Round 38 CB+penalty multi-seed: confirm clean cos-vs-acc dissociation across 3 fixed-feedback methods

CB+penalty 3-seed (4-block d=256, 30ep, lam=1e-2):
- acc 0.360±0.003 (same as DFA+pen, 9pp below SB+pen)
- ||h_L||=5680±178, ||g_L||=1.9e-5 (HEALTHY)
- layer-0 cos +0.652±0.005
- deep cos +0.679±0.008 (4x DFA+pen, 2x SB+pen)
- deep rho +0.464±0.025 (6x DFA+pen)

Final 3-method rescue comparison:
  DFA+pen: acc 0.363, deep cos 0.155, deep rho 0.080
  SB+pen:  acc 0.453, deep cos 0.322, deep rho 0.402
  CB+pen:  acc 0.360, deep cos 0.679, deep rho 0.464

Clean cos-vs-acc dissociation:
- CB has 4x higher deep cos than DFA but SAME accuracy
- SB has intermediate deep cos but HIGHEST accuracy
- Alignment to BP gradient is NECESSARY but NOT SUFFICIENT for usable credit

Paper updates:
- §4 ¶4: now includes all 3 methods with numbers, adds 'cos is necessary but not
  sufficient' framing based on 3-method dissociation
- Appendix K: adds CB+pen 3-seed rows + vanilla CB baseline for comparison
- Main content still 9 pages exactly (within E&D limit)

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

2 files changed, 8 insertions, 3 deletions

diff --git a/paper/main.pdf b/paper/main.pdf
index 834f5da..c2510e5 100644
--- a/paper/main.pdf
+++ b/paper/main.pdf
diff --git a/paper/main.tex b/paper/main.tex
index af04885..f256119 100644
--- a/paper/main.tex
+++ b/paper/main.tex
@@ -91,7 +91,7 @@ A second metric with different numerical failure modes tells the same story. Cos
 
 Per-layer reporting is therefore not cosmetic. In ResMLP under vanilla DFA, the headline aggregate alignment $\Gamma \approx 0.07$--$0.10$ can look mildly positive only because layer $0$ remains strongly aligned while the deep network is not: at the same early checkpoints where layers $1$--$4$ are essentially zero, layer $0$ has cosine $+0.42$, $+0.45$, and $+0.39$ across seeds (Table~\ref{tab:mode_validation}). The resulting average can therefore be driven by the embedding layer even when the interior blocks are effectively unaligned, so aggregate reporting obscures the very distinction needed to separate ``measurement collapse'' from ``poor credit direction.'' This layer-$0$ dominance is specific to the ResMLP DFA setting; on ViT-Mini DFA, all layers are near zero, which strengthens the broader methodological point that alignment should be reported per layer rather than only in aggregate. With the two modes separated observationally, the remaining question is whether intervention can move them independently.
 
-Mode~2 has method-dependent severity within the audited fixed-feedback family once Mode~1 is alleviated. Applying the same per-block scale-control penalty $\lambda{=}10^{-2}$ that rescued DFA to State Bridge on the same 4-block $d{=}256$ ResMLP backbone over $30$ epochs and three seeds gives a converged test accuracy of $0.453 \pm 0.003$ and a deep mean cosine of $+0.322 \pm 0.007$ with deep mean $\rho$ of $+0.402 \pm 0.015$, while DFA under the same intervention reaches only $0.363 \pm 0.001$ with deep cosine $+0.155 \pm 0.025$ and deep $\rho$ $+0.080 \pm 0.011$ (Table~\ref{tab:mode_validation}; Appendix~\ref{app:sb_penalty}). The State Bridge penalty rescue is roughly $24$ percentage points above the vanilla State Bridge baseline of $0.213$ on the same architecture and seed and, more importantly for the paper's central walk-back, exceeds the architecture-matched frozen-blocks shallow baseline of $0.349$ by $+10.4$ percentage points. State Bridge with the penalty intervention is therefore the first audited non-BP method whose trained deep blocks substantively improve over an architecture-matched random-block baseline; the headline accuracy gap is comparable to BP+penalty's $+18.1$ pp over the same shallow baseline. Neither the activation scale nor the deep BP gradient magnitude is silenced under the penalty: $\|h_L\|$ stays at $302 \pm 8$ and $\|g_L\|$ at $\sim\!1.8\times 10^{-4}$, both well within the meaningful-measurement regime, so the recovered deep cosine is computed against an informative reference and not against a numerical floor. Within this rescued regime, deep cosine is positive but does not by itself predict end-task accuracy across methods, which strengthens the broader methodological point that alignment must be reported jointly with measurement validity and a depth-utilization baseline rather than as a single headline number.
+Mode~2 has method-dependent severity within the audited fixed-feedback family once Mode~1 is alleviated. Applying the same per-block scale-control penalty $\lambda{=}10^{-2}$ that rescued DFA to State Bridge and to Credit Bridge on the same 4-block $d{=}256$ ResMLP backbone over $30$ epochs and three seeds gives converged test accuracies of $0.453 \pm 0.003$ (SB) and $0.360 \pm 0.003$ (CB), with deep mean cosines of $+0.322 \pm 0.007$ (SB) and $+0.679 \pm 0.008$ (CB) and deep mean $\rho$ of $+0.402 \pm 0.015$ (SB) and $+0.464 \pm 0.025$ (CB), while DFA under the same intervention reaches $0.363 \pm 0.001$ with deep cosine $+0.155 \pm 0.025$ and deep $\rho$ $+0.080 \pm 0.011$ (Table~\ref{tab:mode_validation}; Appendix~\ref{app:sb_penalty}). The State Bridge penalty rescue is roughly $24$ percentage points above the vanilla State Bridge baseline of $0.213$ on the same architecture and, more importantly for the paper's central walk-back, exceeds the architecture-matched frozen-blocks shallow baseline of $0.349$ by $+10.4$ percentage points. State Bridge with the penalty intervention is therefore the first audited non-BP method whose trained deep blocks substantively improve over an architecture-matched random-block baseline; the headline accuracy gap is comparable to BP+penalty's $+18.1$ pp over the same shallow baseline. Neither the activation scale nor the deep BP gradient magnitude is silenced under the penalty: $\|h_L\|$ stays at $302 \pm 8$ for SB and $5680 \pm 178$ for CB, with $\|g_L\|$ at $\sim\!1.8\times 10^{-4}$ and $\sim\!1.9\times 10^{-5}$ respectively, both well within the meaningful-measurement regime, so the recovered deep cosines are computed against an informative reference and not against a numerical floor. Within this rescued regime, Credit Bridge has roughly $4\times$ the deep cosine of DFA and $2\times$ the deep cosine of State Bridge but reaches the same final accuracy as DFA and $9$ percentage points below State Bridge, so alignment to the BP gradient is a necessary but not sufficient diagnostic of usable credit for depth. This cross-method dissociation strengthens the methodological point that alignment must be reported jointly with measurement validity and a depth-utilization baseline rather than as a single headline number.
 
 \section{Intervention and Cross-Architecture Evidence}
 \label{sec:validation}
@@ -477,16 +477,21 @@ seed & test acc & $\|h_L\|$ & $\|g_L\|$ & deep cos & deep $\rho$ \\
 SB+pen $42$ & $0.4564$ & $302$ & $1.75\times 10^{-4}$ & $+0.312$ & $+0.392$ \\
 SB+pen $123$ & $0.4514$ & $311$ & $1.74\times 10^{-4}$ & $+0.327$ & $+0.424$ \\
 SB+pen $456$ & $0.4509$ & $292$ & $1.92\times 10^{-4}$ & $+0.326$ & $+0.391$ \\
-\midrule
 SB+pen mean & $0.453 \pm 0.003$ & $302 \pm 8$ & $1.80\times 10^{-4}$ & $+0.322 \pm 0.007$ & $+0.402 \pm 0.015$ \\
 \midrule
+CB+pen $42$ & $0.3596$ & $5431$ & $1.88\times 10^{-5}$ & $+0.684$ & $+0.498$ \\
+CB+pen $123$ & $0.3642$ & $5834$ & $1.81\times 10^{-5}$ & $+0.667$ & $+0.452$ \\
+CB+pen $456$ & $0.3562$ & $5775$ & $2.01\times 10^{-5}$ & $+0.685$ & $+0.442$ \\
+CB+pen mean & $0.360 \pm 0.003$ & $5680 \pm 178$ & $1.90\times 10^{-5}$ & $+0.679 \pm 0.008$ & $+0.464 \pm 0.025$ \\
+\midrule
 vanilla SB $42$ & $0.213$ & $9.85\times 10^6$ & $1\times 10^{-8}$ & --- & --- \\
+vanilla CB $42$ & $0.211$ & $6.7\times 10^7$ & $\sim 0$ & --- & --- \\
 DFA+pen mean (3 seeds) & $0.363 \pm 0.001$ & $4.0\times 10^4$ & $9.0\times 10^{-7}$ & $+0.155 \pm 0.025$ & $+0.080 \pm 0.011$ \\
 \bottomrule
 \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. Verifying whether Credit Bridge under the same intervention shows a similar within-class gradation is in-flight at the time of writing; results will be reported as a multi-seed extension of Table~\ref{tab:sb_penalty}.
+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. 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}