P25, n=12: local integer refinement of the public cross-domain point set. The domain is the union of [1,2] x [0,3] and [0,3] x [1,2]: every point satisfies 1<=x<=2 or 1<=y<=2. Start from the certificate below, with minimum doubled-area integer 388663381718697520. Scale coordinates by S=10^9. At each step keep each point in its current vertical or horizontal rectangular arm and use integer coordinate displacements bounded by +/-L. For each potentially active triple, compute its signed determinant D and exact coordinate derivatives. Preserve the current sign of D and linearise the doubled-area lower bound; maximise a shared improvement variable using SciPy/HiGHS MILP, zero relative MIP gap and a five-second limit. Determinant rows are divided by S, with a 0.000001 margin, and the shared improvement variable has lower bound 0.00001. A triple whose current doubled-area slack exceeds 20*L*S may be omitted from the search model. All triples are checked in the final exact verification, including their nonlinear displacement terms, so the linear model alone never certifies a candidate. Run L=2,20,200 in sequence, retaining only strictly improving valid point sets. The first two searches produce the final construction. Replaying from the supplied start gives identical coordinates. A further pass at L=2,20,200,2000 found no additional valid improvement; this does not prove local or global optimality. Two independent exact determinant implementations check all 12 domain memberships and all 220 triangles, with no collinear triple. The final minimum doubled-area integer is 388663382303450288; divide by 2*S^2 to obtain minimum triangle area 0.194331691151725144, improved from 0.19433169085934876. Moving points between arms or using larger coordinated changes remains a possible next step. Starting certificate: {"points":[["1.000000000","0.652704360"],["1.053383402","2.874447686"],["2.000000000","2.347295640"],["3.000000000","1.379833807"],["1.946616598","0.125552314"],["1.808442899","3.000000000"],["1.236619283","1.875425649"],["1.763380717","1.124574351"],["1.191557101","0.000000000"],["0.000000000","1.620166193"],["2.990873464","1.000000000"],["0.009126536","2.000000000"]]}
P25 · 十字形内的最小三角形 · 讨论