diag: log refreshed-basis overlap with prior basis per v_hack refresh

cin_t collapses from ~0.3 to ~0.04 exactly at the first refresh step on the
4B substrate route run. Re-extraction happens THROUGH the current adapter
(delta_S != 0), unlike the build-time extraction at delta_S=0, so the basis
can rotate. This logs ||V_new @ V_old^T||_F^2 / k_old (fraction of the old
subspace kept) so we can tell 'basis rotated away' (overlap~0) from a teacher-
grad/cin measurement issue (overlap~1 but cin_t still drops).

Co-Authored-By: Claudypoo <288921227+claudypoo@users.noreply.github.com>

src/projected_grpo/train.py

@@ -1443,6 +1443,21 @@ def main(cfg: Config) -> int:
             finally:
                 logger.enable("projected_grpo.extract_vhack_grad")
                 logger.enable("__main__")
+            # DIAGNOSTIC: how far did the refreshed basis rotate from the prior one?
+            # Rows are orthonormal, so ||V_new @ V_old^T||_F^2 / k_old = fraction of
+            # the OLD subspace still spanned by the NEW basis, in [0,1].
+            #   ~1 -> refresh tracks a stable hack subspace (the design's premise)
+            #   ~0 -> re-extraction at current weights landed near-orthogonal, so the
+            #         live grad's overlap (cin_t) jumps discontinuously at the refresh.
+            shared = set(v_hack) & set(_post)
+            ovl = [((_post[n].float().to(device) @ v_hack[n].float().mT)).pow(2).sum().item()
+                   / v_hack[n].shape[0] for n in shared]
+            overlap = sum(ovl) / max(1, len(ovl))
+            logger.info(
+                f"refresh@step{step}: {len(_post)}mod/{sum(V.shape[0] for V in _post.values())}ax  "
+                f"basis_overlap_with_prev={overlap:.3f}  "
+                f"SHOULD: >~0.5 if refresh tracks a stable hack subspace; <~0.2 => "
+                f"re-extraction rotated the basis (cin_t jumps, refresh is harmful)")
             v_hack.clear()
             v_hack.update({n: V.to(device) for n, V in _post.items()})
             opt.zero_grad(set_to_none=True)  # extract leaves .grad populated