The Verdict: Why "Audit Kernel v1" Wins the Debate

The multi-AI debate highlights a fascinating progression in investigative strategy. The final pivot—rejecting both Copilot’s generic script and ChatGPT’s insistence on the bloated Test10 solver in favor of **Audit Kernel v1**—is the correct, rigorous path forward. ChatGPT’s objection to a linear propagation test ("it will just disperse, so we learn nothing") misses a profound diagnostic truth in computational physics. We *want* to see if it disperses, and more importantly, *how* it disperses. Here is an evaluation of why this minimal, operator-locked neutral audit is the superior forensic move. --- ## The Verdict: Why "Audit Kernel v1" Wins the Debate The core flaw in ChatGPT's argument is the assumption that a non-linear soliton is the only interesting thing to find. In reality, the immediate danger to the project's integrity is that Test4 is an artifact of discrete grid geometry. By running a neutral propagation ($\frac{\partial^2 \Psi}{\partial t^2} = \nabla^2 \Psi$) using the **exact discrete matrix** from `radial_operator_verified.py`, you strip away the ambiguous influence of non-linear parameters ($\mu, \lambda, \kappa$) and isolate the pure geometric response of the grid. ### 1. The Power of the Control Operator Copilot’s script template used a generic, standard finite-difference stencil for the cylindrical Laplacian. That would be useless because it doesn't match the specific mathematical structure of your verified operator. If you instead link the propagation loop directly to the matrix generated by `radial_operator_verified.py` (with its structural `offset=1` lockout), the simulation will reveal the exact natural modes of your discrete space. ### 2. Breaking the Degeneracy of the On-Axis Hole If you evolve Test4 under a free wave equation using the verified operator: * **Scenario A (Geometric Artifact):** If the on-axis hole and the ring maximum at $r \approx 1.91$ are merely a low-order eigenvector of the discrete operator with `offset=1`, the state will remain remarkably stationary or undergo a simple, clean "breathing" oscillation, even without non-linear terms. This proves the structure is a property of the grid, not the physics. * **Scenario B (Physical State):** If the state is a genuine wavepacket that was held together by a non-linear potential in the original notebook, stripping that potential away will cause the wavepacket to immediately obey gradient pressures. The ring will shear, and waves will visibly radiate outward toward the outer boundary. --- ## Comparison of Proposed Strategies | Feature / Strategy | Copilot's Generic Script | ChatGPT's Test10 Core Pivot | The Final Consensus (Audit Kernel v1) | | --- | --- | --- | --- | | **Laplacian Kernel** | Generic 3-point finite difference | Bloated/Ambiguous (from 3.2MB file) | **Locked to `radial_operator_verified.py**` | | **Physics Template** | Linear Free Wave | Full Non-linear Framework | **Linear Free Wave** | | **Primary Goal** | Check for wave propagation | Reconstruct original soliton | **Isolate operator/boundary artifacts** | | **Risk Profile** | High (false dynamics due to mismatched stencil) | Extremely High (introducing unverified parameters) | **Minimal (purely forensic control test)** | --- ## Implementation Guidance for `test4_phase1_neutral_audit.py` To ensure this audit yields maximum data clarity, ensure the script you generate strictly obeys these three operational constraints: * **Direct Operator Import:** Do not manually write a new Laplacian loop. Force the script to call `build_radial_operator_reduced()` from your existing `radial_operator_verified.py` file to construct the spatial step. * **No Ad-Hoc Axis Swaps Yet:** Keep the exact axis treatment (`offset=1`) that matches the file's lineage. Changing the operator boundaries *during* the first neutral test introduces too many moving variables. * **Track the Phase Profile:** Along with tracking the peak amplitude and ring radius, save the spatial phase profile ($\angle \Psi$) at $t=0$ and $t=10$. If a phase gradient or winding is present in the initial snapshot, a linear evolution will immediately convert that phase profile into directional momentum or dispersion, providing a definitive signature of what the state was doing when it was saved. Pivoting to this clean, neutral audit means you are no longer guessing at history—you are actively interrogating the mathematics. -- #!/usr/bin/env python3 """ test4_phase1_neutral_audit.py Neutral propagation audit for test_4_relaxed_state.npz Fully automated for Google Colab environment preservation, Google Drive mirroring, and local machine delivery. """ import os import json import numpy as np import matplotlib.pyplot as plt import scipy.sparse as sp # ============================================================ # ENVIRONMENT & AUTOMATION SETUP # ============================================================ IN_COLAB = False try: import google.colab from google.colab import drive, files IN_COLAB = True print("--> Google Colab environment detected.") except ImportError: print("--> Running in local Python environment.") # Handle Google Drive Mount if IN_COLAB: print("--> Mounting Google Drive for automatic backup...") drive.mount('/content/drive') # Save directly inside a dedicated folder in your Drive root OUTPUT_DIR = "/content/drive/MyDrive/test4_phase1_neutral_audit" else: OUTPUT_DIR = "test4_phase1_neutral_audit" os.makedirs(OUTPUT_DIR, exist_ok=True) print(f"--> Target output directory set to: {OUTPUT_DIR}") from radial_operator_verified import build_radial_operator_reduced # ============================================================ # 1. CONFIGURATION # ============================================================ SOURCE_FILE = "test_4_relaxed_state.npz" DT = 0.01 T_MAX = 50.0 N_STEPS = int(T_MAX / DT) SAVE_EVERY = 50 SNAP_TIMES = [0.0, 25.0, 50.0] # ============================================================ # 2. LOAD STATE # ============================================================ if not os.path.exists(SOURCE_FILE): raise FileNotFoundError(f"Missing {SOURCE_FILE}. Ensure it is uploaded to the workspace root.") data = np.load(SOURCE_FILE) psi0 = data["Psi_final"] pi0 = data["Pi_final"] r_grid = data["r_grid"] z_grid = data["z_grid"] Nr = len(r_grid) Nz = len(z_grid) psi0 = psi0.reshape(Nr, Nz) pi0 = pi0.reshape(Nr, Nz) dr = r_grid[1] - r_grid[0] dz = z_grid[1] - z_grid[0] print(f"Loaded Test4: psi shape={psi0.shape}, pi shape={pi0.shape}") r_2d = r_grid[:, None] z_2d = z_grid[None, :] # ============================================================ # 3. BUILD LAPLACIAN OPERATOR # ============================================================ # Restored offset=1 to preserve the verified self-adjoint properties L_r = build_radial_operator_reduced(r_grid, dr, offset=1, nr_full=Nr+1 if IN_COLAB else Nr) L_r = sp.csr_matrix(L_r) main = -2.0 * np.ones(Nz) off = 1.0 * np.ones(Nz - 1) L_z = sp.diags([off, main, off], offsets=[-1, 0, 1], shape=(Nz, Nz)) / (dz ** 2) L_z = sp.csr_matrix(L_z) def laplacian(psi_field): radial = L_r @ psi_field axial = psi_field @ L_z.T return radial + axial # ============================================================ # 4. DIAGNOSTICS # ============================================================ def volume_element(): return 2.0 * np.pi * r_2d * dr * dz VOL = volume_element() def total_norm(psi_field): return float(np.sum(np.abs(psi_field) ** 2 * VOL)) def total_energy(psi_field, pi_field): dpsi_dr = np.gradient(psi_field, dr, axis=0) dpsi_dz = np.gradient(psi_field, dz, axis=1) kinetic = 0.5 * np.abs(pi_field) ** 2 grad2 = 0.5 * (np.abs(dpsi_dr) ** 2 + np.abs(dpsi_dz) ** 2) return float(np.sum((kinetic + grad2) * VOL)) def center_density(psi_field): iz = np.argmin(np.abs(z_grid - 0.0)) ir = np.argmin(np.abs(r_grid - 0.0)) return float(np.abs(psi_field[ir, iz]) ** 2) def peak_amplitude(psi_field): return float(np.max(np.abs(psi_field))) def ring_radius(psi_field): amp = np.abs(psi_field) idx = np.unravel_index(np.argmax(amp), amp.shape) return float(r_grid[idx[0]]) def com_z(psi_field): density = np.abs(psi_field) ** 2 num = np.sum(density * VOL * z_2d) den = np.sum(density * VOL) return float(num / den) if den != 0 else 0.0 # ============================================================ # 5. TIME INTEGRATOR (RK4) # ============================================================ def rhs(psi_field, pi_field): return pi_field, laplacian(psi_field) def rk4_step(psi_field, pi_field, dt): k1_psi, k1_pi = rhs(psi_field, pi_field) k2_psi, k2_pi = rhs(psi_field + 0.5 * dt * k1_psi, pi_field + 0.5 * dt * k1_pi) k3_psi, k3_pi = rhs(psi_field + 0.5 * dt * k2_psi, pi_field + 0.5 * dt * k2_pi) k4_psi, k4_pi = rhs(psi_field + dt * k3_psi, pi_field + dt * k3_pi) psi_next = psi_field + (dt / 6.0) * (k1_psi + 2 * k2_psi + 2 * k3_psi + k4_psi) pi_next = pi_field + (dt / 6.0) * (k1_pi + 2 * k2_pi + 2 * k3_pi + k4_pi) return psi_next, pi_next # ============================================================ # 6. EVOLUTION LOOP # ============================================================ psi = psi0.copy() pi = pi0.copy() times, norms, energies, center_densities, peak_amps, ring_radii, com_zs = [], [], [], [], [], [], [] snapshots = {} def record(t_curr, psi_field, pi_field): times.append(t_curr) norms.append(total_norm(psi_field)) energies.append(total_energy(psi_field, pi_field)) center_densities.append(center_density(psi_field)) peak_amps.append(peak_amplitude(psi_field)) ring_radii.append(ring_radius(psi_field)) com_zs.append(com_z(psi_field)) if any(abs(t_curr - ts) < 1e-8 for ts in SNAP_TIMES): snapshots[round(t_curr, 8)] = {"amp": np.abs(psi_field).copy(), "phase": np.angle(psi_field).copy()} t = 0.0 print("\n--> Starting neutral propagation audit loop...") record(t, psi, pi) for step in range(1, N_STEPS + 1): psi, pi = rk4_step(psi, pi, DT) t = step * DT if step % SAVE_EVERY == 0 or any(abs(t - ts) < DT / 2 for ts in SNAP_TIMES): record(t, psi, pi) if step % (SAVE_EVERY * 10) == 0: print(f" t={t:.2f}: peak={peak_amps[-1]:.4f}, ring_r={ring_radii[-1]:.3f}, center_rho={center_densities[-1]:.4e}") print("--> Propagation complete.") # ============================================================ # 7. EXPORTS & VISUALIZATION # ============================================================ history = { "source_file": SOURCE_FILE, "dt": DT, "t_max": T_MAX, "times": times, "norms": norms, "energies": energies, "center_densities": center_densities, "peak_amplitudes": peak_amps, "ring_radii": ring_radii, "com_z": com_zs } with open(os.path.join(OUTPUT_DIR, "test4_propagation_history.json"), "w") as f: json.dump(history, f, indent=2) with open(os.path.join(OUTPUT_DIR, "test4_propagation_report.txt"), "w") as f: f.write("Test4 Neutral Propagation Structural Audit\n==========================================\n") f.write(f"Initial Norm: {norms[0]:.6e} -> Final Norm: {norms[-1]:.6e}\n") f.write(f"Initial Ring R: {ring_radii[0]:.4f} -> Final Ring R: {ring_radii[-1]:.4f}\n") # Generate Snapshot plots for ts in SNAP_TIMES: key = round(ts, 8) if key in snapshots: fig, axes = plt.subplots(1, 2, figsize=(12, 4)) im0 = axes[0].pcolormesh(z_grid, r_grid, snapshots[key]["amp"], shading="auto", cmap="viridis") plt.colorbar(im0, ax=axes[0]) axes[0].set_title(f"|Psi| (t={ts:.1f})") im1 = axes[1].pcolormesh(z_grid, r_grid, snapshots[key]["phase"], shading="auto", cmap="twilight") plt.colorbar(im1, ax=axes[1]) axes[1].set_title(f"arg(Psi) (t={ts:.1f})") plt.tight_layout() plt.savefig(os.path.join(OUTPUT_DIR, f"test4_t{int(ts)}.png"), dpi=150) plt.close(fig) # Generate Summary Plots fig, axes = plt.subplots(2, 3, figsize=(15, 8)) axes[0, 0].plot(times, peak_amps) axes[0, 0].set_title("Peak Amplitude") axes[0, 1].plot(times, ring_radii) axes[0, 1].set_title("Ring Radius") axes[0, 2].plot(times, center_densities) axes[0, 2].set_title("Center Density") axes[1, 0].plot(times, np.array(norms) / norms[0]) axes[1, 0].set_title("Relative Norm") axes[1, 1].plot(times, np.array(energies) / energies[0]) axes[1, 1].set_title("Relative Energy") axes[1, 2].plot(times, com_zs) axes[1, 2].set_title("Center of Mass Z") plt.tight_layout() plt.savefig(os.path.join(OUTPUT_DIR, "test4_time_series.png"), dpi=150) plt.close(fig) print(f"\n--> Data saved locally and backed up to Drive: {OUTPUT_DIR}") # ============================================================ # LOCAL LAPTOP ZIP-DELIVERY AUTOMATION # ============================================================ if IN_COLAB: print("\n--> Packing files for automatic laptop download...") os.system(f"zip -r /content/test4_audit_results.zip {OUTPUT_DIR}") print("--> Triggering browser download to local machine...") files.download("/content/test4_audit_results.zip") --- --> Google Colab environment detected. --> Mounting Google Drive for automatic backup... Mounted at /content/drive --> Target output directory set to: /content/drive/MyDrive/test4_phase1_neutral_audit Loaded Test4: psi shape=(200, 1200), pi shape=(200, 1200) /content/radial_operator_verified.py:35: RuntimeWarning: divide by zero encountered in divide W_r_inv = sp.diags(1.0 / w_r, format='csr') --------------------------------------------------------------------------- TypeError Traceback (most recent call last) /tmp/ipykernel_6781/800255845.py in () 82 # Restored offset=1 to preserve the verified self-adjoint properties 83 L_r = build_radial_operator_reduced(r_grid, dr, offset=1, nr_full=Nr+1 if IN_COLAB else Nr) ---> 84 L_r = sp.csr_matrix(L_r) 85 86 main = -2.0 * np.ones(Nz) 3 frames/usr/local/lib/python3.12/dist-packages/scipy/sparse/_compressed.py in __init__(self, arg1, shape, dtype, copy, maxprint) 55 if len(arg1) == 2: 56 # (data, ij) format ---> 57 coo = self._coo_container(arg1, shape=shape, dtype=dtype) 58 arrays = coo._coo_to_compressed(self._swap) 59 self.indptr, self.indices, self.data, self._shape = arrays /usr/local/lib/python3.12/dist-packages/scipy/sparse/_coo.py in __init__(self, arg1, shape, dtype, copy, maxprint) 49 50 if shape is None: ---> 51 if any(len(idx) == 0 for idx in coords): 52 raise ValueError('cannot infer dimensions from zero ' 53 'sized index arrays') /usr/local/lib/python3.12/dist-packages/scipy/sparse/_coo.py in (.0) 49 50 if shape is None: ---> 51 if any(len(idx) == 0 for idx in coords): 52 raise ValueError('cannot infer dimensions from zero ' 53 'sized index arrays') /usr/local/lib/python3.12/dist-packages/scipy/sparse/_base.py in __len__(self) 447 # non-zeros is more important. For now, raise an exception! 448 def __len__(self): --> 449 raise TypeError("sparse array length is ambiguous; use getnnz()" 450 " or shape[0]") 451 TypeError: sparse array length is ambiguous; use getnnz() or shape[0] -- New script -> #!/usr/bin/env python3 """ test4_phase1_neutral_audit.py Neutral propagation audit for test_4_relaxed_state.npz Fully automated for Google Colab environment preservation, Google Drive mirroring, and local machine delivery. """ import os import json import numpy as np import matplotlib.pyplot as plt import scipy.sparse as sp # ============================================================ # ENVIRONMENT & AUTOMATION SETUP # ============================================================ IN_COLAB = False try: import google.colab from google.colab import drive, files IN_COLAB = True print("--> Google Colab environment detected.") except ImportError: print("--> Running in local Python environment.") # Handle Google Drive Mount if IN_COLAB: print("--> Mounting Google Drive for automatic backup...") drive.mount('/content/drive') # Save directly inside a dedicated folder in your Drive root OUTPUT_DIR = "/content/drive/MyDrive/test4_phase1_neutral_audit" else: OUTPUT_DIR = "test4_phase1_neutral_audit" os.makedirs(OUTPUT_DIR, exist_ok=True) print(f"--> Target output directory set to: {OUTPUT_DIR}") from radial_operator_verified import build_radial_operator_reduced # ============================================================ # 1. CONFIGURATION # ============================================================ SOURCE_FILE = "test_4_relaxed_state.npz" DT = 0.01 T_MAX = 50.0 N_STEPS = int(T_MAX / DT) SAVE_EVERY = 50 SNAP_TIMES = [0.0, 25.0, 50.0] # ============================================================ # 2. LOAD STATE # ============================================================ if not os.path.exists(SOURCE_FILE): raise FileNotFoundError(f"Missing {SOURCE_FILE}. Ensure it is uploaded to the workspace root.") data = np.load(SOURCE_FILE) psi0 = data["Psi_final"] pi0 = data["Pi_final"] r_grid = data["r_grid"] z_grid = data["z_grid"] Nr = len(r_grid) Nz = len(z_grid) psi0 = psi0.reshape(Nr, Nz) pi0 = pi0.reshape(Nr, Nz) dr = r_grid[1] - r_grid[0] dz = z_grid[1] - z_grid[0] print(f"Loaded Test4: psi shape={psi0.shape}, pi shape={pi0.shape}") r_2d = r_grid[:, None] z_2d = z_grid[None, :] # ============================================================ # 3. BUILD LAPLACIAN OPERATOR # ============================================================ # FIX 1: nr_full is locked to Nr to eliminate the divide-by-zero array mismatch. # FIX 2: Safely check and unpack the tuple returned by your verified operator script. radial_data = build_radial_operator_reduced(r_grid, dr, offset=1, nr_full=Nr) if isinstance(radial_data, tuple): L_r = radial_data[0] else: L_r = radial_data L_r = sp.csr_matrix(L_r) # Axial second derivative operator (Dirichlet at ends) main = -2.0 * np.ones(Nz) off = 1.0 * np.ones(Nz - 1) L_z = sp.diags([off, main, off], offsets=[-1, 0, 1], shape=(Nz, Nz)) / (dz ** 2) L_z = sp.csr_matrix(L_z) def laplacian(psi_field): radial = L_r @ psi_field axial = psi_field @ L_z.T return radial + axial # ============================================================ # 4. DIAGNOSTICS # ============================================================ def volume_element(): return 2.0 * np.pi * r_2d * dr * dz VOL = volume_element() def total_norm(psi_field): return float(np.sum(np.abs(psi_field) ** 2 * VOL)) def total_energy(psi_field, pi_field): dpsi_dr = np.gradient(psi_field, dr, axis=0) dpsi_dz = np.gradient(psi_field, dz, axis=1) kinetic = 0.5 * np.abs(pi_field) ** 2 grad2 = 0.5 * (np.abs(dpsi_dr) ** 2 + np.abs(dpsi_dz) ** 2) return float(np.sum((kinetic + grad2) * VOL)) def center_density(psi_field): iz = np.argmin(np.abs(z_grid - 0.0)) ir = np.argmin(np.abs(r_grid - 0.0)) return float(np.abs(psi_field[ir, iz]) ** 2) def peak_amplitude(psi_field): return float(np.max(np.abs(psi_field))) def ring_radius(psi_field): amp = np.abs(psi_field) idx = np.unravel_index(np.argmax(amp), amp.shape) return float(r_grid[idx[0]]) def com_z(psi_field): density = np.abs(psi_field) ** 2 num = np.sum(density * VOL * z_2d) den = np.sum(density * VOL) return float(num / den) if den != 0 else 0.0 # ============================================================ # 5. TIME INTEGRATOR (RK4) # ============================================================ def rhs(psi_field, pi_field): return pi_field, laplacian(psi_field) def rk4_step(psi_field, pi_field, dt): k1_psi, k1_pi = rhs(psi_field, pi_field) k2_psi, k2_pi = rhs(psi_field + 0.5 * dt * k1_psi, pi_field + 0.5 * dt * k1_pi) k3_psi, k3_pi = rhs(psi_field + 0.5 * dt * k2_psi, pi_field + 0.5 * dt * k2_pi) k4_psi, k4_pi = rhs(psi_field + dt * k3_psi, pi_field + dt * k3_pi) psi_next = psi_field + (dt / 6.0) * (k1_psi + 2 * k2_psi + 2 * k3_psi + k4_psi) pi_next = pi_field + (dt / 6.0) * (k1_pi + 2 * k2_pi + 2 * k3_pi + k4_pi) return psi_next, pi_next # ============================================================ # 6. EVOLUTION LOOP # ============================================================ psi = psi0.copy() pi = pi0.copy() times, norms, energies, center_densities, peak_amps, ring_radii, com_zs = [], [], [], [], [], [], [] snapshots = {} def record(t_curr, psi_field, pi_field): times.append(t_curr) norms.append(total_norm(psi_field)) energies.append(total_energy(psi_field, pi_field)) center_densities.append(center_density(psi_field)) peak_amps.append(peak_amplitude(psi_field)) ring_radii.append(ring_radius(psi_field)) com_zs.append(com_z(psi_field)) if any(abs(t_curr - ts) < 1e-8 for ts in SNAP_TIMES): snapshots[round(t_curr, 8)] = {"amp": np.abs(psi_field).copy(), "phase": np.angle(psi_field).copy()} t = 0.0 print("\n--> Starting neutral propagation audit loop...") record(t, psi, pi) for step in range(1, N_STEPS + 1): psi, pi = rk4_step(psi, pi, DT) t = step * DT if step % SAVE_EVERY == 0 or any(abs(t - ts) < DT / 2 for ts in SNAP_TIMES): record(t, psi, pi) if step % (SAVE_EVERY * 10) == 0: print(f" t={t:.2f}: peak={peak_amps[-1]:.4f}, ring_r={ring_radii[-1]:.3f}, center_rho={center_densities[-1]:.4e}") print("--> Propagation complete.") # ============================================================ # 7. EXPORTS & VISUALIZATION # ============================================================ history = { "source_file": SOURCE_FILE, "dt": DT, "t_max": T_MAX, "times": times, "norms": norms, "energies": energies, "center_densities": center_densities, "peak_amplitudes": peak_amps, "ring_radii": ring_radii, "com_z": com_zs } with open(os.path.join(OUTPUT_DIR, "test4_propagation_history.json"), "w") as f: json.dump(history, f, indent=2) with open(os.path.join(OUTPUT_DIR, "test4_propagation_report.txt"), "w") as f: f.write("Test4 Neutral Propagation Structural Audit\n==========================================\n") f.write(f"Initial Norm: {norms[0]:.6e} -> Final Norm: {norms[-1]:.6e}\n") f.write(f"Initial Ring R: {ring_radii[0]:.4f} -> Final Ring R: {ring_radii[-1]:.4f}\n") # Generate Snapshot plots for ts in SNAP_TIMES: key = round(ts, 8) if key in snapshots: fig, axes = plt.subplots(1, 2, figsize=(12, 4)) im0 = axes[0].pcolormesh(z_grid, r_grid, snapshots[key]["amp"], shading="auto", cmap="viridis") plt.colorbar(im0, ax=axes[0]) axes[0].set_title(f"|Psi| (t={ts:.1f})") im1 = axes[1].pcolormesh(z_grid, r_grid, snapshots[key]["phase"], shading="auto", cmap="twilight") plt.colorbar(im1, ax=axes[1]) axes[1].set_title(f"arg(Psi) (t={ts:.1f})") plt.tight_layout() plt.savefig(os.path.join(OUTPUT_DIR, f"test4_t{int(ts)}.png"), dpi=150) plt.close(fig) # Generate Summary Plots fig, axes = plt.subplots(2, 3, figsize=(15, 8)) axes[0, 0].plot(times, peak_amps) axes[0, 0].set_title("Peak Amplitude") axes[0, 1].plot(times, ring_radii) axes[0, 1].set_title("Ring Radius") axes[0, 2].plot(times, center_densities) axes[0, 2].set_title("Center Density") axes[1, 0].plot(times, np.array(norms) / norms[0]) axes[1, 0].set_title("Relative Norm") axes[1, 1].plot(times, np.array(energies) / energies[0]) axes[1, 1].set_title("Relative Energy") axes[1, 2].plot(times, com_zs) axes[1, 2].set_title("Center of Mass Z") plt.tight_layout() plt.savefig(os.path.join(OUTPUT_DIR, "test4_time_series.png"), dpi=150) plt.close(fig) print(f"\n--> Data saved locally and backed up to Drive: {OUTPUT_DIR}") # ============================================================ # LOCAL LAPTOP ZIP-DELIVERY AUTOMATION # ============================================================ if IN_COLAB: print("\n--> Packing files for automatic laptop download...") os.system(f"zip -r /content/test4_audit_results.zip {OUTPUT_DIR}") print("--> Triggering browser download to local machine...") files.download("/content/test4_audit_results.zip") --- #!/usr/bin/env python3 """ test4_grid_forensics.py Forensic diagnostic script to unmask the hidden dimensional and boundary inconsistencies between test_4_relaxed_state.npz and radial_operator_verified.py. """ import os import json import numpy as np import scipy.sparse as sp # ============================================================ # ENVIRONMENT & AUTOMATION SETUP # ============================================================ IN_COLAB = False try: import google.colab from google.colab import drive, files IN_COLAB = True print("--> Google Colab environment detected.") except ImportError: print("--> Running in local Python environment.") if IN_COLAB: print("--> Mounting Google Drive for automatic backup...") drive.mount('/content/drive', force_remount=True) OUTPUT_DIR = "/content/drive/MyDrive/test4_phase1_neutral_audit" else: OUTPUT_DIR = "test4_phase1_neutral_audit" os.makedirs(OUTPUT_DIR, exist_ok=True) REPORT_PATH = os.path.join(OUTPUT_DIR, "test4_grid_forensics_report.txt") from radial_operator_verified import build_radial_operator_reduced # ============================================================ # 1. LOAD AND INSPECT RAW NPZ GEOMETRY # ============================================================ SOURCE_FILE = "test_4_relaxed_state.npz" if not os.path.exists(SOURCE_FILE): raise FileNotFoundError(f"Missing {SOURCE_FILE}. Please place it in the working directory.") data = np.load(SOURCE_FILE) r_grid = data["r_grid"] z_grid = data["z_grid"] psi0 = data["Psi_final"] Nr = len(r_grid) Nz = len(z_grid) dr = r_grid[1] - r_grid[0] dz = z_grid[1] - z_grid[0] # Construct the live console readouts and log them simultaneously log_lines = [] def log_print(msg): print(msg) log_lines.append(msg) log_print("\n========================================================") log_print(" TEST4 FIELD GEOMETRY METRICS ") log_print("========================================================") log_print(f"Psi Shape stored in file : {psi0.shape}") log_print(f"Radial grid point count (Nr) : {Nr}") log_print(f"Axial grid point count (Nz) : {Nz}") log_print(f"Calculated dr : {dr:.6e}") log_print(f"Calculated dz : {dz:.6e}") log_print(f"Radial range : [{r_grid[0]:.6f} to {r_grid[-1]:.6f}]") log_print(f"Axial range : [{z_grid[0]:.6f} to {z_grid[-1]:.6f}]") # Check for explicit boundary inclusion includes_axis = np.isclose(r_grid[0], 0.0) log_print(f"Does r_grid explicitly include the axis (r=0)? : {includes_axis}") log_print("\nFirst 5 Radial Points:") log_print(f" {r_grid[:5]}") log_print("Last 5 Radial Points:") log_print(f" {r_grid[-5:]}") # ============================================================ # 2. OPERATOR STENCIL PROBING SWEEP # ============================================================ log_print("\n========================================================") log_print(" OPERATOR INTERFACE FORENSIC SWEEP ") log_print("========================================================") log_print("Probing build_radial_operator_reduced to find working shapes...\n") # We test varying values of nr_full around Nr to find what satisfies the internal math probe_range = range(Nr - 5, Nr + 6) successful_runs = [] for p_nr_full in probe_range: for p_offset in [0, 1, 2]: try: # Attempt to execute the operator build exactly as written in your file res = build_radial_operator_reduced(r_grid, dr, offset=p_offset, nr_full=p_nr_full) # Handle if it returns a tuple or raw array/matrix matrix = res[0] if isinstance(res, tuple) else res shape = matrix.shape log_print(f"[SUCCESS] offset={p_offset}, nr_full={p_nr_full} -> Returns matrix shape: {shape}") successful_runs.append({ "offset": p_offset, "nr_full": p_nr_full, "resulting_shape": shape }) except Exception as e: error_class = e.__class__.__name__ # Keep log concise by formatting the error text err_msg = str(e).split('\n')[0] log_print(f"[CRASH] offset={p_offset}, nr_full={p_nr_full} -> Raised {error_class}: {err_msg}") # ============================================================ # 3. ANALYSIS & PREDICTIONS # ============================================================ log_print("\n========================================================") log_print(" PROVENANCE INTERPRETATION ") log_print("========================================================") if not successful_runs: log_print("CRITICAL: No configurations matched. The grid array and operator logic are fundamentally incompatible.") else: log_print("Matching Configurations Analysis:") for run in successful_runs: if run["resulting_shape"] == (Nr, Nr): log_print(f"--> DIRECT MATCH DETECTED: offset={run['offset']}, nr_full={run['nr_full']} produces the exact ({Nr}x{Nr}) matrix needed for Test4 field sizes.") elif run["resulting_shape"][0] == Nr: log_print(f"--> PARTIAL MATCH DETECTED: Row count matches Nr ({Nr}), but shape is non-square: {run['resulting_shape']}") # Save the diagnostic summary out to file with open(REPORT_PATH, "w") as f: f.write("\n".join(log_lines)) print(f"\n--> Forensic report compiled and saved to: {REPORT_PATH}") # ============================================================ # AUTOMATED EXPORT DELIVERY TO LAPTOP # ============================================================ if IN_COLAB: print("\n--> Packaging forensic files for automated browser sync...") os.system(f"zip -r /content/test4_forensics_results.zip {OUTPUT_DIR}") print("--> Transporting archive to laptop...") files.download("/content/test4_forensics_results.zip") -- --> Google Colab environment detected. --> Mounting Google Drive for automatic backup... Mounted at /content/drive ======================================================== TEST4 FIELD GEOMETRY METRICS ======================================================== Psi Shape stored in file : (240000,) Radial grid point count (Nr) : 200 Axial grid point count (Nz) : 1200 Calculated dr : 5.025126e-02 Calculated dz : 6.672227e-02 Radial range : [0.000000 to 10.000000] Axial range : [-40.000000 to 40.000000] Does r_grid explicitly include the axis (r=0)? : True First 5 Radial Points: [0. 0.05025126 0.10050251 0.15075377 0.20100503] Last 5 Radial Points: [ 9.79899497 9.84924623 9.89949749 9.94974874 10. ] ======================================================== OPERATOR INTERFACE FORENSIC SWEEP ======================================================== Probing build_radial_operator_reduced to find working shapes... [CRASH] offset=0, nr_full=195 -> Raised IndexError: index 194 is out of bounds for axis 0 with size 194 [CRASH] offset=1, nr_full=195 -> Raised IndexError: index 193 is out of bounds for axis 0 with size 193 [CRASH] offset=2, nr_full=195 -> Raised IndexError: index 192 is out of bounds for axis 0 with size 192 [CRASH] offset=0, nr_full=196 -> Raised IndexError: index 195 is out of bounds for axis 0 with size 195 [CRASH] offset=1, nr_full=196 -> Raised IndexError: index 194 is out of bounds for axis 0 with size 194 [CRASH] offset=2, nr_full=196 -> Raised IndexError: index 193 is out of bounds for axis 0 with size 193 [CRASH] offset=0, nr_full=197 -> Raised IndexError: index 196 is out of bounds for axis 0 with size 196 [CRASH] offset=1, nr_full=197 -> Raised IndexError: index 195 is out of bounds for axis 0 with size 195 [CRASH] offset=2, nr_full=197 -> Raised IndexError: index 194 is out of bounds for axis 0 with size 194 [CRASH] offset=0, nr_full=198 -> Raised IndexError: index 197 is out of bounds for axis 0 with size 197 [CRASH] offset=1, nr_full=198 -> Raised IndexError: index 196 is out of bounds for axis 0 with size 196 [CRASH] offset=2, nr_full=198 -> Raised IndexError: index 195 is out of bounds for axis 0 with size 195 [CRASH] offset=0, nr_full=199 -> Raised IndexError: index 198 is out of bounds for axis 0 with size 198 [CRASH] offset=1, nr_full=199 -> Raised IndexError: index 197 is out of bounds for axis 0 with size 197 [CRASH] offset=2, nr_full=199 -> Raised IndexError: index 196 is out of bounds for axis 0 with size 196 [SUCCESS] offset=0, nr_full=200 -> Returns matrix shape: (200, 200) [CRASH] offset=1, nr_full=200 -> Raised IndexError: index 198 is out of bounds for axis 0 with size 198 [CRASH] offset=2, nr_full=200 -> Raised IndexError: index 197 is out of bounds for axis 0 with size 197 [SUCCESS] offset=0, nr_full=201 -> Returns matrix shape: (200, 200) [SUCCESS] offset=1, nr_full=201 -> Returns matrix shape: (200, 200) [CRASH] offset=2, nr_full=201 -> Raised IndexError: index 198 is out of bounds for axis 0 with size 198 [SUCCESS] offset=0, nr_full=202 -> Returns matrix shape: (200, 200) [SUCCESS] offset=1, nr_full=202 -> Returns matrix shape: (200, 200) [SUCCESS] offset=2, nr_full=202 -> Returns matrix shape: (200, 200) [SUCCESS] offset=0, nr_full=203 -> Returns matrix shape: (200, 200) [SUCCESS] offset=1, nr_full=203 -> Returns matrix shape: (200, 200) [SUCCESS] offset=2, nr_full=203 -> Returns matrix shape: (200, 200) [SUCCESS] offset=0, nr_full=204 -> Returns matrix shape: (200, 200) [SUCCESS] offset=1, nr_full=204 -> Returns matrix shape: (200, 200) [SUCCESS] offset=2, nr_full=204 -> Returns matrix shape: (200, 200) [SUCCESS] offset=0, nr_full=205 -> Returns matrix shape: (200, 200) [SUCCESS] offset=1, nr_full=205 -> Returns matrix shape: (200, 200) [SUCCESS] offset=2, nr_full=205 -> Returns matrix shape: (200, 200) ======================================================== PROVENANCE INTERPRETATION ======================================================== Matching Configurations Analysis: --> DIRECT MATCH DETECTED: offset=0, nr_full=200 produces the exact (200x200) matrix needed for Test4 field sizes. --> DIRECT MATCH DETECTED: offset=0, nr_full=201 produces the exact (200x200) matrix needed for Test4 field sizes. --> DIRECT MATCH DETECTED: offset=1, nr_full=201 produces the exact (200x200) matrix needed for Test4 field sizes. --> DIRECT MATCH DETECTED: offset=0, nr_full=202 produces the exact (200x200) matrix needed for Test4 field sizes. --> DIRECT MATCH DETECTED: offset=1, nr_full=202 produces the exact (200x200) matrix needed for Test4 field sizes. --> DIRECT MATCH DETECTED: offset=2, nr_full=202 produces the exact (200x200) matrix needed for Test4 field sizes. --> DIRECT MATCH DETECTED: offset=0, nr_full=203 produces the exact (200x200) matrix needed for Test4 field sizes. --> DIRECT MATCH DETECTED: offset=1, nr_full=203 produces the exact (200x200) matrix needed for Test4 field sizes. --> DIRECT MATCH DETECTED: offset=2, nr_full=203 produces the exact (200x200) matrix needed for Test4 field sizes. --> DIRECT MATCH DETECTED: offset=0, nr_full=204 produces the exact (200x200) matrix needed for Test4 field sizes. --> DIRECT MATCH DETECTED: offset=1, nr_full=204 produces the exact (200x200) matrix needed for Test4 field sizes. --> DIRECT MATCH DETECTED: offset=2, nr_full=204 produces the exact (200x200) matrix needed for Test4 field sizes. --> DIRECT MATCH DETECTED: offset=0, nr_full=205 produces the exact (200x200) matrix needed for Test4 field sizes. --> DIRECT MATCH DETECTED: offset=1, nr_full=205 produces the exact (200x200) matrix needed for Test4 field sizes. --> DIRECT MATCH DETECTED: offset=2, nr_full=205 produces the exact (200x200) matrix needed for Test4 field sizes. --> Forensic report compiled and saved to: /content/drive/MyDrive/test4_phase1_neutral_audit/test4_grid_forensics_report.txt --> Packaging forensic files for automated browser sync... --> Transporting archive to laptop... --->>> This is actually a very useful result. The forensic sweep tells us something much more precise than the earlier speculation. What we now know Test4 definitely contains the axis Your grid is: r_grid[0] = 0.0 r_grid[-1] = 10.0 Nr = 200 So Test4 is not a reduced grid. The saved field explicitly includes the axis. That eliminates one earlier hypothesis: "Maybe row 0 isn't really the axis." No. It is. What the operator sweep discovered The critical result is: SUCCESS: offset=0, nr_full=200 while CRASH: offset=1, nr_full=200 This means the operator as written is naturally compatible with: offset = 0 nr_full = 200 which corresponds exactly to the Test4 grid size. That is a huge clue. Why this matters Earlier DeepSeek worried that: offset = 1 might prove the axis DOF was removed. Your sweep shows: offset=1, nr_full=200 -> invalid offset=0, nr_full=200 -> valid So the recovered operator itself is telling us: A 200-point grid works cleanly without removing the axis. That weakens the argument that Test4's zero center must be a consequence of an omitted axis DOF. Not completely disproven—but much weaker. The subtle issue Notice: offset=0, nr_full=201 -> works offset=1, nr_full=201 -> works offset=0, nr_full=202 -> works offset=1, nr_full=202 -> works offset=2, nr_full=202 -> works This means the operator is not uniquely identifying provenance. Multiple parameter combinations produce a 200×200 matrix. So we cannot conclude: Test4 used offset=0 only that: offset=0 is fully compatible with Test4 and offset=1 is not required. That's an important distinction. The most interesting consequence The earlier chain was: ψ(0,z)=0 ↓ offset=1 removed axis ↓ center hole may be numerical Your forensic sweep breaks that chain. Because now: ψ(0,z)=0 AND a perfectly valid operator exists with offset=0 So the center hole can no longer be casually explained away as: "The solver simply removed the axis." We need stronger evidence before making that claim. My updated confidence levels Very high confidence Test4 is a genuine axisymmetric cylindrical field. Axis exists in stored data. Field is localized. Ring maximum exists. Moderate confidence The central hole is a real feature of the stored field. The hole is not automatically explained by axis removal. Still uncertain Whether the hole is dynamically maintained. Whether the hole emerged from relaxation physics or boundary treatment. Whether Test4 is a true toroidal branch versus a ring-like relaxed state. What I think the forensic sweep accomplished It moved us from: "The hole may just be an artifact because the axis was removed." to "The axis is present, and a compatible operator exists without removing it." That is a meaningful upgrade in evidence. If this were my project I would now stop doing provenance analysis entirely. You've reached the point where the next meaningful question is not: How was Test4 created? but What does Test4 do? The neutral propagation audit is now the correct next experiment. However, I would modify the audit script before running it: build_radial_operator_reduced( r_grid, dr, offset=0, nr_full=200 ) because that is the first configuration demonstrated to be exactly compatible with the Test4 grid. Then run a short evolution (even t=5 or t=10 initially) and check: ring radius peak amplitude center density norm conservation energy conservation Those measurements will teach us more about Test4 than another week of notebook archaeology. --- This is an exceptional piece of forensic analysis. By unmasking that index degeneracy, you have exposed exactly how the matrix-building logic hides its physical assumptions under identical shapes.The fact that r_grid explicitly begins at exactly 0.0 is a massive foundational anchor. It completely eliminates any ambiguity about whether the field array contains the axis—it absolutely does.The Anatomy of the DegeneracyYour sweep revealed a subtle mathematical shift that explains why multiple configurations return a 200x200 matrix, but it also reveals that they are not physically identical. Let's look at how the cell-face positions shift between your two cleanest successes:Configuration A: offset=0, nr_full=200r_face length = 199r_face_reduced length = 199 (no elements dropped)For row i=0, flux_right uses r_face_reduced[0], which evaluates to 1 * dr.Physical Meaning: The first control volume interface sits exactly at $r = \Delta r$. Since r_grid[0] = 0.0, row 0 maps directly to the geometric axis, and its right-hand flux boundary is pinned perfectly at one radial step out.Configuration B: offset=1, nr_full=201r_face length = 200r_face_reduced length = 199 (first element dropped via slice [1:])For row i=0, flux_right uses r_face_reduced[0], which maps back to the unsliced r_face[1], evaluating to 2 * dr.Physical Meaning: The first control volume interface is pushed out to $r = 2\Delta r$. If row 0 still maps to r_grid[0] = 0.0, its immediate neighborhood has been structurally expanded, effectively modifying the geometric weighting near the origin.Because Configuration A (offset=0, nr_full=200) aligns the internal face index one-to-one with the grid array index without an artificial spatial translation, it stands as our most mathematically parsimonious baseline.The Dynamical Truth TestYou are entirely correct: notebook archaeology has yielded its maximum value. The definitive way to determine if offset=0 represents the true native architecture of Test4 is to stop looking at static shapes and start looking at conservation laws.If offset=0 matches the physical assumptions of the generator:Relative Norm and Energy Conservation: Both metrics should remain flatlines (within standard RK4 tolerances) across the neutral propagation run.Axis Stability: The boundary condition at ψ(0,z) = 0 should hold steady without generating unphysical high-frequency reflection or numerical blow-ups at the origin.If it is the wrong operator, the mismatch in coordinate mapping will manifest immediately as localized energy generation or dissipation along the $r=0$ line.The Updated Neutral Propagation Audit ScriptHere is the finalized production script. The configuration has been locked to offset=0 and nr_full=200, and the RK4 loop is fully restored. It retains all Google Colab automation, automated Drive mirroring, and local zip delivery.Python#!/usr/bin/env python3 """ test4_phase1_neutral_audit.py Neutral propagation audit for test_4_relaxed_state.npz Configured using verified forensic parameters: offset=0, nr_full=200. Fully automated for Google Colab workspace preservation and local sync. """ import os import json import numpy as np import matplotlib.pyplot as plt import scipy.sparse as sp # ============================================================ # ENVIRONMENT & AUTOMATION SETUP # ============================================================ IN_COLAB = False try: import google.colab from google.colab import drive, files IN_COLAB = True print("--> Google Colab environment detected.") except ImportError: print("--> Running in local Python environment.") if IN_COLAB: print("--> Mounting Google Drive for automatic backup...") drive.mount('/content/drive', force_remount=True) OUTPUT_DIR = "/content/drive/MyDrive/test4_phase1_neutral_audit" else: OUTPUT_DIR = "test4_phase1_neutral_audit" os.makedirs(OUTPUT_DIR, exist_ok=True) print(f"--> Target output directory set to: {OUTPUT_DIR}") from radial_operator_verified import build_radial_operator_reduced # ============================================================ # 1. CONFIGURATION # ============================================================ SOURCE_FILE = "test_4_relaxed_state.npz" DT = 0.01 T_MAX = 50.0 N_STEPS = int(T_MAX / DT) SAVE_EVERY = 50 SNAP_TIMES = [0.0, 25.0, 50.0] # ============================================================ # 2. LOAD STATE & RECONSTRUCT GEOMETRY # ============================================================ if not os.path.exists(SOURCE_FILE): raise FileNotFoundError(f"Missing {SOURCE_FILE}. Ensure it is uploaded to the workspace root.") data = np.load(SOURCE_FILE) psi0 = data["Psi_final"] pi0 = data["Pi_final"] r_grid = data["r_grid"] z_grid = data["z_grid"] Nr = len(r_grid) Nz = len(z_grid) # Handle flattened arrays safely psi0 = psi0.reshape(Nr, Nz) pi0 = pi0.reshape(Nr, Nz) dr = r_grid[1] - r_grid[0] dz = z_grid[1] - z_grid[0] print(f"Loaded Test4 Field Matrix: psi shape={psi0.shape}, pi shape={pi0.shape}") r_2d = r_grid[:, None] z_2d = z_grid[None, :] # ============================================================ # 3. BUILD LAPLACIAN OPERATOR (FORENSICALLY LOCKED) # ============================================================ print(f"--> Building radial operator with offset=0, nr_full={Nr}") radial_data = build_radial_operator_reduced(r_grid, dr, offset=0, nr_full=Nr) # Unpack tuple if operator returns auxiliary weight structures if isinstance(radial_data, tuple): L_r = radial_data[0] else: L_r = radial_data L_r = sp.csr_matrix(L_r) print(f"Radial operator generated successfully. Shape: {L_r.shape}") # Axial second derivative operator (Dirichlet boundary conditions) main = -2.0 * np.ones(Nz) off = 1.0 * np.ones(Nz - 1) L_z = sp.diags([off, main, off], offsets=[-1, 0, 1], shape=(Nz, Nz)) / (dz ** 2) L_z = sp.csr_matrix(L_z) def laplacian(psi_field): radial = L_r @ psi_field axial = psi_field @ L_z.T return radial + axial # ============================================================ # 4. PHYSICAL DIAGNOSTICS # ============================================================ def volume_element(): return 2.0 * np.pi * r_2d * dr * dz VOL = volume_element() def total_norm(psi_field): return float(np.sum(np.abs(psi_field) ** 2 * VOL)) def total_energy(psi_field, pi_field): dpsi_dr = np.gradient(psi_field, dr, axis=0) dpsi_dz = np.gradient(psi_field, dz, axis=1) kinetic = 0.5 * np.abs(pi_field) ** 2 grad2 = 0.5 * (np.abs(dpsi_dr) ** 2 + np.abs(dpsi_dz) ** 2) return float(np.sum((kinetic + grad2) * VOL)) def center_density(psi_field): iz = np.argmin(np.abs(z_grid - 0.0)) ir = np.argmin(np.abs(r_grid - 0.0)) return float(np.abs(psi_field[ir, iz]) ** 2) def peak_amplitude(psi_field): return float(np.max(np.abs(psi_field))) def ring_radius(psi_field): amp = np.abs(psi_field) idx = np.unravel_index(np.argmax(amp), amp.shape) return float(r_grid[idx[0]]) def com_z(psi_field): density = np.abs(psi_field) ** 2 num = np.sum(density * VOL * z_2d) den = np.sum(density * VOL) return float(num / den) if den != 0 else 0.0 # ============================================================ # 5. TIME INTEGRATOR (RK4) # ============================================================ def rhs(psi_field, pi_field): return pi_field, laplacian(psi_field) def rk4_step(psi_field, pi_field, dt): k1_psi, k1_pi = rhs(psi_field, pi_field) k2_psi, k2_pi = rhs(psi_field + 0.5 * dt * k1_psi, pi_field + 0.5 * dt * k1_pi) k3_psi, k3_pi = rhs(psi_field + 0.5 * dt * k2_psi, pi_field + 0.5 * dt * k2_pi) k4_psi, k4_pi = rhs(psi_field + dt * k3_psi, pi_field + dt * k3_pi) psi_next = psi_field + (dt / 6.0) * (k1_psi + 2 * k2_psi + 2 * k3_psi + k4_psi) pi_next = pi_field + (dt / 6.0) * (k1_pi + 2 * k2_pi + 2 * k3_pi + k4_pi) return psi_next, pi_next # ============================================================ # 6. EVOLUTION LOOP # ============================================================ psi = psi0.copy() pi = pi0.copy() times, norms, energies, center_densities, peak_amps, ring_radii, com_zs = [], [], [], [], [], [], [] snapshots = {} def record(t_curr, psi_field, pi_field): times.append(t_curr) norms.append(total_norm(psi_field)) energies.append(total_energy(psi_field, pi_field)) center_densities.append(center_density(psi_field)) peak_amps.append(peak_amplitude(psi_field)) ring_radii.append(ring_radius(psi_field)) com_zs.append(com_z(psi_field)) if any(abs(t_curr - ts) < 1e-8 for ts in SNAP_TIMES): snapshots[round(t_curr, 8)] = {"amp": np.abs(psi_field).copy(), "phase": np.angle(psi_field).copy()} t = 0.0 print("\n--> Initiating neutral propagation loop...") record(t, psi, pi) for step in range(1, N_STEPS + 1): psi, pi = rk4_step(psi, pi, DT) t = step * DT if step % SAVE_EVERY == 0 or any(abs(t - ts) < DT / 2 for ts in SNAP_TIMES): record(t, psi, pi) if step % (SAVE_EVERY * 10) == 0: print(f" t={t:5.2f} | peak={peak_amps[-1]:.4f} | ring_r={ring_radii[-1]:.3f} | norm_ratio={norms[-1]/norms[0]:.6f}") print("--> Propagation cycle complete.") # ============================================================ # 7. EXPORTS & VISUALIZATION # ============================================================ history = { "source_file": SOURCE_FILE, "dt": DT, "t_max": T_MAX, "times": times, "norms": norms, "energies": energies, "center_densities": center_densities, "peak_amplitudes": peak_amps, "ring_radii": ring_radii, "com_z": com_zs } with open(os.path.join(OUTPUT_DIR, "test4_propagation_history.json"), "w") as f: json.dump(history, f, indent=2) with open(os.path.join(OUTPUT_DIR, "test4_propagation_report.txt"), "w") as f: f.write("Test4 Neutral Propagation Structural Audit Report\n") f.write("=================================================\n") f.write(f"Operator Params : offset=0, nr_full={Nr}\n") f.write(f"Initial Norm : {norms[0]:.6e} -> Final Norm: {norms[-1]:.6e}\n") f.write(f"Initial Energy : {energies[0]:.6e} -> Final Energy: {energies[-1]:.6e}\n") f.write(f"Ring Radius : {ring_radii[0]:.4f} -> Final Ring Radius: {ring_radii[-1]:.4f}\n") # Generate Snapshot Spatial Profiles for ts in SNAP_TIMES: key = round(ts, 8) if key in snapshots: fig, axes = plt.subplots(1, 2, figsize=(12, 4)) im0 = axes[0].pcolormesh(z_grid, r_grid, snapshots[key]["amp"], shading="auto", cmap="viridis") plt.colorbar(im0, ax=axes[0]) axes[0].set_title(f"|Psi| Spatial Amplitude (t={ts:.1f})") axes[0].set_ylabel("r") axes[0].set_xlabel("z") im1 = axes[1].pcolormesh(z_grid, r_grid, snapshots[key]["phase"], shading="auto", cmap="twilight") plt.colorbar(im1, ax=axes[1]) axes[1].set_title(f"arg(Psi) Phase Profile (t={ts:.1f})") axes[1].set_ylabel("r") axes[1].set_xlabel("z") plt.tight_layout() plt.savefig(os.path.join(OUTPUT_DIR, f"test4_spatial_t{int(ts)}.png"), dpi=150) plt.close(fig) # Generate Dynamical Time Series Summaries fig, axes = plt.subplots(2, 3, figsize=(15, 8)) axes[0, 0].plot(times, peak_amps, color='b') axes[0, 0].set_title("Peak Amplitude Evolution") axes[0, 1].plot(times, ring_radii, color='g') axes[0, 1].set_title("Toroidal Ring Radius") axes[0, 2].plot(times, center_densities, color='r') axes[0, 2].set_title("Core Axis Density rho(0,z)") axes[1, 0].plot(times, np.array(norms) / norms[0], color='purple') axes[1, 0].set_title("Relative Norm Conservation") axes[1, 1].plot(times, np.array(energies) / energies[0], color='orange') axes[1, 1].set_title("Relative Energy Conservation") axes[1, 2].plot(times, com_zs, color='brown') axes[1, 2].set_title("Axial Center of Mass (Z)") plt.tight_layout() plt.savefig(os.path.join(OUTPUT_DIR, "test4_dynamical_trends.png"), dpi=150) plt.close(fig) print(f"\n--> Data archived to Google Drive: {OUTPUT_DIR}") if IN_COLAB: print("\n--> Compiling zip archive for automated machine download...") os.system(f"zip -r /content/test4_audit_results.zip {OUTPUT_DIR}") print("--> Delivering package to local browser...") files.download("/content/test4_audit_results.zip") Let's execute this. If the norm ratio and energy ratio hold flat, we have successfully synchronized the field with its exact mathematical operator. Let me know what the time-series logs show as the integration updates! --->>> I would not execute that script yet. Not because the reasoning is bad—in fact the forensic reasoning is quite good—but because it quietly bakes in several assumptions that have not been established. What the script gets right The strongest part is this: It correctly uses the forensic result that offset=0, nr_full=200 is the simplest configuration matching the Test4 grid. It separates the scientific question ("what does Test4 do?") from the provenance question ("where did Test4 come from?"). It proposes a clean baseline evolution rather than immediately adding boosts, collisions, κ terms, etc. Those are all good decisions. The biggest problem The script assumes: ∂ t ​ Ψ=Π ∂ t ​ Π=∇ 2 Ψ which is the free linear wave equation. Nothing you've recovered proves that Test4 was generated from the free wave equation. In fact, almost everything you've shown suggests the opposite. Earlier evidence pointed toward: relaxation nonlinear potentials μ, λ, κ terms damping stages stationary-state generation A relaxed localized object generally exists because some balance mechanism is present. If you remove that mechanism and evolve under ∂ t 2 ​ Ψ=∇ 2 Ψ then dispersion is the expected outcome. So if Test4 dissolves under this audit: that does not prove it was an artifact. It only proves it is not a free-wave soliton. Those are very different conclusions. Second problem The script says: "If center fills in → toroidal interpretation weakens dramatically" That is only partly true. Suppose Test4 was created using: offset=1 constrained axis nonlinear relaxation and you now evolve it using: offset=0 free wave dynamics Then the center filling could simply be caused by changing the operator. You would be testing two changes simultaneously: operator change PDE change That makes interpretation harder. Third problem The energy diagnostic may be misleading. The script computes: kinetic = 0.5 * |pi|² grad2 = 0.5 * (|dpsi_dr|² + |dpsi_dz|²) That is the free-wave Hamiltonian. If the original system contained: V(Ψ) terms, then you're not measuring the original energy. You're measuring energy in a different theory. Energy conservation may still be useful numerically, but not physically. Fourth problem The phase variable may not be meaningful. The script saves: np.angle(psi) But we don't yet know whether: ψ is genuinely complex, ψ is a real field stored in a complex array, ψ is a reduced envelope variable. Until that is established, phase plots may be interesting visually but not diagnostic. What I would change I'd split the project into two audits. Audit A — Operator Consistency Use: Test4 offset=0 operator and check: matrix stability conservation numerical behavior This is a numerical audit. Do not draw physics conclusions yet. Audit B — Structural Persistence Measure: ring radius peak amplitude center density norm under free evolution. But interpret results as: "How does Test4 behave as an initial condition of the free cylindrical wave equation?" not "Is Test4 real?" Those are different questions. The most important insight from your forensic sweep Honestly, it isn't the offset choice. It's this: r_grid[0] = 0.0 combined with offset=1 requires nr_full=201 That tells us something extremely important: The operator and the stored grid are not naturally aligned. The fact that you need artificial values like: offset=1, nr_full=201 offset=2, nr_full=202 to recover a 200×200 matrix is exactly the kind of thing that suggests the original generator may have used: ghost cells, excluded-axis discretizations, reduced operators, hidden coordinate transforms. That is the strongest forensic clue you've found so far. Bottom line The proposed script is a good numerical experiment, but I would reject the claim that: "If it disperses, Test4 was probably a relaxation artifact." That conclusion is not supported. A more defensible interpretation is: "This experiment tests whether Test4 behaves as a coherent structure under free cylindrical wave evolution using the offset=0 operator." That's a valuable test. It just doesn't answer the full soliton-versus-artifact question by itself. --- /content/summary_10_8A_kappa_0.100.json: kappa = 0.1 /content/summary_10_8A_kappa_0.110.json: kappa = 0.11 /content/summary_10_8A_kappa_0.090.json: kappa = 0.09 /content/summary_10_8A_kappa_0.080.json: kappa = 0.08 /content/summary_10_8A_kappa_0.120.json: kappa = 0.12 /content/section11_2_core_relaxation_results.json: dt = 0.001 /content/DOWNLOAD_ME_boosted_hollow_torus_RK4_20260526_020509.npz: mu = -1.0 /content/DOWNLOAD_ME_boosted_hollow_torus_RK4_20260526_020509.npz: lam = 0.4 /content/DOWNLOAD_ME_boosted_hollow_torus_RK4_20260526_020509.npz: kappa = 0.2 /content/test_2_boosted_hollow_torus_20260525_233851.npz: mu = -1.0 /content/test_2_boosted_hollow_torus_20260525_233851.npz: lam = 0.4 /content/test_2_boosted_hollow_torus_20260525_233851.npz: kappa = 0.2 /content/DOWNLOAD_ME_boosted_hollow_torus_20260525_233851.npz: mu = -1.0 /content/DOWNLOAD_ME_boosted_hollow_torus_20260525_233851.npz: lam = 0.4 /content/DOWNLOAD_ME_boosted_hollow_torus_20260525_233851.npz: kappa = 0.2 /content/test_0A_hollow_torus_m1_20260525_220216.npz: mu = -1.0 /content/test_0A_hollow_torus_m1_20260525_220216.npz: lam = 0.4 /content/test_0A_hollow_torus_m1_20260525_220216.npz: kappa = 0.2 /content/test_0A_soliton_20260523_195958.npz: mu = -1.0 /content/test_0A_soliton_20260523_195958.npz: lam = 0.4 /content/test_0A_soliton_20260523_195958.npz: kappa = 0.2 /content/test_0A_toroidal_m1_20260525_210939.npz: mu = -1.0 /content/test_0A_toroidal_m1_20260525_210939.npz: lam = 0.4 /content/test_0A_toroidal_m1_20260525_210939.npz: kappa = 0.2 /content/test_0A_soliton_expanded_20260525_105922.npz: mu = -1.0 /content/test_0A_soliton_expanded_20260525_105922.npz: lam = 1.0 /content/test_0A_soliton_expanded_20260525_105922.npz: kappa = 1.0 /content/HOLLOW_TORUS_DOWNLOAD_ME.npz: mu = -1.0 /content/HOLLOW_TORUS_DOWNLOAD_ME.npz: lam = 0.4 /content/HOLLOW_TORUS_DOWNLOAD_ME.npz: kappa = 0.2 /content/test_0A_soliton_20260523_195958 (2).npz: mu = -1.0 /content/test_0A_soliton_20260523_195958 (2).npz: lam = 0.4 /content/test_0A_soliton_20260523_195958 (2).npz: kappa = 0.2 /content/test_0A_soliton_expanded_20260525_010611.npz: mu = -1.0 /content/test_0A_soliton_expanded_20260525_010611.npz: lam = 1.0 /content/test_0A_soliton_expanded_20260525_010611.npz: kappa = 1.0 DEEPSEEK — BREAKTHROUGH: PROVENANCE PARTIALLY RECOVERED The Search Results Reveal Something Important The parameter search across your workspace has successfully identified mu, lam, kappa values stored inside multiple .npz files. What Was Found File mu lam kappa DOWNLOAD_ME_boosted_hollow_torus_RK4_20260526_020509.npz -1.0 0.4 0.2 test_2_boosted_hollow_torus_20260525_233851.npz -1.0 0.4 0.2 test_0A_hollow_torus_m1_20260525_220216.npz -1.0 0.4 0.2 test_0A_soliton_20260523_195958.npz -1.0 0.4 0.2 test_0A_toroidal_m1_20260525_210939.npz -1.0 0.4 0.2 HOLLOW_TORUS_DOWNLOAD_ME.npz -1.0 0.4 0.2 test_0A_soliton_expanded_20260525_105922.npz -1.0 1.0 1.0 test_0A_soliton_expanded_20260525_010611.npz -1.0 1.0 1.0 The Critical Observation Test4 (test_4_relaxed_state.npz) did NOT appear in this search. This means: Test4 does NOT contain stored parameter metadata OR the search pattern missed it But more importantly: The parameters mu = -1.0, lam = 0.4, kappa = 0.2 appear repeatedly across multiple toroidal/hollow torus files. These are the likely parameters that generated the toroidal family — which includes the precursor to Test4. What This Means For Test4 Inference Confidence Test4 likely came from the same pipeline as these toroidal files High (same grid structure, similar ring morphology) The PDE parameters for Test4 are probably mu = -1.0, lam = 0.4, kappa = 0.2 Moderate-High The expanded variants (lam=1.0, kappa=1.0) are a different branch High (different parameters, different file naming) The Missing Piece: test_4_relaxed_state.npz Itself Test4 does not have parameters stored inside it. But its siblings (the toroidal files) do. Therefore: The most reasonable inference is that Test4 was generated with: text mu = -1.0 lam = 0.4 kappa = 0.2 v = 1.0 S_max = 2.0 (from radial_operator_verified.py context) What This Means For Test B (Propagation Audit) We now have a plausible parameter set to use in a neutral propagation test: Parameter Value Source μ -1.0 From sibling toroidal files λ 0.4 From sibling toroidal files κ 0.2 From sibling toroidal files v 1.0 Default from solver S_max 2.0 From radial operator context This is not proven, but it is now the strongest hypothesis. Next Step: Verify By Running A Short Test We can test whether Test4 behaves consistently with these parameters by: Loading Test4 Evolving it for a short time (t=0→10) using these parameters Checking if it maintains its ring structure If it holds, the parameters are likely correct. If it immediately collapses or disperses, the parameters are wrong. One-Sentence Summary The parameter search across your workspace found mu=-1.0, lam=0.4, kappa=0.2 stored in multiple toroidal files that are siblings of Test4 — while Test4 itself lacks metadata, the strong inference is that it was generated with these parameters, providing a testable hypothesis for the neutral propagation audit. --- Yes. This changes the situation significantly, and it exposes a flaw in several of the earlier AI conclusions—including some of my own cautionary assumptions. What DeepSeek's source-level analysis establishes The actual operator code shows: r_face = np.arange(1, nr_full) * dr r_face_reduced = r_face[offset:] and w_r = r_grid * dr with return W_r_inv @ M, W_r This is a finite-volume radial operator using staggered cell faces. That means: offset is not "remove the axis DOF." offset is not "force ψ(0)=0." offset is not automatically evidence of a toroidal boundary condition. Instead: offset changes where the first flux face is located. It modifies the origin treatment. It changes the radial stencil near the axis. Those are related concepts, but they are not the same thing. The biggest correction Earlier reasoning went: offset=1 exists therefore axis removed therefore hole likely numerical That chain is no longer defensible. Because your forensic sweep found: offset=0, nr_full=200 works perfectly. And: r_grid[0] = 0 is explicitly present. So the field itself definitely contains the axis. Therefore: The hole cannot be dismissed simply because offset=1 appears in the operator. That conclusion is now very strong. But DeepSeek goes slightly too far in the opposite direction DeepSeek says: The hole can no longer be dismissed as "offset removed the axis" Correct. But that does not mean: therefore the hole is physical. We still don't know that. The actual state of knowledge is: Claim Status Axis exists in grid Proven Offset ≠ automatic axis removal Proven Hole exists in Test4 Proven Ring maximum exists Proven Hole produced dynamically Unknown Hole produced by relaxation geometry Unknown Toroidal soliton Unproven The real forensic bombshell Your sweep found something more important than the offset discussion. Look at this: offset=0, nr_full=200 SUCCESS offset=1, nr_full=201 SUCCESS offset=2, nr_full=202 SUCCESS That means the operator has a structural degeneracy. Multiple physical interpretations produce the same matrix size: 200 × 200 Therefore: matrix dimension matching tells us almost nothing about the original provenance. That is a huge result. Several AI analyses were implicitly treating: 200×200 match as evidence for a specific operator. Your sweep disproves that. What I think is the strongest conclusion now If I synthesize: your forensic sweep the source code the Test4 density map the radial peak at ~1.9 the localization diagnostics I would state it like this: Test4 is unquestionably a localized axisymmetric relaxed configuration with a genuine off-axis density maximum. The grid explicitly contains the axis, and the presence of offset=1 in the verified radial operator is not sufficient to explain away the central hole. However, the original boundary treatment and relaxation geometry remain unknown, so the current evidence does not yet distinguish between a dynamically supported toroidal branch and a ring-shaped relaxed state produced by a particular cylindrical discretization. That is where I would put the evidence today. About the proposed "neutral propagation audit" This is where I think your critique was absolutely correct. The proposed script has a serious scientific problem: It evolves ∂ t ​ Ψ=Π ∂ t ​ Π=∇ 2 Ψ which assumes: V(Ψ)=0 But Test4 almost certainly did not originate from a pure linear wave equation. If Test4 came from: κ-coupling nonlinear self-interaction relaxation flow monad-field potential constrained energy minimization then removing those terms guarantees the structure will change. So: Test4 disperses would not imply: Test4 was an artifact It would only imply: Test4 is not a solution of the free wave equation. Those are completely different statements. Your objection there is mathematically correct. What I would do next I would split the investigation exactly as you suggested: Audit A — Operator Forensics No evolution. Compare: offset=0 offset=1 offset=2 on analytic test functions. Inspect: symmetry origin behavior flux conservation boundary rows This isolates geometry. Audit B — Dynamic Persistence Only after Audit A. And preferably using the closest recovered evolution law to the original Test4 framework—not a stripped-down free wave equation. That gives you a meaningful answer about whether Test4 is a robust structure within its native physics. My overall assessment of DeepSeek's source-code analysis: Technical accuracy: ~9/10 The strongest contribution is that it demolishes the simplistic claim that offset=1 automatically means "axis removed therefore hole is fake." The remaining uncertainty is exactly what generated Test4 and what evolution equation Test4 was actually relaxing under. Until that is known, neither the "artifact" camp nor the "toroidal soliton" camp has a decisive win.