feat: add numeric and symbolic scripts

scripts/seeded_exploration.py

@@ -0,0 +1,69 @@
+"""
+Seeded exploration around the known-good point (3.0, 3.0, 3.0).
+
+Two questions this answers:
+  1) Is 3.0 a STABLE fixed point of the alternating scheme (perturb the witnesses a
+     little, does it converge back to 3.0)?
+  2) Does searching the FULL PPT-mixture body (strictly larger than biseparable states)
+     starting from near this point ever find something BETTER than 3.0?
+
+If nothing beats 3.0 even when explicitly seeded nearby and given many iterations, that
+is now fairly strong evidence -- across both a from-scratch parametrized search (earlier)
+and this SDP-based search over the larger PPT-mixture relaxation -- that 3.0 is the true
+supremum (at least for PPT-mixtures, hence an upper bound on the biseparable one too,
+since biseparable subset PPT-mixtures).
+"""
+import numpy as np
+from sdp_ppt_mixture import solve_fixed_witness_step, true_norms_and_witnesses, CLUSTERS
+
+data = np.load('O_seed.npz')
+O_seed = {name: data[name] for name, _, _ in CLUSTERS}
+
+
+def project_to_unit_opnorm(A):
+    """Rescale A to have operator norm exactly 1 (SVD-based projection)."""
+    U, s, Vt = np.linalg.svd(A)
+    return U @ Vt if s.max() == 0 else A / s.max()
+
+
+def run_seeded(perturbation_strength, n_iters=25, seed=0, verbose=True):
+    rng = np.random.default_rng(seed)
+    O = {}
+    for name, _, _ in CLUSTERS:
+        noise = rng.normal(size=(15, 15)) * perturbation_strength
+        O[name] = project_to_unit_opnorm(O_seed[name] + noise)
+
+    best_min = -np.inf
+    history = []
+    for it in range(n_iters):
+        rho_val, t_val, M_vals = solve_fixed_witness_step(O)
+        norms, O = true_norms_and_witnesses(M_vals)
+        cur_min = min(norms.values())
+        history.append(cur_min)
+        best_min = max(best_min, cur_min)
+        if verbose:
+            nice = {k: round(v, 5) for k, v in norms.items()}
+            print(f"    iter {it:2d}: SDP t={t_val:.5f}  norms={nice}  min={cur_min:.5f}")
+    return best_min, history
+
+
+if __name__ == "__main__":
+    print("=== Stability check: seed EXACTLY at the known optimum (no perturbation) ===")
+    best0, _ = run_seeded(perturbation_strength=0.0, n_iters=10, seed=0)
+    print(f"  best min found: {best0:.6f}  (should stay essentially at 3.0)\n")
+
+    print("=== Perturbation sweep: does it converge back to 3.0, drift, or improve? ===")
+    results = {}
+    for strength in [0.05, 0.1, 0.2, 0.4, 0.7, 1.0]:
+        print(f"--- perturbation strength {strength} ---")
+        best, hist = run_seeded(perturbation_strength=strength, n_iters=25, seed=1, verbose=True)
+        results[strength] = best
+        print(f"  final best: {best:.6f}\n")
+
+    print("=" * 60)
+    for s, v in results.items():
+        print(f"  perturbation {s:.2f} -> best min found = {v:.6f}")
+    overall_best = max(results.values())
+    print()
+    print("Overall best across all perturbed seeded runs:", overall_best)
+    print("Compare: 3.0 (conjectured exact), 11/3 =", 11/3, "(proven upper bound)")