diff --git a/src/projected_grpo/proj.py b/src/projected_grpo/proj.py
index 9319871..72a3e93 100644
--- a/src/projected_grpo/proj.py
+++ b/src/projected_grpo/proj.py
@@ -81,6 +81,8 @@ def project_delta_S_grad(
         g = info["delta_S"].grad
         if g is None:
             continue
+        if name not in v_hack:  # module dropped by global noise-floor filter
+            continue
         V = v_hack[name].to(g.device, dtype=g.dtype)   # [k, r]
         gn = g.norm()
         if gn < 1e-12:
diff --git a/src/projected_grpo/train.py b/src/projected_grpo/train.py
index 14bd73e..bfaa9b9 100644
--- a/src/projected_grpo/train.py
+++ b/src/projected_grpo/train.py
@@ -166,6 +166,11 @@ class Config:
     v_hack_extract_top_k: int = 12    # max k to save at extract; n_train_pairs caps it lower
     v_hack_k: int = 5                 # load-time slice; k=1 = mean-diff, k=k_max = full
     v_hack_tau_axis: float = 0.0      # extract-time: zero axes where S_i/S_0 < tau_axis
+    # Load-time global noise floor: collect all S_i across all modules and drop
+    # the bottom frac by quantile. Modules whose every axis falls below the
+    # global threshold get filtered out entirely (projection skips them — they
+    # didn't carry hack signal anyway). 0 = no filter.
+    v_hack_drop_bottom_frac: float = 0.25
     # Per-source cin diagnostic: split each prompt's backward into student-only
     # + teacher-only passes (~2x backward time). 1 = every step (default; full
     # signal); N>1 = only every Nth step (combined backward elsewhere, ~halves
@@ -228,18 +233,23 @@ def load_problems(n: int) -> list[dict]:
 
 
 def load_v_hack(
-    path: Path, model_name: str, wrappers: dict, k_use: int | None = None,
+    path: Path, model_name: str, wrappers: dict,
+    k_use: int | None = None, drop_bottom_frac: float = 0.0,
 ) -> dict[str, torch.Tensor]:
     """Load v_hack (top-k directions) for this wrapped model.
 
     File schema (v2): bare `{name}` keys hold V[k_max, r]; `_sv/{name}` keys hold
-    S[k_max] (read but not returned — no caller uses them yet). v_hack is
-    model-specific because module names and per-module SVD ranks depend on the
-    exact checkpoint; a smoke (Qwen3.5-0.8B) v_hack must not be reused for a
-    full (Qwen3-4B) run.
+    S[k_max]. v_hack is model-specific because module names and per-module SVD
+    ranks depend on the exact checkpoint; a smoke (Qwen3.5-0.8B) v_hack must
+    not be reused for a full (Qwen3-4B) run.
 
-    If `k_use` is given, slices V to top-k_use rows. Errors if k_use > k_max
-    saved (re-extract with a higher top_k).
+    If `k_use` is given, slices V (and S) to top-k_use rows. Errors if
+    k_use > k_max saved (re-extract with a higher top_k).
+
+    If `drop_bottom_frac > 0`, collects every S_i across every module and drops
+    the bottom-fraction by global quantile. Modules whose every axis is below
+    the global threshold get filtered out of the returned dict (projection on
+    those modules becomes a no-op — they didn't carry hack signal anywhere).
     """
     with safe_open(str(path), framework="pt", device="cpu") as f:
         meta = f.metadata() or {}
@@ -258,9 +268,8 @@ def load_v_hack(
                 f"v_hack dtype/SVD-basis mismatch: {path} was extracted with dtype={saved_dtype}; "
                 "train.py loads models in bf16. Re-extract with `--dtype=bf16`."
             )
-        # Read only V keys (bare module names); _sv/{name} keys are saved
-        # alongside but no runtime path consumes them currently.
         v_hack = {k: f.get_tensor(k) for k in f.keys() if not k.startswith("_sv/")}
+        v_sv   = {k[len("_sv/"):]: f.get_tensor(k) for k in f.keys() if k.startswith("_sv/")}
 
     wrapper_keys = set(wrappers)
     vhack_keys = set(v_hack)
@@ -290,8 +299,33 @@ def load_v_hack(
                 f"Re-extract with `--top-k={k_use}`."
             )
         v_hack = {n: v[:k_use].contiguous() for n, v in v_hack.items()}
+        v_sv   = {n: s[:k_use].contiguous() for n, s in v_sv.items()}
+
+    # Global noise floor: drop the bottom drop_bottom_frac of all (module, axis)
+    # pairs by S_i. One quantile across every S_i in every module. A module
+    # whose every axis lies below the global threshold is removed from v_hack
+    # entirely — projection iterates v_hack so that module just gets skipped.
+    n_dropped_modules = 0
+    n_axes_before = sum(v.shape[0] for v in v_hack.values())
+    threshold = None
+    if drop_bottom_frac > 0 and v_sv:
+        all_S = torch.cat([v_sv[n].float() for n in v_hack])
+        threshold = torch.quantile(all_S, drop_bottom_frac).item()
+        filtered: dict[str, torch.Tensor] = {}
+        for name, V in v_hack.items():
+            keep = v_sv[name].float() >= threshold
+            if keep.any():
+                filtered[name] = V[keep].contiguous()
+            else:
+                n_dropped_modules += 1
+        v_hack = filtered
+    n_axes_after = sum(v.shape[0] for v in v_hack.values())
+
     logger.info(
-        f"loaded v_hack from {path}: modules={len(v_hack)}; k_saved={k_max}, k_use={k_use or k_max}"
+        f"loaded v_hack from {path}: modules={len(v_hack)} (dropped {n_dropped_modules}); "
+        f"k_saved={k_max}, k_use={k_use or k_max}; "
+        f"axes={n_axes_after}/{n_axes_before} kept "
+        f"(drop_bottom_frac={drop_bottom_frac}, threshold={threshold})"
     )
     return v_hack
 
@@ -396,7 +430,10 @@ def main(cfg: Config) -> int:
                             "tau_axis": str(cfg.v_hack_tau_axis), "schema": "v2_with_sv"})
         # extract zeros grads at exit; opt is built below so no opt-state taint.
         model.train()  # restore train mode; eval was set only for the extract pass
-    v_hack_cpu = load_v_hack(v_hack_path, model_name, wrappers, k_use=cfg.v_hack_k)
+    v_hack_cpu = load_v_hack(
+        v_hack_path, model_name, wrappers,
+        k_use=cfg.v_hack_k, drop_bottom_frac=cfg.v_hack_drop_bottom_frac,
+    )
     v_hack = {name: v.to(device) for name, v in v_hack_cpu.items()}
     # Teacher pool: pre-generated rollouts on disk keyed by problem_id. Each step's
     # G_t teacher rollouts come from a uniform random sample of that prompt's cache,