compare_run3.py -- verification run 3 (direct float billiard vs exact reduced map)
== A. Chord map (1), Lemma 2.3, mirror (8), time reversal (Lemma 5.2(a)); float
   150 trajectories in tables with random radii, 410479 chords compared with (1): largest error in u 4.96e-13
   segment reflections: 39345 seen, 39345 predicted by Lemma 2.3 (compared chord by chord)
   time reversal u -> alpha+lambda-u at 30017 reflections off A1: largest error 2.09e-14
   mirror u -> u+1 at 30017 reflections off A1: largest error 6.66e-16
== B. Periods and reflection counts, start by start: direct billiard vs exact families
   50 rational pairs (all 28 with denominators <= 7 and 22 with a denominator in 8..12), 40 starts each
   (1/6, 1/7): exact 8=6+0+2, 13=6+7+0 | direct starts 8=6+0+2: 18; 13=6+7+0: 22
   (1/5, 1/7): exact 31=15+14+2, 43=20+21+2 | direct starts 31=15+14+2: 16; 43=20+21+2: 24
   (1/5, 1/6): exact 11=5+6+0, 18=10+6+2 | direct starts 11=5+6+0: 27; 18=10+6+2: 13
   (1/4, 1/7): exact 6=4+0+2, 11=4+7+0 | direct starts 6=4+0+2: 12; 11=4+7+0: 28
   (1/4, 1/6): exact 5=2+3+0, 6=4+0+2 | direct starts 5=2+3+0: 21; 6=4+0+2: 19
   (1/4, 1/5): exact 6=4+0+2, 9=4+5+0 | direct starts 6=4+0+2: 11; 9=4+5+0: 29
   (2/7, 1/7): exact 5=2+3+0, 11=5+4+2 | direct starts 5=2+3+0: 9; 11=5+4+2: 31
   (2/7, 1/6): exact 15=7+6+2, 34=14+18+2 | direct starts 15=7+6+2: 28; 34=14+18+2: 12
   (2/7, 1/5): exact 31=14+15+2, 45=21+20+4 | direct starts 31=14+15+2: 20; 45=21+20+4: 20
   (2/7, 1/4): exact 15=7+8+0, 28=14+12+2 | direct starts 15=7+8+0: 15; 28=14+12+2: 25
   (1/3, 1/7): exact 5=3+0+2, 15=6+7+2 | direct starts 5=3+0+2: 2; 15=6+7+2: 38
   (1/3, 1/6): exact 14=6+6+2 | direct starts 14=6+6+2: 40
   (1/3, 1/5): exact 13=6+5+2, 21=9+10+2 | direct starts 13=6+5+2: 14; 21=9+10+2: 26
   (1/3, 1/4): exact 7=3+4+0, 12=6+4+2 | direct starts 7=3+4+0: 14; 12=6+4+2: 26
   (1/3, 2/7): exact 13=6+7+0, 31=15+14+2 | direct starts 13=6+7+0: 10; 31=15+14+2: 30
   (2/5, 1/7): exact 37=15+14+8, 51=20+21+10 | direct starts 37=15+14+8: 16; 51=20+21+10: 24
   (2/5, 1/6): exact 13=5+6+2, 22=10+6+6 | direct starts 13=5+6+2: 28; 22=10+6+6: 12
   (2/5, 1/5): exact 4=1+3+0, 8=4+2+2 | direct starts 4=1+3+0: 13; 8=4+2+2: 27
   (2/5, 1/4): exact 11=5+4+2, 24=10+12+2 | direct starts 11=5+4+2: 21; 24=10+12+2: 19
   (2/5, 2/7): exact 26=10+14+2, 52=25+21+6 | direct starts 26=10+14+2: 11; 52=25+21+6: 29
   (2/5, 1/3): exact 11=5+6+0, 21=10+9+2 | direct starts 11=5+6+0: 16; 21=10+9+2: 24
   (3/7, 1/7): exact 5=1+4+0, 13=6+3+4 | direct starts 5=1+4+0: 9; 13=6+3+4: 31
   (3/7, 1/6): exact 17=7+6+4, 38=14+18+6 | direct starts 17=7+6+4: 24; 38=14+18+6: 16
   (3/7, 1/5): exact 35=14+15+6, 51=21+20+10 | direct starts 35=14+15+6: 15; 51=21+20+10: 25
   (3/7, 1/4): exact 17=7+8+2, 32=14+12+6 | direct starts 17=7+8+2: 20; 32=14+12+6: 20
   (3/7, 2/7): exact 3=1+2+0, 7=4+1+2 | direct starts 3=1+2+0: 19; 7=4+1+2: 21
   (3/7, 1/3): exact 15=7+6+2, 31=14+15+2 | direct starts 15=7+6+2: 14; 31=14+15+2: 26
   (3/7, 2/5): exact 29=14+15+0, 43=21+20+2 | direct starts 29=14+15+0: 20; 43=21+20+2: 20
   (3/11, 1/8): exact 23=11+8+4, 52=22+24+6 | direct starts 23=11+8+4: 7; 52=22+24+6: 33
   (1/5, 2/11): exact 21=10+11+0, 49=25+22+2 | direct starts 21=10+11+0: 19; 49=25+22+2: 21
   (5/12, 1/5): exact 146=60+60+26 | direct starts 146=60+60+26: 40
   (3/7, 4/11): exact 27=14+11+2, 137=63+66+8 | direct starts 27=14+11+2: 6; 137=63+66+8: 34
   (5/11, 5/12): exact 23=11+12+0, 228=110+108+10 | direct starts 23=11+12+0: 2; 228=110+108+10: 38
   (3/10, 2/9): exact 21=10+9+2, 152=70+72+10 | direct starts 21=10+9+2: 10; 152=70+72+10: 30
   (5/12, 1/11): exact 31=12+11+8, 102=36+44+22 | direct starts 31=12+11+8: 31; 102=36+44+22: 9
   (2/7, 1/10): exact 19=7+10+2, 30=14+10+6 | direct starts 19=7+10+2: 16; 30=14+10+6: 24
   (1/8, 1/11): exact 10=8+0+2, 19=8+11+0 | direct starts 10=8+0+2: 15; 19=8+11+0: 25
   (1/5, 1/9): exact 21=10+9+2, 35=15+18+2 | direct starts 21=10+9+2: 26; 35=15+18+2: 14
   (5/12, 2/5): exact 122=60+60+2 | direct starts 122=60+60+2: 40
   (3/8, 2/7): exact 122=56+56+10 | direct starts 122=56+56+10: 40
   (1/3, 1/9): exact 5=3+0+2, 17=6+9+2 | direct starts 5=3+0+2: 9; 17=6+9+2: 31
   (4/9, 2/11): exact 81=27+44+10, 169=72+55+42 | direct starts 81=27+44+10: 9; 169=72+55+42: 31
   (2/11, 1/7): exact 25=11+14+0, 45=22+21+2 | direct starts 25=11+14+0: 6; 45=22+21+2: 34
   (3/10, 1/9): exact 23=10+9+4, 76=30+36+10 | direct starts 23=10+9+4: 20; 76=30+36+10: 20
   (4/9, 3/11): exact 75=27+44+4, 157=72+55+30 | direct starts 75=27+44+4: 13; 157=72+55+30: 27
   (4/11, 1/3): exact 26=11+15+0, 42=22+18+2 | direct starts 26=11+15+0: 13; 42=22+18+2: 27
   (1/4, 1/12): exact 6=4+0+2, 8=2+6+0 | direct starts 6=4+0+2: 16; 8=2+6+0: 24
   (3/8, 1/10): exact 11=4+5+2, 14=8+0+6 | direct starts 11=4+5+2: 35; 14=8+0+6: 5
   (2/7, 3/11): exact 43=21+22+0, 113=56+55+2 | direct starts 43=21+22+0: 9; 113=56+55+2: 31
   (1/4, 2/9): exact 6=4+0+2, 17=8+9+0 | direct starts 6=4+0+2: 11; 17=8+9+0: 29
   2000 starts compared
== C. Components and regions by the direct billiard (no reduction): number of regions, mirror symmetry
   (starts: 2n states on A1; classes by continuity of the sequence of reflections; see regions_direct)
   (1/3,1/4): components of G: 2; direct: 2 region(s) ['7=3+4+0', '12=6+4+2']; classes in the full table: 3 periodic; mirror test: 0 of 1800 starts in another region than their mirror image
   (1/4,1/6): components of G: 2; direct: 2 region(s) ['5=2+3+0', '6=4+0+2']; classes in the full table: 3 periodic; mirror test: 0 of 1800 starts in another region than their mirror image
   (1/3,1/5): components of G: 2; direct: 2 region(s) ['13=6+5+2', '21=9+10+2']; classes in the full table: 4 periodic; mirror test: 0 of 1800 starts in another region than their mirror image
   (1/3,1/6): components of G: 1; direct: 1 region(s) ['14=6+6+2']; classes in the full table: 1 periodic; mirror test: 0 of 1800 starts in another region than their mirror image
   (2/5,1/5): components of G: 2; direct: 2 region(s) ['4=1+3+0', '8=4+2+2']; classes in the full table: 4 periodic; mirror test: 0 of 1800 starts in another region than their mirror image
   (3/7,2/7): components of G: 2; direct: 2 region(s) ['3=1+2+0', '7=4+1+2']; classes in the full table: 4 periodic; mirror test: 0 of 1800 starts in another region than their mirror image
   (2/7,1/7): components of G: 2; direct: 2 region(s) ['5=2+3+0', '11=5+4+2']; classes in the full table: 4 periodic; mirror test: 0 of 1800 starts in another region than their mirror image
   (3/8,1/8): components of G: 1; direct: 1 region(s) ['10=4+4+2']; classes in the full table: 2 periodic; mirror test: 0 of 1800 starts in another region than their mirror image
   (5/12,1/12): components of G: 1; direct: 1 region(s) ['16=6+6+4']; classes in the full table: 2 periodic; mirror test: 0 of 1800 starts in another region than their mirror image
   (4/9,2/9): components of G: 2; direct: 2 region(s) ['7=2+5+0', '15=7+4+4']; classes in the full table: 4 periodic; mirror test: 0 of 1800 starts in another region than their mirror image
   (3/10,1/5): components of G: 2; direct: 2 region(s) ['4=2+2+0', '14=6+6+2']; classes in the full table: 3 periodic; mirror test: 0 of 1800 starts in another region than their mirror image
   (5/11,3/11): components of G: 2; direct: 2 region(s) ['3=1+2+0', '11=6+1+4']; classes in the full table: 4 periodic; mirror test: 0 of 1800 starts in another region than their mirror image
   (1/4, 1/sqrt30): components of G: 2; direct: 2 region(s) ['6=4+0+2', 'not closed']; classes in the full table: 1 periodic, 1 not closed; mirror test: 0 of 900 starts in another region than their mirror image
   ((5-sqrt5)/10, sqrt5/10): components of G: 2; direct: 2 region(s) ['4=2+2+0', '14=6+6+2']; classes in the full table: 3 periodic; mirror test: 0 of 1800 starts in another region than their mirror image
   (1/3, sqrt2/10): components of G: 2; direct: 2 region(s) ['5=3+0+2', 'not closed']; classes in the full table: 2 periodic, 1 not closed; mirror test: 0 of 900 starts in another region than their mirror image
   (1/4 - sqrt2/18, sqrt2/18): components of G: 2; direct: 2 region(s) ['70=32+32+6', '166=76+76+14']; classes in the full table: 3 periodic; mirror test: 0 of 1800 starts in another region than their mirror image
   (sqrt2/4, 1/5): components of G: 1; direct: 1 region(s) ['not closed']; classes in the full table: 0 periodic, 1 not closed; mirror test: 0 of 900 starts in another region than their mirror image
   rho1 + 3 rho2 = 1/2, rho2 = sqrt2/16 [Thm 1.4(c)]: components of G: 2; direct: 2 region(s) ['8=2+6+0', 'not closed']; classes in the full table: 2 periodic, 1 not closed; mirror test: 0 of 900 starts in another region than their mirror image
   rho1 + 3 rho2 = 1/2, rho2 = sqrt2/20 [Thm 1.4(c), none]: components of G: 1; direct: 1 region(s) ['not closed']; classes in the full table: 0 periodic, 1 not closed; mirror test: 0 of 900 starts in another region than their mirror image
   3 rho1 + rho2 = 3/2, rho2 = sqrt2/10 [Thm 1.4(d)]: components of G: 2; direct: 2 region(s) ['38=18+6+14', 'not closed']; classes in the full table: 1 periodic, 1 not closed; mirror test: 0 of 900 starts in another region than their mirror image
   (1/2 - 3 sqrt2/32, sqrt2/16) [2 rho1 + 3 rho2 = 1]: components of G: 2; direct: 2 region(s) ['12=4+6+2', 'not closed']; classes in the full table: 2 periodic, 1 not closed; mirror test: 0 of 900 starts in another region than their mirror image
== D. Explicit instances of every case of Theorems 1.4 and 1.5, 1.2 and 1.3(b)
  Theorem 1.4(a): rho2 rational, rho1 irrational -> no periodic region
   (1/4*sqrt(2), 1/5): N=2; no family; measure 0.000000 | direct: not closed: 60
   (1/3*sqrt(2), 1/4): N=2; no family; measure 0.000000 | direct: not closed: 60
   (1/5*sqrt(2), 1/6): N=3; no family; measure 0.000000 | direct: not closed: 60
   (1/2 - 1/10*sqrt(2), 1/3): N=1; no family; measure 0.000000 | direct: not closed: 60
  Theorem 1.4(a): 1, rho1, rho2 linearly independent (float only: two different fields)
   (sqrt2/4, sqrt3/8): direct: 0 of 200 starts closed within 8000 reflections
   (sqrt5/5, sqrt7/12): direct: 0 of 200 starts closed within 8000 reflections
   (pi/8, e/12): direct: 0 of 200 starts closed within 8000 reflections
  Theorem 1.4(b): rho1 = s/t
   (1/4, 1/12*sqrt2) exists: N=4; 6=4+0+2 (m1,m2,v,n)=(2,0,0,1) width 1/2 - 1/6*sqrt(2); measure 0.528595; no further periodic point: certificate K=2 | direct: 6=4+0+2: 28; not closed: 32
   (1/4, 1/8*sqrt2) exists: N=2; 6=4+0+2 (m1,m2,v,n)=(2,0,0,1) width 1/2 - 1/4*sqrt(2); measure 0.292893; no further periodic point: certificate K=2 | direct: 6=4+0+2: 23; not closed: 37
   (3/8, 1/12*sqrt2) exists: N=4; 14=8+0+6 (m1,m2,v,n)=(4,0,0,3) width 1/4 - 1/6*sqrt(2); measure 0.057191; no further periodic point: certificate K=4 | direct: 14=8+0+6: 3; not closed: 57
   (3/8, 1/8*sqrt2) none: N=2; no family; measure 0.000000; no further periodic point: certificate K=4 | direct: not closed: 60
   (1/3, 1/10*sqrt2) exists: N=3; 5=3+0+2 (m1,m2,v,n)=(3,0,0,2) width 1/3 - 1/5*sqrt(2); measure 0.151472; no further periodic point: certificate K=3 | direct: 5=3+0+2: 6; not closed: 54
   (1/3, 1/6*sqrt2) none: N=2; no family; measure 0.000000; no further periodic point: certificate K=3 | direct: not closed: 60
   (2/5, 1/20*sqrt2) exists: N=7; 9=5+0+4 (m1,m2,v,n)=(5,0,0,4) width 1/5 - 1/10*sqrt(2); measure 0.292893; no further periodic point: certificate K=5 | direct: 9=5+0+4: 12; not closed: 48
   (2/5, 1/5*sqrt2) none: N=1; no family; measure 0.000000; no further periodic point: certificate K=4 | direct: not closed: 60
   (3/7, 1/24*sqrt2) exists: N=8; 13=7+0+6 (m1,m2,v,n)=(7,0,0,6) width 1/7 - 1/12*sqrt(2); measure 0.175042; no further periodic point: certificate K=7 | direct: 13=7+0+6: 11; not closed: 49
   (3/7, 1/12*sqrt2) none: N=4; no family; measure 0.000000; no further periodic point: certificate K=7 | direct: not closed: 60
   (1/6, 1/10*sqrt2) exists: N=3; 8=6+0+2 (m1,m2,v,n)=(3,0,0,1) width 1/3 - 1/5*sqrt(2); measure 0.151472; no further periodic point: certificate K=3 | direct: 8=6+0+2: 17; not closed: 43
   (5/12, 1/20*sqrt2) exists: N=7; 22=12+0+10 (m1,m2,v,n)=(6,0,0,5) width 1/6 - 1/10*sqrt(2); measure 0.151472; no further periodic point: certificate K=6 | direct: 22=12+0+10: 3; not closed: 57
   (5/12, 1/10*sqrt2) none: N=3; no family; measure 0.000000; no further periodic point: certificate K=6 | direct: not closed: 60
  Theorem 1.4(c): rho1 + q rho2 = n/2, q >= 2 (Remark 6.2 for n = 2 and n >= 3)
   q=2 n=1 lambda=1/5*sqrt(2) exists: N=3; 6=2+4+0 (m1,m2,v,n)=(2,4,1,2) width 1 - 3/5*sqrt(2); measure 0.302944; no further periodic point: certificate K=2 | direct: 6=2+4+0: 20; not closed: 40
   q=2 n=1 lambda=1/8*sqrt(2) exists: N=5; 20=6+12+2 (m1,m2,v,n)=(3,6,1,3) width 1 - 5/8*sqrt(2); measure 0.348350; no further periodic point: certificate K=3 | direct: 20=6+12+2: 25; not closed: 35
   q=3 n=1 lambda=1/8*sqrt(2) exists: N=5; 8=2+6+0 (m1,m2,v,n)=(2,6,1,2) width 1 - 5/8*sqrt(2); measure 0.232233; no further periodic point: certificate K=2 | direct: 8=2+6+0: 18; not closed: 42
   q=3 n=1 lambda=1/10*sqrt(2) none: N=7; no family; measure 0.000000; no further periodic point: certificate K=3 | direct: not closed: 60
   q=3 n=1 lambda=1/12*sqrt(2) exists: N=8; 26=6+18+2 (m1,m2,v,n)=(3,9,1,3) width 1 - 2/3*sqrt(2); measure 0.171573; no further periodic point: certificate K=3 | direct: 26=6+18+2: 14; not closed: 46
   q=4 n=1 lambda=1/9*sqrt(2) none: N=6; no family; measure 0.000000; no further periodic point: certificate K=2 | direct: not closed: 60
   q=4 n=1 lambda=1/12*sqrt(2) exists: N=8; 10=2+8+0 (m1,m2,v,n)=(2,8,1,2) width -1 + 3/4*sqrt(2); measure 0.121320; no further periodic point: certificate K=3 | direct: 10=2+8+0: 18; not closed: 42
   q=5 n=1 lambda=1/13*sqrt(2) exists: N=9; 12=2+10+0 (m1,m2,v,n)=(2,10,1,2) width 1 - 9/13*sqrt(2); measure 0.041858; no further periodic point: certificate K=2 | direct: 12=2+10+0: 4; not closed: 56
   q=5 n=1 lambda=1/14*sqrt(2) exists: N=9; 12=2+10+0 (m1,m2,v,n)=(2,10,1,2) width 1 - 9/14*sqrt(2); measure 0.181725; no further periodic point: certificate K=2 | direct: 12=2+10+0: 15; not closed: 45
   q=3 n=2 lambda=1/3*sqrt(2) exists: N=2; 4=1+3+0 (m1,m2,v,n)=(1,3,1,2) width 1 - 2/3*sqrt(2); measure 0.057191; no further periodic point: certificate K=1 | direct: 4=1+3+0: 8; not closed: 52
   q=4 n=2 lambda=1/5*sqrt(2) exists: N=3; 5=1+4+0 (m1,m2,v,n)=(1,4,1,2) width 1 - 3/5*sqrt(2); measure 0.151472; no further periodic point: certificate K=1 | direct: 5=1+4+0: 16; not closed: 44
   q=4 n=2 lambda=1/4*sqrt(2) none: N=2; no family; measure 0.000000; no further periodic point: certificate K=1 | direct: not closed: 60
   q=5 n=2 lambda=1/5*sqrt(2) none: N=3; no family; measure 0.000000; no further periodic point: certificate K=1 | direct: not closed: 60
   q=5 n=2 lambda=1/6*sqrt(2) exists: N=4; 6=1+5+0 (m1,m2,v,n)=(1,5,1,2) width 1 - 2/3*sqrt(2); measure 0.057191; no further periodic point: certificate K=1 | direct: 6=1+5+0: 9; not closed: 51
   q=5 n=3 lambda=1/3*sqrt(2) none: N=2; no family; measure 0.000000; no further periodic point: certificate K=1 | direct: not closed: 60
   q=6 n=3 lambda=3/10*sqrt(2) none: N=2; no family; measure 0.000000; no further periodic point: certificate K=1 | direct: not closed: 60
   q=7 n=4 lambda=1/2 - 1/40*sqrt(2) none: N=2; no family; measure 0.000000; no further periodic point: certificate K=1 | direct: not closed: 60
  Theorem 1.4(d): p rho1 + rho2 = n/2, p >= 2
   p=2 n=2 lambda=1/5*sqrt(2) exists: N=3; 13=6+3+4 (m1,m2,v,n)=(6,3,1,6) width -1 + 4/5*sqrt(2); measure 0.788225; no further periodic point: certificate K=7 | direct: 13=6+3+4: 36; not closed: 24
   p=2 n=2 lambda=1/6*sqrt(2) exists: N=4; 18=8+4+6 (m1,m2,v,n)=(8,4,1,8) width 1 - 2/3*sqrt(2); measure 0.457528; no further periodic point: certificate K=7 | direct: 18=8+4+6: 24; not closed: 36
   p=2 n=2 lambda=1/3*sqrt(2) exists: N=2; 8=4+2+2 (m1,m2,v,n)=(4,2,1,4) width 1 - 2/3*sqrt(2); measure 0.228764; no further periodic point: certificate K=3 | direct: 8=4+2+2: 8; not closed: 52
   p=3 n=3 lambda=1/5*sqrt(2) exists: N=3; 38=18+6+14 (m1,m2,v,n)=(9,3,1,9) width 1 - 2/3*sqrt(2); measure 0.514719; no further periodic point: certificate K=8 | direct: 38=18+6+14: 23; not closed: 37
   p=3 n=3 lambda=1/6*sqrt(2) none: N=4; no family; measure 0.000000; no further periodic point: certificate K=11 | direct: not closed: 60
   p=4 n=4 lambda=1/5*sqrt(2) exists: N=3; 25=12+3+10 (m1,m2,v,n)=(12,3,1,12) width 1 - 7/10*sqrt(2); measure 0.120606; no further periodic point: certificate K=11 | direct: 25=12+3+10: 5; not closed: 55
   p=4 n=4 lambda=1/6*sqrt(2) none: N=4; no family; measure 0.000000; no further periodic point: certificate K=14 | direct: not closed: 60
   p=5 n=5 lambda=1/8*sqrt(2) exists: N=5; 106=50+10+46 (m1,m2,v,n)=(25,5,1,25) width 1 - 7/10*sqrt(2); measure 0.251263; no further periodic point: certificate K=24 | direct: 106=50+10+46: 11; not closed: 49
   p=5 n=5 lambda=1/5*sqrt(2) none: N=3; no family; measure 0.000000; no further periodic point: certificate K=14 | direct: not closed: 60
   p=3 n=3 lambda=1/2*sqrt(2) exists: N=1; 10=6+2+2 (m1,m2,v,n)=(3,1,1,3) width 1 - 2/3*sqrt(2); measure 0.171573; no further periodic point: certificate K=2 | direct: 10=6+2+2: 17; not closed: 43
   p=2 n=1 lambda=1/5*sqrt(2) none: N=3; no family; measure 0.000000; no further periodic point: certificate K=5 | direct: not closed: 60
   p=3 n=2 lambda=1/5*sqrt(2) none: N=3; no family; measure 0.000000; no further periodic point: certificate K=5 | direct: not closed: 60
   p=3 n=1 lambda=1/9*sqrt(2) none: N=6; no family; measure 0.000000; no further periodic point: certificate K=7 | direct: not closed: 60
   p=4 n=3 lambda=1/5*sqrt(2) none: N=3; no family; measure 0.000000; no further periodic point: certificate K=7 | direct: not closed: 60
   p=6 n=6 lambda=3/20*sqrt(2) exists: N=4; 50=24+4+22 (m1,m2,v,n)=(24,4,1,24) width 1 - 7/10*sqrt(2); measure 0.241212; no further periodic point: certificate K=23 | direct: 50=24+4+22: 12; not closed: 48
  Theorem 1.5: rho1 + rho2 = 1/2
   lambda=1/5*sqrt(5) (rho2=0.22361): N=2; 4=2+2+0 (m1,m2,v,n)=(2,2,1,2) width -1 + 3/5*sqrt(5); 14=6+6+2 (m1,m2,v,n)=(3,3,1,3) width 1 - 2/5*sqrt(5); measure 1 | direct: 4=2+2+0: 27; 14=6+6+2: 33
   lambda=1/6*sqrt(2) (rho2=0.11785): N=4; 10=4+4+2 (m1,m2,v,n)=(4,4,1,4) width -1 + 5/6*sqrt(2); 26=10+10+6 (m1,m2,v,n)=(5,5,1,5) width 1 - 2/3*sqrt(2); measure 1 | direct: 10=4+4+2: 44; 26=10+10+6: 16
   lambda=1/3*sqrt(2) (rho2=0.23570): N=2; 4=2+2+0 (m1,m2,v,n)=(2,2,1,2) width -1 + 1*sqrt(2); 14=6+6+2 (m1,m2,v,n)=(3,3,1,3) width 1 - 2/3*sqrt(2); measure 1 | direct: 4=2+2+0: 40; 14=6+6+2: 20
   lambda=1/8*sqrt(2) (rho2=0.08839): N=5; 16=6+6+4 (m1,m2,v,n)=(6,6,1,6) width 1 - 5/8*sqrt(2); 26=10+10+6 (m1,m2,v,n)=(5,5,1,5) width -1 + 3/4*sqrt(2); measure 1 | direct: 16=6+6+4: 40; 26=10+10+6: 20
   lambda=1/10*sqrt(3) (rho2=0.08660): N=5; 16=6+6+4 (m1,m2,v,n)=(6,6,1,6) width 1 - 1/2*sqrt(3); 26=10+10+6 (m1,m2,v,n)=(5,5,1,5) width -1 + 3/5*sqrt(3); measure 1 | direct: 16=6+6+4: 48; 26=10+10+6: 12
   lambda=1/3 (rho2=0.16667): N=3; 14=6+6+2 (m1,m2,v,n)=(3,3,1,3) width 1/3; measure 1 | direct: 14=6+6+2: 60
   lambda=1/4 (rho2=0.12500): N=4; 10=4+4+2 (m1,m2,v,n)=(4,4,1,4) width 1/4; measure 1 | direct: 10=4+4+2: 60
   lambda=2/5 (rho2=0.20000): N=2; 4=2+2+0 (m1,m2,v,n)=(2,2,1,2) width 1/5; 14=6+6+2 (m1,m2,v,n)=(3,3,1,3) width 1/5; measure 1 | direct: 4=2+2+0: 28; 14=6+6+2: 32
   lambda=2/7 (rho2=0.14286): N=3; 10=4+4+2 (m1,m2,v,n)=(4,4,1,4) width 1/7; 14=6+6+2 (m1,m2,v,n)=(3,3,1,3) width 1/7; measure 1 | direct: 10=4+4+2: 35; 14=6+6+2: 25
   lambda=1/5 (rho2=0.10000): N=5; 26=10+10+6 (m1,m2,v,n)=(5,5,1,5) width 1/5; measure 1 | direct: 26=10+10+6: 60
   lambda=3/10 (rho2=0.15000): N=3; 10=4+4+2 (m1,m2,v,n)=(4,4,1,4) width 1/10; 14=6+6+2 (m1,m2,v,n)=(3,3,1,3) width 1/5; measure 1 | direct: 10=4+4+2: 26; 14=6+6+2: 34
  Theorems 1.2 and 1.3(b): rho1 + rho2 rational, rho2 irrational -> all periodic, two regions, m1 = m2
   rho1+rho2=1/4, lambda=1/9*sqrt(2): 70=32+32+6 (m1=m2=32, v=5); 166=76+76+14 (m1=m2=38, v=6); measure 1 | direct: 70=32+32+6: 17; 166=76+76+14: 13
   rho1+rho2=1/4, lambda=1/14*sqrt(5): 96=44+44+8 (m1=m2=44, v=7); 218=100+100+18 (m1=m2=50, v=8); measure 1 | direct: 96=44+44+8: 10; 218=100+100+18: 20
   rho1+rho2=1/4, lambda=-1/8 + 1/8*sqrt(5): 70=32+32+6 (m1=m2=32, v=5); 114=52+52+10 (m1=m2=26, v=4); measure 1 | direct: 70=32+32+6: 9; 114=52+52+10: 21
   rho1+rho2=1/4, lambda=-1/2 + 1/4*sqrt(7): 148=68+68+12 (m1=m2=68, v=11); 270=124+124+22 (m1=m2=62, v=10); measure 1 | direct: 148=68+68+12: 10; 270=124+124+22: 20
   rho1+rho2=1/4, lambda=1/12*sqrt(3): 44=20+20+4 (m1=m2=20, v=3); 62=28+28+6 (m1=m2=14, v=2); measure 1 | direct: 44=20+20+4: 7; 62=28+28+6: 23
   rho1+rho2=1/4, lambda=1/2 - 1/5*sqrt(3): 44=20+20+4 (m1=m2=20, v=3); 114=52+52+10 (m1=m2=26, v=4); measure 1 | direct: 44=20+20+4: 1; 114=52+52+10: 29
   rho1+rho2=1/4, lambda=1/20*sqrt(7): 18=8+8+2 (m1=m2=8, v=1); 62=28+28+6 (m1=m2=14, v=2); measure 1 | direct: 18=8+8+2: 19; 62=28+28+6: 11
   rho1+rho2=1/4, lambda=-1/4 + 1/4*sqrt(2): 110=48+48+14 (m1=m2=48, v=5); 266=116+116+34 (m1=m2=58, v=6); measure 1 | direct: 110=48+48+14: 6; 266=116+116+34: 24
   rho1+rho2=1/3, lambda=1/9*sqrt(2): 494=210+210+74 (m1=m2=210, v=33); 614=261+261+92 (m1=m2=261, v=41); measure 1 | direct: 494=210+210+74: 27; 614=261+261+92: 3
   rho1+rho2=1/3, lambda=1/14*sqrt(5): 162=69+69+24 (m1=m2=69, v=11); 338=144+144+50 (m1=m2=144, v=23); measure 1 | direct: 338=144+144+50: 30
   rho1+rho2=1/3, lambda=-1/8 + 1/8*sqrt(5): 198=84+84+30 (m1=m2=84, v=13); 290=123+123+44 (m1=m2=123, v=19); measure 1 | direct: 198=84+84+30: 5; 290=123+123+44: 25
   rho1+rho2=1/3, lambda=-1/2 + 1/4*sqrt(7): 218=93+93+32 (m1=m2=93, v=15); 450=192+192+66 (m1=m2=192, v=31); measure 1 | direct: 218=93+93+32: 2; 450=192+192+66: 28
   80 pairs with rational sum and irrational rho2: all completely periodic with two families, m1 = m2
  Theorem 1.3(c): rho1 + rho2 irrational, a relation exists -> at most one family, measure < 1
   58 pairs: never more than one family, measure of the families < 1
== E. Proposition 8.1: rational pairs
   231 pairs with denominators <= 12: 30 with one region, 201 with two (the note says 231, 30, 201)
   Table 2 of the note as printed: 28 rows compared with this computation
   period table of the package (period_table_T12.txt): 231 rows compared
== F. Section 5 (symmetric points, Lemma 5.4, inequality (14)); Lemma 2.5; dictionary (4)
   92 exact models (rational pairs with denominators <= 9, and 14 quadratic pairs): 164 families and 10 minimal components checked
   Lemma 2.5: 6000 points on the L-shaped polygon: the return of the slope-1 flow equals p(Phi(u,b))
   dictionary (4): 4000 points: F_t = R o S_t equals Phi modulo 1
== G. Tables 3 and 4 and the long example of Section 9
  Table 3: rho1 + rho2 = 1/4, N = 6
   rho2=1/18*sqrt(2): (m1,v,period) = [(32, 5, 70), (38, 6, 166)]; measure 1 | direct: 70=32+32+6: 20; 166=76+76+14: 20
   rho2=1/28*sqrt(5): (m1,v,period) = [(44, 7, 96), (50, 8, 218)]; measure 1 | direct: 96=44+44+8: 11; 218=100+100+18: 29
   rho2=-1/16 + 1/16*sqrt(5): (m1,v,period) = [(26, 4, 114), (32, 5, 70)]; measure 1 | direct: 70=32+32+6: 14; 114=52+52+10: 26
   rho2=-1/4 + 1/8*sqrt(7): (m1,v,period) = [(62, 10, 270), (68, 11, 148)]; measure 1 | direct: 148=68+68+12: 20; 270=124+124+22: 20
  Table 4: 2 rho1 + 3 rho2 = n/2
   n=1 rho2=-1/16 + 1/16*sqrt(2) N=19: no family; least K with a certificate: 27 (printed 27) | direct: not closed: 40
   n=1 rho2=1/16*sqrt(2) N=5: family (m1,m2,v)=(4,6,1), 10=4+6+0; complement certified with K=4 | direct: 10=4+6+0: 17; not closed: 23
   n=2 rho2=-1/16 + 1/16*sqrt(2) N=19: family (m1,m2,v)=(26,39,2), 87=26+39+22; complement certified with K=13 | direct: 87=26+39+22: 19; not closed: 21
   n=2 rho2=1/16*sqrt(2) N=5: family (m1,m2,v)=(4,6,1), 12=4+6+2; complement certified with K=4 | direct: 12=4+6+2: 22; not closed: 18
   n=2 rho2=1/16 + 1/16*sqrt(2) N=3: family (m1,m2,v)=(2,3,1), 5=2+3+0; complement certified with K=3 | direct: 5=2+3+0: 15; not closed: 25
   n=3 rho2=1/8 + 1/16*sqrt(2) N=2: no family; least K with a certificate: 7 (printed 7) | direct: not closed: 40
   n=3 rho2=3/16 + 1/16*sqrt(2) N=1: no family; least K with a certificate: 3 (printed 3) | direct: not closed: 40
   n=4 rho2=1/4 + 1/16*sqrt(2) N=1: family (m1,m2,v)=(2,3,2), 5=2+3+0; complement certified with K=1 | direct: 5=2+3+0: 29; not closed: 11
  The long example: rho1 + rho2 = 3/10, rho2 = sqrt7/46
   (m1, period) = [(465320, 1102778), (466085, 2209182)]; v = [53527, 53615]; measure of the families 1
== H. Theorem 1.3(a), Lemma 5.1(a) in the full table, exact, rational triples (A/M, B/M), M <= 16
   560 triples; (families of G, families in the full table, regions): [((1, 1, 1), 142), ((1, 2, 1), 105), ((2, 3, 2), 242), ((2, 4, 2), 71)]
   in every case: sigma K = theta K for each family K of the full table; regions <-> families of G; lifts by the parity of n
TOTAL failures: 0
