diff --git a/Cargo.lock b/Cargo.lock
index 48db26d..f2366f2 100644
--- a/Cargo.lock
+++ b/Cargo.lock
@@ -112,6 +112,15 @@ dependencies = [
  "xtrain-tensor",
 ]
 
+[[package]]
+name = "xtrain-optim"
+version = "0.1.0"
+dependencies = [
+ "xtrain-autodiff",
+ "xtrain-cuda",
+ "xtrain-tensor",
+]
+
 [[package]]
 name = "xtrain-tensor"
 version = "0.1.0"
diff --git a/Cargo.toml b/Cargo.toml
index 456b6d7..1f91829 100644
--- a/Cargo.toml
+++ b/Cargo.toml
@@ -5,6 +5,7 @@ members = [
     "crates/xtrain-tensor",
     "crates/xtrain-autodiff",
     "crates/xtrain-model",
+    "crates/xtrain-optim",
 ]
 
 [workspace.package]
diff --git a/crates/xtrain-optim/Cargo.toml b/crates/xtrain-optim/Cargo.toml
new file mode 100644
index 0000000..d2aff25
--- /dev/null
+++ b/crates/xtrain-optim/Cargo.toml
@@ -0,0 +1,12 @@
+[package]
+name = "xtrain-optim"
+version.workspace = true
+edition.workspace = true
+
+[dependencies]
+xtrain-tensor = { path = "../xtrain-tensor" }
+xtrain-autodiff = { path = "../xtrain-autodiff" }
+
+[dev-dependencies]
+# Acceptance tests drive the GPU (device selection) directly.
+xtrain-cuda = { path = "../xtrain-cuda" }
diff --git a/crates/xtrain-optim/build.rs b/crates/xtrain-optim/build.rs
new file mode 100644
index 0000000..43ac264
--- /dev/null
+++ b/crates/xtrain-optim/build.rs
@@ -0,0 +1,26 @@
+use std::env;
+use std::path::Path;
+use std::process::Command;
+
+// Per-crate convention (see the other crates): the AdamW *math* is host-only and
+// always compiles, but `AdamW::step(&[Var])` round-trips parameter values/grads
+// through GPU tensors, so that call site is gated behind `not(no_cuda)`. cfg does
+// not propagate across crates, so this crate re-detects nvcc. No CUDA is compiled.
+fn main() {
+    println!("cargo:rustc-check-cfg=cfg(no_cuda)");
+
+    let cuda_path = env::var("CUDA_HOME")
+        .or_else(|_| env::var("CUDA_PATH"))
+        .unwrap_or_else(|_| "/usr/local/cuda".to_string());
+
+    if !nvcc_available(&cuda_path) {
+        println!("cargo:rustc-cfg=no_cuda");
+    }
+}
+
+fn nvcc_available(cuda_path: &str) -> bool {
+    if Command::new("nvcc").arg("--version").output().is_ok() {
+        return true;
+    }
+    Path::new(&format!("{cuda_path}/bin/nvcc")).exists()
+}
diff --git a/crates/xtrain-optim/src/lib.rs b/crates/xtrain-optim/src/lib.rs
new file mode 100644
index 0000000..edd37e7
--- /dev/null
+++ b/crates/xtrain-optim/src/lib.rs
@@ -0,0 +1,154 @@
+//! Hand-written AdamW optimizer (Phase T6).
+//!
+//! AdamW = Adam with **decoupled** weight decay (Loshchilov & Hutter, 2019): the
+//! weight-decay term is applied directly to the parameter, NOT folded into the
+//! gradient (so it does not interact with the adaptive `v` denominator). This
+//! matches `torch.optim.AdamW`.
+//!
+//! Update for parameter `θ` at step `t` (1-indexed), with gradient `g`:
+//! ```text
+//!   m ← β1·m + (1−β1)·g
+//!   v ← β2·v + (1−β2)·g²
+//!   m̂ ← m / (1 − β1ᵗ)          (bias correction)
+//!   v̂ ← v / (1 − β2ᵗ)
+//!   θ ← θ − lr·( m̂ / (√v̂ + ε) + wd·θ )
+//! ```
+//! The `lr·wd·θ` term is the decoupled decay. Note PyTorch applies decay as
+//! `θ ← θ·(1 − lr·wd)` then the Adam step; both are algebraically the same
+//! first-order update — we fold decay into the single subtraction above, which
+//! is what PyTorch's default (`maximize=False`, no `amsgrad`) computes.
+//!
+//! The math operates on flat host `f32` buffers ([`AdamW::step_host`]) so it is
+//! unit-testable on a GPU-less host; [`AdamW::step`] is a thin wrapper that
+//! round-trips each parameter's value/grad through the GPU tensor and is gated
+//! behind `not(no_cuda)`.
+
+/// Per-parameter optimizer state: the first (`m`) and second (`v`) moment
+/// estimates, one f32 per element, kept flat (matching the parameter layout).
+struct ParamState {
+    m: Vec<f32>,
+    v: Vec<f32>,
+}
+
+/// Decoupled-weight-decay Adam. One instance owns the moment state for a fixed
+/// list of parameters, keyed by their index in the slice passed to `step`
+/// (the model's stable `params()` order).
+pub struct AdamW {
+    pub lr: f32,
+    beta1: f32,
+    beta2: f32,
+    eps: f32,
+    weight_decay: f32,
+    /// Global step count (shared across all params for bias correction).
+    t: u64,
+    /// Lazily sized to the parameter list on the first `step`.
+    state: Vec<ParamState>,
+}
+
+impl AdamW {
+    /// PyTorch-default hyperparameters except `lr`/`weight_decay`, which you set
+    /// (β1=0.9, β2=0.999, ε=1e-8).
+    pub fn new(lr: f32, weight_decay: f32) -> Self {
+        Self::with_betas(lr, weight_decay, 0.9, 0.999, 1e-8)
+    }
+
+    pub fn with_betas(lr: f32, weight_decay: f32, beta1: f32, beta2: f32, eps: f32) -> Self {
+        Self {
+            lr,
+            beta1,
+            beta2,
+            eps,
+            weight_decay,
+            t: 0,
+            state: Vec::new(),
+        }
+    }
+
+    /// Current global step (number of `step` calls so far).
+    pub fn step_count(&self) -> u64 {
+        self.t
+    }
+
+    /// Pure-host AdamW step over flat parameter/gradient buffers. `params[i]` is
+    /// updated in place using `grads[i]`; both are the i-th parameter's elements
+    /// in the model's stable order. Lazily allocates moment state on first call.
+    ///
+    /// This is the testable core — no GPU, no autograd. `lr` is passed per call
+    /// so a schedule can vary it each step.
+    pub fn step_host(&mut self, lr: f32, params: &mut [Vec<f32>], grads: &[Vec<f32>]) {
+        assert_eq!(params.len(), grads.len(), "param/grad count mismatch");
+        if self.state.is_empty() {
+            self.state = params
+                .iter()
+                .map(|p| ParamState {
+                    m: vec![0.0; p.len()],
+                    v: vec![0.0; p.len()],
+                })
+                .collect();
+        }
+        assert_eq!(self.state.len(), params.len(), "param count changed");
+
+        self.t += 1;
+        let bc1 = 1.0 - self.beta1.powi(self.t as i32);
+        let bc2 = 1.0 - self.beta2.powi(self.t as i32);
+
+        for (i, (p, g)) in params.iter_mut().zip(grads).enumerate() {
+            assert_eq!(p.len(), g.len(), "param/grad len mismatch at {i}");
+            let st = &mut self.state[i];
+            for j in 0..p.len() {
+                let gj = g[j];
+                st.m[j] = self.beta1 * st.m[j] + (1.0 - self.beta1) * gj;
+                st.v[j] = self.beta2 * st.v[j] + (1.0 - self.beta2) * gj * gj;
+                let mhat = st.m[j] / bc1;
+                let vhat = st.v[j] / bc2;
+                // Decoupled weight decay: decay term uses the *current* param,
+                // matching PyTorch's `p ← p − lr·wd·p` applied alongside the step.
+                p[j] -= lr * (mhat / (vhat.sqrt() + self.eps) + self.weight_decay * p[j]);
+            }
+        }
+    }
+}
+
+#[cfg(not(no_cuda))]
+mod gpu {
+    use super::AdamW;
+    use xtrain_autodiff::tape::Var;
+    use xtrain_tensor::{Device, Tensor};
+
+    impl AdamW {
+        /// Apply one AdamW step to every parameter `Var`, using `lr` for this step
+        /// (so an LR schedule can vary it). Pulls each param's value and `.grad()`
+        /// to the host, runs [`AdamW::step_host`], and writes the updated value
+        /// back with `set_value`. A param with no grad is fed a zero grad, so the
+        /// Adam term vanishes and only decoupled weight decay applies (the model's
+        /// params all receive grads each step, so this is just a safety default).
+        ///
+        /// Does NOT zero grads — the caller does that (matching the GD-step
+        /// template in the T5 overfit test).
+        pub fn step(&mut self, lr: f32, params: &[Var]) {
+            let device = params[0].value().device();
+            let shapes: Vec<Vec<usize>> =
+                params.iter().map(|p| p.value().shape().to_vec()).collect();
+
+            let mut host_params: Vec<Vec<f32>> = params
+                .iter()
+                .map(|p| p.value().to_device(Device::Cpu).as_slice::<f32>().to_vec())
+                .collect();
+            let host_grads: Vec<Vec<f32>> = params
+                .iter()
+                .zip(&host_params)
+                .map(|(p, hp)| match p.grad() {
+                    Some(g) => g.to_device(Device::Cpu).as_slice::<f32>().to_vec(),
+                    None => vec![0.0; hp.len()], // no grad → no update this step
+                })
+                .collect();
+
+            self.step_host(lr, &mut host_params, &host_grads);
+
+            for ((p, data), shape) in params.iter().zip(&host_params).zip(&shapes) {
+                let t = Tensor::from_slice(data, shape).to_device(device);
+                p.set_value(t);
+            }
+        }
+    }
+}
diff --git a/crates/xtrain-optim/tests/adamw_host.rs b/crates/xtrain-optim/tests/adamw_host.rs
new file mode 100644
index 0000000..15e8d33
--- /dev/null
+++ b/crates/xtrain-optim/tests/adamw_host.rs
@@ -0,0 +1,99 @@
+// Host-only unit test for the AdamW *math* (no GPU). Verifies the update against
+// an independent, hand-rolled reference implementation of the same recurrence for
+// several steps with non-trivial weight decay — catching bias-correction and
+// decoupled-decay mistakes. The rigorous vs-PyTorch parity (end-to-end on a real
+// model) lives in xtrain-train; this is the fast local guard on the formula.
+
+use xtrain_optim::AdamW;
+
+// Independent reference: the textbook AdamW recurrence, kept separate from the
+// implementation so a shared bug can't hide.
+struct RefAdamW {
+    b1: f32,
+    b2: f32,
+    eps: f32,
+    wd: f32,
+    t: i32,
+    m: Vec<f32>,
+    v: Vec<f32>,
+}
+
+impl RefAdamW {
+    fn new(n: usize, wd: f32) -> Self {
+        Self {
+            b1: 0.9,
+            b2: 0.999,
+            eps: 1e-8,
+            wd,
+            t: 0,
+            m: vec![0.0; n],
+            v: vec![0.0; n],
+        }
+    }
+    fn step(&mut self, lr: f32, p: &mut [f32], g: &[f32]) {
+        self.t += 1;
+        let bc1 = 1.0 - self.b1.powi(self.t);
+        let bc2 = 1.0 - self.b2.powi(self.t);
+        for i in 0..p.len() {
+            self.m[i] = self.b1 * self.m[i] + (1.0 - self.b1) * g[i];
+            self.v[i] = self.b2 * self.v[i] + (1.0 - self.b2) * g[i] * g[i];
+            let mhat = self.m[i] / bc1;
+            let vhat = self.v[i] / bc2;
+            p[i] -= lr * (mhat / (vhat.sqrt() + self.eps) + self.wd * p[i]);
+        }
+    }
+}
+
+#[test]
+fn adamw_matches_reference_recurrence() {
+    let lr = 0.01;
+    let wd = 0.1;
+    let mut opt = AdamW::new(lr, wd);
+
+    // Two parameters of different sizes (exercises per-param state keying).
+    let mut p_impl = vec![vec![0.5f32, -1.0, 2.0, 0.0], vec![1.5f32, -0.25]];
+    let mut p_ref = p_impl.clone();
+    let mut r0 = RefAdamW::new(4, wd);
+    let mut r1 = RefAdamW::new(2, wd);
+
+    // Deterministic pseudo-grads that change every step.
+    let grad = |step: usize, idx: usize, j: usize| -> f32 {
+        let s = (step * 13 + idx * 7 + j * 3) as f32;
+        (s * 0.123).sin() * 0.5
+    };
+
+    for step in 0..20 {
+        let grads = vec![
+            (0..4).map(|j| grad(step, 0, j)).collect::<Vec<_>>(),
+            (0..2).map(|j| grad(step, 1, j)).collect::<Vec<_>>(),
+        ];
+        opt.step_host(lr, &mut p_impl, &grads);
+        r0.step(lr, &mut p_ref[0], &grads[0]);
+        r1.step(lr, &mut p_ref[1], &grads[1]);
+    }
+
+    assert_eq!(opt.step_count(), 20);
+    for (pi, pr) in p_impl.iter().zip(&p_ref) {
+        for (a, b) in pi.iter().zip(pr) {
+            assert!((a - b).abs() < 1e-6, "impl {a} != ref {b}");
+        }
+    }
+}
+
+#[test]
+fn zero_grad_only_decays() {
+    // With g=0 and wd>0, the step must reduce to pure decoupled decay:
+    // θ ← θ − lr·wd·θ (Adam term is 0/eps = 0).
+    let lr = 0.1;
+    let wd = 0.5;
+    let mut opt = AdamW::new(lr, wd);
+    let mut p = vec![vec![2.0f32]];
+    let g = vec![vec![0.0f32]];
+    opt.step_host(lr, &mut p, &g);
+    let expected = 2.0 - lr * wd * 2.0;
+    assert!(
+        (p[0][0] - expected).abs() < 1e-6,
+        "{} != {expected}",
+        p[0][0]
+    );
+}