diff --git a/configs/training/cifar10_default.yaml b/configs/training/cifar10_default.yaml
index 1507486..a9d292a 100644
--- a/configs/training/cifar10_default.yaml
+++ b/configs/training/cifar10_default.yaml
@@ -1,7 +1,7 @@
 # @package _global_
 
 training:
-  num_epochs: 100  # CIFAR-10 needs more epochs than MNIST
+  num_epochs: 3  # CIFAR-10 needs more epochs than MNIST
   step_size: 0.05  # ODE solver step size (20 steps for inference)
   n_samples: 4  # Generate multiple samples at once
   log_interval: 10  # Log every 10 epochs
diff --git a/configs/training/cifar10_improved.yaml b/configs/training/cifar10_improved.yaml
index b84ddba..9e7c9ba 100644
--- a/configs/training/cifar10_improved.yaml
+++ b/configs/training/cifar10_improved.yaml
@@ -1,7 +1,7 @@
 # @package _global_
 
 training:
-  num_epochs: 300  # 3x longer than default (100 epochs)
+  num_epochs: 3  # 3x longer than default (100 epochs)
   step_size: 0.01  # 100 ODE steps for high-quality inference (vs 20 steps)
   n_samples: 9  # Generate 3x3 grid of samples
   log_interval: 10  # Log every 10 epochs
diff --git a/experiments/_common.py b/experiments/_common.py
index 35d1ef7..5694b3d 100644
--- a/experiments/_common.py
+++ b/experiments/_common.py
@@ -3,6 +3,7 @@
 import matplotlib.pyplot as plt
 import numpy as np
 from tinygrad.nn.state import safe_load
+from tinygrad.tensor import Tensor as T
 
 
 def detect_time_embed_dim(model_path: str, in_channels: int) -> int:
@@ -43,3 +44,59 @@ def save_grid_figure(x_np: np.ndarray, title: str, path: str, figsize=(4, 4.2)):
     plt.tight_layout()
     plt.savefig(path, dpi=150, bbox_inches="tight")
     plt.close(fig)
+
+
+def per_sample_norm_mean(x: T) -> float:
+    """Mean over the batch of the per-sample L2 norm."""
+    flat = x.reshape(x.shape[0], -1)
+    return ((flat * flat).sum(axis=-1) + 1e-20).sqrt().mean().numpy().item()
+
+
+def normalize_per_sample(x: T) -> T:
+    """Normalizes each sample in the batch to unit L2 norm."""
+    flat = x.reshape(x.shape[0], -1)
+    mag = ((flat * flat).sum(axis=-1) + 1e-20).sqrt()
+    view_shape = (x.shape[0],) + (1,) * (len(x.shape) - 1)
+    return x / mag.reshape(view_shape)
+
+
+def compute_fid_vs_real(
+    generated_np: np.ndarray, dataset_name: str, n_samples: int, weights_dir: str = "weights"
+) -> float | None:
+    """FID of generated images against real images freshly sampled from `dataset_name`."""
+    from tinyflow.dataloader import CIFAR10Loader, FashionMNISTLoader, MNISTLoader
+    from tinyflow.metrics import calculate_fid, get_feature_extractor
+
+    classifier = get_feature_extractor(dataset_name, weights_dir=weights_dir)
+    if not classifier._weights_loaded:
+        print(f"Warning: FID classifier weights not found for {dataset_name}, skipping FID.")
+        return None
+
+    if dataset_name == "mnist":
+        dataloader = MNISTLoader(
+            path="dataset/mnist/trainset/trainingSet/*/*.jpg",
+            batch_size=64,
+            shuffle=True,
+            flatten=False,
+        )
+    elif dataset_name == "fashion_mnist":
+        dataloader = FashionMNISTLoader(
+            path="dataset/fashion_mnist", batch_size=64, shuffle=True, flatten=False, train=True
+        )
+    elif dataset_name == "cifar10":
+        dataloader = CIFAR10Loader(
+            path="dataset/cifar10/cifar-10-batches-py", batch_size=64, shuffle=True, train=True
+        )
+    else:
+        raise ValueError(f"Unknown dataset: {dataset_name}")
+
+    real_images = []
+    collected = 0
+    for batch_images, _ in dataloader:
+        real_images.append(batch_images)
+        collected += len(batch_images)
+        if collected >= n_samples:
+            break
+    real_images = np.concatenate(real_images, axis=0)[:n_samples]
+
+    return calculate_fid(real_images, generated_np, feature_extractor=classifier, batch_size=64)
diff --git a/experiments/adaptive_sampler.py b/experiments/adaptive_sampler.py
new file mode 100644
index 0000000..cec10a6
--- /dev/null
+++ b/experiments/adaptive_sampler.py
@@ -0,0 +1,250 @@
+"""
+Experiment 1 (HOMOTOPY_EXPERIMENTS.md): Condition-Number-Adaptive Sampler.
+
+Replaces fixed-step Euler integration with curvature-aware step sizing:
+
+    kappa ~= ||v_theta(x + eps*u, t) - v_theta(x, t)|| / eps      (u: random unit dir)
+    dt    <- dt_base / (1 + beta * kappa)
+
+Compares, at a matched NFE budget, three configs on MNIST:
+    reference   fixed uniform Euler at --reference-nfe (gold standard, e.g. 50)
+    fixed       fixed uniform Euler at the tested budget (control)
+    adaptive    curvature-adaptive Euler at the tested budget
+
+Usage:
+    uv run experiments/adaptive_sampler.py
+    uv run experiments/adaptive_sampler.py --nfe-budgets 10 20 --beta 5.0
+"""
+
+import argparse
+import os
+
+import matplotlib.pyplot as plt
+import mlflow
+import numpy as np
+from _common import (
+    compute_fid_vs_real,
+    detect_time_embed_dim,
+    normalize_per_sample,
+    per_sample_norm_mean,
+    save_grid_figure,
+    schedule_grid,
+)
+from tinygrad.tensor import Tensor as T
+
+from tinyflow.nn import UNetTinygrad
+from tinyflow.trainer import BaseTrainer
+from tinyflow.utils import preprocess_time_mnist
+
+plt.style.use("ggplot")
+mlflow.set_tracking_uri("sqlite:///mlflow.db")
+
+
+def integrate_fixed(model, preprocess, x0, N: int):
+    x = x0
+    ts = schedule_grid(N, 1.0)
+    nfe = 0
+    for i in range(len(ts) - 1):
+        t = (T.zeros(1) + float(ts[i])).contiguous()
+        t_in = preprocess(t, x)
+        v = model(x, t_in).realize()
+        nfe += 1
+        h = float(ts[i + 1] - ts[i])
+        x = (x + h * v).realize()
+    return x, {"steps": N, "nfe": nfe}
+
+
+def integrate_adaptive(
+    model,
+    preprocess,
+    x0,
+    nfe_target: int,
+    beta: float,
+    eps_x: float,
+    kappa_every: int,
+    max_steps: int = 500,
+):
+    """Runs to completion (t reaches 1), unlike a fixed-N loop. `nfe_target`
+    only sets the nominal dt_base (1/nfe_target); curvature can make the
+    *actual* NFE spent come out higher or lower than that nominal target —
+    that actual count is what should be compared against a fixed baseline."""
+    x = x0
+    t_val = 0.0
+    nfe = 0
+    steps = 0
+    kappa = 0.0
+    dt_base = 1.0 / nfe_target
+    log = {"t": [], "dt": [], "kappa": []}
+
+    while t_val < 1.0 - 1e-9 and steps < max_steps:
+        t = (T.zeros(1) + t_val).contiguous()
+        t_in = preprocess(t, x)
+        v = model(x, t_in).realize()
+        nfe += 1
+
+        if steps % kappa_every == 0:
+            u_hat = normalize_per_sample(T.randn(*x.shape).realize())
+            v_pert = model(x + eps_x * u_hat, t_in).realize()
+            nfe += 1
+            kappa = per_sample_norm_mean((v_pert - v) * (1.0 / eps_x))
+
+        dt = dt_base / (1.0 + beta * kappa)
+        dt = min(dt, 1.0 - t_val)
+        x = (x + dt * v).realize()
+        t_val += dt
+        steps += 1
+        log["t"].append(t_val)
+        log["dt"].append(dt)
+        log["kappa"].append(kappa)
+
+    return x, {"steps": steps, "nfe": nfe, **log}
+
+
+def make_dt_plot(log: dict, budget: int) -> plt.Figure:
+    fig, axes = plt.subplots(1, 2, figsize=(11, 4))
+    axes[0].plot(log["t"], log["dt"], marker="o", markersize=3)
+    axes[0].set_xlabel("t")
+    axes[0].set_ylabel("dt")
+    axes[0].set_title(f"Adaptive step size (budget NFE={budget})")
+
+    axes[1].plot(log["t"], log["kappa"], marker="o", markersize=3, color="C1")
+    axes[1].set_xlabel("t")
+    axes[1].set_ylabel("kappa (curvature estimate)")
+    axes[1].set_title("Curvature along trajectory")
+
+    plt.tight_layout()
+    return fig
+
+
+def main():
+    parser = argparse.ArgumentParser()
+    parser.add_argument("--model-path", default="weights/model_mnist_unet_linear.safetensors")
+    parser.add_argument(
+        "--nfe-budgets",
+        type=int,
+        nargs="+",
+        default=[10, 20],
+        help="Nominal NFE targets (sets dt_base=1/target); adaptive runs to "
+        "completion so actual NFE spent may differ, and the fixed baseline "
+        "is matched to that actual value post-hoc.",
+    )
+    parser.add_argument("--reference-nfe", type=int, default=50)
+    parser.add_argument("--beta", type=float, default=5.0)
+    parser.add_argument("--kappa-every", type=int, default=4)
+    parser.add_argument("--eps-x", type=float, default=1e-2)
+    parser.add_argument("--n-samples", type=int, default=256)
+    parser.add_argument("--seed", type=int, default=42)
+    parser.add_argument("--experiment", default="homotopy_experiments")
+    args = parser.parse_args()
+
+    if not os.path.exists(args.model_path):
+        raise FileNotFoundError(f"Model file not found: {args.model_path}")
+
+    in_ch = 1
+    time_embed_dim = detect_time_embed_dim(args.model_path, in_ch)
+    model = UNetTinygrad(in_ch, in_ch, time_embed_dim=time_embed_dim)
+    BaseTrainer.load_model(model, args.model_path)
+    T.training = False
+
+    mlflow.set_experiment(args.experiment)
+    with mlflow.start_run(run_name="1_adaptive_sampler") as parent:
+        mlflow.set_tags({"exp": "1_adaptive_sampler", "dataset": "mnist"})
+        mlflow.log_params(
+            {
+                "model_path": args.model_path,
+                "reference_nfe": args.reference_nfe,
+                "beta": args.beta,
+                "kappa_every": args.kappa_every,
+                "eps_x": args.eps_x,
+                "n_samples": args.n_samples,
+                "seed": args.seed,
+                "nfe_budgets": ",".join(str(b) for b in args.nfe_budgets),
+            }
+        )
+
+        # gold-standard reference
+        T.manual_seed(args.seed)
+        x0_ref = T.randn(args.n_samples, 1, 28, 28).realize()
+        with mlflow.start_run(run_name=f"reference_NFE{args.reference_nfe}", nested=True):
+            x_ref, info = integrate_fixed(model, preprocess_time_mnist, x0_ref, args.reference_nfe)
+            x_ref_np = x_ref.numpy()
+            fid_ref = compute_fid_vs_real(x_ref_np, "mnist", args.n_samples)
+            mlflow.log_params({"config": "reference", **info})
+            if fid_ref is not None:
+                mlflow.log_metric("fid", fid_ref)
+            grid_path = "/tmp/adaptive_sampler_reference.png"
+            save_grid_figure(x_ref_np, f"Reference NFE={args.reference_nfe}", grid_path)
+            mlflow.log_artifact(grid_path)
+            print(f"reference NFE={args.reference_nfe}: fid={fid_ref}")
+
+        for target in args.nfe_budgets:
+            # adaptive runs to completion; the target only sets the nominal
+            # dt_base, actual NFE spent is discovered here
+            T.manual_seed(args.seed)
+            x0 = T.randn(args.n_samples, 1, 28, 28).realize()
+
+            with mlflow.start_run(run_name=f"adaptive_target{target}", nested=True):
+                x_adapt, info = integrate_adaptive(
+                    model,
+                    preprocess_time_mnist,
+                    x0,
+                    target,
+                    args.beta,
+                    args.eps_x,
+                    args.kappa_every,
+                )
+                actual_nfe = info["nfe"]
+                x_adapt_np = x_adapt.numpy()
+                fid_adapt = compute_fid_vs_real(x_adapt_np, "mnist", args.n_samples)
+                mlflow.log_params(
+                    {
+                        "config": "adaptive",
+                        "nfe_target": target,
+                        "steps": info["steps"],
+                        "nfe": actual_nfe,
+                        "beta": args.beta,
+                        "kappa_every": args.kappa_every,
+                    }
+                )
+                if fid_adapt is not None:
+                    mlflow.log_metric("fid", fid_adapt)
+                mlflow.log_metric("mean_dt", float(np.mean(info["dt"])))
+                mlflow.log_metric("mean_kappa", float(np.mean(info["kappa"])))
+
+                grid_path = f"/tmp/adaptive_sampler_adaptive_target{target}.png"
+                save_grid_figure(
+                    x_adapt_np, f"Adaptive target={target} (actual NFE={actual_nfe})", grid_path
+                )
+                mlflow.log_artifact(grid_path)
+
+                dt_fig = make_dt_plot(info, target)
+                mlflow.log_figure(dt_fig, f"dt_profile_target{target}.png")
+                plt.close(dt_fig)
+
+                print(
+                    f"adaptive target={target} -> actual steps={info['steps']}, "
+                    f"actual nfe={actual_nfe}: fid={fid_adapt}"
+                )
+
+            # fixed baseline at the SAME actual NFE the adaptive run spent
+            T.manual_seed(args.seed)
+            x0 = T.randn(args.n_samples, 1, 28, 28).realize()
+
+            with mlflow.start_run(run_name=f"fixed_NFE{actual_nfe}", nested=True):
+                x_fixed, info = integrate_fixed(model, preprocess_time_mnist, x0, actual_nfe)
+                x_fixed_np = x_fixed.numpy()
+                fid_fixed = compute_fid_vs_real(x_fixed_np, "mnist", args.n_samples)
+                mlflow.log_params({"config": "fixed", "matched_to_target": target, **info})
+                if fid_fixed is not None:
+                    mlflow.log_metric("fid", fid_fixed)
+                grid_path = f"/tmp/adaptive_sampler_fixed_NFE{actual_nfe}.png"
+                save_grid_figure(x_fixed_np, f"Fixed NFE={actual_nfe}", grid_path)
+                mlflow.log_artifact(grid_path)
+                print(f"fixed NFE={actual_nfe} (matched to target={target}): fid={fid_fixed}")
+
+        print(f"parent run: {parent.info.run_id}")
+        print("view with: uv run mlflow ui  (tracking uri: sqlite:///mlflow.db)")
+
+
+if __name__ == "__main__":
+    main()
diff --git a/experiments/curvature_profile.py b/experiments/curvature_profile.py
index f87f994..cca2ca8 100644
--- a/experiments/curvature_profile.py
+++ b/experiments/curvature_profile.py
@@ -19,7 +19,12 @@
 import matplotlib.pyplot as plt
 import mlflow
 import numpy as np
-from _common import detect_time_embed_dim, schedule_grid
+from _common import (
+    detect_time_embed_dim,
+    normalize_per_sample,
+    per_sample_norm_mean,
+    schedule_grid,
+)
 from tinygrad.tensor import Tensor as T
 from tqdm import tqdm
 
@@ -75,18 +80,6 @@ def build_model(dataset: str, model_path: str):
     return UNetTinygrad(in_ch, in_ch, time_embed_dim=detect_time_embed_dim(model_path, in_ch))
 
 
-def per_sample_norm_mean(x: T) -> float:
-    flat = x.reshape(x.shape[0], -1)
-    return ((flat * flat).sum(axis=-1) + 1e-20).sqrt().mean().numpy().item()
-
-
-def normalize_per_sample(x: T) -> T:
-    flat = x.reshape(x.shape[0], -1)
-    mag = ((flat * flat).sum(axis=-1) + 1e-20).sqrt()
-    view_shape = (x.shape[0],) + (1,) * (len(x.shape) - 1)
-    return x / mag.reshape(view_shape)
-
-
 def profile(
     dataset: str, model_path: str, num_steps: int, batch_size: int, eps_x: float, eps_t: float
 ):
diff --git a/experiments/homotopy_runner.py b/experiments/homotopy_runner.py
new file mode 100644
index 0000000..70e1729
--- /dev/null
+++ b/experiments/homotopy_runner.py
@@ -0,0 +1,47 @@
+"""
+Runner for the Flow Matching x Homotopy Continuation experiment plan
+(see HOMOTOPY_EXPERIMENTS.md). Dispatches by experiment number to the
+per-experiment script; unrecognized args are forwarded unchanged.
+
+All experiments log to the same MLflow experiment ("homotopy_experiments",
+sqlite:///mlflow.db) so results are comparable across runs.
+
+Usage:
+    uv run experiments/homotopy_runner.py 1
+    uv run experiments/homotopy_runner.py 1 --nfe-budgets 10 20 --beta 5.0
+    uv run experiments/homotopy_runner.py 3   # not implemented yet
+"""
+
+import argparse
+import subprocess
+import sys
+from pathlib import Path
+
+EXPERIMENTS = {
+    1: ("adaptive_sampler.py", "Condition-Number-Adaptive Sampler"),
+    2: (None, "Predictor-Corrector Sampling"),
+    3: (None, "Mode-Matched Base Distribution"),
+    4: (None, "Implicit Homotopy Generation (IHG)"),
+}
+
+
+def main():
+    parser = argparse.ArgumentParser(
+        description="Dispatch one experiment from HOMOTOPY_EXPERIMENTS.md"
+    )
+    parser.add_argument("experiment", type=int, choices=sorted(EXPERIMENTS))
+    args, rest = parser.parse_known_args()
+
+    script, name = EXPERIMENTS[args.experiment]
+    if script is None:
+        print(f"Experiment {args.experiment} ({name}) is not implemented yet.")
+        print("See HOMOTOPY_EXPERIMENTS.md for the plan.")
+        sys.exit(1)
+
+    script_path = Path(__file__).parent / script
+    print(f"Running experiment {args.experiment}: {name}")
+    subprocess.run([sys.executable, str(script_path), *rest], check=True)
+
+
+if __name__ == "__main__":
+    main()