15-board symmetry-quotient certificate for 300*T + 144*P <= 625*M
=====================================================================

Board labels:
  D0,D1,D2 are the three D-points.
  A0..A3, B0..B3, C0..C3 are the three 4-point columns.
Variable types:
  m = missing z=3 quad, t = present z=2 triple, p = present z=1 pair.
Symmetry group:
  S3 on D-points, S3 on columns, and in each column the flips 0<->1 and 2<->3.
  Its order is 2304.

After quotienting by this group:
  m-orbits: 13, t-orbits: 9, p-orbits: 5; total variable orbits: 27.
  residual/closure constraints collapse to 239 orbit inequalities.
  The LP optimum is 0.

Orbit representatives and orbit sizes:
  0  m size 24  {D0,A0,B0,C0}
  1  m size 72  {D0,A0,B0,C2}
  2  m size 72  {D0,A0,B2,C2}
  3  m size 24  {D0,A2,B2,C2}
  4  m size 12  {A0,A1,B0,C0}
  5  m size 24  {A0,A1,B0,C2}
  6  m size 12  {A0,A1,B2,C2}
  7  m size 48  {A0,A2,B0,C0}
  8  m size 96  {A0,A2,B0,C2}
  9  m size 48  {A0,A2,B2,C2}
 10  m size 12  {A0,B0,C2,C3}
 11  m size 24  {A0,B2,B3,C2}
 12  m size 12  {A2,A3,B2,C2}
 13  t size 36  {D0,A0,B0}
 14  t size 72  {D0,A0,B2}
 15  t size 36  {D0,A2,B2}
 16  t size 12  {A0,A1,B0}
 17  t size 12  {A0,A1,B2}
 18  t size 48  {A0,A2,B0}
 19  t size 48  {A0,A2,B2}
 20  t size 12  {A0,B2,B3}
 21  t size 12  {A2,A3,B2}
 22  p size 18  {D0,A0}
 23  p size 18  {D0,A2}
 24  p size 3   {A0,A1}
 25  p size 12  {A0,A2}
 26  p size 3   {A2,A3}

Twenty nonzero quotient residual inequalities, with integer multipliers:
Rows have the form: trace-orbit coefficient + missing-orbit coefficients <= 0.
Multiplying these rows by the listed weights gives a vector that coefficient-wise dominates
  objective = -625*sum(size(m_i)m_i)+300*sum(size(t_i)t_i)+144*sum(size(p_i)p_i).
Since all variables are nonnegative and every listed row is valid, the objective is <=0.

 1. weight 3300:  t15 - m1 - m3 - m4 <= 0
 2. weight 1146:  p22 - m0 - m3 - m8 <= 0
 3. weight 7746:  t14 - m1 - m2 - m8 <= 0
 4. weight 10800: t13 - m1 - m2 - m9 <= 0
 5. weight 2592:  p23 - m1 - m7 - m9 <= 0
 6. weight 1776:  t19 - m7 - m9 <= 0
 7. weight 8724:  t19 - 2*m8 <= 0
 8. weight 7500:  t15 - 2*m1 - m10 <= 0
 9. weight 13854: t14 - m0 - m2 - m11 <= 0
10. weight 3900:  t19 - m6 - m8 <= 0
11. weight 14400: t18 - m8 - m9 <= 0
12. weight 3600:  t20 - 2*m8 <= 0
13. weight 3600:  t21 - m7 - m8 <= 0
14. weight 1446:  p22 - 2*m1 - m12 <= 0
15. weight 432:   p24 - m8 - m9 <= 0
16. weight 2454:  t17 - m5 - m12 <= 0
17. weight 1146:  t17 - m8 - m11 <= 0
18. weight 3600:  t16 - m6 - m12 <= 0
19. weight 1728:  p25 <= 0
20. weight 432:   p26 <= 0

Coefficient check:
  For every t-orbit and p-orbit, the combined coefficient equals its positive objective coefficient.
  For every m-orbit, the combined coefficient is at least the negative objective coefficient.
  The only slack occurs on missing-orbits m1,m2,m3,m4,m5,m7,m8, all in the safe direction.

This quotient certificate can be expanded to a full original-board certificate by applying every
symmetry in the group and dividing the multiplier by the orbit multiplicity in the standard way.
