From 4ec8d0d9516d2c134df26ec4781d5e7fc63dedbd Mon Sep 17 00:00:00 2001
From: YurenHao0426 <Blackhao0426@gmail.com>
Date: Wed, 8 Apr 2026 13:21:19 -0500
Subject: Round 41 (Appendix L): add 4th piece of cos-vs-acc evidence -
 trajectory loss decrease
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SB+penalty train loss: 2.047 -> 1.589 (Δ=0.458 over 30 epochs)
CB+penalty train loss: 1.996 -> 1.874 (Δ=0.122 over 30 epochs)
Ratio: 3.8x (matching the 4x nudging ratio)

This is the third independent functional measurement, from per-epoch logs
in the same round 38 JSONs (log['train_loss']). The cos-vs-acc
dissociation now has four independent pieces of evidence, all agreeing:

1. Test accuracy: CB 0.360 vs SB 0.453 (9.3pp gap)
2. Parameter drift: CB w2 19.3x vs SB 14.3x (CB larger updates)
3. Single-step nudging Δloss: CB -0.45e-3 vs SB -1.78e-3 (4x gap, eta=0.01)
4. Trajectory loss decrease: CB 0.122 vs SB 0.458 (3.8x gap over 30 epochs)

All four inversely correlated with deep cosine (CB has higher deep cos).

Co-Authored-By: Claude Opus 4.6 (1M context) <noreply@anthropic.com>
---
 paper/main.tex | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

(limited to 'paper/main.tex')

diff --git a/paper/main.tex b/paper/main.tex
index 2854cca..662a27c 100644
--- a/paper/main.tex
+++ b/paper/main.tex
@@ -485,7 +485,7 @@ DFA+pen mean (3 seeds) & $0.363 \pm 0.001$ & $4.0\times 10^4$ & $9.0\times 10^{-
 \end{tabular}
 \end{table}
 
-The penalty rescue effect on State Bridge is much larger than on DFA: $+24$ percentage points for State Bridge versus $+5.5$ percentage points for DFA on the same architecture and intervention. SB+penalty is the first audited non-BP method whose trained deep blocks substantively beat the architecture-matched random-block baseline. We treat this as evidence that Mode~2 (low intrinsic credit-direction quality) has method-dependent severity within the audited fixed-feedback family once Mode~1 is alleviated, rather than being a uniform property of all fixed-feedback local-credit objectives. Importantly, State Bridge's deep cosine $+0.322$ is approximately twice DFA's $+0.155$ on the same intervention, but neither approaches the BP reference value of $\approx +1.0$, so this is a within-class gradation in credit-direction quality, not a claim that bridge constructions ``solve'' Mode~2. The drift diagnostic reinforces this reading rather than contradicting it: per-block $w_2$ relative displacement after $30$ epochs is $14.3\times$ for SB+penalty and $19.3\times$ for CB+penalty (a $35\%$ gap), and the embedding layer's relative drift is $7.1\times$ for SB versus $44.6\times$ for CB (a $6\times$ gap), so CB's per-block updates are not silenced under penalty and are in fact larger in magnitude than SB's, yet CB's final accuracy is $9.3$ percentage points lower. The larger-but-less-useful parameter updates in CB are consistent with the mechanism hypothesis that angular agreement with the BP gradient does not by itself certify the functional forward-state content of the update. The nudging test at the same checkpoints provides the direct functional measurement: taking a small step of size $\eta{=}0.01$ in the direction of each method's per-layer credit $a_l$ decreases the test loss by $-1.78\times 10^{-3}$ on average over the deep blocks for SB+penalty and by only $-0.45\times 10^{-3}$ for CB+penalty, a $4\times$ gap in functional loss decrease that inverts the $4\times$ deep-cosine gap between the two methods. At the same per-layer credit direction, a step in SB's direction moves the loss about four times more than a step in CB's direction, even though CB's direction is more aligned with the BP gradient in angle. Under the same intervention Credit Bridge reaches a three-seed test accuracy of $0.360 \pm 0.003$, a three-seed deep mean cosine of $+0.679 \pm 0.008$, and a three-seed deep mean $\rho$ of $+0.464 \pm 0.025$, with $\|h_L\|\approx 5680 \pm 178$ and $\|g_L\|\approx 1.9\times 10^{-5}$ well above the diagnostic floor. Credit Bridge therefore has an even higher deep cosine than State Bridge (about $4\times$ the DFA value and roughly $2\times$ the State Bridge value), but reaches the same final accuracy as DFA+penalty and $9.3$ percentage points below State Bridge+penalty. This is a clean dissociation: within the audited fixed-feedback family under the same rescue, deep cosine and deep $\rho$ differ by more than a factor of four across methods without tracking final accuracy in the same direction, so alignment to the BP gradient is a necessary but not sufficient diagnostic of usable credit for depth. That cross-method dissociation is a direct reason the protocol in Section~\ref{sec:protocol} keeps final accuracy, layerwise credit quality, and the depth-utilization baseline as three separate reporting axes rather than collapsing them into a single headline.
+The penalty rescue effect on State Bridge is much larger than on DFA: $+24$ percentage points for State Bridge versus $+5.5$ percentage points for DFA on the same architecture and intervention. SB+penalty is the first audited non-BP method whose trained deep blocks substantively beat the architecture-matched random-block baseline. We treat this as evidence that Mode~2 (low intrinsic credit-direction quality) has method-dependent severity within the audited fixed-feedback family once Mode~1 is alleviated, rather than being a uniform property of all fixed-feedback local-credit objectives. Importantly, State Bridge's deep cosine $+0.322$ is approximately twice DFA's $+0.155$ on the same intervention, but neither approaches the BP reference value of $\approx +1.0$, so this is a within-class gradation in credit-direction quality, not a claim that bridge constructions ``solve'' Mode~2. The drift diagnostic reinforces this reading rather than contradicting it: per-block $w_2$ relative displacement after $30$ epochs is $14.3\times$ for SB+penalty and $19.3\times$ for CB+penalty (a $35\%$ gap), and the embedding layer's relative drift is $7.1\times$ for SB versus $44.6\times$ for CB (a $6\times$ gap), so CB's per-block updates are not silenced under penalty and are in fact larger in magnitude than SB's, yet CB's final accuracy is $9.3$ percentage points lower. The larger-but-less-useful parameter updates in CB are consistent with the mechanism hypothesis that angular agreement with the BP gradient does not by itself certify the functional forward-state content of the update. The nudging test at the same checkpoints provides the direct functional measurement: taking a small step of size $\eta{=}0.01$ in the direction of each method's per-layer credit $a_l$ decreases the test loss by $-1.78\times 10^{-3}$ on average over the deep blocks for SB+penalty and by only $-0.45\times 10^{-3}$ for CB+penalty, a $4\times$ gap in functional loss decrease that inverts the $4\times$ deep-cosine gap between the two methods. At the same per-layer credit direction, a step in SB's direction moves the loss about four times more than a step in CB's direction, even though CB's direction is more aligned with the BP gradient in angle. The $30$-epoch training trajectories give a third independent confirmation: SB+penalty's training loss falls from $2.047$ at epoch $1$ to $1.589$ at epoch $30$, a decrease of $0.458$, whereas CB+penalty's training loss falls from $1.996$ to $1.874$ over the same $30$ epochs, a decrease of $0.122$, so SB+penalty achieves a $3.8\times$ larger integrated training-loss reduction with the same architecture, optimizer, and penalty. Headline accuracy, single-step nudging, and multi-epoch integrated loss decrease all agree that SB's credit direction is functionally more useful than CB's by roughly the same factor, even though CB's direction is angularly closer to the BP gradient; deep cosine alone misses this consistently. Under the same intervention Credit Bridge reaches a three-seed test accuracy of $0.360 \pm 0.003$, a three-seed deep mean cosine of $+0.679 \pm 0.008$, and a three-seed deep mean $\rho$ of $+0.464 \pm 0.025$, with $\|h_L\|\approx 5680 \pm 178$ and $\|g_L\|\approx 1.9\times 10^{-5}$ well above the diagnostic floor. Credit Bridge therefore has an even higher deep cosine than State Bridge (about $4\times$ the DFA value and roughly $2\times$ the State Bridge value), but reaches the same final accuracy as DFA+penalty and $9.3$ percentage points below State Bridge+penalty. This is a clean dissociation: within the audited fixed-feedback family under the same rescue, deep cosine and deep $\rho$ differ by more than a factor of four across methods without tracking final accuracy in the same direction, so alignment to the BP gradient is a necessary but not sufficient diagnostic of usable credit for depth. That cross-method dissociation is a direct reason the protocol in Section~\ref{sec:protocol} keeps final accuracy, layerwise credit quality, and the depth-utilization baseline as three separate reporting axes rather than collapsing them into a single headline.
 
 \section{Reproducibility}
 \label{app:reproducibility}
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