OWR-1275-010, verification run 2 (independent code), Python 3.13.5
== Part A: types and winding numbers
  n=3: 2 types; by m {3: 2}; by k [1, 1]; geometric check failures 0
  n=4: 10 types; by m {3: 4, 4: 6}; by k [1, 8, 1]; geometric check failures 0
  n=5: 64 types; by m {3: 10, 4: 30, 5: 24}; by k [1, 31, 31, 1]; geometric check failures 0
  n=6: 476 types; by m {3: 20, 4: 120, 5: 216, 6: 120}; by k [1, 87, 300, 87, 1]; geometric check failures 0
  n=7: 4206 types; by m {3: 42, 4: 420, 5: 1344, 6: 1680, 7: 720}; by k [1, 225, 1877, 1877, 225, 1]; geometric check failures 0
  n=8: 43086 types; by m {3: 84, 4: 1386, 5: 7056, 6: 15120, 7: 14400, 8: 5040}; by k [1, 561, 9585, 22792, 9585, 561, 1]; geometric check failures 0
  n=9: 502748 types; by m {3: 170, 4: 4410, 5: 33768, 6: 110880, 7: 177120, 8: 136080, 9: 40320}; by k [1, 1375, 43975, 206023, 206023, 43975, 1375, 1]; geometric check failures 0
== Part B: Borel-Moore cellular homology of Delta'_k with geometric incidence numbers
  n=3 k=1: 1 cells, 0 facet incidences, dd=0: True; H^* over GF(2),GF(3),GF(10007): [{0: 1}, {0: 1}, {0: 1}]  expected {0: 1}  OK  (0.0s)
  n=3 k=2: 1 cells, 0 facet incidences, dd=0: True; H^* over GF(2),GF(3),GF(10007): [{0: 1}, {0: 1}, {0: 1}]  expected {0: 1}  OK  (0.0s)
  n=4 k=1: 1 cells, 0 facet incidences, dd=0: True; H^* over GF(2),GF(3),GF(10007): [{0: 1}, {0: 1}, {0: 1}]  expected {0: 1}  OK  (0.0s)
  n=4 k=2: 8 cells, 8 facet incidences, dd=0: True; H^* over GF(2),GF(3),GF(10007): [{0: 1, 1: 1}, {0: 1, 1: 1}, {0: 1, 1: 1}]  expected {0: 1, 1: 1}  OK  (0.0s)
  n=4 k=3: 1 cells, 0 facet incidences, dd=0: True; H^* over GF(2),GF(3),GF(10007): [{0: 1}, {0: 1}, {0: 1}]  expected {0: 1}  OK  (0.0s)
  n=5 k=1: 1 cells, 0 facet incidences, dd=0: True; H^* over GF(2),GF(3),GF(10007): [{0: 1}, {0: 1}, {0: 1}]  expected {0: 1}  OK  (0.0s)
  n=5 k=2: 31 cells, 50 facet incidences, dd=0: True; H^* over GF(2),GF(3),GF(10007): [{0: 1}, {0: 1}, {0: 1}]  expected {0: 1}  OK  (0.0s)
  n=5 k=3: 31 cells, 50 facet incidences, dd=0: True; H^* over GF(2),GF(3),GF(10007): [{0: 1}, {0: 1}, {0: 1}]  expected {0: 1}  OK  (0.0s)
  n=5 k=4: 1 cells, 0 facet incidences, dd=0: True; H^* over GF(2),GF(3),GF(10007): [{0: 1}, {0: 1}, {0: 1}]  expected {0: 1}  OK  (0.0s)
  n=6 k=1: 1 cells, 0 facet incidences, dd=0: True; H^* over GF(2),GF(3),GF(10007): [{0: 1}, {0: 1}, {0: 1}]  expected {0: 1}  OK  (0.0s)
  n=6 k=2: 87 cells, 162 facet incidences, dd=0: True; H^* over GF(2),GF(3),GF(10007): [{0: 1}, {0: 1}, {0: 1}]  expected {0: 1}  OK  (0.0s)
  n=6 k=3: 300 cells, 756 facet incidences, dd=0: True; H^* over GF(2),GF(3),GF(10007): [{0: 1, 3: 1}, {0: 1, 3: 1}, {0: 1, 3: 1}]  expected {0: 1, 3: 1}  OK  (0.0s)
  n=6 k=4: 87 cells, 162 facet incidences, dd=0: True; H^* over GF(2),GF(3),GF(10007): [{0: 1}, {0: 1}, {0: 1}]  expected {0: 1}  OK  (0.0s)
  n=6 k=5: 1 cells, 0 facet incidences, dd=0: True; H^* over GF(2),GF(3),GF(10007): [{0: 1}, {0: 1}, {0: 1}]  expected {0: 1}  OK  (0.0s)
  n=7 k=1: 1 cells, 0 facet incidences, dd=0: True; H^* over GF(2),GF(3),GF(10007): [{0: 1}, {0: 1}, {0: 1}]  expected {0: 1}  OK  (0.0s)
  n=7 k=2: 225 cells, 476 facet incidences, dd=0: True; H^* over GF(2),GF(3),GF(10007): [{0: 1}, {0: 1}, {0: 1}]  expected {0: 1}  OK  (0.0s)
  n=7 k=3: 1877 cells, 5754 facet incidences, dd=0: True; H^* over GF(2),GF(3),GF(10007): [{0: 1}, {0: 1}, {0: 1}]  expected {0: 1}  OK  (0.2s)
  n=7 k=4: 1877 cells, 5754 facet incidences, dd=0: True; H^* over GF(2),GF(3),GF(10007): [{0: 1}, {0: 1}, {0: 1}]  expected {0: 1}  OK  (0.2s)
  n=7 k=5: 225 cells, 476 facet incidences, dd=0: True; H^* over GF(2),GF(3),GF(10007): [{0: 1}, {0: 1}, {0: 1}]  expected {0: 1}  OK  (0.0s)
  n=7 k=6: 1 cells, 0 facet incidences, dd=0: True; H^* over GF(2),GF(3),GF(10007): [{0: 1}, {0: 1}, {0: 1}]  expected {0: 1}  OK  (0.0s)
  n=8 k=1: 1 cells, 0 facet incidences, dd=0: True; H^* over GF(2),GF(3),GF(10007): [{0: 1}, {0: 1}, {0: 1}]  expected {0: 1}  OK  (0.0s)
  n=8 k=2: 561 cells, 1336 facet incidences, dd=0: True; H^* over GF(2),GF(3),GF(10007): [{0: 1}, {0: 1}, {0: 1}]  expected {0: 1}  OK  (0.1s)
  n=8 k=3: 9585 cells, 33376 facet incidences, dd=0: True; H^* over GF(2),GF(3),GF(10007): [{0: 1}, {0: 1}, {0: 1}]  expected {0: 1}  OK  (1.3s)
  n=8 k=4: 22792 cells, 88232 facet incidences, dd=0: True; H^* over GF(2),GF(3),GF(10007): [{0: 1, 5: 1}, {0: 1, 5: 1}, {0: 1, 5: 1}]  expected {0: 1, 5: 1}  OK  (3.7s)
  n=8 k=5: 9585 cells, 33376 facet incidences, dd=0: True; H^* over GF(2),GF(3),GF(10007): [{0: 1}, {0: 1}, {0: 1}]  expected {0: 1}  OK  (1.6s)
  n=8 k=6: 561 cells, 1336 facet incidences, dd=0: True; H^* over GF(2),GF(3),GF(10007): [{0: 1}, {0: 1}, {0: 1}]  expected {0: 1}  OK  (0.1s)
  n=8 k=7: 1 cells, 0 facet incidences, dd=0: True; H^* over GF(2),GF(3),GF(10007): [{0: 1}, {0: 1}, {0: 1}]  expected {0: 1}  OK  (0.0s)
  n=9 k=1: 1 cells, 0 facet incidences, dd=0: True; H^* over GF(2),GF(3),GF(10007): [{0: 1}, {0: 1}, {0: 1}]  expected {0: 1}  OK  (0.3s)
  n=9 k=2: 1375 cells, 3654 facet incidences, dd=0: True; H^* over GF(2),GF(3),GF(10007): [{0: 1}, {0: 1}, {0: 1}]  expected {0: 1}  OK  (0.5s)
  n=9 k=3: 43975 cells, 170262 facet incidences, dd=0: True; H^* over GF(2),GF(3),GF(10007): [{0: 1}, {0: 1}, {0: 1}]  expected {0: 1}  OK  (8.7s)
  n=9 k=4: 206023 cells, 922122 facet incidences, dd=0: True; H^* over GF(2),GF(3),GF(10007): [{0: 1}, {0: 1}, {0: 1}]  expected {0: 1}  OK  (49.0s)
  n=9 k=5: 206023 cells, 922122 facet incidences, dd=0: True; H^* over GF(2),GF(3),GF(10007): [{0: 1}, {0: 1}, {0: 1}]  expected {0: 1}  OK  (52.7s)
  n=9 k=6: 43975 cells, 170262 facet incidences, dd=0: True; H^* over GF(2),GF(3),GF(10007): [{0: 1}, {0: 1}, {0: 1}]  expected {0: 1}  OK  (10.8s)
  n=9 k=7: 1375 cells, 3654 facet incidences, dd=0: True; H^* over GF(2),GF(3),GF(10007): [{0: 1}, {0: 1}, {0: 1}]  expected {0: 1}  OK  (0.6s)
  n=9 k=8: 1 cells, 0 facet incidences, dd=0: True; H^* over GF(2),GF(3),GF(10007): [{0: 1}, {0: 1}, {0: 1}]  expected {0: 1}  OK  (0.3s)
== Part C: pi^{-1}(E_n) and pi^{-1}(O_n) (expected S^{n/2-2}; duality in dimension n/2)
  n=4 E: 2 cells; H^* over GF(2),GF(3),GF(10007): [{0: 2}, {0: 2}, {0: 2}]  expected {0: 2}  OK
  n=4 O: 2 cells; H^* over GF(2),GF(3),GF(10007): [{0: 2}, {0: 2}, {0: 2}]  expected {0: 2}  OK
  n=6 E: 12 cells; H^* over GF(2),GF(3),GF(10007): [{0: 1, 1: 1}, {0: 1, 1: 1}, {0: 1, 1: 1}]  expected {0: 1, 1: 1}  OK
  n=6 O: 12 cells; H^* over GF(2),GF(3),GF(10007): [{0: 1, 1: 1}, {0: 1, 1: 1}, {0: 1, 1: 1}]  expected {0: 1, 1: 1}  OK
  n=8 E: 74 cells; H^* over GF(2),GF(3),GF(10007): [{0: 1, 2: 1}, {0: 1, 2: 1}, {0: 1, 2: 1}]  expected {0: 1, 2: 1}  OK
  n=8 O: 74 cells; H^* over GF(2),GF(3),GF(10007): [{0: 1, 2: 1}, {0: 1, 2: 1}, {0: 1, 2: 1}]  expected {0: 1, 2: 1}  OK
  n=10 E: 540 cells; H^* over GF(2),GF(3),GF(10007): [{0: 1, 3: 1}, {0: 1, 3: 1}, {0: 1, 3: 1}]  expected {0: 1, 3: 1}  OK
  n=12 E: 4682 cells; H^* over GF(2),GF(3),GF(10007): [{0: 1, 4: 1}, {0: 1, 4: 1}, {0: 1, 4: 1}]  expected {0: 1, 4: 1}  OK
== Part D: PGL(2,R): reflection on types; Delta'_{n/2}/rho via the anti-invariant subcomplex
  n=3: rho is an involution on all 2 types exchanging k and n-k: True; commutes with the facet relation: True; fixed types: 0
  n=4: rho is an involution on all 10 types exchanging k and n-k: True; commutes with the facet relation: True; fixed types: 0
  n=5: rho is an involution on all 64 types exchanging k and n-k: True; commutes with the facet relation: True; fixed types: 0
  n=6: rho is an involution on all 476 types exchanging k and n-k: True; commutes with the facet relation: True; fixed types: 0
  n=7: rho is an involution on all 4206 types exchanging k and n-k: True; commutes with the facet relation: True; fixed types: 0
  n=8: rho is an involution on all 43086 types exchanging k and n-k: True; commutes with the facet relation: True; fixed types: 0
  n=4: 8 middle types, 4 orbits; twisted complex dd=0: True; H^*(Delta'/rho) over GF(2),GF(3),GF(10007): [{0: 1, 1: 1}, {0: 1, 1: 1}, {0: 1, 1: 1}]  OK (RP^1)
     integral cohomology H^i = (rank, torsion): {0: (1, []), 1: (1, [])}; expected H^*(RP^1; Z) = {0: (1, []), 1: (1, [])}  OK (0.0s)
  n=6: 300 middle types, 150 orbits; twisted complex dd=0: True; H^*(Delta'/rho) over GF(2),GF(3),GF(10007): [{0: 1, 1: 1, 2: 1, 3: 1}, {0: 1, 3: 1}, {0: 1, 3: 1}]  OK (RP^3)
     integral cohomology H^i = (rank, torsion): {0: (1, []), 2: (0, [2]), 3: (1, [])}; expected H^*(RP^3; Z) = {0: (1, []), 2: (0, [2]), 3: (1, [])}  OK (0.0s)
  n=8: 22792 middle types, 11396 orbits; twisted complex dd=0: True; H^*(Delta'/rho) over GF(2),GF(3),GF(10007): [{0: 1, 1: 1, 2: 1, 3: 1, 4: 1, 5: 1}, {0: 1, 5: 1}, {0: 1, 5: 1}]  OK (RP^5)
     integral cohomology H^i = (rank, torsion): {0: (1, []), 2: (0, [2]), 4: (0, [2]), 5: (1, [])}; expected H^*(RP^5; Z) = {0: (1, []), 2: (0, [2]), 4: (0, [2]), 5: (1, [])}  OK (0.2s)
== Part E: Lemma 6.1 by interval dynamic programming (independent of the ear argument)
  n=3: 1 coincidence patterns (>= 3 values), without good triangulation: 0, without good ear: 0 (0.0s)
  n=4: 3 coincidence patterns (>= 3 values), without good triangulation: 0, without good ear: 0 (0.0s)
  n=5: 11 coincidence patterns (>= 3 values), without good triangulation: 0, without good ear: 0 (0.0s)
  n=6: 40 coincidence patterns (>= 3 values), without good triangulation: 0, without good ear: 0 (0.0s)
  n=7: 162 coincidence patterns (>= 3 values), without good triangulation: 0, without good ear: 0 (0.0s)
  n=8: 714 coincidence patterns (>= 3 values), without good triangulation: 0, without good ear: 0 (0.0s)
  n=9: 3425 coincidence patterns (>= 3 values), without good triangulation: 0, without good ear: 0 (0.0s)
  n=10: 17721 coincidence patterns (>= 3 values), without good triangulation: 0, without good ear: 0 (0.1s)
  n=11: 98253 coincidence patterns (>= 3 values), without good triangulation: 0, without good ear: 0 (0.8s)
  total for n <= 10: 22077 (paper: 22,077)
== Part F: friezes (odd n), exact arithmetic
  n=3: 300 random configurations; windings seen {1: 164, 2: 136}; M(c)!=eps*Id: 0; eps!=(-1)^k: 0; SL(2,Z) non-invariance: 0; inverse construction failures: 0
  n=5: 300 random configurations; windings seen {1: 15, 2: 146, 3: 130, 4: 9}; M(c)!=eps*Id: 0; eps!=(-1)^k: 0; SL(2,Z) non-invariance: 0; inverse construction failures: 0
  n=7: 300 random configurations; windings seen {2: 26, 3: 119, 4: 134, 5: 20, 6: 1}; M(c)!=eps*Id: 0; eps!=(-1)^k: 0; SL(2,Z) non-invariance: 0; inverse construction failures: 0
  n=9: 300 random configurations; windings seen {2: 1, 3: 29, 4: 116, 5: 122, 6: 32}; M(c)!=eps*Id: 0; eps!=(-1)^k: 0; SL(2,Z) non-invariance: 0; inverse construction failures: 0
  n=11: 300 random configurations; windings seen {3: 3, 4: 37, 5: 109, 6: 104, 7: 39, 8: 8}; M(c)!=eps*Id: 0; eps!=(-1)^k: 0; SL(2,Z) non-invariance: 0; inverse construction failures: 0
  n=5: 5 Conway-Coxeter quiddities (triangulations); M(c) != -Id: 0; winding k != 1: 0
  n=7: 42 Conway-Coxeter quiddities (triangulations); M(c) != -Id: 0; winding k != 1: 0
  n=9: 429 Conway-Coxeter quiddities (triangulations); M(c) != -Id: 0; winding k != 1: 0
  n=11: 4862 Conway-Coxeter quiddities (triangulations); M(c) != -Id: 0; winding k != 1: 0
== Part G: n = 4 in the slice C_4 = {(inf, 0, x, y)} (Example 4.3)
  windings of the arcs: {'a': 3, 'b': 2, 'c': 2, 'd': 2, 'e': 2, 'f': 1}  special points: {'E+': 2, 'E-': 2, 'O+': 2, 'O-': 2}
  arcs met by the charts at the special points: {'E+': 'bd', 'E-': 'ce', 'O+': 'bc', 'O-': 'de'}
  incidence graph: 8 vertices of degree 2, connected: True
== Part H: Lemma 4.1 (<=): f_m -> f in E, g_m f_m -> f' in O, exact; b_j = infinity allowed
  max over trials of m^4 * (squared chordal distance), n=4..10 and m=10,100,1000:
    n=4: ['80.3', '81', '81']
    n=6: ['80.3', '81', '81']
    n=8: ['80.3', '81', '81']
    n=10: ['80.3', '81', '81']
  (bounded in m, so both distances are O(1/m^2)); entries b_j = infinity used: 141
== Part I: Lemma 5.1 (open stars), exact random tests
  n=5: 5028 sample points, 4434 in the open star; star membership == side conditions: 5028/5028; midpoints of star points in the star: 23106/23106
  n=6: 4734 sample points, 3930 in the open star; star membership == side conditions: 4734/4734; midpoints of star points in the star: 18276/18276
  n=7: 4591 sample points, 3650 in the open star; star membership == side conditions: 4591/4591; midpoints of star points in the star: 15654/15654
== Part J: Proposition 6.2 (Fock-Goncharov coordinates), exact random tests
  n=4: 150 random (triangulation, coordinates); failures: 0
  n=5: 150 random (triangulation, coordinates); failures: 0
  n=6: 150 random (triangulation, coordinates); failures: 0
  n=7: 150 random (triangulation, coordinates); failures: 0
  n=8: 150 random (triangulation, coordinates); failures: 0
  n=4: x_{13} = 1/x_{02} on the overlap: True
== Summary: 0 failed checks; total time 136s
ALL CHECKS PASSED
EXIT=0
