Fix precision: ||g_L|| = 7.2e-4 (mean), not 7.4e-4 (rounding)

2 files changed, 1 insertions, 1 deletions

diff --git a/paper/main.pdf b/paper/main.pdf
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--- a/paper/main.pdf
+++ b/paper/main.pdf
diff --git a/paper/main.tex b/paper/main.tex
index f331b2c..1bbf7eb 100644
--- a/paper/main.tex
+++ b/paper/main.tex
@@ -80,7 +80,7 @@ Mode~1 has two parts. The activation-growth part~(a) is a scale pathology of fix
 
 We tested this mechanism story against four natural alternative attributions, all of which it survives. \emph{Not residual-skip-driven:} on the same ResMLP-d256 with terminal LN kept and the additive skip removed ($h_{l+1}{=}F_l(h_l)$), DFA still inflates $\|h_L\|$ from $\sim\!5$ to $\sim\!2.2{\times}10^{4}$ in three epochs and converges to $\|h_L\|{\approx}1.06{\times}10^{8}$ and $\|g_L\|{\approx}1.09{\times}10^{-10}$ at $100$ epochs, both already at the diagnostic floor (Appendix~\ref{app:no_residual}). \emph{Not task-signal-driven:} replacing labels by i.i.d.\ random class targets refreshed every minibatch on the same backbone, DFA still reaches $\|h_L\|{\approx}1.67{\times}10^{8}$ and $\|g_L\|{\approx}8{\times}10^{-12}$ at $100$ epochs while accuracy stays at chance (Appendix~\ref{app:random_targets}). \emph{Not DFA-specific:} the same random-target ablation also drives $\|h_L\|$ from $9$ to $6.2{\times}10^{3}$ for State Bridge and $2.0{\times}10^{4}$ for Credit Bridge in three epochs, again at chance accuracy, so all three audited fixed-feedback methods exhibit data-agnostic activation growth (Appendix~\ref{app:random_targets}). \emph{Not shared by EP:} under the same random-target protocol, EP keeps $\|h_L\|{\approx}586$ at five epochs of training, $25\times$ smaller than DFA's three-epoch value on the same architecture, consistent with EP's bounded behavior on real labels and confirming that the random-target assay separates the explosion-prone fixed-feedback class from EP's energy-based local objective.
 
-The matched same-backbone causal control for diagnostic~(b) is removing terminal LayerNorm. On the same ResMLP-d256 with the residual skip intact, $100$ epochs of DFA, three seeds, the residual stream still inflates to $\|h_L\|\!\approx\!1.21\times 10^7$, but the deepest hidden-layer BP gradient remains at $\|g_L\|\!\approx\!7.4\times 10^{-4}$ (four orders of magnitude above the diagnostic~(b) floor), and the final test accuracy is $0.327\pm 0.013$, statistically indistinguishable from vanilla DFA's $0.308\pm 0.014$. Removing terminal LayerNorm therefore preserves Mode~1~(a) but cleanly eliminates Mode~1~(b) on the same architecture, while leaving final task accuracy essentially unchanged. Combined with the broader cross-architecture pattern (StudentNet and the BatchNorm CNN, which lack terminal LayerNorm, never trigger diagnostic~(b); ViT-Mini with a terminal LN does, by epochs 2--3 (Figure~\ref{fig:temporal_cross_arch})), terminal LayerNorm is necessary for Mode~1~(b) in the audited residual ResMLP and ViT-Mini setting. The collapse is also not a late-epoch curiosity: $\|g_L\|$ drops from $9.8\times 10^{-4}$ at epoch~0 to $6.7\times 10^{-8}$ by epoch~4 in the temporal replay across three seeds, so the protocol fires within the first $11$ epochs of a $100$-epoch run and is actionable as an early-stop criterion rather than a post hoc explanation. Once measurement degeneracy is identified, the next question is whether poor deep credit remains even before collapse.
+The matched same-backbone causal control for diagnostic~(b) is removing terminal LayerNorm. On the same ResMLP-d256 with the residual skip intact, $100$ epochs of DFA, three seeds, the residual stream still inflates to $\|h_L\|\!\approx\!1.21\times 10^7$, but the deepest hidden-layer BP gradient remains at $\|g_L\|\!\approx\!7.2\times 10^{-4}$ (four orders of magnitude above the diagnostic~(b) floor), and the final test accuracy is $0.327\pm 0.013$, statistically indistinguishable from vanilla DFA's $0.308\pm 0.014$. Removing terminal LayerNorm therefore preserves Mode~1~(a) but cleanly eliminates Mode~1~(b) on the same architecture, while leaving final task accuracy essentially unchanged. Combined with the broader cross-architecture pattern (StudentNet and the BatchNorm CNN, which lack terminal LayerNorm, never trigger diagnostic~(b); ViT-Mini with a terminal LN does, by epochs 2--3 (Figure~\ref{fig:temporal_cross_arch})), terminal LayerNorm is necessary for Mode~1~(b) in the audited residual ResMLP and ViT-Mini setting. The collapse is also not a late-epoch curiosity: $\|g_L\|$ drops from $9.8\times 10^{-4}$ at epoch~0 to $6.7\times 10^{-8}$ by epoch~4 in the temporal replay across three seeds, so the protocol fires within the first $11$ epochs of a $100$-epoch run and is actionable as an early-stop criterion rather than a post hoc explanation. Once measurement degeneracy is identified, the next question is whether poor deep credit remains even before collapse.
 
 \section{Failure Mode 2: Low Intrinsic Credit-Direction Quality}
 \label{sec:mode2}