paper v2.30: fix layer-0 cosine numbers + add per-seed appendix M

Found a numerical error in §4 ¶3: the layer-0 vanilla DFA cosines were
listed as +0.42, +0.45, +0.39 across seeds 42/123/456 but the actual
re-measurement on the saved early-epoch checkpoints gives +0.421, +0.436,
+0.418 (the s456 value was off by 0.03). The deep-mean numbers in
Table 2 (-0.008 ± 0.013) were already correct.

Changes:
- §4 ¶3: layer-0 trio updated to +0.42, +0.44, +0.42 across seeds and
  cite now points to a new per-seed appendix.
- New Appendix M (Layer-0 Dominance): 6-row table of per-seed per-layer
  cosines on vanilla DFA early checkpoints (3 seeds × ep 1, 2), with
  per-layer ||g||. Documents the layer-0 dominance pattern that drives
  the headline aggregate Γ on these checkpoints.
- results/vanilla_dfa_early_ckpts/per_layer_cos_3seed.json: machine-
  readable dump of all 6 measurements for future audit.
- §7 compressed (~30 words trimmed across the closing paragraph) and
  Figure 3 width 0.92 → 0.82 to keep main content at exactly 9 pages
  after the appendix addition.

Verified: 9 pages main + refs on p10, 18 total, 0 overfull boxes.

Co-Authored-By: Claude Opus 4.6 (1M context) <noreply@anthropic.com>

1 files changed, 98 insertions, 0 deletions

diff --git a/results/vanilla_dfa_early_ckpts/per_layer_cos_3seed.json b/results/vanilla_dfa_early_ckpts/per_layer_cos_3seed.json
new file mode 100644
index 0000000..b1c9de6
--- /dev/null
+++ b/results/vanilla_dfa_early_ckpts/per_layer_cos_3seed.json
@@ -0,0 +1,98 @@
+{
+  "s42_ep1": {
+    "per_layer_cos": [
+      0.4208017587661743,
+      0.004838745109736919,
+      -0.027907393872737885,
+      -0.038808517158031464,
+      -0.03807452693581581
+    ],
+    "per_layer_g_norm_median": [
+      1.2788983667633147e-06,
+      6.714879532410123e-07,
+      6.794414844080165e-07,
+      6.840860464762955e-07,
+      6.860727808088996e-07
+    ]
+  },
+  "s42_ep2": {
+    "per_layer_cos": [
+      0.4369885325431824,
+      -0.001831148169003427,
+      -0.039548110216856,
+      -0.05499383062124252,
+      -0.05395658314228058
+    ],
+    "per_layer_g_norm_median": [
+      5.726975587094785e-07,
+      1.5392966190574953e-07,
+      1.5485885285215772e-07,
+      1.5658912388971657e-07,
+      1.576423045435149e-07
+    ]
+  },
+  "s123_ep1": {
+    "per_layer_cos": [
+      0.43564510345458984,
+      0.008119785226881504,
+      -0.032616909593343735,
+      0.015984557569026947,
+      0.0165080688893795
+    ],
+    "per_layer_g_norm_median": [
+      1.1976044334005564e-06,
+      6.545930091306218e-07,
+      6.567059358530969e-07,
+      6.61169508475723e-07,
+      6.618395786972542e-07
+    ]
+  },
+  "s123_ep2": {
+    "per_layer_cos": [
+      0.4596974849700928,
+      0.005399191286414862,
+      -0.03668719530105591,
+      0.0029763614293187857,
+      0.0030912011861801147
+    ],
+    "per_layer_g_norm_median": [
+      5.662583362209261e-07,
+      1.351014162764841e-07,
+      1.350312288650457e-07,
+      1.36037257902899e-07,
+      1.3582508984200103e-07
+    ]
+  },
+  "s456_ep1": {
+    "per_layer_cos": [
+      0.4175308644771576,
+      0.010558882728219032,
+      -0.026187893003225327,
+      0.007104482501745224,
+      0.006094767712056637
+    ],
+    "per_layer_g_norm_median": [
+      1.0160714509765967e-06,
+      3.9226171111295116e-07,
+      3.8446077610387874e-07,
+      3.8279964087450935e-07,
+      3.864832649469463e-07
+    ]
+  },
+  "s456_ep2": {
+    "per_layer_cos": [
+      0.40920183062553406,
+      0.002915931399911642,
+      -0.038979340344667435,
+      0.0008528914768248796,
+      6.853311788290739e-06
+    ],
+    "per_layer_g_norm_median": [
+      4.5268734538694844e-07,
+      8.855825228692993e-08,
+      8.533425699397412e-08,
+      8.582557597947016e-08,
+      8.653171335026855e-08
+    ]
+  }
+}
\ No newline at end of file