A rule (1) on float trajectories, random real radii        cases= 466488  failures=0  runs=200, wrong segment count on a chord: 0, max error in u: 6.8e-14
B formula (2) vs the float billiard (rational pairs)       cases=    600  failures=0  pairs=60
B first-return map G of (3) vs the chord map (exact)       cases=   2400  failures=0  
C Lemma 2.5: slope-one flow on the L-shaped surface        cases=  12000  failures=0  
  source Ex. 1: (1/3, 1/4)             N=2 delta=0: 7 = 3+4+0 (m1,m2,v,n)=(3, 4, 2, 4) measure 0.5000; 12 = 6+4+2 (m1,m2,v,n)=(3, 2, 1, 3) measure 0.5000 ; measure of the families = 1 exactly ; float: [(7, 3, 4, 0), (12, 6, 4, 2)]
  source Ex. 2: (1/4, 1/6)             N=3 delta=0: 5 = 2+3+0 (m1,m2,v,n)=(2, 3, 1, 2) measure 0.6667; 6 = 4+0+2 (m1,m2,v,n)=(2, 0, 0, 1) measure 0.3333 ; measure of the families = 1 exactly ; float: [(5, 2, 3, 0), (6, 4, 0, 2)]
  source Ex. 2: (1/3, 1/5)             N=2 delta=0.2000: 13 = 6+5+2 (m1,m2,v,n)=(6, 5, 2, 6) measure 0.4000; 21 = 9+10+2 (m1,m2,v,n)=(9, 10, 4, 10) measure 0.6000 ; measure of the families = 1 exactly ; float: [(13, 6, 5, 2), (21, 9, 10, 2)]
  source Ex. 3: (1/4, 1/sqrt 30)       N=2 delta=0.2697: 6 = 4+0+2 (m1,m2,v,n)=(2, 0, 0, 1) measure 0.2697 ; measure of the families 0.269703 ; rest certified non-periodic (K=2, sums in [-6,-4]) ; float: [(6, 4, 0, 2)]
  [DR] 4.3: ((5-sqrt5)/10, sqrt5/10)   N=2 delta=0.1056: 4 = 2+2+0 (m1,m2,v,n)=(2, 2, 1, 2) measure 0.6833; 14 = 6+6+2 (m1,m2,v,n)=(3, 3, 1, 3) measure 0.3167 ; measure of the families = 1 exactly ; float: [(4, 2, 2, 0), (14, 6, 6, 2)]
  [DR] 8.2: r1+10r2=5 (alpha+10lam=10, N=1): Theorem 1.4(c) predicts no family; float: 0 of 40 starts closed within 50000 reflections
D examples of the sources (Table 1)                        cases=      6  failures=0  
E Theorems 1.2, 1.3(b): rational sum, irrational rho2      cases=    164  failures=0  measure 1, two families, m1 = m2, g | m1 in all
E Theorems 1.2, 1.3(c): irrational sum                     cases=    171  failures=0  never measure 1; number of families: [(0, 72), (1, 99)]
F Theorem 1.5 (rho1 + rho2 = 1/2): families, arcs, periods cases=    436  failures=0  
F Theorem 1.5: float billiard inside the families          cases=     12  failures=0  
G Theorem 1.4(b) (rho1 rational): family, arc, period t+2s cases=    319  failures=0  with a family: 106; rest certified in all
G Theorem 1.4(c) (rho1 + q rho2 = n/2) and Remark 6.2      cases=    165  failures=0  with a family: 57; (n, family?) counts [((1, False), 33), ((1, True), 31), ((2, False), 18), ((2, True), 26), ((3, False), 57)]
G Theorem 1.4(d) (p rho1 + rho2 = n/2): closed form        cases=    147  failures=0  with a family: 41; rest certified in all
G Theorem 1.4: float billiard inside the families          cases=      7  failures=0  
H Proposition 8.1: permutation of the M arcs vs germ following cases=    231  failures=0  denominators <= 12; pairs with 1 / 2 families: 30 / 201
H Table 2 as printed (denominators <= 7)                   cases=     28  failures=0  
I Theorem 1.3(a) (rational pairs, full table, exact)       cases=    325  failures=0  (families of G, families in phase space, regions): [((1, 1, 1), 78), ((1, 2, 1), 60), ((2, 3, 2), 144), ((2, 4, 2), 43)]
J Theorem 1.4(e) on relations with p, q >= 2 (samples)     cases=    112  failures=0  families found: 68
K Lemma 5.2(a) and (8) on the floating-point table         cases=   2000  failures=0  max error 6.4e-14
K Lemmas 5.2(b), 5.3, 5.4 (rational pairs, M <= 14)        cases=    325  failures=0  (families, symmetric elements outside the families): [((1, 2), 86), ((1, 3), 52), ((2, 0), 43), ((2, 1), 144)]
K Lemmas 5.2(b), 5.3, 5.4 (irrational pairs)               cases=     76  failures=0  (families, symmetric elements outside the families): [((0, 5), 18), ((1, 3), 42), ((2, 1), 16)]
  Table 3  rho2=sqrt2/18      N=6  (m1, v, period data): [(32, 5, (70, 32, 32, 6)), (38, 6, (166, 76, 76, 14))]
  Table 3  rho2=sqrt5/28      N=6  (m1, v, period data): [(44, 7, (96, 44, 44, 8)), (50, 8, (218, 100, 100, 18))]
  Table 3  rho2=(sqrt5-1)/16  N=6  (m1, v, period data): [(26, 4, (114, 52, 52, 10)), (32, 5, (70, 32, 32, 6))]
  Table 3  rho2=(sqrt7-2)/8   N=6  (m1, v, period data): [(62, 10, (270, 124, 124, 22)), (68, 11, (148, 68, 68, 12))]
  Table 4  n=1 rho2=0.02589 N=19  family: none ; certificate for the complement: K=27
  Table 4  n=1 rho2=0.08839 N= 5  family: [(4, 6, 1, (10, 4, 6, 0))] ; certificate for the complement: K=4
  Table 4  n=2 rho2=0.02589 N=19  family: [(26, 39, 2, (87, 26, 39, 22))] ; certificate for the complement: K=13
  Table 4  n=2 rho2=0.08839 N= 5  family: [(4, 6, 1, (12, 4, 6, 2))] ; certificate for the complement: K=4
  Table 4  n=2 rho2=0.15089 N= 3  family: [(2, 3, 1, (5, 2, 3, 0))] ; certificate for the complement: K=3
  Table 4  n=3 rho2=0.21339 N= 2  family: none ; certificate for the complement: K=7
  Table 4  n=3 rho2=0.27589 N= 1  family: none ; certificate for the complement: K=3
  Table 4  n=4 rho2=0.33839 N= 1  family: [(2, 3, 2, (5, 2, 3, 0))] ; certificate for the complement: K=1
  Section 9  rho1+rho2=3/10, rho2=sqrt7/46: (m1, m2, v, period) = [(465320, 465320, 53527, 1102778), (466085, 466085, 53615, 2209182)] ; total measure 1 exactly
L Tables 3 and 4 and the long example, as printed          cases=     13  failures=0  
TOTAL failures: 0
