The Entropy-Replacement Theorem

The headline of Paper III. For Haar-random bulk states, the difference between the two observers’ von Neumann entropies and the Shannon entropy difference of their bulk-marginal diagonals is provably small in mean square – one power of $d$ below the signal variance.

$\big[S(\rho_{R_A}) - S(\rho_{R_B})\big] \;-\; \big[H(P_A) - H(P_B)\big] \;=\; O_{L^2}\!\big(d^{-2}\,d_M^{-1}\big).$

Why this is the headline. Paper II established that the averaged observer state collapses to the diagonal of the bulk marginal. It did not establish that the entropy – a non-linear functional – is therefore governed by the diagonal. The entropy-replacement theorem is the missing step. It is what Paper II was missing.

Shape of the proof. Expand the true entropy around the bulk-marginal diagonal. The error splits into two pieces: an off-diagonal part $F_{\rm off}$ (coming from coherences thrown away by the diagonal model) and a diagonal-to-bulk part $F_{\rm diag}$ (the difference between the actual reduced diagonal and the bulk-marginal diagonal we replace it with). Both must be $O(d^{-4}\,d_M^{-2})$ – one power of $d$ below the signal variance $\Theta(d^{-3}\,d_M^{-2})$.

• $F_{\rm off}$ is closed by a resolvent representation, a linear bound, and a fourth-moment estimate (Appendix C.1–C.5).
• $F_{\rm diag}$ is closed by a centered-operator identity that routes it through the same base moment as $F_{\rm off}$ (Appendix C.6). No numerical constant survives inside the proof.

The full Haar-class proof is unconditional. The product-class version is conditional on a parallel replacement principle in a regime where probabilities are concentrated; that conditional is named in plain sight on Result III.