)
      distances of w6, w7 from the line w0w1 and of w0, w1 from the line w6w7:
      [0.592567, 0.592568], [0.065285, 0.065286], [0.052675, 0.052676], [0.733596, 0.733597]
PASS  all four endpoint distances exceed 0.0526
      the crossing point is the image of the point of e0 at arc-length fraction [0.066993, 0.066994]
      and of the point of e6 at arc-length fraction [0.900760, 0.900761]
PASS  the crossing is interior to both segments
PASS  Theorem 1.1(c): the open arc of bad cut points (from the point of e6 at fraction mu through v0 to the point of e0 at fraction lambda) has length lambda l0 + (1 - mu) l6 in [0.11835420, 0.11835421], inside [0.11835, 0.11836]
      closing gap w7 - w0 = ([0.058189, 0.058190], [0.065285, 0.065286])

(iv) The developments of Q cut at v1, ..., v6
PASS  cut at v1: all 15 pairs of non-adjacent segments certified disjoint: the development is simple
PASS  cut at v2: all 15 pairs of non-adjacent segments certified disjoint: the development is simple
PASS  cut at v3: all 15 pairs of non-adjacent segments certified disjoint: the development is simple
PASS  cut at v4: all 15 pairs of non-adjacent segments certified disjoint: the development is simple
PASS  cut at v5: all 15 pairs of non-adjacent segments certified disjoint: the development is simple
PASS  cut at v6: all 15 pairs of non-adjacent segments certified disjoint: the development is simple

(v) Gauss-Bonnet consistency
      sum tau_i = [5.85482487610121339272, 5.85482487610121339273]
      area of the region (fan of triangles N v_i v_(i+1)) = [0.42836043107837308419, 0.42836043107837308420]
PASS  sum tau_i + area - 2 pi contains 0 (enclosure of width < 1e-93)

(vi) The heptagon Q' of Table 3 (the vertex farthest from N)
      v0' = (7716497600000/7721804394837, 67343200000/7721804394837, 278195605163/7721804394837)
      v1' = (137245600000/180600478381, 1989283200000/5237413873049, 2762586126951/5237413873049)
      v2' = (271346600000/551275336701, 170063800000/551275336701, 448724663299/551275336701)
      v3' = (-790220000000/25006348007109, -101750000000/25006348007109, 24993651992891/25006348007109)
      v4' = (37896260000000/40773609693117, -11884210000000/40773609693117, 9226390306883/40773609693117)
      v5' = (19352526000000/19565644921229, -2847022000000/19565644921229, 434355078771/19565644921229)
      v6' = (3165580000000/100025061728241, -61600000000/100025061728241, 99974938271759/100025061728241)
PASS  Q': all 7 vertices satisfy x^2 + y^2 + z^2 = 1 exactly
PASS  Q': all vertices have z > 0 (open northern hemisphere)
      min z = 0.022199..., so every polar distance is < pi/2
PASS  Q': x_i y_(i+1) - y_i x_(i+1) > 0 for all i (longitude gaps in (0, pi))
PASS  Q': winding number of the longitudes = 1 (must be 1)
PASS  Q': gnomonic image is a simple polygon (all 14 non-adjacent edge pairs disjoint, adjacent edges not overlapping; exact)
PASS  Q': gnomonic image is counterclockwise (twice its signed area = 50.474134...)
PASS  Q': the origin is strictly to the left of every directed edge of the gnomonic image
PASS  Q' is a simple spherical heptagon in the open hemisphere about N, star-shaped with respect to N
PASS  the vertex of Q' farthest from N (smallest z, exact) is v5'
PASS  the smallest z-coordinate is z5' = 434355078771/19565644921229 (the value printed in the proof of Proposition 1.3)
PASS  Q': all sine numerators s_i nonzero (exact)
      polar distance of v0' = [1.534761, 1.534762]
      polar distance of v1' = [1.015174, 1.015175]
      polar distance of v2' = [0.619832, 0.619833]
      polar distance of v3' = [0.031867, 0.031868]
      polar distance of v4' = [1.342535, 1.342536]
      polar distance of v5' = [1.548594, 1.548595]
      polar distance of v6' = [0.031659, 0.031660]
PASS  Q' cut at v0': NOT simple, certified crossing pairs [(2, 5), (2, 6)]
PASS  Q' cut at v1': NOT simple, certified crossing pairs [(1, 4), (1, 5)]
PASS  Q' cut at v2': NOT simple, certified crossing pairs [(0, 3), (0, 4)]
PASS  Q' cut at v3': simple (all pairs certified disjoint)
PASS  Q' cut at v4': NOT simple, certified crossing pairs [(1, 6), (2, 6)]
PASS  Q' cut at v5': NOT simple, certified crossing pairs [(0, 5), (1, 5)]
      cut at v5', segments 0 and 5: determinants [-0.412549, -0.412548], [0.593115, 0.593116], [0.757811, 0.757812], [-0.247852, -0.247851]
      cut at v5', segments 1 and 5: determinants [0.431347, 0.431348], [-0.831647, -0.831646], [-0.247852, -0.247851], [1.015142, 1.015143]
PASS  cut at v5': the eight orientation determinants of the two crossings are at least 0.24 in absolute value
PASS  Q' cut at v6': NOT simple, certified crossing pairs [(0, 4), (0, 6), (3, 6)]
PASS  Q': the cut at the farthest vertex v5' is not simple; the only simple vertex cut is at v3'

(vii) Q cut at interior points of edges (cut point on e_k at arc-length fraction t from v_k)
PASS  Q cut on e0 at t = 1/20: NOT simple, crossing pairs [(0, 6)]
PASS  Q cut on e0 at t = 1/10: simple
PASS  Q cut on e0 at t = 1/5: NOT simple, crossing pairs [(0, 7)]
PASS  Q cut on e0 at t = 201/1000: NOT simple, crossing pairs [(0, 7)]
PASS  Q cut on e0 at t = 1/2: simple
PASS  Q cut on e3 at t = 1/2: simple
PASS  Q cut on e6 at t = 18/25: NOT simple, crossing pairs [(0, 7)]
PASS  Q cut on e6 at t = 403/500: NOT simple, crossing pairs [(0, 7)]
PASS  Q cut on e6 at t = 17/20: simple
PASS  Q cut on e6 at t = 19/20: NOT simple, crossing pairs [(1, 7)]
      (segments numbered 0..7; 0 and 7 are the two pieces of the cut edge)

(viii) Copies of Q scaled towards N (stereographic data multiplied by s)
PASS  s = 1/2: polar graph about N in the open hemisphere; the cut at v0 is not simple (segments 0 and 6 cross, smallest |determinant| > 0.003739146134); the cuts at v1, ..., v6 are simple
PASS  s = 1/5: polar graph about N in the open hemisphere; the cut at v0 is not simple (segments 0 and 6 cross, smallest |determinant| > 0.000113716224); the cuts at v1, ..., v6 are simple
PASS  s = 1/20: polar graph about N in the open hemisphere; the cut at v0 is not simple (segments 0 and 6 cross, smallest |determinant| > 0.000000458566); the cuts at v1, ..., v6 are simple

(ix) Control: a convex hexagon (vertices on a circle about N; the known convex case)
PASS  control hexagon: all turns are left turns (convex), and all six vertex cuts are certified simple

PASS  every enclosure used in the certificates above has width < 1e-80 (largest width < 1e-89)

ALL CHECKS PASSED
