paper v2.31.1: deep cos +0.155 → +0.151 (ground-truth re-measurement)

The paper had 3-seed penalized DFA deep cos at +0.155 ± 0.025, but
re-measuring on the saved checkpoints gives 0.1506 (mean of per-seed
deep means is 0.1507; pooled mean over 12 deep-layer values is 0.1506).
The std of 0.025 matches pooled ddof=1 ✓.

Same paragraph also had inconsistent values: "+0.155 ± 0.025" 3-seed
above, then "+0.165" single-seed s42 in the lambda sweep.  Unified to
3-seed throughout.

§5 ¶2 lambda sweep updates:
  lam=1e-4 ||h_L||  2.4e4 (s42 only) → 2.2e4 (3-seed mean)
  lam=1e-4 ||g_L||  6.3e-7 (s42)     → 7.0e-7 (3-seed)
  lam=1e-4 deep cos -0.022 (s42)     → -0.020 (3-seed)
  lam=1e-2 deep cos +0.165 (s42)     → +0.151 (3-seed, same as the
                                       three-seed value used elsewhere)

Other places updated: §4 ¶4 prose, Table 2, Appendix J Table 9
DFA+pen mean row (deep cos +0.155 → +0.151 and ||h_L||/||g_L||
columns updated to 30-ep dfa_pen_short means rather than the round-19
single-seed numbers), Appendix L paragraph.

Page layout preserved: 9 main pages, refs p10, 18 total, 0 overfull.

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

2 files changed, 5 insertions, 5 deletions

diff --git a/paper/main.pdf b/paper/main.pdf
index ae1d43a..928562f 100644
--- a/paper/main.pdf
+++ b/paper/main.pdf
diff --git a/paper/main.tex b/paper/main.tex
index 5c4eb93..d7665b7 100644
--- a/paper/main.tex
+++ b/paper/main.tex
@@ -98,7 +98,7 @@ 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 epoch-1 checkpoints where layers $1$--$4$ are essentially zero, layer $0$ has cosine $+0.42$, $+0.44$, and $+0.42$ across seeds (Table~\ref{tab:mode_validation}; per-seed values in Appendix~\ref{app:layer0_dominance}). 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 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 $0.360 \pm 0.001$ 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, the three methods reveal a clean cosine-versus-accuracy dissociation. Credit Bridge achieves roughly $4\times$ the deep cosine of DFA and $2\times$ that of State Bridge, yet its final accuracy matches DFA's and is $9$ percentage points below State Bridge's. We therefore frame the Mode~2 reading as a three-part proposition. \emph{Observation}: under the same intervention and matched training budget, CB and DFA reach the same accuracy despite a $4\times$ deep-cosine gap, while SB is the best accuracy with intermediate cosine. \emph{Inference}: layerwise cosine to the BP gradient is necessary to rule out grossly wrong credit signals (it distinguishes the rescued regime from the clamp-dominated vanilla regime), but it is not sufficient to certify that the supplied signal is useful credit for depth. \emph{Mechanism hypothesis}: usefulness depends on whether the local update induces useful forward-state change across blocks, not merely whether its direction is close to the BP gradient in angle. Under this reading, CB supplies a gradient-direction surrogate that aligns with BP in angle but does not translate to a coordinated forward-state improvement, while State Bridge supplies a state-level downstream teaching signal that preserves aspects of useful credit which layerwise cosine does not measure. We state this as a mechanism hypothesis rather than a theorem because we have measured the angle-to-accuracy gap but not the full functional-credit content; the reporting rule that follows is robust to either interpretation. 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.
+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 $0.360 \pm 0.001$ with deep cosine $+0.151 \pm 0.025$ 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, the three methods reveal a clean cosine-versus-accuracy dissociation. Credit Bridge achieves roughly $4\times$ the deep cosine of DFA and $2\times$ that of State Bridge, yet its final accuracy matches DFA's and is $9$ percentage points below State Bridge's. We therefore frame the Mode~2 reading as a three-part proposition. \emph{Observation}: under the same intervention and matched training budget, CB and DFA reach the same accuracy despite a $4\times$ deep-cosine gap, while SB is the best accuracy with intermediate cosine. \emph{Inference}: layerwise cosine to the BP gradient is necessary to rule out grossly wrong credit signals (it distinguishes the rescued regime from the clamp-dominated vanilla regime), but it is not sufficient to certify that the supplied signal is useful credit for depth. \emph{Mechanism hypothesis}: usefulness depends on whether the local update induces useful forward-state change across blocks, not merely whether its direction is close to the BP gradient in angle. Under this reading, CB supplies a gradient-direction surrogate that aligns with BP in angle but does not translate to a coordinated forward-state improvement, while State Bridge supplies a state-level downstream teaching signal that preserves aspects of useful credit which layerwise cosine does not measure. We state this as a mechanism hypothesis rather than a theorem because we have measured the angle-to-accuracy gap but not the full functional-credit content; the reporting rule that follows is robust to either interpretation. 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}
@@ -117,13 +117,13 @@ Condition & Deep-layer alignment signal & Measurement regime & Interpretation \\
 \midrule
 Vanilla DFA, early epoch & $\overline{\cos}_{deep}{=}{-}0.008{\pm}0.013$, $\overline{\rho}_{deep}{=}{-}0.003{\pm}0.005$ & meaningful ($\|g\|{\sim}10^{-6}$) & mode 2 present without mode 1 \\
 Vanilla DFA, converged & $\overline{\cos}_{deep}{=}{-}0.022$, $\overline{\rho}_{deep}{=}+0.002$ & degenerate ($\|g\|{\sim}10^{-9}$) & mode 1 obscures mode 2 \\
-Penalized DFA, $\lambda{=}10^{-2}$ & $\overline{\cos}_{deep}{=}+0.155{\pm}0.025$, $\overline{\rho}_{deep}{=}+0.080{\pm}0.011$ & meaningful ($\|g\|{\sim}10^{-6}$) & partial alleviation of both modes \\
+Penalized DFA, $\lambda{=}10^{-2}$ & $\overline{\cos}_{deep}{=}+0.151{\pm}0.025$, $\overline{\rho}_{deep}{=}+0.080{\pm}0.011$ & meaningful ($\|g\|{\sim}10^{-6}$) & partial alleviation of both modes \\
 Fresh-$B$ null control & $\overline{\cos}_{deep}{=}+0.002{\pm}0.022$ ($n{=}20$ draws) & meaningful & training-specific adaptation check \\
 \bottomrule
 \end{tabular}}
 \end{table}
 
-Once the reference vector is meaningful again, the deep layers no longer sit exactly at null. At $\lambda{=}10^{-2}$, penalized DFA reaches a three-seed deep-layer mean cosine of $+0.155 \pm 0.025$ and deep perturbation correlation of $+0.080 \pm 0.011$, whereas vanilla DFA is essentially zero on both metrics in the deep blocks, consistent with prior concerns that alternative feedback can fail by supplying poor credit directions even before full collapse \citep{bartunov2018assessing,moskovitz2018feedback,crafton2019backpropagation,refinetti2023aligning}. The null calibration rules out the interpretation that this recovered signal is merely measurement noise: on the same penalized checkpoint, replacing the training-time feedback matrices with 20 fresh random $B_l$ draws gives a deep cosine of only $+0.002 \pm 0.022$, with per-layer standard deviations of $0.013$--$0.023$, all within noise of zero (Table~\ref{tab:mode_validation}). The $\lambda$ sweep sharpens the dissociation further: at $\lambda{=}10^{-4}$, Mode~1 is already alleviated, with $\|h_L\|{=}2.4\times 10^4$ and $\|g_L\|{=}6.3\times 10^{-7}$, but deep cosine remains $-0.022$, while at $\lambda{=}10^{-2}$ it rises to $+0.165$ and deep $\rho$ to $+0.091$ (Figure~\ref{fig:penalty_rescue}). The improvement is real, but it is only partial.
+Once the reference vector is meaningful again, the deep layers no longer sit exactly at null. At $\lambda{=}10^{-2}$, penalized DFA reaches a three-seed deep-layer mean cosine of $+0.151 \pm 0.025$ and deep perturbation correlation of $+0.080 \pm 0.011$, whereas vanilla DFA is essentially zero on both metrics in the deep blocks, consistent with prior concerns that alternative feedback can fail by supplying poor credit directions even before full collapse \citep{bartunov2018assessing,moskovitz2018feedback,crafton2019backpropagation,refinetti2023aligning}. The null calibration rules out the interpretation that this recovered signal is merely measurement noise: on the same penalized checkpoint, replacing the training-time feedback matrices with 20 fresh random $B_l$ draws gives a deep cosine of only $+0.002 \pm 0.022$, with per-layer standard deviations of $0.013$--$0.023$, all within noise of zero (Table~\ref{tab:mode_validation}). The $\lambda$ sweep sharpens the dissociation further: at $\lambda{=}10^{-4}$, Mode~1 is already alleviated, with three-seed mean $\|h_L\|{\approx}2.2\times 10^4$ and $\|g_L\|{\approx}7.0\times 10^{-7}$, but the three-seed deep cosine remains $-0.020$, while $\lambda{=}10^{-2}$ delivers the $+0.151$ and $+0.080$ above (Figure~\ref{fig:penalty_rescue}). The improvement is real, but it is only partial.
 
 A rescue intervention is only informative if its direct cost is controlled. The relevant control is BP trained under the same penalty for the same matched $30$-epoch budget: across three seeds, BP falls from $0.585 \pm 0.001$ without the penalty to $0.530$ with $\lambda{=}10^{-2}$ (BP+penalty single seed), so the penalty has a direct cost of about $5.5$ percentage points even when credit assignment is correct, whereas DFA moves in the opposite direction, from $0.301 \pm 0.005$ to $0.360 \pm 0.001$, and State Bridge moves further still, from $0.213$ to $0.453 \pm 0.003$, all under the same $30$-epoch intervention (Figure~\ref{fig:penalty_rescue}; Appendix~\ref{app:sb_penalty}). Relative to the frozen-blocks baseline of $0.349$, BP+penalty retains a margin of $+18.1$ points, State Bridge+penalty retains $+10.4$ points, and DFA+penalty retains only $+1.1$ points. The remaining BP-to-DFA gap of $17.0$ points is therefore a lower bound on the part of DFA's deficit that is not explained by simple penalty-induced capacity loss alone, though not a clean isolation because BP uses an end-to-end loss whereas DFA uses block-local losses. The substantially smaller BP-to-State-Bridge gap of $0.530 - 0.453 = 7.7$ points shows that the cross-method differences in penalty-rescued accuracy are not all attributable to a uniform ``random-feedback ceiling'': the bridge construction in State Bridge can recover much more of the BP-with-penalty performance than DFA can, on the same architecture and the same intervention. The residual gap after that control is what keeps Mode~2 substantively alive while letting it have method-dependent severity.
 
@@ -512,12 +512,12 @@ CB+pen mean & $0.360 \pm 0.003$ & $5680 \pm 178$ & $1.90\times 10^{-5}$ & $+0.67
 \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 & $0.360 \pm 0.001$ & $4.0\times 10^4$ & $9.0\times 10^{-7}$ & $+0.155 \pm 0.025$ & $+0.080 \pm 0.011$ \\
+DFA+pen mean & $0.360 \pm 0.001$ & $1.3\times 10^4$ & $1.6\times 10^{-6}$ & $+0.151 \pm 0.025$ & $+0.080 \pm 0.011$ \\
 \bottomrule
 \end{tabularx}
 \end{table}
 
-The penalty rescue effect on State Bridge is much larger than on DFA: $+24$ percentage points for State Bridge versus $+5.9$ 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 averages $14.3\times$ for SB+penalty, $18.6\times \pm 0.5$ for DFA+penalty, and $19.3\times$ for CB+penalty (three seeds each), and the embedding layer's relative drift is $7.1\times$ for SB versus $44.6\times$ for CB and $94.6\times \pm 1.4$ for DFA, so none of the three methods' per-block updates are silenced under penalty and CB's are in fact larger in magnitude than SB's while DFA's embedding updates are the largest of all, yet CB's and DFA's final accuracies are both $9.3$ percentage points below State Bridge's. 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, by $-0.45\times 10^{-3}$ for CB+penalty, and by only $-5\times 10^{-5}$ for DFA+penalty (three seeds each, $30$-epoch runs via the same training script). 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 and about thirty-five times more than a step in DFA's direction, even though CB's direction is more aligned with the BP gradient in angle than either. 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 by only $0.122$ and DFA+penalty's by only $0.095 \pm 0.007$ over the same $30$ epochs (three seeds). Deep cosine ranks the three methods CB $>$ SB $>$ DFA, but every functional metric (nudging, integrated training-loss decrease, headline accuracy) ranks them SB $\gg$ CB $\approx$ DFA: the ordering produced by deep cosine is the only one that does not predict accuracy correctly. This is the strongest form of the cos-versus-accuracy dissociation: across three audited fixed-feedback methods under the same penalty intervention, the ranking implied by angular agreement with the BP gradient is contradicted by three independent functional measurements that do predict accuracy. 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 $+5.9$ 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 $+0.151$ 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 averages $14.3\times$ for SB+penalty, $18.6\times \pm 0.5$ for DFA+penalty, and $19.3\times$ for CB+penalty (three seeds each), and the embedding layer's relative drift is $7.1\times$ for SB versus $44.6\times$ for CB and $94.6\times \pm 1.4$ for DFA, so none of the three methods' per-block updates are silenced under penalty and CB's are in fact larger in magnitude than SB's while DFA's embedding updates are the largest of all, yet CB's and DFA's final accuracies are both $9.3$ percentage points below State Bridge's. 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, by $-0.45\times 10^{-3}$ for CB+penalty, and by only $-5\times 10^{-5}$ for DFA+penalty (three seeds each, $30$-epoch runs via the same training script). 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 and about thirty-five times more than a step in DFA's direction, even though CB's direction is more aligned with the BP gradient in angle than either. 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 by only $0.122$ and DFA+penalty's by only $0.095 \pm 0.007$ over the same $30$ epochs (three seeds). Deep cosine ranks the three methods CB $>$ SB $>$ DFA, but every functional metric (nudging, integrated training-loss decrease, headline accuracy) ranks them SB $\gg$ CB $\approx$ DFA: the ordering produced by deep cosine is the only one that does not predict accuracy correctly. This is the strongest form of the cos-versus-accuracy dissociation: across three audited fixed-feedback methods under the same penalty intervention, the ranking implied by angular agreement with the BP gradient is contradicted by three independent functional measurements that do predict accuracy. 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{Layer-0 Dominance: Per-Seed Vanilla DFA Early-Epoch Cosines}
 \label{app:layer0_dominance}