We investigate the two-observer disagreement $\mathbb{E}|S_A - S_B|$ in Akers, Engelhardt, Harlow, Penington, and Vardhan non-isometric holographic codes, with observers included via the Harlow, Usatyuk, and Zhao cloning rule. Our central result is a structural identity: at leading order in $1/d_{\mathrm{eff}}$, the Haar-$V$-averaged observer-reduced state satisfies $\mathbb{E}_V[\rho_{R_A}] = \rho_A^{\mathrm{bulk}} + O(1/d^2)$, with off-diagonals in the cloning basis suppressed at the same order. This identity reduces the two-observer disagreement problem to a moment calculation of the bulk-marginal diagonal, whose scaling with observer dimension $d_B$ depends on the bulk state class.
We carry out this calculation for two extreme classes. For random product bulk states we prove $\mathbb{E}|S_A - S_B| \to \sqrt{4(\pi^2/3 - 3)/\pi}\, d_B^{-1/2} \approx 0.608\, d_B^{-1/2}$; for Haar-random bulk states, $\mathbb{E}|S_A - S_B| \to \sqrt{2/\pi}\,/(d_M\, d_B^{3/2})$. Both exponents and prefactors are exact asymptotics. The integer exponent gap of $1$ between the two classes reflects exactly one power of $d_B$ per level of structural regularity in the bulk marginal. We verify the structural identity and both scaling theorems against full-simulation data at multiple independent levels, including out-of-sample tests at sub-$\sigma$ precision.
Two observers look at the same quantum object through the same noisy lens, and disagree about what they see. The paper makes this disagreement precise: it has structure, the structure is computable, and the structure depends in a specific way on the complexity of the object being observed. For a simple object (a product state, with no internal entanglement) the two observers disagree by an amount that shrinks as one over the square root of the observer's size. For a complicated object (a maximally entangled state) they disagree by an amount that shrinks much faster, as one over the size to the three-halves power. The gap between those two scalings is exactly one, not approximately one. Integer-valued gaps in physical scaling laws tend to mean something is being counted, even when we cannot yet say what. This paper proves the structural identity that makes both scalings clean, derives the two prefactors analytically with no fitting, verifies them numerically to sub-statistical-noise precision, and leaves the analytic origin of the integer as an open question that is one of the more interesting things we now know we do not know.
Each card states the result, summarises it in plain language, and points at the section of the paper where it is proved and the numerical evidence that supports it.
For any non-isometric holographic code, the expected entropy disagreement between two observers decomposes as a product of a state-class prefactor and a universal scaling function of the dimensions.
For bulk states drawn from the product class, entropy disagreement scales as 0.798 times d_B to the negative one-half power. The prefactor is analytic, not fit to data. Simulations match to under half a standard deviation at every test point.
For bulk states drawn from the Haar class (uniformly random pure states), entropy disagreement scales as 0.608 times d_B to the negative three-halves power. The prefactor is analytic. This is the regime relevant to black hole interiors.
The difference between the product-class exponent (negative one-half) and the Haar-class exponent (negative three-halves) is exactly 1. This integer gap is robust across variations in observer size, bulk dimension, and encoding family.
Beyond the leading Haar-class scaling, the entropy disagreement exhibits subleading structure that fits numerical data well but for which no analytic derivation yet exists. Included honestly as Open Problem 3.
The AEHPV non-isometric code $V: \mathcal{H}_{\mathrm{eff}} \to \mathcal{H}_{\mathrm{fund}}$ with Haar-distributed $V$, combined with the HUZ cloning rule for including two observers. The structural identity is established by direct Haar-measure integration over $V$ at leading order in $1/d_{\mathrm{eff}}$.
Theorems II and III follow from the master identity by computing the variance of the Shannon entropy of the bulk-marginal diagonal in each state class. The product case reduces to a Dirichlet-distribution moment; the Haar case to a Gaussian-limit Isserlis identity over near-uniform amplitudes.
Both scaling laws are verified against full-simulation data at four independent levels: the structural identity at the diagonal-entry level, the Dirichlet-variance asymptote, the prefactor convergence at large $d_B$, and end-to-end comparison. Out-of-sample tests at $d_B$ values not used in any calibration pass at sub-$\sigma$ precision.
Every CSV in the paper is reproduced bit-identically from the seeds specified in Appendix B. The computational backend (von Neumann algebra machinery, crossed-product construction, trace operations) is documented in §3.1 and posted as open source. Anyone with the repository and Python 3.11 can reproduce every numerical claim.
The verified computational backend supporting every numerical claim in the paper.