The Marginal Was Not the Diagonal

The first thing that went wrong after Paper I was a sentence we had been proud of.

Paper I’s foundation was a structural identity: average over the random code, and what an observer’s reduced state collapses to is the bulk marginal – the reduced description of the bulk state on the observer’s slice. We wrote $\mathbb{E}_V[\rho_{R_A}] = \rho_A^{\mathrm{bulk}} + O(1/d^2)$ and built two scaling laws on top of it.

When we sat down to re-derive that identity from scratch – not to defend it, but to try to break it – it broke in a specific and instructive way. The first-moment Haar integral over the code does not return the full marginal. It returns the diagonal of the marginal, in the observer’s cloning basis. Every off-diagonal entry – every coherence between two distinct pointer values – gets multiplied by a Kronecker delta that is zero unless the two indices agree. They vanish. Not approximately; at first moment, exactly.

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

Why does it happen? Two effects conspire. The HUZ cloning rule copies the observer into a chosen pointer basis, which privileges that basis. Then the Haar average over the code scrambles the phases of anything off-diagonal in it, and scrambled phases average to zero. The cloning picks the basis; the randomness erases everything that is not aligned with it. The diagonal is what is left.

Is this a disaster? No – it is a gift, once you stop being embarrassed by it. The diagonal of a density matrix is an ordinary probability distribution. By proving the average observer state is exactly that diagonal, we turned the reduction to a moment calculation from an approximation into an identity. The scaling laws now had a precise object to be laws of: the bulk-marginal diagonal $P_X$.

But it sharpened a question we had been able to ignore while the foundation was vague. If the average observer state is the diagonal, fine – but the actual observer entropy is computed from the actual state, not its average, and the actual state has off-diagonal structure and normalization wobble that the diagonal throws away. Does the disagreement between two diagonals actually equal the disagreement between two true observer entropies?

Paper I never had to ask. Paper II asked it and could not yet answer. That is the whole story of the next several weeks, and it has its own posts.

The small lesson: when a foundation is stated loosely, it can carry a paper a long way before anyone notices the looseness is load-bearing. Re-derive your foundations on purpose, hostile to yourself. The marginal was not the diagonal, and the difference was the rest of the programme.