== set-up check, ranks 1..9: printed families vs independently enumerated unit rhombi
r=1: points=3 (formula 3), boundary=3 (3r=3), interior=0, printed inequalities=0 (3r(r-1)/2=0), unit rhombi=0, families==unit rhombi each once: True
r=2: points=6 (formula 6), boundary=6 (3r=6), interior=0, printed inequalities=3 (3r(r-1)/2=3), unit rhombi=3, families==unit rhombi each once: True
r=3: points=10 (formula 10), boundary=9 (3r=9), interior=1, printed inequalities=9 (3r(r-1)/2=9), unit rhombi=9, families==unit rhombi each once: True
r=4: points=15 (formula 15), boundary=12 (3r=12), interior=3, printed inequalities=18 (3r(r-1)/2=18), unit rhombi=18, families==unit rhombi each once: True
r=5: points=21 (formula 21), boundary=15 (3r=15), interior=6, printed inequalities=30 (3r(r-1)/2=30), unit rhombi=30, families==unit rhombi each once: True
r=6: points=28 (formula 28), boundary=18 (3r=18), interior=10, printed inequalities=45 (3r(r-1)/2=45), unit rhombi=45, families==unit rhombi each once: True
r=7: points=36 (formula 36), boundary=21 (3r=21), interior=15, printed inequalities=63 (3r(r-1)/2=63), unit rhombi=63, families==unit rhombi each once: True
r=8: points=45 (formula 45), boundary=24 (3r=24), interior=21, printed inequalities=84 (3r(r-1)/2=84), unit rhombi=84, families==unit rhombi each once: True
r=9: points=55 (formula 55), boundary=27 (3r=27), interior=28, printed inequalities=108 (3r(r-1)/2=108), unit rhombi=108, families==unit rhombi each once: True

== rank 6 (the example of the claim): M is 63 x 73; b* = (0^18, c), c in [-1, 0, 1]; distribution of w=-c: [(-1, 10), (0, 33), (1, 2)]
   point h~* = (phi, c - R phi): M h~* = b* and h~* >= 0 exactly: True; nonintegral coordinates of h~*: 25 (of them in the h-part: 4)
   exact simplex phase 1: feasible, 74 pivots, 0.0s
   exact min/max (certificates verified against M, b*) for 73 coordinates: all min = max = coordinate of h~*: True; total pivots 530; 0.4s
   coordinates forced to a non-integer value: 25; in the h-part: [('(1, 4, 1)', '1/2'), ('(2, 1, 3)', '1/2'), ('(2, 3, 1)', '1/2'), ('(3, 1, 2)', '1/2')]
   sign convention: R row = (tips) - (short diagonal), so (R h) = -excess(h); R h <= c  <=>  excess(h) >= w = -c; c = ceil(R phi) = -floor(excess(phi)) [asserted in code]
   rank 6 (example), reduced form WITHOUT h>=0, n=10 interior variables, 45 inequalities: phi in P(0,w): True; every coordinate has exact min = max = phi: True; forced non-integers: [((1, 4, 1), '1/2'), ((2, 1, 3), '1/2'), ((2, 3, 1), '1/2'), ((3, 1, 2), '1/2')]; 0.5s
   remark: sum over rhombus rows of the column of an interior point (must be 1 for all columns of M to lie on a hyperplane l=1 with l=1 on the slack columns): [-2, -1, 0]
   Proposition 4 of the claim: all fifteen printed inequalities are rhombus inequalities of my system with my right-hand sides w, and the printed linear forms are correct: True
   sanity search: integer points of P(0,w) with all interior values in [-3,3]: 0 (nodes 134)

== encodings: B' = U B with U = (corner value, consecutive differences around the boundary), det U = 1 (unimodular: integral b <-> integral b', b = 0 <-> b' = 0; M' = diag(U, I) M, all maximal minors change by the factor det U)

== rank 7: extension built by own code; rhombi with 0/1/2 points on S={k=0}: [45, 6, 12]; support of phi': [(1, 4, 2), (2, 1, 4), (2, 3, 2), (3, 1, 3)]
   phi' satisfies all inequalities excess >= w': True; number of rhombi with w' < floor(excess(phi')): 4
   rank 7 (extension, right-hand side w' of Lemma 5), reduced form WITHOUT h>=0, n=15 interior variables, 63 inequalities: phi in P(0,w): True; every coordinate has exact min = max = phi: True; forced non-integers: [((1, 4, 2), '1/2'), ((2, 1, 4), '1/2'), ((2, 3, 2), '1/2'), ((3, 1, 3), '1/2')]; 0.6s
   literal form M (84 x 99), b* = (0, -w'): nonempty (point verified exactly); exact (min, max) of the support coordinates over {h~>=0 : M h~ = b*}: [((1, 4, 2), '1/2', '1/2'), ((2, 1, 4), '1/2', '1/2'), ((2, 3, 2), '1/2', '1/2'), ((3, 1, 3), '1/2', '1/2')]; 0.6s

== rank 8: extension built by own code; rhombi with 0/1/2 points on S={k=0}: [63, 7, 14]; support of phi': [(1, 4, 3), (2, 1, 5), (2, 3, 3), (3, 1, 4)]
   phi' satisfies all inequalities excess >= w': True; number of rhombi with w' < floor(excess(phi')): 11
   rank 8 (extension, right-hand side w' of Lemma 5), reduced form WITHOUT h>=0, n=21 interior variables, 84 inequalities: phi in P(0,w): True; every coordinate has exact min = max = phi: True; forced non-integers: [((1, 4, 3), '1/2'), ((2, 1, 5), '1/2'), ((2, 3, 3), '1/2'), ((3, 1, 4), '1/2')]; 1.0s
   literal form M (108 x 129), b* = (0, -w'): nonempty (point verified exactly); exact (min, max) of the support coordinates over {h~>=0 : M h~ = b*}: [((1, 4, 3), '1/2', '1/2'), ((2, 1, 5), '1/2', '1/2'), ((2, 3, 3), '1/2', '1/2'), ((3, 1, 4), '1/2', '1/2')]; 1.1s

total time 1.1s
