The Centered-Operator Trick

The fix for the diagonal-to-bulk bound is short, which is how you know it is right. The failed route – splitting the matrix, trusting the script – was long and full of caveats. The working route is one identity and a reuse.

Here is the move. Instead of attacking the covariance matrix $M$ directly, you rewrite the diagonal-to-bulk error using centered block operators. For each observer $X$ and each pointer value $a$, define $\widetilde Q_X^{(a)} = Q_X^{(a)} - p_X^a\, Q,$ the block operator with its trace piece subtracted off, and collect them into $G_X = \sum_a \delta_a^X\,\widetilde Q_X^{(a)}$. The centering is the whole trick: it kills the non-zero-trace part that was spoiling everything. With it, the leading diagonal-to-bulk fluctuation becomes a single trace against the projector fluctuation, $L_A - L_B = -\frac{d}{\rho}\,\operatorname{Tr}\!\big[(P - \rho I)(G_A - G_B)\big] + \text{remainder},$ and the rank-projector variance formula bounds its mean square by $\mathbb{E}_V[(L_A-L_B)^2 \mid \psi] \le C\,(d^2/D)\,\operatorname{Tr}[(G_A - G_B)^2]$.

Now the reuse. That object, $\mathbb{E}_\psi \operatorname{Tr}[(G_A - G_B)^2]$, is not new. It is the same base moment we had already proved, in closed form, to control the off-diagonal error $F_{\mathrm{off}}$ – bounded by $C_0/(d^4 d_M)$. So the diagonal-to-bulk piece and the off-diagonal piece are now bounded by the same quantity through the same projector estimate. Put $D = d^2 d_M$ in and you get $\mathbb{E}[(L_A - L_B)^2] = O(d^{-4} d_M^{-2})$ – exactly the order required – with no new constant, and nothing left to a script.

That is the difference between the two routes, stated cleanly. The failed route needed a numerical constant because it treated the diagonal-to-bulk error as a separate problem with its own off-diagonal block to estimate. The working route shows it was never a separate problem: centred correctly, it is the off-diagonal error in different clothing, and the bound we already had covers both. One identity collapses two problems into one.

We verified the representation before we trusted it – the correlation between the centered-operator form and the direct fluctuation is better than 0.999 in simulation – but the verification only checks the proof; it is not part of it. That is the line Scholar drew, and drawing it is what made the theorem unconditional.

When a bound resists every direct attack, the question to ask is not “how do I estimate this harder,” but “is this secretly the same as something I have already bounded?” Centering the operators made the secret visible. That is the trick, and it closed Appendix C.