The Haar-Class Law, Unconditional

For Haar-random (maximally scrambled) bulk states, the disagreement falls off as the observer size to the negative three-halves power. The prefactor is analytic, not fit. Crucially, this is now unconditional: the diagonal-to-bulk piece is closed by the centered-operator reduction with no constant left to numerics.

Theorem 2Haar ClassUnconditionalNo numerical constant

The result, in full

The Haar-Class Law, Unconditional

For Haar-random (maximally scrambled) bulk states, the disagreement law is now unconditional – proved end-to-end with no numerical constant inside the proof.

\mathbb{E}|S_A - S_B| \;=\; \sqrt{\tfrac{2}{\pi}}\,\cdot\,\frac{1}{d_M\, d_B^{3/2}}\,(1+o(1)) \;\approx\; \frac{0.798}{d_M}\, d_B^{-3/2}.

This is the regime relevant to black-hole interiors: states with maximal global scrambling. The exponent $-3/2$ is exact; the prefactor $\sqrt{2/\pi}$ is exact and analytic, not fit. The grouped-Dirichlet covariance shortcut that gave the right prefactor for the wrong reason – twice – is corrected against the exact Dirichlet covariance (negative within an observer, exactly zero across observers).

What changed from Paper II. Paper II proved this law in the diagonal model. It did not prove that the diagonal model governs the true von Neumann entropy. Paper III proves both: first the entropy-replacement theorem (Result I), then specialisation to Haar via a grouped-Dirichlet/Gaussian-limit moment calculation. The result is unconditional in the regime physicists most care about.

The closed appendix. Appendix C.6 – the diagonal-to-bulk fluctuation – is closed by the centered-operator identity. There is no scratch_Mdom.py-style constant sitting inside the proof; that script is preserved in the reproducibility bundle as a labelled diagnostic only, marker of the route we did not take.