feat: add numeric and symbolic scripts

scripts/diagnostic_seeded_run_1.py

@@ -0,0 +1,55 @@
+"""
+DIAGNOSTIC: run a single SDP step seeded with the EXACT optimal dual witnesses of the
+known-good point rho_mix = 0.5*(Bell_AB x Bell_CD) + 0.5*(Bell_AC x Bell_BD), which is a
+manifestly valid PPT-mixture state (explicit convex combination of two product states)
+scoring EXACTLY (3.0, 3.0, 3.0) for (||M_AB||_*, ||M_AC||_*, ||M_AD||_*).
+
+Since rho_mix is feasible and, for these SPECIFIC witnesses O_S = U_S V_S^T (from its own
+SVD), achieves t = sum_S tr(O_S^T M_S(rho_mix)) = 3.0 exactly, any correct implementation
+of
+
+    max_{rho in PPT-mixtures} min_S tr(O_S^T M_S(rho))
+
+MUST return an optimal value >= 3.0 (the SDP maximizes over a set containing rho_mix).
+
+If the reported optimal t comes back < 3.0 here, that is conclusive evidence of an
+implementation bug in the SDP construction itself (not just a weakness of the alternating
+heuristic / random restarts).
+
+Run this BEFORE re-running the full alternating_search -- it isolates the problem.
+"""
+import numpy as np
+import cvxpy as cp
+from sdp_ppt_mixture import solve_fixed_witness_step, CLUSTERS
+
+data = np.load('O_seed.npz')
+O = {name: data[name] for name, _, _ in CLUSTERS}
+
+print("Loaded seed witnesses (each should have operator norm 1):")
+for name, Om in O.items():
+    print(f"  ||O_{name}||_op =", np.linalg.norm(Om, ord=2))
+
+print()
+print("Solving ONE SDP step with these witnesses...")
+rho_val, t_val, M_vals = solve_fixed_witness_step(O, verbose=True)
+
+print()
+print("SDP optimal t =", t_val)
+print("Expected: t >= 3.0 (since rho_mix itself is feasible and scores exactly 3.0 here)")
+print()
+norms, _ = __import__('sdp_ppt_mixture').true_norms_and_witnesses(M_vals)
+print("True nuclear norms of the returned optimal rho:", norms)
+
+if t_val < 2.99:
+    print()
+    print("!!! t < 3.0 -- there IS an implementation bug in the SDP construction. !!!")
+    print("Next diagnostic step: check prob.status, and manually verify PSD/PPT of")
+    print("rho_val's 7 constituent blocks (may need to re-solve while keeping rho_gammas")
+    print("accessible, i.e. return them from solve_fixed_witness_step for inspection).")
+else:
+    print()
+    print("t >= 3.0 as expected: the SDP construction is correct.")
+    print("The earlier runs' convergence to 7/3 was the alternating heuristic getting")
+    print("stuck in a (large-basin) symmetric fixed point from random starts -- not a bug.")
+    print("Fix: warm-start alternating_search from these O_seed witnesses (or from a small")
+    print("random perturbation of them) instead of purely random O's.")