The Structural Identity

$\mathbb{E}_V[\rho_{R_A}] \;=\; \operatorname{diag}\!\big(\rho_A^{\mathrm{bulk}}\big) + O(1/d^2).$

Averaged over the random code, the observer’s reduced state equals the diagonal of the bulk marginal in the cloning basis. The off-diagonal coherences across distinct pointer values are projected out by the combination of HUZ cloning and Haar averaging – not by hand, by the physics.

Three papers in one line.

• Paper I stated this identity against the full bulk marginal – too loose.
• Paper II corrected it to the diagonal – the load-bearing first-moment lemma.
• Paper III proved the entropy-replacement theorem (Result I) that lets this averaged-state collapse govern the true observer von Neumann entropy.

It is the foundation. Everything else in the paper rests on it. The first-moment Haar integration that produces the Kronecker-delta collapse is direct and is verified at the level of individual diagonal entries (eighteen entries; all agree to within first-moment numerics). The downstream consequence – that the entropy disagreement reduces to a moment calculation of the bulk-marginal diagonal – is the reason the rest of the paper exists.