diff --git a/2026_tdl_challenge/outputs/2026-07-27_05-00-00/results.json b/2026_tdl_challenge/outputs/2026-07-27_05-00-00/results.json
new file mode 100644
index 000000000..dbdbae400
--- /dev/null
+++ b/2026_tdl_challenge/outputs/2026-07-27_05-00-00/results.json
@@ -0,0 +1,5272 @@
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+      "test_best_rerun_accuracy": null,
+      "test_best_rerun_mse": 20381.810546875,
+      "test_triangles_total_structural": 60093.0,
+      "test_mse_by_total_triangles": 0.3391711272007555,
+      "ood_test": {
+        "h_lo__d_lo__pl_lo": {
+          "test_best_rerun_accuracy": null,
+          "test_best_rerun_mse": 37342.09375,
+          "test_triangles_total_structural": 1388.0,
+          "test_mse_by_total_triangles": 26.90352575648415
+        },
+        "h_lo__d_lo__pl_hi": {
+          "test_best_rerun_accuracy": null,
+          "test_best_rerun_mse": 42938.55078125,
+          "test_triangles_total_structural": 190.0,
+          "test_mse_by_total_triangles": 225.99237253289473
+        },
+        "h_lo__d_hi__pl_lo": {
+          "test_best_rerun_accuracy": null,
+          "test_best_rerun_mse": 903981.5625,
+          "test_triangles_total_structural": 13185.0,
+          "test_mse_by_total_triangles": 68.56136234357224
+        },
+        "h_lo__d_hi__pl_hi": {
+          "test_best_rerun_accuracy": null,
+          "test_best_rerun_mse": 10427.005859375,
+          "test_triangles_total_structural": 2967.0,
+          "test_mse_by_total_triangles": 3.514326208080553
+        },
+        "h_mid__d_lo__pl_lo": {
+          "test_best_rerun_accuracy": null,
+          "test_best_rerun_mse": 21393.744140625,
+          "test_triangles_total_structural": 7485.0,
+          "test_mse_by_total_triangles": 2.8582156500501004
+        },
+        "h_mid__d_lo__pl_hi": {
+          "test_best_rerun_accuracy": null,
+          "test_best_rerun_mse": 32005.9609375,
+          "test_triangles_total_structural": 1158.0,
+          "test_mse_by_total_triangles": 27.638999082469777
+        },
+        "h_mid__d_hi__pl_lo": {
+          "test_best_rerun_accuracy": null,
+          "test_best_rerun_mse": 584962.9375,
+          "test_triangles_total_structural": 43064.0,
+          "test_mse_by_total_triangles": 13.58357183494334
+        },
+        "h_mid__d_hi__pl_hi": {
+          "test_best_rerun_accuracy": null,
+          "test_best_rerun_mse": 7434.94189453125,
+          "test_triangles_total_structural": 14262.0,
+          "test_mse_by_total_triangles": 0.5213113093907762
+        },
+        "h_hi__d_lo__pl_lo": {
+          "test_best_rerun_accuracy": null,
+          "test_best_rerun_mse": 10889.7724609375,
+          "test_triangles_total_structural": 19509.0,
+          "test_mse_by_total_triangles": 0.5581922426027731
+        },
+        "h_hi__d_lo__pl_hi": {
+          "test_best_rerun_accuracy": null,
+          "test_best_rerun_mse": 12028.462890625,
+          "test_triangles_total_structural": 5625.0,
+          "test_mse_by_total_triangles": 2.1383934027777776
+        },
+        "h_hi__d_hi__pl_lo": {
+          "test_best_rerun_accuracy": null,
+          "test_best_rerun_mse": 44471.203125,
+          "test_triangles_total_structural": 88403.0,
+          "test_mse_by_total_triangles": 0.503050836792869
+        }
+      },
+      "output_dir": "/scratch/work/francoj3/challengetopo/TopoBench/logs/train/runs/notebook_gu_grid_2026-07-27_05-00-00__triangle_counting__11__h_hi__d_hi__pl_hi__s42"
+    },
+    {
+      "experiment": "triangle_counting",
+      "wandb_project": "challenge_triangle_counting",
+      "wandb_run_name": "smcn_difflift_hom_0.9-1__deg_4-5__gamma_4-5__s43",
+      "train_seed": 43,
+      "homophily": "h_hi",
+      "avg_degree": "d_hi",
+      "power_law": "pl_hi",
+      "run_slug": "h_hi__d_hi__pl_hi",
+      "test_loss": 12489.291015625,
+      "test_best_rerun_accuracy": null,
+      "test_best_rerun_mse": 12640.27734375,
+      "test_triangles_total_structural": 60093.0,
+      "test_mse_by_total_triangles": 0.21034525391892567,
+      "ood_test": {
+        "h_lo__d_lo__pl_lo": {
+          "test_best_rerun_accuracy": null,
+          "test_best_rerun_mse": 40216.453125,
+          "test_triangles_total_structural": 1388.0,
+          "test_mse_by_total_triangles": 28.974389859510087
+        },
+        "h_lo__d_lo__pl_hi": {
+          "test_best_rerun_accuracy": null,
+          "test_best_rerun_mse": 19595.33203125,
+          "test_triangles_total_structural": 190.0,
+          "test_mse_by_total_triangles": 103.13332648026316
+        },
+        "h_lo__d_hi__pl_lo": {
+          "test_best_rerun_accuracy": null,
+          "test_best_rerun_mse": 915462.1875,
+          "test_triangles_total_structural": 13185.0,
+          "test_mse_by_total_triangles": 69.43209613196815
+        },
+        "h_lo__d_hi__pl_hi": {
+          "test_best_rerun_accuracy": null,
+          "test_best_rerun_mse": 51040.74609375,
+          "test_triangles_total_structural": 2967.0,
+          "test_mse_by_total_triangles": 17.2028129739636
+        },
+        "h_mid__d_lo__pl_lo": {
+          "test_best_rerun_accuracy": null,
+          "test_best_rerun_mse": 38349.328125,
+          "test_triangles_total_structural": 7485.0,
+          "test_mse_by_total_triangles": 5.123490731462926
+        },
+        "h_mid__d_lo__pl_hi": {
+          "test_best_rerun_accuracy": null,
+          "test_best_rerun_mse": 15991.88671875,
+          "test_triangles_total_structural": 1158.0,
+          "test_mse_by_total_triangles": 13.809919446243523
+        },
+        "h_mid__d_hi__pl_lo": {
+          "test_best_rerun_accuracy": null,
+          "test_best_rerun_mse": 462450.34375,
+          "test_triangles_total_structural": 43064.0,
+          "test_mse_by_total_triangles": 10.738676011285529
+        },
+        "h_mid__d_hi__pl_hi": {
+          "test_best_rerun_accuracy": null,
+          "test_best_rerun_mse": 23570.748046875,
+          "test_triangles_total_structural": 14262.0,
+          "test_mse_by_total_triangles": 1.652695838372949
+        },
+        "h_hi__d_lo__pl_lo": {
+          "test_best_rerun_accuracy": null,
+          "test_best_rerun_mse": 25921.947265625,
+          "test_triangles_total_structural": 19509.0,
+          "test_mse_by_total_triangles": 1.3287173748334102
+        },
+        "h_hi__d_lo__pl_hi": {
+          "test_best_rerun_accuracy": null,
+          "test_best_rerun_mse": 18001.5625,
+          "test_triangles_total_structural": 5625.0,
+          "test_mse_by_total_triangles": 3.200277777777778
+        },
+        "h_hi__d_hi__pl_lo": {
+          "test_best_rerun_accuracy": null,
+          "test_best_rerun_mse": 56375.19140625,
+          "test_triangles_total_structural": 88403.0,
+          "test_mse_by_total_triangles": 0.6377067679405676
+        }
+      },
+      "output_dir": "/scratch/work/francoj3/challengetopo/TopoBench/logs/train/runs/notebook_gu_grid_2026-07-27_05-00-00__triangle_counting__11__h_hi__d_hi__pl_hi__s43"
+    },
+    {
+      "experiment": "triangle_counting",
+      "wandb_project": "challenge_triangle_counting",
+      "wandb_run_name": "smcn_difflift_hom_0.9-1__deg_4-5__gamma_4-5__s44",
+      "train_seed": 44,
+      "homophily": "h_hi",
+      "avg_degree": "d_hi",
+      "power_law": "pl_hi",
+      "run_slug": "h_hi__d_hi__pl_hi",
+      "test_loss": 9318.8623046875,
+      "test_best_rerun_accuracy": null,
+      "test_best_rerun_mse": 9319.6318359375,
+      "test_triangles_total_structural": 60093.0,
+      "test_mse_by_total_triangles": 0.1550868127059308,
+      "ood_test": {
+        "h_lo__d_lo__pl_lo": {
+          "test_best_rerun_accuracy": null,
+          "test_best_rerun_mse": 9270.857421875,
+          "test_triangles_total_structural": 1388.0,
+          "test_mse_by_total_triangles": 6.67929209068804
+        },
+        "h_lo__d_lo__pl_hi": {
+          "test_best_rerun_accuracy": null,
+          "test_best_rerun_mse": 6164.3955078125,
+          "test_triangles_total_structural": 190.0,
+          "test_mse_by_total_triangles": 32.444186883223686
+        },
+        "h_lo__d_hi__pl_lo": {
+          "test_best_rerun_accuracy": null,
+          "test_best_rerun_mse": 483439.78125,
+          "test_triangles_total_structural": 13185.0,
+          "test_mse_by_total_triangles": 36.665891638225254
+        },
+        "h_lo__d_hi__pl_hi": {
+          "test_best_rerun_accuracy": null,
+          "test_best_rerun_mse": 18734.0078125,
+          "test_triangles_total_structural": 2967.0,
+          "test_mse_by_total_triangles": 6.314124641894169
+        },
+        "h_mid__d_lo__pl_lo": {
+          "test_best_rerun_accuracy": null,
+          "test_best_rerun_mse": 10441.5234375,
+          "test_triangles_total_structural": 7485.0,
+          "test_mse_by_total_triangles": 1.394993111222445
+        },
+        "h_mid__d_lo__pl_hi": {
+          "test_best_rerun_accuracy": null,
+          "test_best_rerun_mse": 5023.5380859375,
+          "test_triangles_total_structural": 1158.0,
+          "test_mse_by_total_triangles": 4.338115790965026
+        },
+        "h_mid__d_hi__pl_lo": {
+          "test_best_rerun_accuracy": null,
+          "test_best_rerun_mse": 358684.9375,
+          "test_triangles_total_structural": 43064.0,
+          "test_mse_by_total_triangles": 8.329113354542077
+        },
+        "h_mid__d_hi__pl_hi": {
+          "test_best_rerun_accuracy": null,
+          "test_best_rerun_mse": 11656.814453125,
+          "test_triangles_total_structural": 14262.0,
+          "test_mse_by_total_triangles": 0.8173337858031833
+        },
+        "h_hi__d_lo__pl_lo": {
+          "test_best_rerun_accuracy": null,
+          "test_best_rerun_mse": 7624.9775390625,
+          "test_triangles_total_structural": 19509.0,
+          "test_mse_by_total_triangles": 0.39084409959826233
+        },
+        "h_hi__d_lo__pl_hi": {
+          "test_best_rerun_accuracy": null,
+          "test_best_rerun_mse": 4412.05810546875,
+          "test_triangles_total_structural": 5625.0,
+          "test_mse_by_total_triangles": 0.7843658854166666
+        },
+        "h_hi__d_hi__pl_lo": {
+          "test_best_rerun_accuracy": null,
+          "test_best_rerun_mse": 47380.71875,
+          "test_triangles_total_structural": 88403.0,
+          "test_mse_by_total_triangles": 0.535962792552289
+        }
+      },
+      "output_dir": "/scratch/work/francoj3/challengetopo/TopoBench/logs/train/runs/notebook_gu_grid_2026-07-27_05-00-00__triangle_counting__11__h_hi__d_hi__pl_hi__s44"
+    }
+  ]
+}
diff --git a/configs/model/cell/smcn.yaml b/configs/model/cell/smcn.yaml
new file mode 100644
index 000000000..65c60a064
--- /dev/null
+++ b/configs/model/cell/smcn.yaml
@@ -0,0 +1,41 @@
+_target_: topobench.model.TBModel
+
+model_name: smcn
+model_domain: cell
+
+feature_encoder:
+  _target_: topobench.nn.encoders.${model.feature_encoder.encoder_name}
+  encoder_name: AllCellFeatureEncoder
+  in_channels: ${infer_in_channels:${dataset},${oc.select:transforms,null}}
+  out_channels: 64
+  proj_dropout: 0.0
+
+backbone:
+  _target_: topobench.nn.backbones.cell.smcn.SMCN
+  in_channels: ${model.feature_encoder.out_channels}
+  sub_channels: 32
+  n_cin_layers: 1
+  n_scl_layers: 3
+  num_mlp_layers: 2
+  max_rank_out: 1
+  max_dist: 10
+  dropout: 0.0
+
+backbone_wrapper:
+  _target_: topobench.nn.wrappers.SMCNWrapper
+  _partial_: true
+  wrapper_name: SMCNWrapper
+  out_channels: ${model.feature_encoder.out_channels}
+  num_cell_dimensions: ${infer_num_cell_dimensions:${oc.select:model.feature_encoder.selected_dimensions,null},${model.feature_encoder.in_channels}}
+
+readout:
+  _target_: topobench.nn.readouts.${model.readout.readout_name}
+  readout_name: PropagateSignalDown
+  num_cell_dimensions: ${infer_num_cell_dimensions:${oc.select:model.feature_encoder.selected_dimensions,null},${model.feature_encoder.in_channels}}
+  hidden_dim: ${model.feature_encoder.out_channels}
+  out_channels: ${dataset.parameters.num_classes}
+  task_level: ${define_task_level:${dataset.parameters.task_level},${dataset.split_params.learning_setting}}
+  pooling_type: sum
+
+# compile model for faster training with pytorch 2.0
+compile: false
diff --git a/configs/model/cell/smcn_difflift.yaml b/configs/model/cell/smcn_difflift.yaml
new file mode 100644
index 000000000..25b0ee0c8
--- /dev/null
+++ b/configs/model/cell/smcn_difflift.yaml
@@ -0,0 +1,43 @@
+_target_: topobench.model.TBModel
+
+model_name: smcn_difflift
+model_domain: cell
+
+feature_encoder:
+  _target_: topobench.nn.encoders.${model.feature_encoder.encoder_name}
+  encoder_name: AllCellFeatureEncoder
+  in_channels: ${infer_in_channels:${dataset},${oc.select:transforms,null}}
+  out_channels: 64
+  proj_dropout: 0.0
+
+backbone:
+  _target_: topobench.nn.backbones.cell.smcn.SMCN
+  in_channels: ${model.feature_encoder.out_channels}
+  sub_channels: 32
+  n_cin_layers: 1
+  n_scl_layers: 3
+  num_mlp_layers: 2
+  max_rank_out: 1
+  max_dist: 10
+  dropout: 0.0
+  learned_lifting: true # 2-cells selected end-to-end (DiffLift)
+  sharpening: 10.0
+
+backbone_wrapper:
+  _target_: topobench.nn.wrappers.SMCNWrapper
+  _partial_: true
+  wrapper_name: SMCNWrapper
+  out_channels: ${model.feature_encoder.out_channels}
+  num_cell_dimensions: ${infer_num_cell_dimensions:${oc.select:model.feature_encoder.selected_dimensions,null},${model.feature_encoder.in_channels}}
+
+readout:
+  _target_: topobench.nn.readouts.${model.readout.readout_name}
+  readout_name: PropagateSignalDown
+  num_cell_dimensions: ${infer_num_cell_dimensions:${oc.select:model.feature_encoder.selected_dimensions,null},${model.feature_encoder.in_channels}}
+  hidden_dim: ${model.feature_encoder.out_channels}
+  out_channels: ${dataset.parameters.num_classes}
+  task_level: ${define_task_level:${dataset.parameters.task_level},${dataset.split_params.learning_setting}}
+  pooling_type: sum
+
+# compile model for faster training with pytorch 2.0
+compile: false
diff --git a/test/nn/backbones/cell/test_smcn.py b/test/nn/backbones/cell/test_smcn.py
new file mode 100644
index 000000000..0f8f0b21e
--- /dev/null
+++ b/test/nn/backbones/cell/test_smcn.py
@@ -0,0 +1,337 @@
+"""Unit tests for the SMCN backbone."""
+
+import pytest
+import torch
+
+from topobench.nn.backbones.cell.smcn import SMCN
+from topobench.nn.backbones.cell.smcn_utils.layers import (
+    BagInit,
+    BagPool,
+    CINBlock,
+    SafeBatchNorm,
+    SCLLayer,
+    TwoCellInit,
+)
+from topobench.nn.backbones.cell.smcn_utils.structures import (
+    adjacency_from_incidence,
+    build_smcn_structures,
+    hop_distance_buckets,
+)
+from topobench.nn.liftings.difflift import (
+    CellScorer,
+    DiffLift,
+    DiffLiftEncoder,
+    EdgeSampler,
+)
+
+
+def _two_triangles():
+    """Two triangles sharing an edge: 4 nodes, 5 edges, 2 two-cells.
+
+    Returns
+    -------
+    tuple
+        Incidence matrices and batch vectors of a single graph.
+    """
+    i1 = torch.zeros(4, 5)
+    for e, (a, b) in enumerate([(0, 1), (0, 2), (1, 2), (1, 3), (2, 3)]):
+        i1[a, e] = 1
+        i1[b, e] = 1
+    i2 = torch.zeros(5, 2)
+    i2[[0, 1, 2], 0] = 1
+    i2[[2, 3, 4], 1] = 1
+    b0 = torch.zeros(4, dtype=torch.long)
+    b1 = torch.zeros(5, dtype=torch.long)
+    b2 = torch.zeros(2, dtype=torch.long)
+    return i1, i2, b0, b1, b2
+
+
+def _structures():
+    """Structures of the two-triangle fixture.
+
+    Returns
+    -------
+    SMCNStructures
+        The assembled structures.
+    """
+    i1, i2, b0, b1, b2 = _two_triangles()
+    return build_smcn_structures(
+        i1.to_sparse(), i2.to_sparse(), b0, b1, b2
+    )
+
+
+def test_adjacency_pairs():
+    """Adjacency pairs and bridges of the fixture are exact."""
+    s = _structures()
+    assert s.a01_pairs.shape[1] == 10  # 5 edges, both directions
+    assert s.a12_pairs.shape[1] == 12  # 2 cells x 6 ordered pairs
+    pairs = set(
+        map(
+            tuple,
+            torch.cat([s.a01_pairs, s.a01_bridge.unsqueeze(0)]).t().tolist(),
+        )
+    )
+    assert (0, 1, 0) in pairs and (1, 0, 0) in pairs
+    assert (2, 3, 4) in pairs and (3, 2, 4) in pairs
+
+
+def test_adjacency_invalid_incidence():
+    """A node-edge incidence column with one entry is rejected."""
+    bad = torch.zeros(3, 2)
+    bad[0, 0] = 1
+    bad[1, 0] = 1
+    bad[2, 1] = 1  # dangling edge
+    with pytest.raises(ValueError):
+        adjacency_from_incidence(bad, expected_size=2)
+
+
+def test_bag_layout_and_marking():
+    """Bag rows follow row(u, e) = u * n1 + e with min-endpoint marking."""
+    s = _structures()
+    assert s.bag_low_index.tolist()[:5] == [0, 0, 0, 0, 0]
+    assert s.bag_high_index.tolist()[:5] == [0, 1, 2, 3, 4]
+    # row(u=3, e0=(0,1)): d(3,0)=2, d(3,1)=1 -> 1
+    assert int(s.bag_marking[15]) == 1
+    # row(u=0, e0=(0,1)): endpoint -> 0
+    assert int(s.bag_marking[0]) == 0
+    assert s.bag_low_adj_pairs.shape[1] == 50  # 10 pairs x 5 copies
+    assert s.bag_inc_pairs.shape[1] == 50  # 10 incidences x 5 copies
+
+
+def test_hop_distance_buckets():
+    """Distances are exact up to the cutoff, then bucketed."""
+    adj = torch.zeros(16, 16, dtype=torch.bool)
+    for i in range(14):
+        adj[i, i + 1] = adj[i + 1, i] = True
+    dist = hop_distance_buckets(adj, max_dist=10)
+    assert int(dist[0, 5]) == 5
+    assert int(dist[0, 14]) == 10  # connected but farther
+    assert int(dist[0, 15]) == 11  # disconnected
+
+
+def test_batch_offsets():
+    """Batched structures equal per-graph structures plus offsets."""
+    i1, i2, b0, b1, b2 = _two_triangles()
+    single = _structures()
+    double = build_smcn_structures(
+        torch.block_diag(i1, i1).to_sparse(),
+        torch.block_diag(i2, i2).to_sparse(),
+        torch.cat([b0, b0 + 1]),
+        torch.cat([b1, b1 + 1]),
+        torch.cat([b2, b2 + 1]),
+    )
+    assert torch.equal(double.bag_marking[:20], single.bag_marking)
+    assert torch.equal(double.bag_marking[20:], single.bag_marking)
+    assert torch.equal(
+        double.bag_low_index[20:] - 4, single.bag_low_index
+    )
+    assert torch.equal(
+        double.bag_inc_pairs[:, 50:] - 20, single.bag_inc_pairs
+    )
+
+
+def test_layers_shapes_and_grads():
+    """Every layer produces the documented shapes and gradients."""
+    s = _structures()
+    dim = 16
+    x_0 = torch.randn(4, dim, requires_grad=True)
+    x_1 = torch.randn(5, dim)
+    x_2 = torch.zeros(2, dim)
+    x_2 = TwoCellInit(dim)(x_0, x_2, s.inc02_pairs)
+    assert x_2.shape == (2, dim)
+    x_0b, x_1b, x_2b = CINBlock(dim)(x_0, x_1, x_2, s)
+    bag = BagInit(dim)(x_0b, x_1b, s)
+    assert bag.shape == (20, dim)
+    bag = SCLLayer(dim, 12, edge_dim=dim)(bag, x_1b, s)
+    assert bag.shape == (20, 12)
+    bag = SCLLayer(12, dim, edge_dim=dim)(bag, x_1b, s)
+    p0, p1 = BagPool()(bag, s, 4, 5)
+    assert p0.shape == (4, dim) and p1.shape == (5, dim)
+    (p0.sum() + p1.sum()).backward()
+    assert x_0.grad is not None and x_0.grad.abs().sum() > 0
+
+
+def test_cin_block_max_rank():
+    """The reduced block leaves higher ranks untouched."""
+    s = _structures()
+    dim = 8
+    x = [torch.randn(n, dim) for n in (4, 5, 2)]
+    block = CINBlock(dim, max_rank=0)
+    y0, y1, y2 = block(*x, s)
+    assert torch.equal(y1, x[1]) and torch.equal(y2, x[2])
+    with pytest.raises(ValueError):
+        CINBlock(dim, max_rank=3)
+
+
+@pytest.mark.parametrize("learned_lifting", [False, True])
+def test_backbone_forward_backward(learned_lifting):
+    """The backbone runs and back-propagates in both variants."""
+    i1, i2, b0, b1, b2 = _two_triangles()
+    dim = 16
+    model = SMCN(
+        dim,
+        sub_channels=12,
+        n_scl_layers=2,
+        learned_lifting=learned_lifting,
+    )
+    x_0 = torch.randn(4, dim, requires_grad=True)
+    y0, y1, y2 = model(
+        x_0,
+        torch.randn(5, dim),
+        torch.zeros(2, dim),
+        i1.to_sparse(),
+        i2.to_sparse(),
+        b0,
+        b1,
+        b2,
+    )
+    assert y0.shape == (4, dim)
+    assert y1.shape == (5, dim)
+    assert y2.shape == (2, dim)
+    (y0.sum() + y1.sum() + y2.sum()).backward()
+    assert x_0.grad is not None and x_0.grad.abs().sum() > 0
+
+
+def test_backbone_no_two_cells():
+    """A tree graph (no cycles) runs through both variants."""
+    i1 = torch.zeros(4, 3)
+    for e, (a, b) in enumerate([(0, 1), (1, 2), (2, 3)]):
+        i1[a, e] = 1
+        i1[b, e] = 1
+    for learned in (False, True):
+        model = SMCN(8, n_scl_layers=2, learned_lifting=learned)
+        _, _, y2 = model(
+            torch.randn(4, 8),
+            torch.randn(3, 8),
+            torch.zeros(0, 8),
+            i1.to_sparse(),
+            torch.zeros(3, 0).to_sparse(),
+            torch.zeros(4, dtype=torch.long),
+            torch.zeros(3, dtype=torch.long),
+            torch.zeros(0, dtype=torch.long),
+        )
+        assert y2.shape == (0, 8)
+
+
+def test_safe_batch_norm_single_row():
+    """Single rows pass through in training; larger batches normalize."""
+    bn = SafeBatchNorm(8)
+    bn.train()
+    x1 = torch.randn(1, 8)
+    assert torch.equal(bn(x1), x1)
+    x4 = torch.randn(4, 8)
+    ref = torch.nn.BatchNorm1d(8)
+    ref.train()
+    assert torch.allclose(bn(x4), ref(x4))
+
+
+@pytest.mark.parametrize("learned_lifting", [False, True])
+def test_backbone_single_two_cell_training(learned_lifting):
+    """A batch whose graphs total a single 2-cell trains without error."""
+    i1 = torch.zeros(3, 3)
+    for e, (a, b) in enumerate([(0, 1), (0, 2), (1, 2)]):
+        i1[a, e] = 1
+        i1[b, e] = 1
+    i2 = torch.ones(3, 1)
+    model = SMCN(
+        8, sub_channels=6, n_scl_layers=2, learned_lifting=learned_lifting
+    )
+    model.train()
+    x_0 = torch.randn(3, 8, requires_grad=True)
+    y0, y1, y2 = model(
+        x_0,
+        torch.randn(3, 8),
+        torch.zeros(1, 8),
+        i1.to_sparse(),
+        i2.to_sparse(),
+        torch.zeros(3, dtype=torch.long),
+        torch.zeros(3, dtype=torch.long),
+        torch.zeros(1, dtype=torch.long),
+    )
+    assert y2.shape == (1, 8)
+    (y0.sum() + y1.sum() + y2.sum()).backward()
+    assert x_0.grad is not None
+
+
+def test_cell_scorer_rescue():
+    """The scorer gates cells and the rescue keeps one per graph."""
+    i1, i2, b0, b1, b2 = _two_triangles()
+    s = build_smcn_structures(i1.to_sparse(), i2.to_sparse(), b0, b1, b2)
+    z = torch.randn(4, 8, requires_grad=True)
+    scorer = CellScorer(8)
+    torch.nn.init.constant_(scorer.score[2].bias, -100.0)
+    torch.nn.init.zeros_(scorer.score[2].weight)
+    gate = scorer(z, s.inc02_pairs, b2)
+    assert gate.shape == (2,)
+    assert gate.detach().sum() >= 1.0
+    gate.sum().backward()
+    assert z.grad is not None
+
+
+def test_backbone_invalid_scl_layers():
+    """Fewer than two SCL layers is rejected."""
+    with pytest.raises(ValueError):
+        SMCN(8, n_scl_layers=1)
+
+
+def test_cell_scorer_hypergraph_and_stochastic():
+    """The scorer is domain-agnostic and supports Bernoulli sampling."""
+    membership = torch.tensor(
+        [[0, 1, 2, 1, 2, 3, 4], [0, 0, 0, 1, 1, 2, 2]]
+    )
+    hyperedge_batch = torch.tensor([0, 0, 1])
+    z = torch.randn(5, 8, requires_grad=True)
+    gate = CellScorer(8)(z, membership, hyperedge_batch)
+    assert gate.shape == (3,)
+    assert set(gate.detach().unique().tolist()) <= {0.0, 1.0}
+    gate.sum().backward()
+    assert z.grad is not None
+    stochastic = CellScorer(8, stochastic=True, rescue=False).train()
+    gate = stochastic(z.detach(), membership, hyperedge_batch)
+    assert set(gate.detach().unique().tolist()) <= {0.0, 1.0}
+
+
+def test_difflift_encoder_and_edge_sampler():
+    """Learned edges are new, deduplicated, and carry gradients."""
+    i1, _, b0, _, _ = _two_triangles()
+    s = build_smcn_structures(
+        i1.to_sparse(), torch.zeros(5, 0).to_sparse(), b0,
+        torch.zeros(5, dtype=torch.long), torch.zeros(0, dtype=torch.long),
+    )
+    x = torch.randn(4, 6, requires_grad=True)
+    z = DiffLiftEncoder(6, hidden_channels=16)(x, s.a01_pairs)
+    assert z.shape == (4, 16)
+    sampler = EdgeSampler(16, k_min=1, k_max=3)
+    pairs, gate = sampler(z, s.a01_pairs, b0)
+    assert pairs.shape[0] == 2 and gate.shape == (pairs.shape[1],)
+    observed = {
+        tuple(sorted(p)) for p in s.a01_pairs.t().tolist()
+    }
+    for a, b in pairs.t().tolist():
+        assert (a, b) not in observed  # only new edges
+    if gate.numel():
+        gate.sum().backward()
+        assert x.grad is not None
+    with pytest.raises(ValueError):
+        EdgeSampler(16, k_min=3, k_max=2)
+
+
+def test_difflift_full_recipe():
+    """The complete lifting returns a consistent gated complex."""
+    i1, _, b0, _, _ = _two_triangles()
+    s = build_smcn_structures(
+        i1.to_sparse(), torch.zeros(5, 0).to_sparse(), b0,
+        torch.zeros(5, dtype=torch.long), torch.zeros(0, dtype=torch.long),
+    )
+    x = torch.randn(4, 6, requires_grad=True)
+    lift = DiffLift(6, hidden_channels=16, max_cell_length=6)
+    out = lift(x, s.a01_pairs, b0)
+    n_cells = out["cell_batch"].numel()
+    assert out["cell_gate"].shape == (n_cells,)
+    assert out["x_2"].shape == (n_cells, 6)
+    assert out["cell_membership"].shape[0] == 2
+    total = out["x_2"].sum() + out["cell_gate"].sum()
+    if out["new_edge_gate"].numel():
+        total = total + out["new_edge_gate"].sum()
+    total.backward()
+    assert x.grad is not None and x.grad.abs().sum() > 0
diff --git a/test/nn/wrappers/cell/test_cell_wrappers.py b/test/nn/wrappers/cell/test_cell_wrappers.py
index 12d11f0cc..6e0642f81 100644
--- a/test/nn/wrappers/cell/test_cell_wrappers.py
+++ b/test/nn/wrappers/cell/test_cell_wrappers.py
@@ -6,12 +6,14 @@
 from ...._utils.flow_mocker import FlowMocker
 from unittest.mock import MagicMock
 
+from topobench.nn.backbones.cell.smcn import SMCN
 from topobench.nn.wrappers import (
     AbstractWrapper,
     CCCNWrapper,
     CANWrapper,
     CCXNWrapper,
-    CWNWrapper
+    CWNWrapper,
+    SMCNWrapper,
 )
 from topomodelx.nn.cell.can import CAN
 from topomodelx.nn.cell.ccxn import CCXN
@@ -94,3 +96,25 @@ def test_CWNWrapper(self, sg1_cell_lifted):
 
         for key in ["labels", "batch_0", "x_0", "x_1", "x_2"]:
             assert key in out
+
+    def test_SMCNWrapper(self, sg1_cell_lifted):
+        """Test SMCNWrapper.
+
+        Parameters
+        ----------
+        sg1_cell_lifted : torch_geometric.data.Data
+            A fixture of simple graph 1 lifted with CellCycleLifting.
+        """
+        data = sg1_cell_lifted
+        out_dim = data.x_0.shape[1]
+        import torch
+        data.x_1 = torch.randn(data.incidence_1.shape[1], out_dim)
+        data.x_2 = torch.randn(data.incidence_2.shape[1], out_dim)
+        wrapper = SMCNWrapper(
+            SMCN(out_dim, n_scl_layers=2),
+            out_channels=out_dim,
+            num_cell_dimensions=3,
+        )
+        out = wrapper(data)
+        for key in ["labels", "batch_0", "x_0", "x_1", "x_2"]:
+            assert key in out
diff --git a/test/pipeline/test_pipeline.py b/test/pipeline/test_pipeline.py
index a61165ae9..d1cb5ffdb 100644
--- a/test/pipeline/test_pipeline.py
+++ b/test/pipeline/test_pipeline.py
@@ -7,7 +7,7 @@
 
 
 DATASET = "graph/MUTAG"  # ADD YOUR DATASET HERE
-MODELS = ["graph/gcn", "cell/topotune", "simplicial/topotune"]  # ADD ONE OR SEVERAL MODELS
+MODELS   = ["cell/smcn", "cell/smcn_difflift"]  # ADD ONE OR SEVERAL MODELS
 
 
 class TestPipeline:
diff --git a/topobench/nn/backbones/cell/smcn.py b/topobench/nn/backbones/cell/smcn.py
new file mode 100644
index 000000000..73fc34212
--- /dev/null
+++ b/topobench/nn/backbones/cell/smcn.py
@@ -0,0 +1,198 @@
+"""Scalable Multi-Cellular Network (SMCN) backbone.
+
+SMCN extends higher-order message passing with features indexed by
+(node, edge) pairs — a bag holding one marked copy of the node set per
+edge, processed with subgraph-GNN-style updates (SCL layers). This
+mitigates provable expressivity limitations of standard topological
+message passing (diameter, orientability, homology; Theorem 4.3 of the
+paper). The assembly follows the model the authors evaluate on graph
+benchmarks: CIN-style blocks, bag initialization with distance marking,
+a stack of SCL layers over the (0, 1) pair space (Eq. 55; the
+GNN-SSWL+ instantiation), and sum-pooling back to the cochains.
+
+The optional ``learned_lifting`` mode selects the 2-cells produced by
+the cycle lifting with a straight-through scorer (DiffLift), learning
+which cycles enter the complex end-to-end.
+
+References
+----------
+Eitan et al. "Topological Blindspots: Understanding and Extending
+Topological Deep Learning Through the Lens of Expressivity." ICLR 2025.
+https://arxiv.org/abs/2408.05486 (official implementation:
+https://github.com/yoavgelberg/SMCN)
+Franco et al. "Differentiable Lifting for Topological Neural Networks."
+https://openreview.net/forum?id=eC89CbINIw
+"""
+
+import torch
+from torch import nn
+
+from topobench.nn.backbones.cell.smcn_utils.layers import (
+    BagInit,
+    BagPool,
+    CINBlock,
+    SCLLayer,
+    TwoCellInit,
+)
+from topobench.nn.backbones.cell.smcn_utils.structures import (
+    build_smcn_structures,
+)
+from topobench.nn.liftings.difflift import (
+    CellScorer,
+    DiffLiftEncoder,
+)
+
+
+class SMCN(nn.Module):
+    """SMCN backbone operating on 2-dimensional cell complexes.
+
+    Parameters
+    ----------
+    in_channels : int
+        Feature dimension of all cochains (as produced by the feature
+        encoder); also the CIN-block width.
+    sub_channels : int, optional
+        Width of the subcomplex (SCL) layers. Default is ``in_channels``.
+    n_cin_layers : int, optional
+        Number of CIN blocks before the bag. Default is 1.
+    n_scl_layers : int, optional
+        Number of SCL layers (at least 2: the first maps into
+        ``sub_channels``, the last maps back). Default is 3.
+    num_mlp_layers : int, optional
+        Depth of the CIN convolution MLPs. Default is 2.
+    max_rank_out : int, optional
+        Highest rank updated by the final reduced CIN block. Default 1.
+    max_dist : int, optional
+        Distance cutoff of the bag marking. Default is 10.
+    dropout : float, optional
+        Dropout applied between stages. Default is 0.0.
+    learned_lifting : bool, optional
+        If True, candidate 2-cells are gated by a straight-through
+        scorer (DiffLift) instead of all being kept. Default is False.
+    sharpening : float, optional
+        Logit sharpening of the 2-cell scorer. Default is 10.0.
+
+    Raises
+    ------
+    ValueError
+        If ``n_scl_layers`` is smaller than 2.
+    """
+
+    def __init__(
+        self,
+        in_channels,
+        sub_channels=None,
+        n_cin_layers=1,
+        n_scl_layers=3,
+        num_mlp_layers=2,
+        max_rank_out=1,
+        max_dist=10,
+        dropout=0.0,
+        learned_lifting=False,
+        sharpening=10.0,
+    ):
+        super().__init__()
+        if n_scl_layers < 2:
+            raise ValueError("n_scl_layers must be at least 2")
+        sub_channels = sub_channels or in_channels
+        self.out_channels = in_channels
+        self.max_dist = max_dist
+        self.dropout = nn.Dropout(dropout)
+        self.learned_lifting = learned_lifting
+
+        if learned_lifting:
+            self.lift_encoder = DiffLiftEncoder(in_channels)
+            self.select = CellScorer(32, sharpening=sharpening)
+
+        self.two_cell_init = TwoCellInit(in_channels, num_mlp_layers)
+        self.cin_blocks = nn.ModuleList(
+            CINBlock(in_channels, num_mlp_layers) for _ in range(n_cin_layers)
+        )
+        self.bag_init = BagInit(in_channels, max_dist)
+        dims = (
+            [(in_channels, sub_channels)]
+            + [(sub_channels, sub_channels)] * (n_scl_layers - 2)
+            + [(sub_channels, in_channels)]
+        )
+        self.scl_layers = nn.ModuleList(
+            SCLLayer(d_in, d_out, edge_dim=in_channels, max_dist=max_dist)
+            for d_in, d_out in dims
+        )
+        self.bag_pool = BagPool()
+        self.final_block = CINBlock(
+            in_channels, num_mlp_layers, max_rank=max_rank_out
+        )
+
+    def forward(
+        self,
+        x_0,
+        x_1,
+        x_2,
+        incidence_1,
+        incidence_2,
+        batch_0,
+        batch_1,
+        batch_2,
+    ):
+        """Run the SMCN forward pass.
+
+        Parameters
+        ----------
+        x_0 : torch.Tensor
+            Node features ``[n0, in_channels]``.
+        x_1 : torch.Tensor
+            Edge features ``[n1, in_channels]``.
+        x_2 : torch.Tensor
+            2-cell features ``[n2, in_channels]``.
+        incidence_1 : torch.Tensor
+            Node-edge incidence ``[n0, n1]`` (sparse or dense).
+        incidence_2 : torch.Tensor
+            Edge-2-cell incidence ``[n1, n2]`` (sparse or dense).
+        batch_0 : torch.Tensor
+            Graph index of every node, ``[n0]``.
+        batch_1 : torch.Tensor
+            Graph index of every edge, ``[n1]``.
+        batch_2 : torch.Tensor
+            Graph index of every 2-cell, ``[n2]``.
+
+        Returns
+        -------
+        tuple of torch.Tensor
+            Updated ``(x_0, x_1, x_2)``.
+        """
+        with torch.no_grad():
+            structures = build_smcn_structures(
+                incidence_1,
+                incidence_2,
+                batch_0,
+                batch_1,
+                batch_2,
+                max_dist=self.max_dist,
+            )
+        s = structures
+
+        gate = None
+        if self.learned_lifting:
+            z = self.lift_encoder(x_0, s.a01_pairs)
+            gate = self.select(z, s.inc02_pairs, batch_2)
+
+        x_2 = self.two_cell_init(x_0, x_2, s.inc02_pairs)
+        if gate is not None:
+            x_2 = gate.unsqueeze(-1) * x_2
+
+        a12_gate = gate[s.a12_bridge] if gate is not None else None
+        for block in self.cin_blocks:
+            x_0, x_1, x_2 = block(x_0, x_1, x_2, s, a12_gate=a12_gate)
+            if gate is not None:
+                x_2 = gate.unsqueeze(-1) * x_2
+            x_0, x_1 = self.dropout(x_0), self.dropout(x_1)
+
+        x_bag = self.bag_init(x_0, x_1, s)
+        for layer in self.scl_layers:
+            x_bag = self.dropout(layer(x_bag, x_1, s))
+        x_0, x_1 = self.bag_pool(x_bag, s, x_0.size(0), x_1.size(0))
+
+        x_0, x_1, x_2 = self.final_block(x_0, x_1, x_2, s, a12_gate=a12_gate)
+        if gate is not None:
+            x_2 = gate.unsqueeze(-1) * x_2
+        return x_0, x_1, x_2
diff --git a/topobench/nn/backbones/cell/smcn_utils/__init__.py b/topobench/nn/backbones/cell/smcn_utils/__init__.py
new file mode 100644
index 000000000..94f3fbda3
--- /dev/null
+++ b/topobench/nn/backbones/cell/smcn_utils/__init__.py
@@ -0,0 +1 @@
+"""Utility modules for the SMCN backbone."""
diff --git a/topobench/nn/backbones/cell/smcn_utils/layers.py b/topobench/nn/backbones/cell/smcn_utils/layers.py
new file mode 100644
index 000000000..18dd3457c
--- /dev/null
+++ b/topobench/nn/backbones/cell/smcn_utils/layers.py
@@ -0,0 +1,505 @@
+"""Neural building blocks of the SMCN backbone.
+
+These layers implement the components of Scalable Multi-Cellular
+Networks: the CIN-style higher-order message-passing block, the marked
+subcomplex-bag layers (SCL), and the bag initialization/pooling. The
+semantics follow the official SMCN implementation
+(https://github.com/yoavgelberg/SMCN); the code here is written for the
+batched structures of
+:mod:`topobench.nn.backbones.cell.smcn_utils.structures`.
+
+References
+----------
+Eitan et al. "Topological Blindspots: Understanding and Extending
+Topological Deep Learning Through the Lens of Expressivity." ICLR 2025.
+https://arxiv.org/abs/2408.05486
+Bodnar et al. "Weisfeiler and Lehman Go Cellular: CW Networks."
+NeurIPS 2021. https://arxiv.org/abs/2106.12575
+"""
+
+import torch
+from torch import nn
+from torch_geometric.nn import MessagePassing
+from torch_geometric.nn.conv import GINConv
+
+
+class SafeBatchNorm(nn.BatchNorm1d):
+    """Batch norm that passes single-row inputs through unchanged.
+
+    A batch can contain a single 2-cell in total (sparse graphs under
+    the cycle lifting), and batch statistics are undefined for one row,
+    so :class:`torch.nn.BatchNorm1d` raises during training. Evaluation
+    mode is unaffected: it uses the running statistics.
+    """
+
+    def forward(self, x):
+        """Normalize ``x``, skipping single-row batches in training.
+
+        Parameters
+        ----------
+        x : torch.Tensor
+            Input features of shape ``[n, dim]``.
+
+        Returns
+        -------
+        torch.Tensor
+            Normalized features (or ``x`` itself when ``n < 2`` in
+            training mode).
+        """
+        if self.training and x.size(0) < 2:
+            return x
+        return super().forward(x)
+
+
+def homp_mlp(dim, num_layers=2):
+    """Build the width-preserving MLP used by the CIN-block convolutions.
+
+    Parameters
+    ----------
+    dim : int
+        Input, hidden, and output dimension.
+    num_layers : int, optional
+        Number of Linear-BatchNorm-ReLU stages. Default is 2.
+
+    Returns
+    -------
+    torch.nn.Sequential
+        The MLP.
+    """
+    layers = []
+    for _ in range(num_layers):
+        layers += [
+            nn.Linear(dim, dim),
+            SafeBatchNorm(dim),
+            nn.ReLU(),
+        ]
+    return nn.Sequential(*layers)
+
+
+def scl_mlp(in_dim, hidden_dim):
+    """Build the two-stage MLP used by the subcomplex convolutions.
+
+    Parameters
+    ----------
+    in_dim : int
+        Input dimension.
+    hidden_dim : int
+        Hidden and output dimension.
+
+    Returns
+    -------
+    torch.nn.Sequential
+        The MLP.
+    """
+    return nn.Sequential(
+        nn.Linear(in_dim, hidden_dim),
+        SafeBatchNorm(hidden_dim),
+        nn.ReLU(),
+        nn.Linear(hidden_dim, hidden_dim),
+        SafeBatchNorm(hidden_dim),
+        nn.ReLU(),
+    )
+
+
+class BridgeGIN(MessagePassing):
+    """GIN convolution whose messages carry the mediating-cell features.
+
+    Messages between two cells adjacent through a common higher-rank
+    cell (the *bridge*) concatenate the source features with the bridge
+    features, pass through a linear layer and a ReLU, and are
+    sum-aggregated before the usual GIN update.
+
+    Parameters
+    ----------
+    mlp : torch.nn.Module
+        Update network applied after aggregation; its first layer
+        defines the input width.
+    edge_dim : int
+        Dimension of the bridge features.
+    train_eps : bool, optional
+        Whether the GIN epsilon is learnable. Default is True.
+    """
+
+    def __init__(self, mlp, edge_dim, train_eps=True):
+        super().__init__(aggr="add")
+        self.mlp = mlp
+        in_dim = mlp[0].in_features
+        self.lin = nn.Linear(in_dim + edge_dim, in_dim)
+        if train_eps:
+            self.eps = nn.Parameter(torch.zeros(1))
+        else:
+            self.register_buffer("eps", torch.zeros(1))
+
+    def forward(self, x, edge_index, edge_attr, edge_weight=None):
+        """Run the bridge-aware GIN update.
+
+        Parameters
+        ----------
+        x : torch.Tensor
+            Cell features of shape ``[n, in_dim]``.
+        edge_index : torch.Tensor
+            Adjacency pairs of shape ``[2, P]``.
+        edge_attr : torch.Tensor
+            Bridge features of shape ``[P, edge_dim]``.
+        edge_weight : torch.Tensor, optional
+            Multiplicative per-pair gate of shape ``[P]`` (used by the
+            learned-lifting variant). Default is None.
+
+        Returns
+        -------
+        torch.Tensor
+            Updated features of shape ``[n, out_dim]``.
+        """
+        out = self.propagate(
+            edge_index, x=x, edge_attr=edge_attr, edge_weight=edge_weight
+        )
+        out = out + (1 + self.eps) * x
+        return self.mlp(out)
+
+    def message(self, x_j, edge_attr, edge_weight):
+        """Compute one bridge-aware message.
+
+        Parameters
+        ----------
+        x_j : torch.Tensor
+            Source-cell features of shape ``[P, in_dim]``.
+        edge_attr : torch.Tensor
+            Bridge features of shape ``[P, edge_dim]``.
+        edge_weight : torch.Tensor or None
+            Optional per-pair gate of shape ``[P]``.
+
+        Returns
+        -------
+        torch.Tensor
+            Messages of shape ``[P, in_dim]``.
+        """
+        msg = self.lin(torch.cat([x_j, edge_attr], dim=-1)).relu()
+        if edge_weight is not None:
+            msg = msg * edge_weight.unsqueeze(-1)
+        return msg
+
+
+class ConcatHead(nn.Module):
+    """Merge concatenated branch outputs back to the embedding width.
+
+    Parameters
+    ----------
+    num_branches : int
+        Number of concatenated branches.
+    dim : int
+        Embedding dimension of each branch and of the output.
+    """
+
+    def __init__(self, num_branches, dim):
+        super().__init__()
+        self.head = nn.Sequential(
+            nn.Linear(num_branches * dim, dim),
+            SafeBatchNorm(dim),
+            nn.ReLU(),
+        )
+
+    def forward(self, parts):
+        """Concatenate the branch outputs and apply the head.
+
+        Parameters
+        ----------
+        parts : list of torch.Tensor
+            Branch outputs, each of shape ``[n, dim]``.
+
+        Returns
+        -------
+        torch.Tensor
+            Merged features of shape ``[n, dim]``.
+        """
+        return self.head(torch.cat(parts, dim=-1))
+
+
+class MarkingEmbedding(nn.Module):
+    """Embedding of the bucketed hop-distance marking of bag rows.
+
+    Parameters
+    ----------
+    dim : int
+        Embedding dimension.
+    max_dist : int, optional
+        Distance cutoff used by the marking buckets. Default is 10.
+    """
+
+    def __init__(self, dim, max_dist=10):
+        super().__init__()
+        self.embed = nn.Embedding(max_dist + 2, dim)
+
+    def forward(self, marking):
+        """Embed the marking buckets.
+
+        Parameters
+        ----------
+        marking : torch.Tensor
+            Bucketed distances of shape ``[R]``.
+
+        Returns
+        -------
+        torch.Tensor
+            Embeddings of shape ``[R, dim]``.
+        """
+        return self.embed(marking)
+
+
+class TwoCellInit(nn.Module):
+    """Initialize 2-cell features from their constituent nodes.
+
+    A GIN update over the node-to-2-cell incidence, so every 2-cell
+    aggregates the features of its boundary nodes (CIN-style
+    initialization).
+
+    Parameters
+    ----------
+    dim : int
+        Embedding dimension.
+    num_mlp_layers : int, optional
+        Depth of the update MLP. Default is 2.
+    """
+
+    def __init__(self, dim, num_mlp_layers=2):
+        super().__init__()
+        self.conv = GINConv(homp_mlp(dim, num_mlp_layers), train_eps=True)
+
+    def forward(self, x_0, x_2, inc02_pairs):
+        """Compute the initial 2-cell features.
+
+        Parameters
+        ----------
+        x_0 : torch.Tensor
+            Node features of shape ``[n0, dim]``.
+        x_2 : torch.Tensor
+            Incoming 2-cell features of shape ``[n2, dim]``.
+        inc02_pairs : torch.Tensor
+            Node-2-cell incidence pairs of shape ``[2, I]``.
+
+        Returns
+        -------
+        torch.Tensor
+            Initialized 2-cell features of shape ``[n2, dim]``.
+        """
+        return self.conv(x=(x_0, x_2), edge_index=inc02_pairs)
+
+
+class CINBlock(nn.Module):
+    """One CIN-style higher-order message-passing block.
+
+    Updates the three cochains sequentially (each update sees the ones
+    already computed in this block, as in the reference): nodes from
+    edge-mediated adjacency, edges from node incidence and 2-cell
+    mediated adjacency, and 2-cells from edge incidence.
+
+    Parameters
+    ----------
+    dim : int
+        Embedding dimension of all ranks.
+    num_mlp_layers : int, optional
+        Depth of the convolution MLPs. Default is 2.
+    max_rank : int, optional
+        Highest rank updated by the block (0, 1, or 2). Default is 2.
+    """
+
+    def __init__(self, dim, num_mlp_layers=2, max_rank=2):
+        super().__init__()
+        if max_rank not in (0, 1, 2):
+            raise ValueError("max_rank must be 0, 1, or 2")
+        self.max_rank = max_rank
+        self.node_conv = BridgeGIN(homp_mlp(dim, num_mlp_layers), edge_dim=dim)
+        self.node_head = ConcatHead(1, dim)
+        if max_rank >= 1:
+            self.edge_inc_conv = GINConv(
+                homp_mlp(dim, num_mlp_layers), train_eps=True
+            )
+            self.edge_adj_conv = BridgeGIN(
+                homp_mlp(dim, num_mlp_layers), edge_dim=dim
+            )
+            self.edge_head = ConcatHead(2, dim)
+        if max_rank >= 2:
+            self.cell_conv = GINConv(
+                homp_mlp(dim, num_mlp_layers), train_eps=True
+            )
+            self.cell_head = ConcatHead(1, dim)
+
+    def forward(self, x_0, x_1, x_2, structures, a12_gate=None):
+        """Run the block.
+
+        Parameters
+        ----------
+        x_0 : torch.Tensor
+            Node features ``[n0, dim]``.
+        x_1 : torch.Tensor
+            Edge features ``[n1, dim]``.
+        x_2 : torch.Tensor
+            2-cell features ``[n2, dim]``.
+        structures : SMCNStructures
+            Batch connectivity structures.
+        a12_gate : torch.Tensor, optional
+            Per-pair gate on the 2-cell-mediated edge adjacency (used by
+            the learned-lifting variant). Default is None.
+
+        Returns
+        -------
+        tuple of torch.Tensor
+            Updated ``(x_0, x_1, x_2)``.
+        """
+        s = structures
+        x_0 = self.node_head(
+            [self.node_conv(x_0, s.a01_pairs, edge_attr=x_1[s.a01_bridge])]
+        )
+        if self.max_rank >= 1:
+            x_1 = self.edge_head(
+                [
+                    self.edge_inc_conv(x=(x_0, x_1), edge_index=s.inc01_pairs),
+                    self.edge_adj_conv(
+                        x_1,
+                        s.a12_pairs,
+                        edge_attr=x_2[s.a12_bridge],
+                        edge_weight=a12_gate,
+                    ),
+                ]
+            )
+        if self.max_rank >= 2:
+            x_2 = self.cell_head(
+                [self.cell_conv(x=(x_1, x_2), edge_index=s.inc12_pairs)]
+            )
+        return x_0, x_1, x_2
+
+
+class BagInit(nn.Module):
+    """Initialize the subcomplex-bag features.
+
+    Every bag row ``(u, e)`` concatenates the features of node ``u``,
+    edge ``e``, and the embedded distance marking, merged by a linear
+    head.
+
+    Parameters
+    ----------
+    dim : int
+        Embedding dimension.
+    max_dist : int, optional
+        Marking distance cutoff. Default is 10.
+    """
+
+    def __init__(self, dim, max_dist=10):
+        super().__init__()
+        self.marking = MarkingEmbedding(dim, max_dist)
+        self.head = ConcatHead(3, dim)
+
+    def forward(self, x_0, x_1, structures):
+        """Build the initial bag features.
+
+        Parameters
+        ----------
+        x_0 : torch.Tensor
+            Node features ``[n0, dim]``.
+        x_1 : torch.Tensor
+            Edge features ``[n1, dim]``.
+        structures : SMCNStructures
+            Batch structures with the bag indices.
+
+        Returns
+        -------
+        torch.Tensor
+            Bag features of shape ``[R, dim]``.
+        """
+        s = structures
+        return self.head(
+            [
+                x_0[s.bag_low_index],
+                x_1[s.bag_high_index],
+                self.marking(s.bag_marking),
+            ]
+        )
+
+
+class SCLLayer(nn.Module):
+    """One subcomplex (SCL) layer.
+
+    The bag features are updated by the sum of three branches: a GIN
+    over the marked-row broadcast edges, a bridge-aware GIN over the
+    replicated node adjacency (bridge features taken from the edge
+    cochain), and a re-embedded distance marking.
+
+    Parameters
+    ----------
+    in_dim : int
+        Input bag-feature dimension.
+    hidden_dim : int
+        Hidden and output dimension.
+    edge_dim : int
+        Dimension of the (frozen) edge features used as bridges.
+    max_dist : int, optional
+        Marking distance cutoff. Default is 10.
+    """
+
+    def __init__(self, in_dim, hidden_dim, edge_dim, max_dist=10):
+        super().__init__()
+        self.inc_conv = GINConv(scl_mlp(in_dim, hidden_dim), train_eps=True)
+        self.low_conv = BridgeGIN(
+            scl_mlp(in_dim, hidden_dim), edge_dim=edge_dim
+        )
+        self.marking = MarkingEmbedding(hidden_dim, max_dist)
+
+    def forward(self, x_bag, x_1, structures):
+        """Run the layer.
+
+        Parameters
+        ----------
+        x_bag : torch.Tensor
+            Bag features of shape ``[R, in_dim]``.
+        x_1 : torch.Tensor
+            Edge features used as bridge attributes, ``[n1, edge_dim]``.
+        structures : SMCNStructures
+            Batch structures.
+
+        Returns
+        -------
+        torch.Tensor
+            Updated bag features of shape ``[R, hidden_dim]``.
+        """
+        s = structures
+        out = self.inc_conv(x=x_bag, edge_index=s.bag_inc_pairs)
+        out = out + self.low_conv(
+            x_bag,
+            s.bag_low_adj_pairs,
+            edge_attr=x_1[s.bag_low_adj_bridge],
+        )
+        return out + self.marking(s.bag_marking)
+
+
+class BagPool(nn.Module):
+    """Sum-pool the bag features back to nodes and edges."""
+
+    def forward(self, x_bag, structures, num_nodes, num_edges):
+        """Pool the bag.
+
+        Parameters
+        ----------
+        x_bag : torch.Tensor
+            Bag features of shape ``[R, dim]``.
+        structures : SMCNStructures
+            Batch structures with the bag indices.
+        num_nodes : int
+            Total number of nodes in the batch.
+        num_edges : int
+            Total number of edges in the batch.
+
+        Returns
+        -------
+        x_0 : torch.Tensor
+            Node features of shape ``[num_nodes, dim]``.
+        x_1 : torch.Tensor
+            Edge features of shape ``[num_edges, dim]``.
+        """
+        s = structures
+        dim = x_bag.size(1)
+        x_0 = x_bag.new_zeros(num_nodes, dim).index_add_(
+            0, s.bag_low_index, x_bag
+        )
+        x_1 = x_bag.new_zeros(num_edges, dim).index_add_(
+            0, s.bag_high_index, x_bag
+        )
+        return x_0, x_1
diff --git a/topobench/nn/backbones/cell/smcn_utils/structures.py b/topobench/nn/backbones/cell/smcn_utils/structures.py
new file mode 100644
index 000000000..75f28d78f
--- /dev/null
+++ b/topobench/nn/backbones/cell/smcn_utils/structures.py
@@ -0,0 +1,370 @@
+"""Batched structure builders for the SMCN backbone.
+
+SMCN processes, besides the usual cell-complex neighborhoods, a *bag of
+marked subcomplexes* over the rank pair (0, 1): one copy of the node set
+per edge, marked with hop distances to that edge. The reference
+implementation precomputes these structures per sample; here they are
+built on the fly from the batched incidence matrices, which keeps the
+data pipeline unchanged and works for any batch composition.
+
+The construction follows the official SMCN implementation
+(https://github.com/yoavgelberg/SMCN, ``data/utils.py``): bag rows are
+laid out as ``row(u, e) = u * n_edges + e`` within each graph, the bag
+low-adjacency replicates the node adjacency once per edge copy, and the
+marking of row ``(u, e)`` is the shortest-path distance from node ``u``
+to the closest endpoint of edge ``e``, bucketed as in the reference
+(distances above ``max_dist`` map to ``max_dist``, disconnected pairs to
+``max_dist + 1``).
+
+References
+----------
+Eitan et al. "Topological Blindspots: Understanding and Extending
+Topological Deep Learning Through the Lens of Expressivity." ICLR 2025.
+https://arxiv.org/abs/2408.05486
+"""
+
+from dataclasses import dataclass
+
+import torch
+
+
+@dataclass
+class SMCNStructures:
+    """Connectivity and subcomplex-bag structures for one batch.
+
+    All index tensors are global with respect to the batch: cell indices
+    are offset per graph, and bag-row indices are offset by the number of
+    bag rows of the preceding graphs.
+
+    Attributes
+    ----------
+    a01_pairs : torch.Tensor
+        Ordered node pairs adjacent through an edge, shape ``[2, P0]``.
+    a01_bridge : torch.Tensor
+        Mediating edge index of every pair in ``a01_pairs``, shape ``[P0]``.
+    a12_pairs : torch.Tensor
+        Ordered edge pairs adjacent through a 2-cell, shape ``[2, P1]``.
+    a12_bridge : torch.Tensor
+        Mediating 2-cell index of every pair in ``a12_pairs``, shape ``[P1]``.
+    inc01_pairs : torch.Tensor
+        Node-edge incidence pairs, shape ``[2, I0]``.
+    inc12_pairs : torch.Tensor
+        Edge-2-cell incidence pairs, shape ``[2, I1]``.
+    inc02_pairs : torch.Tensor
+        Node-2-cell incidence pairs, shape ``[2, I2]``.
+    bag_low_index : torch.Tensor
+        Node index of every bag row, shape ``[R]``.
+    bag_high_index : torch.Tensor
+        Edge index of every bag row, shape ``[R]``.
+    bag_marking : torch.Tensor
+        Bucketed hop-distance marking of every bag row, shape ``[R]``.
+    bag_low_adj_pairs : torch.Tensor
+        Bag-row pairs of the replicated node adjacency, shape ``[2, Q]``.
+    bag_low_adj_bridge : torch.Tensor
+        Mediating edge index of every bag adjacency pair, shape ``[Q]``.
+    bag_inc_pairs : torch.Tensor
+        Bag-row pairs broadcasting each marked row to the rows of the
+        same node, shape ``[2, S]``.
+    bag_batch : torch.Tensor
+        Graph index of every bag row, shape ``[R]``.
+    """
+
+    a01_pairs: torch.Tensor
+    a01_bridge: torch.Tensor
+    a12_pairs: torch.Tensor
+    a12_bridge: torch.Tensor
+    inc01_pairs: torch.Tensor
+    inc12_pairs: torch.Tensor
+    inc02_pairs: torch.Tensor
+    bag_low_index: torch.Tensor
+    bag_high_index: torch.Tensor
+    bag_marking: torch.Tensor
+    bag_low_adj_pairs: torch.Tensor
+    bag_low_adj_bridge: torch.Tensor
+    bag_inc_pairs: torch.Tensor
+    bag_batch: torch.Tensor
+
+
+def sparse_pairs(matrix):
+    """Return the (row, col) index pairs of a sparse or dense matrix.
+
+    Parameters
+    ----------
+    matrix : torch.Tensor
+        Sparse (COO/CSR) or dense matrix.
+
+    Returns
+    -------
+    torch.Tensor
+        Long tensor of shape ``[2, nnz]`` with row/column indices.
+    """
+    if matrix.is_sparse or matrix.layout == torch.sparse_csr:
+        coo = matrix.to_sparse_coo().coalesce()
+        return coo.indices().long()
+    return torch.nonzero(matrix, as_tuple=False).t().long()
+
+
+def adjacency_from_incidence(incidence, expected_size=None):
+    """Build ordered cell pairs mediated by a higher-rank cell.
+
+    Two rank-``r`` cells are adjacent whenever they are both contained in
+    a common rank-``(r + 1)`` cell; that common cell is the *bridge* and
+    its features are used as edge features by the message passing.
+
+    Parameters
+    ----------
+    incidence : torch.Tensor
+        Incidence matrix of shape ``[n_low, n_high]`` (sparse or dense).
+    expected_size : int, optional
+        If given, every column must have exactly this many nonzero
+        entries (e.g. 2 for a node-edge incidence). Default is None.
+
+    Returns
+    -------
+    pairs : torch.Tensor
+        Ordered pairs of low-rank cells, shape ``[2, P]``.
+    bridge : torch.Tensor
+        Mediating high-rank cell of each pair, shape ``[P]``.
+
+    Raises
+    ------
+    ValueError
+        If ``expected_size`` is given and some column violates it.
+    """
+    idx = sparse_pairs(incidence)
+    if idx.numel() == 0:
+        empty = idx.new_zeros(0)
+        return idx.new_zeros(2, 0), empty
+    rows, cols = idx[0], idx[1]
+    order = torch.argsort(cols, stable=True)
+    rows, cols = rows[order], cols[order]
+    counts = torch.bincount(cols, minlength=int(cols.max()) + 1)
+    counts = counts[counts > 0]
+    if expected_size is not None and not bool((counts == expected_size).all()):
+        raise ValueError(
+            "every column of the incidence must have exactly "
+            f"{expected_size} nonzero entries"
+        )
+    pairs_src, pairs_dst, bridges = [], [], []
+    unique_cols = torch.unique_consecutive(cols)
+    start = 0
+    sizes = counts.tolist()
+    for col, size in zip(unique_cols.tolist(), sizes, strict=False):
+        members = rows[start : start + size]
+        start += size
+        if size < 2:
+            continue
+        grid_a = members.repeat_interleave(size)
+        grid_b = members.repeat(size)
+        keep = grid_a != grid_b
+        pairs_src.append(grid_a[keep])
+        pairs_dst.append(grid_b[keep])
+        bridges.append(torch.full_like(grid_a[keep], fill_value=col))
+    if not pairs_src:
+        empty = idx.new_zeros(0)
+        return idx.new_zeros(2, 0), empty
+    pairs = torch.stack([torch.cat(pairs_src), torch.cat(pairs_dst)], dim=0)
+    return pairs, torch.cat(bridges)
+
+
+def hop_distance_buckets(adjacency, max_dist):
+    """Compute bucketed all-pairs hop distances of one graph.
+
+    Follows the reference marking semantics: exact distances up to
+    ``max_dist``; nodes that are connected but farther than ``max_dist``
+    receive ``max_dist``; disconnected pairs receive ``max_dist + 1``.
+
+    Parameters
+    ----------
+    adjacency : torch.Tensor
+        Dense boolean adjacency of shape ``[n, n]``.
+    max_dist : int
+        Largest exact distance to resolve.
+
+    Returns
+    -------
+    torch.Tensor
+        Long tensor of shape ``[n, n]`` with values in
+        ``[0, max_dist + 1]``.
+    """
+    n = adjacency.size(0)
+    device = adjacency.device
+    dist = torch.full((n, n), max_dist + 1, dtype=torch.long, device=device)
+    dist.fill_diagonal_(0)
+    reach = torch.eye(n, dtype=torch.bool, device=device)
+    adj = adjacency.bool() | reach
+    frontier = reach
+    for d in range(1, max_dist + 1):
+        frontier = (frontier.float() @ adj.float()) > 0
+        newly = frontier & (dist == max_dist + 1)
+        dist[newly] = d
+    # Distinguish "connected but farther" from "disconnected" by closing
+    # the reachability transitively (log-doubling).
+    closure = frontier
+    steps = max(1, (n - 1).bit_length())
+    for _ in range(steps):
+        closure = (closure.float() @ closure.float()) > 0
+    far = closure & (dist == max_dist + 1)
+    dist[far] = max_dist
+    return dist
+
+
+def build_smcn_structures(
+    incidence_1,
+    incidence_2,
+    batch_0,
+    batch_1,
+    batch_2,
+    max_dist=10,
+):
+    """Build all SMCN connectivity/bag structures for a batch.
+
+    Parameters
+    ----------
+    incidence_1 : torch.Tensor
+        Block-diagonal node-edge incidence ``[n_nodes, n_edges]``.
+    incidence_2 : torch.Tensor
+        Block-diagonal edge-2-cell incidence ``[n_edges, n_cells]``.
+    batch_0 : torch.Tensor
+        Graph index of every node, shape ``[n_nodes]``.
+    batch_1 : torch.Tensor
+        Graph index of every edge, shape ``[n_edges]``.
+    batch_2 : torch.Tensor
+        Graph index of every 2-cell, shape ``[n_cells]``.
+    max_dist : int, optional
+        Marking distance cutoff. Default is 10.
+
+    Returns
+    -------
+    SMCNStructures
+        The assembled batch structures.
+    """
+    device = batch_0.device
+    num_graphs = int(batch_0.max()) + 1 if batch_0.numel() else 0
+
+    inc01 = sparse_pairs(incidence_1)
+    inc12 = (
+        sparse_pairs(incidence_2)
+        if incidence_2 is not None and incidence_2.numel()
+        else batch_0.new_zeros(2, 0)
+    )
+    a01_pairs, a01_bridge = adjacency_from_incidence(
+        incidence_1, expected_size=2
+    )
+    a12_pairs, a12_bridge = adjacency_from_incidence(incidence_2)
+
+    # Node-to-2-cell incidence via the edge memberships.
+    if inc12.numel():
+        dense_1 = incidence_1
+        if dense_1.is_sparse or dense_1.layout == torch.sparse_csr:
+            dense_1 = dense_1.to_dense()
+        dense_2 = incidence_2
+        if dense_2.is_sparse or dense_2.layout == torch.sparse_csr:
+            dense_2 = dense_2.to_dense()
+        membership = (dense_1.abs() @ dense_2.abs()) > 0
+        inc02 = torch.nonzero(membership, as_tuple=False).t().long()
+    else:
+        inc02 = batch_0.new_zeros(2, 0)
+
+    n0 = torch.bincount(batch_0, minlength=num_graphs)
+    n1 = torch.bincount(batch_1, minlength=num_graphs)
+    node_off = torch.cumsum(n0, 0) - n0
+    edge_off = torch.cumsum(n1, 0) - n1
+    rows_per_graph = n0 * n1
+    row_off = torch.cumsum(rows_per_graph, 0) - rows_per_graph
+
+    bag_low, bag_high, bag_mark, bag_batch = [], [], [], []
+    bl_pairs, bl_bridge, bi_pairs = [], [], []
+
+    pair_graph = batch_1[a01_bridge] if a01_bridge.numel() else a01_bridge
+    inc_graph = batch_1[inc01[1]] if inc01.numel() else inc01.new_zeros(0)
+
+    for g in range(num_graphs):
+        nl, nh = int(n0[g]), int(n1[g])
+        if nl == 0 or nh == 0:
+            continue
+        n_offset, e_offset, r_offset = (
+            int(node_off[g]),
+            int(edge_off[g]),
+            int(row_off[g]),
+        )
+        rows = torch.arange(nl * nh, device=device)
+        bag_low.append(rows // nh + n_offset)
+        bag_high.append(rows % nh + e_offset)
+        bag_batch.append(
+            torch.full((nl * nh,), g, dtype=torch.long, device=device)
+        )
+
+        # Local structures of this graph.
+        pair_mask = pair_graph == g
+        loc_pairs = a01_pairs[:, pair_mask] - n_offset
+        loc_bridge = a01_bridge[pair_mask] - e_offset
+        inc_mask = inc_graph == g
+        loc_inc = inc01[:, inc_mask].clone()
+        loc_inc[0] -= n_offset
+        loc_inc[1] -= e_offset
+
+        # Distance marking: min over the endpoints of each edge copy.
+        adj = torch.zeros(nl, nl, dtype=torch.bool, device=device)
+        adj[loc_pairs[0], loc_pairs[1]] = True
+        dist = hop_distance_buckets(adj, max_dist)
+        endpoints = torch.full((nh, 2), -1, dtype=torch.long, device=device)
+        edge_sorted = torch.argsort(loc_inc[1], stable=True)
+        sorted_nodes = loc_inc[0][edge_sorted]
+        endpoints[:, 0] = sorted_nodes[0::2]
+        endpoints[:, 1] = sorted_nodes[1::2]
+        marking = dist[:, endpoints].min(dim=-1).values  # [nl, nh]
+        bag_mark.append(marking.reshape(-1))
+
+        # Bag low-adjacency: node adjacency replicated once per edge copy.
+        copies = torch.arange(nh, device=device)
+        src = (loc_pairs[0].unsqueeze(1) * nh + copies).reshape(-1)
+        dst = (loc_pairs[1].unsqueeze(1) * nh + copies).reshape(-1)
+        bl_pairs.append(torch.stack([src, dst], dim=0) + r_offset)
+        bl_bridge.append((loc_bridge + e_offset).repeat_interleave(nh))
+
+        # Bag incidence: marked row (u, e) broadcasts to all rows of u.
+        marked = loc_inc[0] * nh + loc_inc[1]
+        targets = (loc_inc[0].unsqueeze(1) * nh + copies).reshape(-1)
+        sources = marked.repeat_interleave(nh)
+        bi_pairs.append(torch.stack([sources, targets], dim=0) + r_offset)
+
+    def _cat(parts, like, width=None):
+        """Concatenate chunks, or make an empty tensor when none exist.
+
+        Parameters
+        ----------
+        parts : list of torch.Tensor
+            Per-graph chunks to concatenate along the last dimension.
+        like : torch.Tensor
+            Tensor providing the dtype and device for the empty case.
+        width : int, optional
+            First dimension of the empty tensor, used for pair tensors.
+
+        Returns
+        -------
+        torch.Tensor
+            Concatenated tensor, or an empty one if ``parts`` is empty.
+        """
+        if parts:
+            return torch.cat(parts, dim=-1)
+        if width is not None:
+            return like.new_zeros(width, 0)
+        return like.new_zeros(0)
+
+    return SMCNStructures(
+        a01_pairs=a01_pairs,
+        a01_bridge=a01_bridge,
+        a12_pairs=a12_pairs,
+        a12_bridge=a12_bridge,
+        inc01_pairs=inc01,
+        inc12_pairs=inc12,
+        inc02_pairs=inc02,
+        bag_low_index=_cat(bag_low, batch_0),
+        bag_high_index=_cat(bag_high, batch_0),
+        bag_marking=_cat(bag_mark, batch_0),
+        bag_low_adj_pairs=_cat(bl_pairs, batch_0, width=2),
+        bag_low_adj_bridge=_cat(bl_bridge, batch_0),
+        bag_inc_pairs=_cat(bi_pairs, batch_0, width=2),
+        bag_batch=_cat(bag_batch, batch_0),
+    )
diff --git a/topobench/nn/liftings/__init__.py b/topobench/nn/liftings/__init__.py
new file mode 100644
index 000000000..2d3495501
--- /dev/null
+++ b/topobench/nn/liftings/__init__.py
@@ -0,0 +1,28 @@
+"""Learnable lifting modules applied inside models.
+
+Unlike :mod:`topobench.transforms.liftings`, which compute a fixed
+complex during preprocessing, the liftings in this package are
+trainable components: the topology they produce is differentiable and
+is learned end to end with the downstream network.
+
+A backbone plugs a learnable lifting in by instantiating it (or its
+pieces) in its constructor and multiplying the gates it produces into
+the affected features and messages. The components work for cell
+complexes and hypergraphs alike: gates can select 2-cells, learned
+edges, or candidate hyperedges. ``model=cell/smcn_difflift`` is a
+working example of the plug-in pattern.
+"""
+
+from topobench.nn.liftings.difflift import (
+    CellScorer,
+    DiffLift,
+    DiffLiftEncoder,
+    EdgeSampler,
+)
+
+__all__ = [
+    "CellScorer",
+    "DiffLift",
+    "DiffLiftEncoder",
+    "EdgeSampler",
+]
diff --git a/topobench/nn/liftings/difflift.py b/topobench/nn/liftings/difflift.py
new file mode 100644
index 000000000..ba988e129
--- /dev/null
+++ b/topobench/nn/liftings/difflift.py
@@ -0,0 +1,427 @@
+"""Differentiable lifting (DiffLift) modules.
+
+Implements the DiffLift recipe for learning graph liftings end-to-end
+(Franco et al.): a GNN computes node embeddings (Step 1), candidate
+cells are elicited per dimension (Step 2), and a permutation-invariant
+scorer accepts or rejects each candidate (Step 3), with gradients
+propagated through the discrete decisions by the straight-through
+estimator. Both the stochastic (Bernoulli) sampling of the paper and
+its deterministic thresholded variant are provided.
+
+The components are modular:
+
+- :class:`DiffLiftEncoder` — Step 1, node embeddings.
+- :class:`CellScorer` — Step 3, multiset scorer for any candidate cell
+  given node-to-cell membership pairs (works for hyperedges, 1-cells,
+  or 2-cells alike).
+- :class:`EdgeSampler` — the ``D = 1`` iteration for cell complexes:
+  adaptive-size kNN candidate edges added on top of the observed ones.
+- :class:`DiffLift` — the full graph-to-cell-complex recipe
+  (``D_max = 2``): learned edges, then cycle-basis candidates of the
+  augmented graph scored as 2-cells.
+
+The recipe is not tied to a target domain: the scorer accepts any
+candidate given its node membership, so the same components lift
+graphs to cell complexes (:class:`DiffLift`) or select hyperedges for
+a hypergraph model (pass candidate hyperedge memberships to
+:class:`CellScorer`).
+
+References
+----------
+Franco et al. "Differentiable Lifting for Topological Neural Networks."
+https://openreview.net/forum?id=eC89CbINIw
+"""
+
+import networkx as nx
+import torch
+import torch.nn.functional as F
+from torch import nn
+from torch_geometric.nn.conv import GINConv
+
+
+class DiffLiftEncoder(nn.Module):
+    """Node-embedding GNN of the lifting (Step 1 of the recipe).
+
+    Parameters
+    ----------
+    in_channels : int
+        Dimension of the input node features.
+    hidden_channels : int, optional
+        Embedding dimension. Default is 32.
+    num_layers : int, optional
+        Number of GIN layers. Default is 1.
+    """
+
+    def __init__(self, in_channels, hidden_channels=32, num_layers=1):
+        super().__init__()
+        self.convs = nn.ModuleList()
+        dim = in_channels
+        for _ in range(num_layers):
+            mlp = nn.Sequential(
+                nn.Linear(dim, hidden_channels),
+                nn.ReLU(),
+                nn.Linear(hidden_channels, hidden_channels),
+            )
+            self.convs.append(GINConv(mlp, train_eps=True))
+            dim = hidden_channels
+
+    def forward(self, x, edge_pairs):
+        """Embed the nodes.
+
+        Parameters
+        ----------
+        x : torch.Tensor
+            Node features ``[n, in_channels]``.
+        edge_pairs : torch.Tensor
+            Graph connectivity as node pairs ``[2, P]``.
+
+        Returns
+        -------
+        torch.Tensor
+            Node embeddings ``[n, hidden_channels]``.
+        """
+        z = x
+        for conv in self.convs:
+            z = conv(z, edge_pairs)
+        return z
+
+
+class CellScorer(nn.Module):
+    """Accept/reject scorer over candidate cells (Step 3 of the recipe).
+
+    The acceptance probability of a candidate is a learned function of
+    the multiset of its member-node embeddings (mean pooling followed by
+    an MLP). The forward pass returns hard 0/1 gates with
+    straight-through gradients; decisions are either thresholded (the
+    deterministic variant of the paper) or sampled from a Bernoulli.
+
+    Parameters
+    ----------
+    in_channels : int
+        Dimension of the node embeddings.
+    hidden_channels : int, optional
+        Hidden width of the scoring MLP. Default is 32.
+    sharpening : float, optional
+        Multiplier applied to the logits before the sigmoid. Default is
+        10.0.
+    stochastic : bool, optional
+        If True, sample decisions from a Bernoulli; otherwise threshold
+        the probability at 0.5. Default is False.
+    rescue : bool, optional
+        If True, force-keep the highest-scoring candidate of any graph
+        whose candidates were all rejected. Default is True.
+    """
+
+    def __init__(
+        self,
+        in_channels,
+        hidden_channels=32,
+        sharpening=10.0,
+        stochastic=False,
+        rescue=True,
+    ):
+        super().__init__()
+        self.sharpening = sharpening
+        self.stochastic = stochastic
+        self.rescue = rescue
+        self.score = nn.Sequential(
+            nn.Linear(in_channels, hidden_channels),
+            nn.ReLU(),
+            nn.Linear(hidden_channels, 1),
+        )
+
+    def forward(self, z, membership_pairs, cell_batch):
+        """Gate every candidate cell.
+
+        Parameters
+        ----------
+        z : torch.Tensor
+            Node embeddings ``[n, in_channels]``.
+        membership_pairs : torch.Tensor
+            Node-to-cell membership pairs ``[2, I]``.
+        cell_batch : torch.Tensor
+            Graph index of every candidate cell ``[n_cells]``.
+
+        Returns
+        -------
+        torch.Tensor
+            Gates ``[n_cells]``: hard 0/1 forward values with soft
+            gradients.
+        """
+        n_cells = cell_batch.size(0)
+        if n_cells == 0:
+            return z.new_zeros(0)
+        pooled = z.new_zeros(n_cells, z.size(1)).index_add_(
+            0, membership_pairs[1], z[membership_pairs[0]]
+        )
+        counts = torch.bincount(membership_pairs[1], minlength=n_cells)
+        pooled = pooled / counts.clamp(min=1).unsqueeze(1)
+        probs = torch.sigmoid(self.sharpening * self.score(pooled).squeeze(-1))
+        if self.stochastic and self.training:
+            hard = torch.bernoulli(probs)
+        else:
+            hard = (probs > 0.5).float()
+        if self.rescue:
+            num_graphs = int(cell_batch.max()) + 1
+            best = probs.new_full((num_graphs,), -1.0)
+            best = best.scatter_reduce(
+                0, cell_batch, probs, reduce="amax", include_self=True
+            )
+            kept = hard.new_zeros(num_graphs).scatter_add_(0, cell_batch, hard)
+            lift = (kept == 0)[cell_batch] & (probs == best[cell_batch])
+            hard = torch.where(lift, torch.ones_like(hard), hard)
+        return hard + (probs - probs.detach())
+
+
+class EdgeSampler(nn.Module):
+    """Learned 1-cells: the ``D = 1`` iteration for cell complexes.
+
+    For each node, a neighborhood size ``k_v`` is drawn from a
+    categorical distribution parameterized by its embedding
+    (Gumbel-softmax with hard samples), candidate edges connect the node
+    to its ``k_v`` nearest neighbors in embedding space, and a pair
+    scorer gates each candidate. Learned edges are *added* to the
+    observed ones, never removing them.
+
+    Parameters
+    ----------
+    in_channels : int
+        Dimension of the node embeddings.
+    k_min : int, optional
+        Smallest neighborhood size. Default is 1.
+    k_max : int, optional
+        Largest neighborhood size. Default is 3.
+    sharpening : float, optional
+        Logit sharpening of the pair scorer. Default is 10.0.
+    stochastic : bool, optional
+        Sample the pair decisions from a Bernoulli. Default is False.
+    """
+
+    def __init__(
+        self,
+        in_channels,
+        k_min=1,
+        k_max=3,
+        sharpening=10.0,
+        stochastic=False,
+    ):
+        super().__init__()
+        if not 1 <= k_min <= k_max:
+            raise ValueError("need 1 <= k_min <= k_max")
+        self.k_min = k_min
+        self.k_max = k_max
+        self.k_logits = nn.Linear(in_channels, k_max - k_min + 1)
+        self.pair_scorer = CellScorer(
+            in_channels,
+            sharpening=sharpening,
+            stochastic=stochastic,
+            rescue=False,
+        )
+
+    def forward(self, z, edge_pairs, batch):
+        """Propose and gate new edges.
+
+        Parameters
+        ----------
+        z : torch.Tensor
+            Node embeddings ``[n, d]``.
+        edge_pairs : torch.Tensor
+            Observed edges as node pairs ``[2, P]`` (both directions).
+        batch : torch.Tensor
+            Graph index of every node ``[n]``.
+
+        Returns
+        -------
+        new_pairs : torch.Tensor
+            Undirected candidate edges ``[2, E_new]`` (one direction),
+            disjoint from the observed edges.
+        gate : torch.Tensor
+            Straight-through gate of each candidate ``[E_new]``.
+        """
+        from torch_cluster import knn_graph
+
+        n = z.size(0)
+        # Adaptive neighborhood sizes (Gumbel-softmax, hard samples).
+        k_probs = F.gumbel_softmax(self.k_logits(z), tau=1.0, hard=True)
+        k_values = self.k_min + k_probs.argmax(dim=-1)  # [n]
+
+        candidates = knn_graph(z.detach(), k=self.k_max, batch=batch)
+        src, dst = candidates[0], candidates[1]
+        # Keep each source only among its k_dst nearest neighbours: the
+        # kNN output lists neighbours per target node in order, so rank
+        # them and compare against the sampled k of the target.
+        order = torch.argsort(dst, stable=True)
+        src, dst = src[order], dst[order]
+        ranks = torch.arange(src.size(0), device=z.device)
+        first = torch.zeros(n, dtype=torch.long, device=z.device)
+        counts = torch.bincount(dst, minlength=n)
+        first[1:] = torch.cumsum(counts, 0)[:-1]
+        ranks = ranks - first[dst]
+        keep = ranks < k_values[dst]
+
+        # Drop candidates that already exist as observed edges and
+        # deduplicate the two orientations.
+        a = torch.minimum(src[keep], dst[keep])
+        b = torch.maximum(src[keep], dst[keep])
+        cand = torch.unique(torch.stack([a, b], dim=0), dim=1)
+        if edge_pairs.numel():
+            ea = torch.minimum(edge_pairs[0], edge_pairs[1])
+            eb = torch.maximum(edge_pairs[0], edge_pairs[1])
+            existing = set(
+                map(tuple, torch.stack([ea, eb], dim=0).t().tolist())
+            )
+            mask = torch.tensor(
+                [
+                    (int(x), int(y)) not in existing
+                    for x, y in cand.t().tolist()
+                ],
+                dtype=torch.bool,
+                device=z.device,
+            )
+            cand = cand[:, mask]
+        if cand.numel() == 0:
+            return cand, z.new_zeros(0)
+        # Score each candidate pair from its two endpoints.
+        members = torch.cat([cand[0], cand[1]])
+        cells = torch.arange(cand.size(1), device=z.device).repeat(2)
+        gate = self.pair_scorer(
+            z,
+            torch.stack([members, cells], dim=0),
+            batch[cand[0]],
+        )
+        # Gumbel gradient path for the neighborhood sizes.
+        k_grad = k_probs.sum() - k_probs.sum().detach()
+        return cand, gate + 0.0 * k_grad
+
+
+class DiffLift(nn.Module):
+    """Complete graph-to-cell-complex lifting (``D_max = 2``).
+
+    Runs the full recipe: node embeddings, learned 1-cells added to the
+    observed edges, cycle-basis candidates of the augmented graph, and
+    gated 2-cells, with scaled-sum feature lifting.
+
+    Parameters
+    ----------
+    in_channels : int
+        Dimension of the input node features.
+    hidden_channels : int, optional
+        Embedding dimension of the lifting GNN. Default is 32.
+    k_min : int, optional
+        Smallest learned-edge neighborhood size. Default is 1.
+    k_max : int, optional
+        Largest learned-edge neighborhood size. Default is 3.
+    max_cell_length : int, optional
+        Longest cycle admitted as a 2-cell candidate. Default is 10.
+    sharpening : float, optional
+        Logit sharpening of the scorers. Default is 10.0.
+    stochastic : bool, optional
+        Sample accept/reject decisions instead of thresholding.
+        Default is False.
+    """
+
+    def __init__(
+        self,
+        in_channels,
+        hidden_channels=32,
+        k_min=1,
+        k_max=3,
+        max_cell_length=10,
+        sharpening=10.0,
+        stochastic=False,
+    ):
+        super().__init__()
+        self.max_cell_length = max_cell_length
+        self.encoder = DiffLiftEncoder(in_channels, hidden_channels)
+        self.edge_sampler = EdgeSampler(
+            hidden_channels,
+            k_min=k_min,
+            k_max=k_max,
+            sharpening=sharpening,
+            stochastic=stochastic,
+        )
+        self.cell_scorer = CellScorer(
+            hidden_channels,
+            sharpening=sharpening,
+            stochastic=stochastic,
+        )
+
+    def forward(self, x, edge_pairs, batch):
+        """Lift a batch of graphs to gated 2-dimensional cell complexes.
+
+        Parameters
+        ----------
+        x : torch.Tensor
+            Node features ``[n, in_channels]``.
+        edge_pairs : torch.Tensor
+            Observed edges as node pairs ``[2, P]`` (both directions).
+        batch : torch.Tensor
+            Graph index of every node ``[n]``.
+
+        Returns
+        -------
+        dict
+            ``new_edge_pairs`` ``[2, E_new]`` and ``new_edge_gate``
+            ``[E_new]`` (learned 1-cells); ``cell_membership``
+            ``[2, I]`` node-to-2-cell pairs, ``cell_gate``
+            ``[n_cells]``, and ``cell_batch`` ``[n_cells]``; ``x_2``
+            ``[n_cells, in_channels]`` scaled-sum features of the
+            2-cells.
+        """
+        z = self.encoder(x, edge_pairs)
+        new_pairs, edge_gate = self.edge_sampler(z, edge_pairs, batch)
+
+        # Candidate 2-cells: cycle basis of the augmented graph.
+        graph = nx.Graph()
+        graph.add_nodes_from(range(x.size(0)))
+        graph.add_edges_from(edge_pairs.t().tolist())
+        graph.add_edges_from(new_pairs.t().tolist())
+        cycles = [
+            c
+            for c in nx.cycle_basis(graph)
+            if 3 <= len(c) <= self.max_cell_length
+        ]
+        cycles.sort(key=lambda c: (len(c), tuple(sorted(c))))
+
+        if cycles:
+            members = torch.tensor(
+                [v for c in cycles for v in c],
+                dtype=torch.long,
+                device=x.device,
+            )
+            cells = torch.tensor(
+                [i for i, c in enumerate(cycles) for _ in c],
+                dtype=torch.long,
+                device=x.device,
+            )
+            membership = torch.stack([members, cells], dim=0)
+            cell_batch = batch[
+                torch.tensor(
+                    [c[0] for c in cycles],
+                    dtype=torch.long,
+                    device=x.device,
+                )
+            ]
+        else:
+            membership = edge_pairs.new_zeros(2, 0)
+            cell_batch = batch.new_zeros(0)
+
+        cell_gate = self.cell_scorer(z, membership, cell_batch)
+
+        # Scaled-sum feature lifting for the accepted 2-cells.
+        n_cells = cell_batch.size(0)
+        x_2 = x.new_zeros(n_cells, x.size(1))
+        if n_cells:
+            x_2 = x_2.index_add_(0, membership[1], x[membership[0]])
+            sizes = torch.bincount(membership[1], minlength=n_cells)
+            x_2 = x_2 / sizes.clamp(min=1).unsqueeze(1)
+            x_2 = cell_gate.unsqueeze(-1) * x_2
+
+        return {
+            "new_edge_pairs": new_pairs,
+            "new_edge_gate": edge_gate,
+            "cell_membership": membership,
+            "cell_gate": cell_gate,
+            "cell_batch": cell_batch,
+            "x_2": x_2,
+        }
diff --git a/topobench/nn/wrappers/cell/smcn_wrapper.py b/topobench/nn/wrappers/cell/smcn_wrapper.py
new file mode 100644
index 000000000..9cfe9c755
--- /dev/null
+++ b/topobench/nn/wrappers/cell/smcn_wrapper.py
@@ -0,0 +1,66 @@
+"""Wrapper for the SMCN model."""
+
+import torch
+
+from topobench.nn.wrappers.base import AbstractWrapper
+
+
+class SMCNWrapper(AbstractWrapper):
+    r"""Wrapper for the SMCN model.
+
+    This wrapper defines the forward pass of the model. The SMCN model
+    returns the embeddings of the cells of rank 0, 1, and 2.
+    """
+
+    def forward(self, batch):
+        r"""Forward pass for the SMCN wrapper.
+
+        Parameters
+        ----------
+        batch : torch_geometric.data.Data
+            Batch object containing the batched data.
+
+        Returns
+        -------
+        dict
+            Dictionary containing the updated model output.
+        """
+
+        def _batch_vector(rank, size):
+            """Return the batch vector of a rank, or zeros if absent.
+
+            Parameters
+            ----------
+            rank : int
+                Cell rank whose batch vector is requested.
+            size : int
+                Number of cells of that rank in the batch.
+
+            Returns
+            -------
+            torch.Tensor
+                Long tensor assigning each cell to its graph.
+            """
+            vec = batch.get(f"batch_{rank}", None)
+            if not torch.is_tensor(vec):
+                vec = torch.zeros(
+                    size, dtype=torch.long, device=batch.x_0.device
+                )
+            return vec
+
+        x_0, x_1, x_2 = self.backbone(
+            batch.x_0,
+            batch.x_1,
+            batch.x_2,
+            batch.incidence_1,
+            batch.incidence_2,
+            _batch_vector(0, batch.x_0.size(0)),
+            _batch_vector(1, batch.x_1.size(0)),
+            _batch_vector(2, batch.x_2.size(0)),
+        )
+
+        model_out = {"labels": batch.y, "batch_0": batch.batch_0}
+        model_out["x_0"] = x_0
+        model_out["x_1"] = x_1
+        model_out["x_2"] = x_2
+        return model_out