diff --git a/.gitattributes b/.gitattributes
index 380c5c1923de8aba2f261d9f5d40a3d4d7cb3e52..4f0f86c6d2805478e381ae946a4fedc142b17283 100644
--- a/.gitattributes
+++ b/.gitattributes
@@ -69510,3 +69510,5 @@ system/decomposeParDict filter=lfs diff=lfs merge=lfs -text
 system/fvSchemes filter=lfs diff=lfs merge=lfs -text
 system/fvSolution filter=lfs diff=lfs merge=lfs -text
 three_regime_overlap_v1/subsets/hopf/reference_vtk/centeredSquare_CN09_graded_Re100_internal_final_reference.vtk filter=lfs diff=lfs merge=lfs -text
+three_regime_overlap_v1/subsets/periodic/reference_vtk/centeredSquare_CN09_graded_Re100_internal_final_reference.vtk filter=lfs diff=lfs merge=lfs -text
+three_regime_overlap_v1/subsets/steady/reference_vtk/centeredSquare_CN09_graded_Re100_internal_final_reference.vtk filter=lfs diff=lfs merge=lfs -text
diff --git a/three_regime_overlap_v1/subsets/periodic/pod/rom/rank999_ru28_rp26/semi_intrusive_galerkin_tensors_Re50_150_N100_rank999_ru28_rp26_compact.npz b/three_regime_overlap_v1/subsets/periodic/pod/rom/rank999_ru28_rp26/semi_intrusive_galerkin_tensors_Re50_150_N100_rank999_ru28_rp26_compact.npz
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@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:c3b0996906b83140e272d371c52644b285e5a68e7cc6ebc7f4fe1267dc19471a
+size 717594
diff --git a/three_regime_overlap_v1/subsets/periodic/pod/rom/rank99_ru13_rp12/ROM_BUILD_REPORT_Re50_150_N100_rank99_ru13_rp12.md b/three_regime_overlap_v1/subsets/periodic/pod/rom/rank99_ru13_rp12/ROM_BUILD_REPORT_Re50_150_N100_rank99_ru13_rp12.md
new file mode 100644
index 0000000000000000000000000000000000000000..155f05be86824374a16d29798696810c54bc3a43
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+++ b/three_regime_overlap_v1/subsets/periodic/pod/rom/rank99_ru13_rp12/ROM_BUILD_REPORT_Re50_150_N100_rank99_ru13_rp12.md
@@ -0,0 +1,50 @@
+# CenteredSquare ROM Build Report - rank99
+
+- Dataset: `/home/ray/Desktop/centeredSquare/three_regime_overlap_v1/subsets/periodic`
+- Output dir: `/home/ray/Desktop/centeredSquare/three_regime_overlap_v1/subsets/periodic/pod/rom/rank99_ru13_rp12`
+- Ranks: `ru=13`, `rp=12`
+- Re points: `27`
+- Cells: `9400`
+- Derivative backend: PyVista `compute_derivative()` on reference VTK cell data
+- Reference VTK: `/home/ray/Desktop/centeredSquare/three_regime_overlap_v1/subsets/periodic/reference_vtk/centeredSquare_CN09_graded_Re100_internal_final_reference.vtk`
+- VTK cells/points: `9400` / `19306`
+- Max VTK/POD cell-center delta: `1.097595e-06`
+
+## Velocity ROM
+
+```text
+da/dt = c(Re) + A(Re) a + H(a,a) + P b
+```
+
+- `G_u` shape: `[13, 13]`
+- `c_all` shape: `[27, 13]`
+- `A_all` shape: `[27, 13, 13]`
+- `H` shape: `[13, 13, 13]`
+- `P` shape: `[13, 12]`
+- `cond(G_u)`: `1.000000e+00`
+- `max|G_u-I|`: `1.768155e-09`
+- `||H||_F`: `5.757600e+00`
+- `||P||_F`: `1.272682e-01`
+
+## Pressure Poisson Surrogate
+
+```text
+L b(t) = c^p(Re) + A^p(Re) a(t) + H^p(a(t),a(t))
+b(t) = c_tilde(Re) + A_tilde(Re) a(t) + H_tilde(a(t),a(t))
+```
+
+- `L` shape: `[12, 12]`
+- `H_p` shape: `[12, 13, 13]`
+- `H_tilde` shape: `[12, 13, 13]`
+- `rank(L)`: `12/12`
+- `eig(L) min/max`: `-9.641928e+00` / `-1.207659e-01`
+- `rel ||L H_tilde-H_p||`: `1.162153e-15`
+
+## Files
+
+- `semi_intrusive_galerkin_tensors_Re50_150_N100_rank99_ru13_rp12_compact.npz`
+- `pressure_poisson_surrogate_tensors_Re50_150_N100_rank99_ru13_rp12.npz`
+- `pod_rank_pack_Re50_150_N100_rank99_ru13_rp12.npz`
+- `manifest.json`
+
+- Finite arrays: `True`
diff --git a/three_regime_overlap_v1/subsets/periodic/pod/rom/rank99_ru13_rp12/manifest.json b/three_regime_overlap_v1/subsets/periodic/pod/rom/rank99_ru13_rp12/manifest.json
new file mode 100644
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--- /dev/null
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@@ -0,0 +1,57 @@
+{
+  "A_all_shape": [
+    27,
+    13,
+    13
+  ],
+  "G_u_shape": [
+    13,
+    13
+  ],
+  "H_p_shape": [
+    12,
+    13,
+    13
+  ],
+  "H_shape": [
+    13,
+    13,
+    13
+  ],
+  "H_tilde_shape": [
+    12,
+    13,
+    13
+  ],
+  "L_shape": [
+    12,
+    12
+  ],
+  "P_shape": [
+    13,
+    12
+  ],
+  "c_all_shape": [
+    27,
+    13
+  ],
+  "cond_G_u": 1.0000000055822922,
+  "dataset": "/home/ray/Desktop/centeredSquare/three_regime_overlap_v1/subsets/periodic",
+  "eig_L_max": -0.12076588035917687,
+  "eig_L_min": -9.641927870469912,
+  "finite": true,
+  "label": "rank99",
+  "max_abs_G_u_minus_I": 1.7681548447612272e-09,
+  "norm_H_fro": 5.7576001094848515,
+  "norm_P_fro": 0.12726822886271236,
+  "num_cells": 9400,
+  "num_re": 27,
+  "output_dir": "/home/ray/Desktop/centeredSquare/three_regime_overlap_v1/subsets/periodic/pod/rom/rank99_ru13_rp12",
+  "pod_rank_pack": "/home/ray/Desktop/centeredSquare/three_regime_overlap_v1/subsets/periodic/pod/rom/rank99_ru13_rp12/pod_rank_pack_Re50_150_N100_rank99_ru13_rp12.npz",
+  "pressure_H_tilde_relative_residual": 1.162153444099533e-15,
+  "pressure_tensor_file": "/home/ray/Desktop/centeredSquare/three_regime_overlap_v1/subsets/periodic/pod/rom/rank99_ru13_rp12/pressure_poisson_surrogate_tensors_Re50_150_N100_rank99_ru13_rp12.npz",
+  "rank_L": 12,
+  "rp": 12,
+  "ru": 13,
+  "velocity_tensor_file": "/home/ray/Desktop/centeredSquare/three_regime_overlap_v1/subsets/periodic/pod/rom/rank99_ru13_rp12/semi_intrusive_galerkin_tensors_Re50_150_N100_rank99_ru13_rp12_compact.npz"
+}
diff --git a/three_regime_overlap_v1/subsets/periodic/pod/rom/rank99_ru13_rp12/pod_rank_pack_Re50_150_N100_rank99_ru13_rp12.npz b/three_regime_overlap_v1/subsets/periodic/pod/rom/rank99_ru13_rp12/pod_rank_pack_Re50_150_N100_rank99_ru13_rp12.npz
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+version https://git-lfs.github.com/spec/v1
+oid sha256:40882ce91b73d7e55a1b7e1105938f932d3ab2fcbaf9bdfd3b0f979cae1fc2e3
+size 1850134
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diff --git a/three_regime_overlap_v1/subsets/periodic/pod/validation_projection.json b/three_regime_overlap_v1/subsets/periodic/pod/validation_projection.json
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--- /dev/null
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+{
+  "cases": [
+    {
+      "Re": 99.0,
+      "finite": true,
+      "pressure_weighted_rel_l2": 0.029076308189194724,
+      "tag": "Re099p000000",
+      "velocity_weighted_rel_l2": 0.030274491037997234
+    },
+    {
+      "Re": 101.5,
+      "finite": true,
+      "pressure_weighted_rel_l2": 0.03183054048030602,
+      "tag": "Re101p500000",
+      "velocity_weighted_rel_l2": 0.03210985146385977
+    },
+    {
+      "Re": 110.344827586207,
+      "finite": true,
+      "pressure_weighted_rel_l2": 0.031673651419743466,
+      "tag": "Re110p344828",
+      "velocity_weighted_rel_l2": 0.0274536011904289
+    },
+    {
+      "Re": 125.862068965517,
+      "finite": true,
+      "pressure_weighted_rel_l2": 0.03145744479483429,
+      "tag": "Re125p862069",
+      "velocity_weighted_rel_l2": 0.02975572602379019
+    },
+    {
+      "Re": 141.379310344828,
+      "finite": true,
+      "pressure_weighted_rel_l2": 0.02750998771454404,
+      "tag": "Re141p379310",
+      "velocity_weighted_rel_l2": 0.03015597543278394
+    },
+    {
+      "Re": 150.0,
+      "finite": true,
+      "pressure_weighted_rel_l2": 0.04513396188047213,
+      "tag": "Re150p000000",
+      "velocity_weighted_rel_l2": 0.04869137741530765
+    }
+  ]
+}
diff --git a/three_regime_overlap_v1/subsets/periodic/pod/weighted_pod_pressure.npz b/three_regime_overlap_v1/subsets/periodic/pod/weighted_pod_pressure.npz
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+version https://git-lfs.github.com/spec/v1
+oid sha256:efbf4f01f72103724dfca75f68514268a09119b7d0556739cbbeddf29c03ab43
+size 2225233
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+version https://git-lfs.github.com/spec/v1
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diff --git a/three_regime_overlap_v1/subsets/periodic/reference_vtk/README.md b/three_regime_overlap_v1/subsets/periodic/reference_vtk/README.md
new file mode 100644
index 0000000000000000000000000000000000000000..4a8017ff069bb3f10daf8fd8556133d7896edf98
--- /dev/null
+++ b/three_regime_overlap_v1/subsets/periodic/reference_vtk/README.md
@@ -0,0 +1,10 @@
+# Reference VTK
+
+This directory intentionally keeps a single final-time internal-field VTK file for ROM derivative/operator construction.
+
+- Source case: /home/ray/Desktop/centeredSquare/runs/cn09_graded_20260708_000142/Re100
+- Source VTK: /home/ray/Desktop/centeredSquare/runs/cn09_graded_20260708_000142/Re100/VTK/Re100_25000.vtk
+- Copied file: /home/ray/Desktop/centeredSquare/dataset_Re50_150_N100_npz/reference_vtk/centeredSquare_CN09_graded_Re100_internal_final_reference.vtk
+- Mesh/numerics: centeredSquare CN09 + compatible graded mesh
+- Contents: OpenFOAM internal mesh VTK with U/p fields from the reference Re100 run
+- Note: formal NPZ production outputs remain in cases_npz/probes; this reference VTK is intentionally kept separately.
diff --git a/three_regime_overlap_v1/subsets/periodic/reference_vtk/centeredSquare_CN09_graded_Re100_internal_final_reference.vtk b/three_regime_overlap_v1/subsets/periodic/reference_vtk/centeredSquare_CN09_graded_Re100_internal_final_reference.vtk
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@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
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diff --git a/three_regime_overlap_v1/subsets/steady/RUN_SUMMARY.json b/three_regime_overlap_v1/subsets/steady/RUN_SUMMARY.json
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--- /dev/null
+++ b/three_regime_overlap_v1/subsets/steady/RUN_SUMMARY.json
@@ -0,0 +1,41 @@
+{
+  "domain_count": 69,
+  "heldout_Re": [
+    60.0,
+    85.0,
+    95.1,
+    95.3
+  ],
+  "heldout_count": 4,
+  "pod": [
+    {
+      "analysis_rank": 256,
+      "captured": 0.9991934666163927,
+      "field": "velocity",
+      "path": "/home/ray/Desktop/centeredSquare/three_regime_overlap_v1/subsets/steady/pod/weighted_pod_velocity.npz",
+      "rank_99": 3,
+      "rank_999": 5,
+      "stored_modes": 5
+    },
+    {
+      "analysis_rank": 256,
+      "captured": 0.9994876533554901,
+      "field": "pressure",
+      "path": "/home/ray/Desktop/centeredSquare/three_regime_overlap_v1/subsets/steady/pod/weighted_pod_pressure.npz",
+      "rank_99": 2,
+      "rank_999": 4,
+      "stored_modes": 4
+    }
+  ],
+  "regime": "steady",
+  "train_count": 60,
+  "train_snapshots": 7560,
+  "validation_Re": [
+    55.0,
+    75.0,
+    90.0,
+    94.5,
+    95.25
+  ],
+  "validation_count": 5
+}
diff --git a/three_regime_overlap_v1/subsets/steady/RUN_SUMMARY.md b/three_regime_overlap_v1/subsets/steady/RUN_SUMMARY.md
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@@ -0,0 +1,45 @@
+# CenteredSquare regime subset
+
+```json
+{
+  "regime": "steady",
+  "domain_count": 69,
+  "train_count": 60,
+  "validation_count": 5,
+  "heldout_count": 4,
+  "train_snapshots": 7560,
+  "validation_Re": [
+    55.0,
+    75.0,
+    90.0,
+    94.5,
+    95.25
+  ],
+  "heldout_Re": [
+    60.0,
+    85.0,
+    95.1,
+    95.3
+  ],
+  "pod": [
+    {
+      "field": "velocity",
+      "path": "/home/ray/Desktop/centeredSquare/three_regime_overlap_v1/subsets/steady/pod/weighted_pod_velocity.npz",
+      "rank_99": 3,
+      "rank_999": 5,
+      "stored_modes": 5,
+      "analysis_rank": 256,
+      "captured": 0.9991934666163927
+    },
+    {
+      "field": "pressure",
+      "path": "/home/ray/Desktop/centeredSquare/three_regime_overlap_v1/subsets/steady/pod/weighted_pod_pressure.npz",
+      "rank_99": 2,
+      "rank_999": 4,
+      "stored_modes": 4,
+      "analysis_rank": 256,
+      "captured": 0.9994876533554901
+    }
+  ]
+}
+```
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diff --git a/three_regime_overlap_v1/subsets/steady/pod/build_rom.log b/three_regime_overlap_v1/subsets/steady/pod/build_rom.log
new file mode 100644
index 0000000000000000000000000000000000000000..d44cdb7c74b64b6a793154366bf68e203f9d4636
--- /dev/null
+++ b/three_regime_overlap_v1/subsets/steady/pod/build_rom.log
@@ -0,0 +1,146 @@
+/usr/lib/python3/dist-packages/paramiko/pkey.py:82: CryptographyDeprecationWarning: TripleDES has been moved to cryptography.hazmat.decrepit.ciphers.algorithms.TripleDES and will be removed from cryptography.hazmat.primitives.ciphers.algorithms in 48.0.0.
+  "cipher": algorithms.TripleDES,
+/usr/lib/python3/dist-packages/paramiko/transport.py:261: CryptographyDeprecationWarning: TripleDES has been moved to cryptography.hazmat.decrepit.ciphers.algorithms.TripleDES and will be removed from cryptography.hazmat.primitives.ciphers.algorithms in 48.0.0.
+  "class": algorithms.TripleDES,
+{
+  "bundle": "/home/ray/Desktop/centeredSquare/three_regime_overlap_v1/subsets/steady/pod/rom/centeredSquare_Re50_150_N100_rank99_rank999_ROM_and_POD.tar.gz",
+  "dataset": "/home/ray/Desktop/centeredSquare/three_regime_overlap_v1/subsets/steady",
+  "derivative_operator": {
+    "backend": "pyvista_vtk_compute_derivative_cell_data",
+    "center_atol": 5e-06,
+    "max_center_delta": 1.097595216492664e-06,
+    "n_cells": 9400,
+    "n_points": 19306,
+    "preference": "cell",
+    "reference_vtk": "/home/ray/Desktop/centeredSquare/three_regime_overlap_v1/subsets/steady/reference_vtk/centeredSquare_CN09_graded_Re100_internal_final_reference.vtk"
+  },
+  "output_root": "/home/ray/Desktop/centeredSquare/three_regime_overlap_v1/subsets/steady/pod/rom",
+  "rank_specs": [
+    {
+      "label": "rank99",
+      "rp": 2,
+      "ru": 3
+    },
+    {
+      "label": "rank999",
+      "rp": 4,
+      "ru": 5
+    }
+  ],
+  "summaries": [
+    {
+      "A_all_shape": [
+        60,
+        3,
+        3
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+        3,
+        3
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+        2,
+        3,
+        3
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+      "H_shape": [
+        3,
+        3,
+        3
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+      "H_tilde_shape": [
+        2,
+        3,
+        3
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+        2,
+        2
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+      "P_shape": [
+        3,
+        2
+      ],
+      "c_all_shape": [
+        60,
+        3
+      ],
+      "cond_G_u": 1.0000000014990225,
+      "dataset": "/home/ray/Desktop/centeredSquare/three_regime_overlap_v1/subsets/steady",
+      "eig_L_max": -0.06521244606446581,
+      "eig_L_min": -2.6958786766478964,
+      "finite": true,
+      "label": "rank99",
+      "max_abs_G_u_minus_I": 8.446097155001553e-10,
+      "norm_H_fro": 0.04250069940953678,
+      "norm_P_fro": 0.1005150244322599,
+      "num_cells": 9400,
+      "num_re": 60,
+      "output_dir": "/home/ray/Desktop/centeredSquare/three_regime_overlap_v1/subsets/steady/pod/rom/rank99_ru3_rp2",
+      "pod_rank_pack": "/home/ray/Desktop/centeredSquare/three_regime_overlap_v1/subsets/steady/pod/rom/rank99_ru3_rp2/pod_rank_pack_Re50_150_N100_rank99_ru3_rp2.npz",
+      "pressure_H_tilde_relative_residual": 2.823524667818694e-16,
+      "pressure_tensor_file": "/home/ray/Desktop/centeredSquare/three_regime_overlap_v1/subsets/steady/pod/rom/rank99_ru3_rp2/pressure_poisson_surrogate_tensors_Re50_150_N100_rank99_ru3_rp2.npz",
+      "rank_L": 2,
+      "rp": 2,
+      "ru": 3,
+      "velocity_tensor_file": "/home/ray/Desktop/centeredSquare/three_regime_overlap_v1/subsets/steady/pod/rom/rank99_ru3_rp2/semi_intrusive_galerkin_tensors_Re50_150_N100_rank99_ru3_rp2_compact.npz"
+    },
+    {
+      "A_all_shape": [
+        60,
+        5,
+        5
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+      "G_u_shape": [
+        5,
+        5
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+        4,
+        5,
+        5
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+        5,
+        5,
+        5
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+      "H_tilde_shape": [
+        4,
+        5,
+        5
+      ],
+      "L_shape": [
+        4,
+        4
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+      "P_shape": [
+        5,
+        4
+      ],
+      "c_all_shape": [
+        60,
+        5
+      ],
+      "cond_G_u": 1.0000000025580378,
+      "dataset": "/home/ray/Desktop/centeredSquare/three_regime_overlap_v1/subsets/steady",
+      "eig_L_max": -0.06258772017125865,
+      "eig_L_min": -7.029072691743215,
+      "finite": true,
+      "label": "rank999",
+      "max_abs_G_u_minus_I": 9.508294152027474e-10,
+      "norm_H_fro": 0.3274356314687946,
+      "norm_P_fro": 0.10200345492871064,
+      "num_cells": 9400,
+      "num_re": 60,
+      "output_dir": "/home/ray/Desktop/centeredSquare/three_regime_overlap_v1/subsets/steady/pod/rom/rank999_ru5_rp4",
+      "pod_rank_pack": "/home/ray/Desktop/centeredSquare/three_regime_overlap_v1/subsets/steady/pod/rom/rank999_ru5_rp4/pod_rank_pack_Re50_150_N100_rank999_ru5_rp4.npz",
+      "pressure_H_tilde_relative_residual": 1.523240010899727e-15,
+      "pressure_tensor_file": "/home/ray/Desktop/centeredSquare/three_regime_overlap_v1/subsets/steady/pod/rom/rank999_ru5_rp4/pressure_poisson_surrogate_tensors_Re50_150_N100_rank999_ru5_rp4.npz",
+      "rank_L": 4,
+      "rp": 4,
+      "ru": 5,
+      "velocity_tensor_file": "/home/ray/Desktop/centeredSquare/three_regime_overlap_v1/subsets/steady/pod/rom/rank999_ru5_rp4/semi_intrusive_galerkin_tensors_Re50_150_N100_rank999_ru5_rp4_compact.npz"
+    }
+  ]
+}
diff --git a/three_regime_overlap_v1/subsets/steady/pod/pod_energy_report.csv b/three_regime_overlap_v1/subsets/steady/pod/pod_energy_report.csv
new file mode 100644
index 0000000000000000000000000000000000000000..d7cbcb7148bec6eb3772a9e9631738d50c9d32ea
--- /dev/null
+++ b/three_regime_overlap_v1/subsets/steady/pod/pod_energy_report.csv
@@ -0,0 +1,3 @@
+field,rank_99,rank_999,stored_modes,analysis_rank,captured
+velocity,3,5,5,256,0.9991934666163927
+pressure,2,4,4,256,0.9994876533554901
diff --git a/three_regime_overlap_v1/subsets/steady/pod/rom/ROM_PhysicsGeneralizable_manifest.json b/three_regime_overlap_v1/subsets/steady/pod/rom/ROM_PhysicsGeneralizable_manifest.json
new file mode 100644
index 0000000000000000000000000000000000000000..29abb9a11ef046fdee8bd28e19e79c0f143722f5
--- /dev/null
+++ b/three_regime_overlap_v1/subsets/steady/pod/rom/ROM_PhysicsGeneralizable_manifest.json
@@ -0,0 +1,142 @@
+{
+  "bundle": "/home/ray/Desktop/centeredSquare/three_regime_overlap_v1/subsets/steady/pod/rom/centeredSquare_Re50_150_N100_rank99_rank999_ROM_and_POD.tar.gz",
+  "dataset": "/home/ray/Desktop/centeredSquare/three_regime_overlap_v1/subsets/steady",
+  "derivative_operator": {
+    "backend": "pyvista_vtk_compute_derivative_cell_data",
+    "center_atol": 5e-06,
+    "max_center_delta": 1.097595216492664e-06,
+    "n_cells": 9400,
+    "n_points": 19306,
+    "preference": "cell",
+    "reference_vtk": "/home/ray/Desktop/centeredSquare/three_regime_overlap_v1/subsets/steady/reference_vtk/centeredSquare_CN09_graded_Re100_internal_final_reference.vtk"
+  },
+  "output_root": "/home/ray/Desktop/centeredSquare/three_regime_overlap_v1/subsets/steady/pod/rom",
+  "rank_specs": [
+    {
+      "label": "rank99",
+      "rp": 2,
+      "ru": 3
+    },
+    {
+      "label": "rank999",
+      "rp": 4,
+      "ru": 5
+    }
+  ],
+  "summaries": [
+    {
+      "A_all_shape": [
+        60,
+        3,
+        3
+      ],
+      "G_u_shape": [
+        3,
+        3
+      ],
+      "H_p_shape": [
+        2,
+        3,
+        3
+      ],
+      "H_shape": [
+        3,
+        3,
+        3
+      ],
+      "H_tilde_shape": [
+        2,
+        3,
+        3
+      ],
+      "L_shape": [
+        2,
+        2
+      ],
+      "P_shape": [
+        3,
+        2
+      ],
+      "c_all_shape": [
+        60,
+        3
+      ],
+      "cond_G_u": 1.0000000014990225,
+      "dataset": "/home/ray/Desktop/centeredSquare/three_regime_overlap_v1/subsets/steady",
+      "eig_L_max": -0.06521244606446581,
+      "eig_L_min": -2.6958786766478964,
+      "finite": true,
+      "label": "rank99",
+      "max_abs_G_u_minus_I": 8.446097155001553e-10,
+      "norm_H_fro": 0.04250069940953678,
+      "norm_P_fro": 0.1005150244322599,
+      "num_cells": 9400,
+      "num_re": 60,
+      "output_dir": "/home/ray/Desktop/centeredSquare/three_regime_overlap_v1/subsets/steady/pod/rom/rank99_ru3_rp2",
+      "pod_rank_pack": "/home/ray/Desktop/centeredSquare/three_regime_overlap_v1/subsets/steady/pod/rom/rank99_ru3_rp2/pod_rank_pack_Re50_150_N100_rank99_ru3_rp2.npz",
+      "pressure_H_tilde_relative_residual": 2.823524667818694e-16,
+      "pressure_tensor_file": "/home/ray/Desktop/centeredSquare/three_regime_overlap_v1/subsets/steady/pod/rom/rank99_ru3_rp2/pressure_poisson_surrogate_tensors_Re50_150_N100_rank99_ru3_rp2.npz",
+      "rank_L": 2,
+      "rp": 2,
+      "ru": 3,
+      "velocity_tensor_file": "/home/ray/Desktop/centeredSquare/three_regime_overlap_v1/subsets/steady/pod/rom/rank99_ru3_rp2/semi_intrusive_galerkin_tensors_Re50_150_N100_rank99_ru3_rp2_compact.npz"
+    },
+    {
+      "A_all_shape": [
+        60,
+        5,
+        5
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+      "G_u_shape": [
+        5,
+        5
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+      "H_p_shape": [
+        4,
+        5,
+        5
+      ],
+      "H_shape": [
+        5,
+        5,
+        5
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+      "H_tilde_shape": [
+        4,
+        5,
+        5
+      ],
+      "L_shape": [
+        4,
+        4
+      ],
+      "P_shape": [
+        5,
+        4
+      ],
+      "c_all_shape": [
+        60,
+        5
+      ],
+      "cond_G_u": 1.0000000025580378,
+      "dataset": "/home/ray/Desktop/centeredSquare/three_regime_overlap_v1/subsets/steady",
+      "eig_L_max": -0.06258772017125865,
+      "eig_L_min": -7.029072691743215,
+      "finite": true,
+      "label": "rank999",
+      "max_abs_G_u_minus_I": 9.508294152027474e-10,
+      "norm_H_fro": 0.3274356314687946,
+      "norm_P_fro": 0.10200345492871064,
+      "num_cells": 9400,
+      "num_re": 60,
+      "output_dir": "/home/ray/Desktop/centeredSquare/three_regime_overlap_v1/subsets/steady/pod/rom/rank999_ru5_rp4",
+      "pod_rank_pack": "/home/ray/Desktop/centeredSquare/three_regime_overlap_v1/subsets/steady/pod/rom/rank999_ru5_rp4/pod_rank_pack_Re50_150_N100_rank999_ru5_rp4.npz",
+      "pressure_H_tilde_relative_residual": 1.523240010899727e-15,
+      "pressure_tensor_file": "/home/ray/Desktop/centeredSquare/three_regime_overlap_v1/subsets/steady/pod/rom/rank999_ru5_rp4/pressure_poisson_surrogate_tensors_Re50_150_N100_rank999_ru5_rp4.npz",
+      "rank_L": 4,
+      "rp": 4,
+      "ru": 5,
+      "velocity_tensor_file": "/home/ray/Desktop/centeredSquare/three_regime_overlap_v1/subsets/steady/pod/rom/rank999_ru5_rp4/semi_intrusive_galerkin_tensors_Re50_150_N100_rank999_ru5_rp4_compact.npz"
+    }
+  ]
+}
diff --git a/three_regime_overlap_v1/subsets/steady/pod/rom/centeredSquare_Re50_150_N100_rank99_rank999_ROM_and_POD.tar.gz b/three_regime_overlap_v1/subsets/steady/pod/rom/centeredSquare_Re50_150_N100_rank99_rank999_ROM_and_POD.tar.gz
new file mode 100644
index 0000000000000000000000000000000000000000..52fcc866444a95d4b3f41bfcd6aad7de97e77a7d
--- /dev/null
+++ b/three_regime_overlap_v1/subsets/steady/pod/rom/centeredSquare_Re50_150_N100_rank99_rank999_ROM_and_POD.tar.gz
@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:94aa0a863be1d899ccaad7f72d924efe98b357c30e01a06915598643b0cd6457
+size 3182179
diff --git a/three_regime_overlap_v1/subsets/steady/pod/rom/rank999_ru5_rp4/ROM_BUILD_REPORT_Re50_150_N100_rank999_ru5_rp4.md b/three_regime_overlap_v1/subsets/steady/pod/rom/rank999_ru5_rp4/ROM_BUILD_REPORT_Re50_150_N100_rank999_ru5_rp4.md
new file mode 100644
index 0000000000000000000000000000000000000000..466c97b311f4e5896a342923f2419eddf517f00d
--- /dev/null
+++ b/three_regime_overlap_v1/subsets/steady/pod/rom/rank999_ru5_rp4/ROM_BUILD_REPORT_Re50_150_N100_rank999_ru5_rp4.md
@@ -0,0 +1,50 @@
+# CenteredSquare ROM Build Report - rank999
+
+- Dataset: `/home/ray/Desktop/centeredSquare/three_regime_overlap_v1/subsets/steady`
+- Output dir: `/home/ray/Desktop/centeredSquare/three_regime_overlap_v1/subsets/steady/pod/rom/rank999_ru5_rp4`
+- Ranks: `ru=5`, `rp=4`
+- Re points: `60`
+- Cells: `9400`
+- Derivative backend: PyVista `compute_derivative()` on reference VTK cell data
+- Reference VTK: `/home/ray/Desktop/centeredSquare/three_regime_overlap_v1/subsets/steady/reference_vtk/centeredSquare_CN09_graded_Re100_internal_final_reference.vtk`
+- VTK cells/points: `9400` / `19306`
+- Max VTK/POD cell-center delta: `1.097595e-06`
+
+## Velocity ROM
+
+```text
+da/dt = c(Re) + A(Re) a + H(a,a) + P b
+```
+
+- `G_u` shape: `[5, 5]`
+- `c_all` shape: `[60, 5]`
+- `A_all` shape: `[60, 5, 5]`
+- `H` shape: `[5, 5, 5]`
+- `P` shape: `[5, 4]`
+- `cond(G_u)`: `1.000000e+00`
+- `max|G_u-I|`: `9.508294e-10`
+- `||H||_F`: `3.274356e-01`
+- `||P||_F`: `1.020035e-01`
+
+## Pressure Poisson Surrogate
+
+```text
+L b(t) = c^p(Re) + A^p(Re) a(t) + H^p(a(t),a(t))
+b(t) = c_tilde(Re) + A_tilde(Re) a(t) + H_tilde(a(t),a(t))
+```
+
+- `L` shape: `[4, 4]`
+- `H_p` shape: `[4, 5, 5]`
+- `H_tilde` shape: `[4, 5, 5]`
+- `rank(L)`: `4/4`
+- `eig(L) min/max`: `-7.029073e+00` / `-6.258772e-02`
+- `rel ||L H_tilde-H_p||`: `1.523240e-15`
+
+## Files
+
+- `semi_intrusive_galerkin_tensors_Re50_150_N100_rank999_ru5_rp4_compact.npz`
+- `pressure_poisson_surrogate_tensors_Re50_150_N100_rank999_ru5_rp4.npz`
+- `pod_rank_pack_Re50_150_N100_rank999_ru5_rp4.npz`
+- `manifest.json`
+
+- Finite arrays: `True`
diff --git a/three_regime_overlap_v1/subsets/steady/pod/rom/rank999_ru5_rp4/manifest.json b/three_regime_overlap_v1/subsets/steady/pod/rom/rank999_ru5_rp4/manifest.json
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--- /dev/null
+++ b/three_regime_overlap_v1/subsets/steady/pod/rom/rank999_ru5_rp4/manifest.json
@@ -0,0 +1,57 @@
+{
+  "A_all_shape": [
+    60,
+    5,
+    5
+  ],
+  "G_u_shape": [
+    5,
+    5
+  ],
+  "H_p_shape": [
+    4,
+    5,
+    5
+  ],
+  "H_shape": [
+    5,
+    5,
+    5
+  ],
+  "H_tilde_shape": [
+    4,
+    5,
+    5
+  ],
+  "L_shape": [
+    4,
+    4
+  ],
+  "P_shape": [
+    5,
+    4
+  ],
+  "c_all_shape": [
+    60,
+    5
+  ],
+  "cond_G_u": 1.0000000025580378,
+  "dataset": "/home/ray/Desktop/centeredSquare/three_regime_overlap_v1/subsets/steady",
+  "eig_L_max": -0.06258772017125865,
+  "eig_L_min": -7.029072691743215,
+  "finite": true,
+  "label": "rank999",
+  "max_abs_G_u_minus_I": 9.508294152027474e-10,
+  "norm_H_fro": 0.3274356314687946,
+  "norm_P_fro": 0.10200345492871064,
+  "num_cells": 9400,
+  "num_re": 60,
+  "output_dir": "/home/ray/Desktop/centeredSquare/three_regime_overlap_v1/subsets/steady/pod/rom/rank999_ru5_rp4",
+  "pod_rank_pack": "/home/ray/Desktop/centeredSquare/three_regime_overlap_v1/subsets/steady/pod/rom/rank999_ru5_rp4/pod_rank_pack_Re50_150_N100_rank999_ru5_rp4.npz",
+  "pressure_H_tilde_relative_residual": 1.523240010899727e-15,
+  "pressure_tensor_file": "/home/ray/Desktop/centeredSquare/three_regime_overlap_v1/subsets/steady/pod/rom/rank999_ru5_rp4/pressure_poisson_surrogate_tensors_Re50_150_N100_rank999_ru5_rp4.npz",
+  "rank_L": 4,
+  "rp": 4,
+  "ru": 5,
+  "velocity_tensor_file": "/home/ray/Desktop/centeredSquare/three_regime_overlap_v1/subsets/steady/pod/rom/rank999_ru5_rp4/semi_intrusive_galerkin_tensors_Re50_150_N100_rank999_ru5_rp4_compact.npz"
+}
diff --git a/three_regime_overlap_v1/subsets/steady/pod/rom/rank999_ru5_rp4/pod_rank_pack_Re50_150_N100_rank999_ru5_rp4.npz b/three_regime_overlap_v1/subsets/steady/pod/rom/rank999_ru5_rp4/pod_rank_pack_Re50_150_N100_rank999_ru5_rp4.npz
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+version https://git-lfs.github.com/spec/v1
+oid sha256:1d2137b362b48f64ae2d402af406bc0172d53c1e6b8a2da88fe2d82e730f2169
+size 59047
diff --git a/three_regime_overlap_v1/subsets/steady/pod/rom/rank99_ru3_rp2/ROM_BUILD_REPORT_Re50_150_N100_rank99_ru3_rp2.md b/three_regime_overlap_v1/subsets/steady/pod/rom/rank99_ru3_rp2/ROM_BUILD_REPORT_Re50_150_N100_rank99_ru3_rp2.md
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@@ -0,0 +1,50 @@
+# CenteredSquare ROM Build Report - rank99
+
+- Dataset: `/home/ray/Desktop/centeredSquare/three_regime_overlap_v1/subsets/steady`
+- Output dir: `/home/ray/Desktop/centeredSquare/three_regime_overlap_v1/subsets/steady/pod/rom/rank99_ru3_rp2`
+- Ranks: `ru=3`, `rp=2`
+- Re points: `60`
+- Cells: `9400`
+- Derivative backend: PyVista `compute_derivative()` on reference VTK cell data
+- Reference VTK: `/home/ray/Desktop/centeredSquare/three_regime_overlap_v1/subsets/steady/reference_vtk/centeredSquare_CN09_graded_Re100_internal_final_reference.vtk`
+- VTK cells/points: `9400` / `19306`
+- Max VTK/POD cell-center delta: `1.097595e-06`
+
+## Velocity ROM
+
+```text
+da/dt = c(Re) + A(Re) a + H(a,a) + P b
+```
+
+- `G_u` shape: `[3, 3]`
+- `c_all` shape: `[60, 3]`
+- `A_all` shape: `[60, 3, 3]`
+- `H` shape: `[3, 3, 3]`
+- `P` shape: `[3, 2]`
+- `cond(G_u)`: `1.000000e+00`
+- `max|G_u-I|`: `8.446097e-10`
+- `||H||_F`: `4.250070e-02`
+- `||P||_F`: `1.005150e-01`
+
+## Pressure Poisson Surrogate
+
+```text
+L b(t) = c^p(Re) + A^p(Re) a(t) + H^p(a(t),a(t))
+b(t) = c_tilde(Re) + A_tilde(Re) a(t) + H_tilde(a(t),a(t))
+```
+
+- `L` shape: `[2, 2]`
+- `H_p` shape: `[2, 3, 3]`
+- `H_tilde` shape: `[2, 3, 3]`
+- `rank(L)`: `2/2`
+- `eig(L) min/max`: `-2.695879e+00` / `-6.521245e-02`
+- `rel ||L H_tilde-H_p||`: `2.823525e-16`
+
+## Files
+
+- `semi_intrusive_galerkin_tensors_Re50_150_N100_rank99_ru3_rp2_compact.npz`
+- `pressure_poisson_surrogate_tensors_Re50_150_N100_rank99_ru3_rp2.npz`
+- `pod_rank_pack_Re50_150_N100_rank99_ru3_rp2.npz`
+- `manifest.json`
+
+- Finite arrays: `True`
diff --git a/three_regime_overlap_v1/subsets/steady/pod/rom/rank99_ru3_rp2/manifest.json b/three_regime_overlap_v1/subsets/steady/pod/rom/rank99_ru3_rp2/manifest.json
new file mode 100644
index 0000000000000000000000000000000000000000..f41b67e110c50ff5923cfb7fba82958993b526b4
--- /dev/null
+++ b/three_regime_overlap_v1/subsets/steady/pod/rom/rank99_ru3_rp2/manifest.json
@@ -0,0 +1,57 @@
+{
+  "A_all_shape": [
+    60,
+    3,
+    3
+  ],
+  "G_u_shape": [
+    3,
+    3
+  ],
+  "H_p_shape": [
+    2,
+    3,
+    3
+  ],
+  "H_shape": [
+    3,
+    3,
+    3
+  ],
+  "H_tilde_shape": [
+    2,
+    3,
+    3
+  ],
+  "L_shape": [
+    2,
+    2
+  ],
+  "P_shape": [
+    3,
+    2
+  ],
+  "c_all_shape": [
+    60,
+    3
+  ],
+  "cond_G_u": 1.0000000014990225,
+  "dataset": "/home/ray/Desktop/centeredSquare/three_regime_overlap_v1/subsets/steady",
+  "eig_L_max": -0.06521244606446581,
+  "eig_L_min": -2.6958786766478964,
+  "finite": true,
+  "label": "rank99",
+  "max_abs_G_u_minus_I": 8.446097155001553e-10,
+  "norm_H_fro": 0.04250069940953678,
+  "norm_P_fro": 0.1005150244322599,
+  "num_cells": 9400,
+  "num_re": 60,
+  "output_dir": "/home/ray/Desktop/centeredSquare/three_regime_overlap_v1/subsets/steady/pod/rom/rank99_ru3_rp2",
+  "pod_rank_pack": "/home/ray/Desktop/centeredSquare/three_regime_overlap_v1/subsets/steady/pod/rom/rank99_ru3_rp2/pod_rank_pack_Re50_150_N100_rank99_ru3_rp2.npz",
+  "pressure_H_tilde_relative_residual": 2.823524667818694e-16,
+  "pressure_tensor_file": "/home/ray/Desktop/centeredSquare/three_regime_overlap_v1/subsets/steady/pod/rom/rank99_ru3_rp2/pressure_poisson_surrogate_tensors_Re50_150_N100_rank99_ru3_rp2.npz",
+  "rank_L": 2,
+  "rp": 2,
+  "ru": 3,
+  "velocity_tensor_file": "/home/ray/Desktop/centeredSquare/three_regime_overlap_v1/subsets/steady/pod/rom/rank99_ru3_rp2/semi_intrusive_galerkin_tensors_Re50_150_N100_rank99_ru3_rp2_compact.npz"
+}
diff --git a/three_regime_overlap_v1/subsets/steady/pod/rom/rank99_ru3_rp2/pod_rank_pack_Re50_150_N100_rank99_ru3_rp2.npz b/three_regime_overlap_v1/subsets/steady/pod/rom/rank99_ru3_rp2/pod_rank_pack_Re50_150_N100_rank99_ru3_rp2.npz
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+version https://git-lfs.github.com/spec/v1
+oid sha256:c713ecd72b56a21a674eef7f2c3bf90659c5c7143915990f3e089db90a32421e
+size 542853
diff --git a/three_regime_overlap_v1/subsets/steady/pod/rom/rank99_ru3_rp2/pressure_poisson_surrogate_tensors_Re50_150_N100_rank99_ru3_rp2.npz b/three_regime_overlap_v1/subsets/steady/pod/rom/rank99_ru3_rp2/pressure_poisson_surrogate_tensors_Re50_150_N100_rank99_ru3_rp2.npz
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diff --git a/three_regime_overlap_v1/subsets/steady/pod/validation_projection.json b/three_regime_overlap_v1/subsets/steady/pod/validation_projection.json
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+{
+  "cases": [
+    {
+      "Re": 55.0,
+      "finite": true,
+      "pressure_weighted_rel_l2": 0.01679596050099605,
+      "tag": "Re055p000000",
+      "velocity_weighted_rel_l2": 0.02724162896289555
+    },
+    {
+      "Re": 75.0,
+      "finite": true,
+      "pressure_weighted_rel_l2": 0.029144066599716946,
+      "tag": "Re075p000000",
+      "velocity_weighted_rel_l2": 0.02064078117640558
+    },
+    {
+      "Re": 90.0,
+      "finite": true,
+      "pressure_weighted_rel_l2": 0.02504822204697472,
+      "tag": "Re090p000000",
+      "velocity_weighted_rel_l2": 0.02590074387731547
+    },
+    {
+      "Re": 94.5,
+      "finite": true,
+      "pressure_weighted_rel_l2": 0.02876769704830787,
+      "tag": "Re094p500000",
+      "velocity_weighted_rel_l2": 0.04045847908859154
+    },
+    {
+      "Re": 95.25,
+      "finite": true,
+      "pressure_weighted_rel_l2": 0.03122233360314666,
+      "tag": "Re095p250000",
+      "velocity_weighted_rel_l2": 0.05021150676388498
+    }
+  ]
+}
diff --git a/three_regime_overlap_v1/subsets/steady/pod/weighted_pod_pressure.npz b/three_regime_overlap_v1/subsets/steady/pod/weighted_pod_pressure.npz
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+version https://git-lfs.github.com/spec/v1
+oid sha256:ff2f11d6333fa516b9443a7901f8227d5e7d43ea61e0639616615d960f842f4d
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+version https://git-lfs.github.com/spec/v1
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diff --git a/three_regime_overlap_v1/subsets/steady/reference_vtk/README.md b/three_regime_overlap_v1/subsets/steady/reference_vtk/README.md
new file mode 100644
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+++ b/three_regime_overlap_v1/subsets/steady/reference_vtk/README.md
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+# Reference VTK
+
+This directory intentionally keeps a single final-time internal-field VTK file for ROM derivative/operator construction.
+
+- Source case: /home/ray/Desktop/centeredSquare/runs/cn09_graded_20260708_000142/Re100
+- Source VTK: /home/ray/Desktop/centeredSquare/runs/cn09_graded_20260708_000142/Re100/VTK/Re100_25000.vtk
+- Copied file: /home/ray/Desktop/centeredSquare/dataset_Re50_150_N100_npz/reference_vtk/centeredSquare_CN09_graded_Re100_internal_final_reference.vtk
+- Mesh/numerics: centeredSquare CN09 + compatible graded mesh
+- Contents: OpenFOAM internal mesh VTK with U/p fields from the reference Re100 run
+- Note: formal NPZ production outputs remain in cases_npz/probes; this reference VTK is intentionally kept separately.
diff --git a/three_regime_overlap_v1/subsets/steady/reference_vtk/centeredSquare_CN09_graded_Re100_internal_final_reference.vtk b/three_regime_overlap_v1/subsets/steady/reference_vtk/centeredSquare_CN09_graded_Re100_internal_final_reference.vtk
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+version https://git-lfs.github.com/spec/v1
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