feat: add numeric and symbolic scripts

scripts/dps_hierarchy/04_tiles_noise_scan.py

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+"""
+04_tiles_noise_scan.py
+
+Noise-robustness comparison: find the critical white-noise fraction p_c at
+which each criterion stops detecting entanglement of
+
+    rho(p) = p * rho_Tiles + (1-p) * I/9
+
+Reproduces (see the chat for the full discussion/derivation):
+  - plain shadow-map / de Vicente Bloch-representation criterion:
+        p_c ~ 0.9493  (tolerance ~5.07%)
+  - DPS level 2:
+        p_c ~ 0.951   (tolerance ~4.9%)
+
+i.e. DPS level 2 barely improves on the much cheaper order-1 correlation
+criterion for THIS state -- see 05_local_filtering.py for the much bigger
+lever (local filtering).
+
+Expected runtime: seconds for the shadow-map part; a few minutes for the
+DPS-2 bisection (12-16 SDP solves).
+"""
+import cvxpy as cp
+from common import noisy_tiles, shadow_map_nuclear_norm
+from dps_hierarchy import build_dps_problem, dps_feasible
+
+SOLVER = cp.SCS
+
+print("Shadow-map criterion threshold:")
+lo, hi = 0.0, 1.0
+for _ in range(40):
+    mid = (lo + hi) / 2
+    if shadow_map_nuclear_norm(noisy_tiles(mid)) > 1:
+        hi = mid
+    else:
+        lo = mid
+print(f"  p_c = {hi:.5f}  (tolerance {100 * (1 - hi):.2f}%)")
+
+print("\nDPS level-2 threshold:")
+prob, rho_param, sigma = build_dps_problem(d=3, k=2)
+lo, hi = 0.0, 1.0
+for i in range(16):
+    mid = (lo + hi) / 2
+    ok = dps_feasible(prob, rho_param, noisy_tiles(mid), solver=SOLVER,
+                       eps=1e-7, warm_start=True)
+    if ok:
+        lo = mid
+    else:
+        hi = mid
+    print(f"  [{i + 1}] p={mid:.5f}  feasible={ok}")
+print(f"  p_c = {hi:.5f}  (tolerance {100 * (1 - hi):.2f}%)")