Complexity-Sensitive Complementarity in Non-Isometric Holographic Codes (revised, Paper II)

Unified manuscript draft. Combines all section drafts into a single reading document. Order reflects final paper structure. Author’s notes and meta-text preserved from individual drafts have been removed; section content is unchanged.

Abstract

We investigate the two-observer disagreement $\mathbb{E}|S_A - S_B|$ in Akers–Engelhardt–Harlow–Penington–Vardhan non-isometric holographic codes, with observers included via the Harlow–Usatyuk–Zhao cloning rule. Our central result is a structural identity: at leading order in $1/d_{\rm eff}$, the Haar-$V$-averaged observer-reduced state equals the diagonal of the bulk $A$-marginal in the observer’s cloning basis, $\mathbb{E}_V[\rho_{R_A}] = \mathrm{diag}(\rho_A^{\rm bulk}) + O(1/d^2)$. Bulk-marginal coherences across distinct pointer values are projected out by the combination of HUZ cloning and Haar-$V$ averaging. 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.

§1. Introduction

1.1 Observer complementarity in non-isometric codes

In recent work, observer-dependent entropies have emerged as a central diagnostic of bulk reconstruction in non-isometric holographic codes. The essential tension is that different observer-inclusion rules – each motivated by different physical considerations and each a natural construction – give rise to different von Neumann entropies on the same bulk state. This disagreement is the quantitative content of observer complementarity: a bulk state has multiple coexisting entropic interpretations, and the gap between them is a feature, not a bug, of the non-isometric-code framework.

Two influential lines of work have crystalized specific versions of this picture. First, Akers, Engelhardt, Harlow, Penington, and Vardhan (AEHPV) [2207.06536] established the non-isometric-code framework itself: a random code $V: \mathcal{H}_{\rm eff} \to \mathcal{H}_{\rm fund}$ with $d_{\rm eff} > d_{\rm fund}$ and Haar-distributed on its domain, with the ratio $\rho \equiv d_{\rm fund}/d_{\rm eff}$ quantifying the non-isometry. Second, Harlow, Usatyuk, and Zhao (HUZ) [2501.02359] proposed a specific rule for including an observer: clone the observer in a chosen pointer basis onto an external reference register, then trace out the fundamental Hilbert space. This gives a precise notion of “observer-accessible entropy” for any bulk state. HUZ verified that errors in the resulting observer-description are exponentially small in the observer dimension, scaling as $E_{\rm ovl} \sim 1/d_{\rm Ob}$, to leading order.

Engelhardt, Gesteau, and Harlow (EGH) [2507.06046] then applied this framework to the Antonini–Sasieta–Swingle–Rath cosmological setup, finding a quantitative gap between the AdS-boundary-observer and closed-universe-observer SWAP-test coefficients that is the operational signature of observer complementarity in non-trivial holographic configurations. Higginbotham’s subsequent refinement [2512.17993, JHEP03(2026)183] identified that EGH’s specific SWAP observables are suboptimal and derived improved bounds.

1.2 The question

The present paper addresses a question that is adjacent to, but distinct from, all of the above: suppose the bulk Hilbert space decomposes into two observer factors, $\mathcal{H}_{\rm eff} = \mathcal{H}_A \otimes \mathcal{H}_B \otimes \mathcal{H}_C$, and both observers $A$ and $B$ are included via HUZ cloning. What is the typical von Neumann-entropy disagreement $|S_A - S_B|$, as a function of observer dimension $d_B$?

This question differs from the single-observer HUZ setting because it probes a joint moment of the Haar-$V$ ensemble, involving both $\rho_{R_A}$ and $\rho_{R_B}$ simultaneously. It differs from EGH’s SWAP-test gap because the observable is the von Neumann entropy, not a second-Rényi-like trace expression. And it differs from the quantum-reference-frame entropies of [2412.15502, 2603.23598] because observers are included via HUZ cloning rather than by crossed-product construction.

A priori, one might naïvely expect $\mathbb{E}|S_A - S_B| \sim 1/d_B$ – single observer HUZ inheritance scaled by the factor-of-two observer count – with a universal scaling exponent. As we show, this is decisively wrong for typical bulk states, and whether it is right or wrong depends on bulk-state complexity in a quantitatively specific way.

1.3 Main result

The central technical contribution of this paper is a structural identity that unifies the two-observer entropy problem at leading order in $1/d_{\rm eff}$. Let $|\Psi\rangle$ be the HUZ-included state on $\mathcal{H}_{\rm fund} \otimes \mathcal{H}_{R_A} \otimes \mathcal{H}_{R_B}$ and $\rho_{R_A} = \mathrm{Tr}_{{\rm fund}, R_B}\, |\Psi\rangle\langle\Psi|$ the observer-$A$ reduced state.

Main Theorem (structural identity). For any bulk state $|\psi\rangle \in \mathcal{H}_A \otimes \mathcal{H}_B \otimes \mathcal{H}_C$, the Haar-$V$ expectation satisfies $\boxed{\;\;\mathbb{E}_V[\rho_{R_A}] \;=\; \mathrm{diag}(\rho_A^{\rm bulk}) + O(1/d^2),\;\;}$ where $\rho_A^{\rm bulk} = \mathrm{Tr}_{BC}(|\psi\rangle\langle\psi|)$ and $\mathrm{diag}(\cdot)$ denotes the diagonal operator in the cloning basis on $\mathcal{H}_{R_A}$. The bulk-marginal coherences across distinct pointer values are projected out by the $\delta_{aa'}$ structure of the first-moment Haar contraction.

This identity, proved in §3, reduces the two-observer disagreement problem to computing the variance of the Shannon entropy of the bulk-marginal diagonal – a random-matrix calculation that depends on the bulk state class. Applied to two natural extreme classes, we obtain the following scaling theorems (proved in §§4–5, stated formally there):

• Product class (bulk state $|\psi_A\rangle \otimes |\psi_B\rangle \otimes |\psi_C\rangle$ with each factor Haar): $\rho_A^{\rm bulk}$ is rank-1, its diagonal has $\mathrm{Dirichlet}(1,\ldots,1)$ amplitudes, and $\mathbb{E}|S_A - S_B| \approx 0.608\, d_B^{-1/2}$.
• Haar class (bulk state Haar on $\mathcal{H}_{\rm eff}$): $\rho_A^{\rm bulk}$ is near maximally mixed with Dirichlet fluctuations, and $\mathbb{E}|S_A - S_B| \approx (0.798/d_M)\, d_B^{-3/2}$.

The exponents $-1/2$ and $-3/2$ are exact asymptotics, not power-law fits. The prefactors $\sqrt{4(\pi^2/3 - 3)/\pi}$ and $\sqrt{2/\pi}/d_M$ are derived in closed form from the bulk-marginal moment computation in each case.

The integer exponent gap $\alpha_P - \alpha_H = 1$ is a direct consequence of the Dirichlet-variance hierarchy separating rank-1 and near-maximally-mixed bulk marginals. It reflects exactly one power of $d_B$ per unit of structural regularity in the bulk marginal.

The structural identity and both scaling theorems are verified at multiple independent levels – the structural identity directly (across 18 diagonal entries at $d_B \in \{4, 6, 8\}$), the Dirichlet-variance asymptote, the prefactor convergence, the Gaussian-limit ratio, and end-to-end comparison with full HUZ-plus-$V$ simulation data. Out-of-sample tests at $d_B$ values not used in any calibration pass at sub-$\sigma$ precision.

1.4 Physical interpretation

The structural identity has a direct physical reading. Observer-cloning under HUZ is a specific way of extracting the classical pointer record of an observer’s state into an external register. Haar-averaging over the non-isometric code $V$ erases the off-diagonal coherences of this record (in the cloning basis) and leaves only the diagonal – which is, up to the stated subleading correction, exactly the bulk $A$-marginal’s diagonal. In this sense, the Haar-$V$-averaged observer record is a classical readout of the bulk marginal.

When this observation is applied to the two-observer disagreement $|S_A - S_B|$, a natural physical picture emerges: the two classical readouts differ in proportion to how noisy the classical readout is for the given bulk state. For low-complexity (rank-1 bulk marginal) states, the readout carries only a few macroscopic modes with significant Dirichlet noise, and the two observers’ entropic snapshots differ by $O(d_B^{-1/2})$. For high-complexity (near-maximally-mixed bulk marginal) states, the readout is nearly uniform with tiny fluctuations, and the two snapshots agree to within $O(d_B^{-3/2})$.

This refines the observer-complementarity discussion at the entropy level. At the inner-product level, HUZ’s $1/d_{\rm Ob}$ bound on observer-reconstruction errors is state-independent. At the entropy level, the analog has a built-in state-class sensitivity that is visible only when both observers are included simultaneously.

1.5 Organization of the paper

Section 2 fixes notation and reviews the AEHPV+HUZ framework in a form that supports the two-observer analysis. Section 3 proves the structural identity (Theorem 3.2), the common technical core of both theorems. Sections 4 and 5 prove Theorems 1 and 2 respectively. Section 6 gives the physical interpretation, including the integer exponent gap and the conjectured rank-$r$ interpolation. Section 7 assembles the numerical evidence, including all data points from the Phase 5 and Phase 6 scans and three out-of-sample tests. Section 8 positions the result relative to the observer-complementarity, non-isometric-code, holographic-complexity, and quantum-reference-frame literatures. Section 9 concludes with a summary and outlook. Appendix A presents generalized EGH formulas for arbitrary complex bulk states, a technical result obtained in the course of this program. Appendix B documents reproducibility information (seed conventions, sample sizes, code availability).

§2. Setup

This section fixes notation and conventions. We follow the AEHPV non-isometric-code framework [AEHPV 2207.06536], adapted to the two-observer scenario introduced by EGH 2507.06046 and HUZ 2501.02359. Readers familiar with these constructions may skip to §3.

2.1 Non-isometric maps and the AEHPV framework

The bulk effective theory and the fundamental (boundary) theory are two finite-dimensional Hilbert spaces connected by a linear map:

When $d_{\rm eff} > d_{\rm fund}$, $V$ is non-isometric: there are bulk “null states” in the kernel of $V$. Following AEHPV, we take $V$ to be the first $d_{\rm fund}$ rows of a Haar-random unitary on $U(d_{\rm eff})$. This ensures $V V^\dagger = I_{\rm fund}$ exactly, while $V^\dagger V$ is a Haar-random rank-$d_{\rm fund}$ projector on $\mathcal{H}_{\rm eff}$. The non-isometry parameter is

Throughout this paper we fix $\rho = 1/2$.

2.2 Observer-included states via HUZ cloning

The HUZ 2501.02359 rule specifies an observer’s perspective on a bulk state by appending an external reference that clones the observer’s pointer states. In the single-observer case, the bulk factorizes as $\mathcal{H}_{\rm eff} = \mathcal{H}_{\rm Ob} \otimes \mathcal{H}_M$ (observer and matter), and a reference register $\mathcal{H}_R \cong \mathcal{H}_{\rm Ob}$ is added. The cloning isometry $\mathrm{Clone}_{\rm Ob \to R}: |{\rm ob}, m\rangle \otimes |0\rangle_R \mapsto |{\rm ob}, m\rangle \otimes |{\rm ob}\rangle_R$ produces the HUZ map

Applied to any bulk state $|\psi\rangle \otimes |0\rangle_R$ and normalized by post-selection, this produces a state on $\mathcal{H}_{\rm fund} \otimes \mathcal{H}_R$. The observer-accessible reduced state and its entropy are

2.3 The two-observer scenario

We consider the setup of EGH 2507.06046 and its natural refinement to two independent observers. The bulk effective space factorizes as

where $A$ and $B$ are two independent observer factors and $C$ is a matter register. Two auxiliary reference registers $R_A$ and $R_B$ of dimensions $d_A, d_B$ are introduced, and the two-observer HUZ map

is applied to the bulk state $|\psi\rangle \otimes |0\rangle_{R_A} \otimes |0\rangle_{R_B}$. The post-selection-normalized state is denoted $|\Psi\rangle \in \mathcal{H}_{\rm fund} \otimes \mathcal{H}_{R_A} \otimes \mathcal{H}_{R_B}$.

The two observer-dependent entropies are

and similarly for $\rho_{R_B}$. The central quantity of this paper is the Haar-averaged disagreement

where the expectation is taken over $V$ (Haar on $U(d_{\rm eff})$) and optionally over bulk states $|\psi\rangle$ drawn from a specified class.

2.4 Bulk state classes

The theorems of this paper apply to two distinct bulk-state classes, each defining an ensemble over $\mathcal{H}_{\rm eff}$:

• Product class (P). Bulk states of the form $|\psi\rangle = |\psi_A\rangle \otimes |\psi_B\rangle \otimes |\psi_C\rangle$, with each factor Haar-distributed on its respective Hilbert space. Such states have no bulk entanglement across the $A$/$B$/$C$ partition; the $A$-marginal $\rho_A^{\rm bulk}$ is a rank-1 pure state.
• Haar class (H). Bulk states drawn uniformly from the unit sphere of $\mathcal{H}_{\rm eff}$ (Haar measure on $\mathbb{C}^{d_{\rm eff}}$). Such states are generic – high-entanglement, maximally non-product in the sense of the Schmidt decomposition. The marginals $\rho_A^{\rm bulk}, \rho_B^{\rm bulk}$ are close to maximally mixed with small Dirichlet-type fluctuations.

These two classes anchor the extremes of a natural complexity spectrum and are the focus of the theorems that follow. Intermediate classes (Schmidt-rank-$r$ bulk states) are a natural target for follow-up work, discussed in §6.5.

2.5 Parameters

Unless stated otherwise, all numerical work uses $d_A = d_B \equiv d_B$ (so the setup is symmetric under observer exchange), $d_M = 4$, and $\rho = 1/2$ (so $d_{\rm fund} = d_B^2 d_M / 2$). The “scanned dimension” is $d_B$. All averages $\mathbb{E}[\cdot]$ refer to the joint measure over $V$ and bulk states; except where noted the two averages are independent.

§3. The structural identity

The central technical observation of this paper is that the Haar-$V$ expectation of the first-observer reduced state has an especially simple form at leading order in $1/d_{\rm eff}$: it is the diagonal, in the cloning basis on $\mathcal{H}_{R_A}$, of the bulk $A$-marginal density matrix. Theorems 1 and 2 then follow from computing this diagonal for two different bulk state classes.

3.1 Setup

Throughout we take $V: \mathcal{H}_{\rm eff} \to \mathcal{H}_{\rm fund}$ to be an AEHPV non-isometric map with $d_{\rm fund} = \rho\, d_{\rm eff}$, $\rho \in (0,1)$ fixed. Concretely, $V$ is the first $d_{\rm fund}$ rows of a Haar-random unitary on $\mathcal{H}_{\rm eff}$. The effective Hilbert space factorizes as

with $\mathcal{H}_A, \mathcal{H}_B$ the two observer factors and $\mathcal{H}_C$ a matter register. Under the two-observer HUZ rule, both observers are cloned in their respective pointer bases, producing an auxiliary reference pair $(R_A, R_B)$ with $d_{R_A} = d_{R_B} = d$. The post-$V$, post-cloning normalized state is

Using index notation $\psi = \psi_{abc}$ for the bulk-state components in the $(A,B,C)$ basis, and writing $|v_{ab}\rangle \equiv V(|a\rangle \otimes |b\rangle \otimes |\psi_C\rangle)$ when $|\psi\rangle$ factorizes across $C$ (we will treat the general case shortly), the unnormalized state is

The observer-$A$ reduced state is obtained by tracing out $\mathcal{H}_{\rm fund}$ and $\mathcal{H}_{R_B}$, yielding

The norm is

3.2 Haar-$V$ expectation

The map $V$ is drawn from the Haar measure on the first $d_{\rm fund}$ rows of $U(d_{\rm eff})$; equivalently, $V^\dagger V$ is a uniformly random rank-$d_{\rm fund}$ orthogonal projector on $\mathcal{H}_{\rm eff}$. A basic moment identity gives

Applied to (3.1) with $x = |a',b,c'\rangle$ and $y = |a,b,c\rangle$, the bra–ket factorizes as

so substituting into (3.1):

where $\mathrm{diag}(\rho_A^{\rm bulk})$ denotes the diagonal of the bulk $A$-marginal in the cloning basis, regarded as an operator whose off-diagonal entries vanish.

The norm expectation is

where $\rho_A^{\rm bulk} = \mathrm{Tr}_{BC}(|\psi\rangle\langle\psi|)$ is the bulk $A$-marginal density matrix.

A subtlety: equations (3.4)–(3.5) are ratio-of-expectations statements, not the quantity of physical interest $\mathbb{E}_V[(\rho_{R_A})_{aa'}] = \mathbb{E}_V[(\rho_{R_A}^{\rm unnorm})_{aa'} / \|\Psi\|^2]$. These differ by fluctuations in $\|\Psi\|^2$. The following concentration estimate closes the gap.

Lemma 3.1 (Norm concentration). With $|\psi\rangle$ of unit norm, $\mathrm{Var}(\|\Psi\|^2) = O\bigl(\rho(1-\rho)\, S_2(|\psi|^2) / d_{\rm eff}\bigr)$, where $S_2(|\psi|^2) = \sum_{abc} |\psi_{abc}|^4$ is the Rényi-2 probability of the flat distribution over $(a,b,c)$.

Sketch. From (3.2), $\|\Psi\|^2$ is a weighted diagonal sum of $V^\dagger V$ entries. Using the joint second moment $\mathbb{E}_V\bigl[(V^\dagger V)_{xy}\, (V^\dagger V)_{x'y'}\bigr]$ and the symmetry that $V^\dagger V$ is a uniformly random rank-$d_{\rm fund}$ projector (hence has joint diagonal distribution Dirichlet with fixed sum), one obtains the claimed variance bound directly. A detailed accounting gives $\mathrm{Var}(\|\Psi\|^2) \leq 4\rho(1-\rho)/(d^4 d_M)$ for the state classes of interest. $\square$

Consequently $\|\Psi\|^2$ concentrates around $\rho$ with relative fluctuation $\sigma(\|\Psi\|^2)/\mathbb{E}[\|\Psi\|^2] = O(1/d^2)$, and

3.3 Why the off-diagonals collapse

Equation (3.4) carries an important content: the Haar-$V$ average sends the off-diagonal entries of $\rho_{R_A}$ in the cloning basis to zero at leading order, regardless of whether the bulk marginal $\rho_A^{\rm bulk}$ has off-diagonals. The mechanism is the $\delta_{aa'}$ factor in (3.3a): the bra and ket sharing the same value of $a$ is the only configuration in which $V^\dagger V$ contributes at first moment.

This is the precise sense in which the HUZ cloning construction, combined with Haar-random $V$, acts as pointer-basis decoherence on the observer reference register. The cloning map records the value of $a$ classically into $R_A$; the subsequent Haar-$V$ average then erases coherences across distinct $a$ values, because at first moment those coherences require correlating $V^\dagger V$ between different eigenvectors of the $a$-pointer projector, and (3.3) prohibits such correlation at order $\rho$.

The bulk-marginal off-diagonals do not vanish – they remain present in the bulk state – but they are not transmitted to the observer’s reference register when that register has been HUZ-cloned in the pointer basis and filtered through a Haar-random non-isometric code.

The variance of the off-diagonals around their (zero) mean is a separate, subleading phenomenon. From the joint second moment of $V^\dagger V$, one obtains

so off-diagonals of $\rho_{R_A}$ in a single $V$ instance have fluctuations of size $\sim 1/\sqrt{d_{\rm eff}\, d_B}$, not zero. The mean is zero at this order; the fluctuations are a separate phenomenon that contributes to $\mathrm{Var}(S)$ but not to $\mathbb{E}[S]$ at the order we work to.

We can now state the identity formally.

Theorem 3.2 (Structural identity). For any bulk state $|\psi\rangle$ and the two-observer HUZ setup above, in the Haar-$V$ measure,

where $\mathrm{diag}(\cdot)$ denotes the diagonal operator in the cloning basis on $\mathcal{H}_{R_A}$. The bulk-marginal off-diagonals are suppressed in expectation by the $\delta_{aa'}$ structure of the first-moment Haar contraction; their residual instance-by-instance fluctuations are $O(1/\sqrt{d_{\rm eff}\,d_B})$ per entry and do not contribute to $\mathbb{E}[S(\rho_{R_A})]$ at the order of Theorems 4.2 and 5.2.

Remark. The analogous statement holds for observer $B$ by symmetry of the construction: $\mathbb{E}_V[\rho_{R_B}] = \mathrm{diag}(\rho_B^{\rm bulk}) + O(1/d^2)$ in the cloning basis on $\mathcal{H}_{R_B}$.

3.4 Numerical verification

Figure 2 verifies Theorem 3.2 directly. Panel (a) shows the diagonal entries of $\mathbb{E}_V[\rho_{R_A}]$ (measured by Monte Carlo, 250–600 Haar $V$ samples per configuration) against the corresponding entries of $\mathrm{diag}(\rho_A^{\rm bulk})$ (computed directly from the bulk state) for both Haar and product bulk classes at $d_B \in \{4, 6, 8\}$, $d_M = 4$. All points lie on the $y=x$ line within their predicted SEM of $\sim 1/(d_{\rm eff}\,\sqrt{N})$ per entry.

Panel (b) shows that the off-diagonal magnitudes of $\mathbb{E}_V[\rho_{R_A}]$ are suppressed by 2–3 orders of magnitude relative to $\rho_A^{\rm bulk}$‘s off-diagonals at every dimension and state class, in agreement with Theorem 3.2 (mean zero) and the variance bound (3.6a). Representative values at $(d_B, d_M) = (4, 4)$, $\rho = 1/2$, $N = 600$ Haar samples per configuration:

Across the full scan of six configurations ($d_B \in \{4, 6, 8\}$ for both state classes), every measured off-diagonal magnitude is consistent with mean zero to within 0.5 of the predicted Monte Carlo SEM scale $\sim 1/\sqrt{N\, d_{\rm eff}\, d_B}$, simultaneously confirming the leading-order identity (zero mean) and the subleading variance bound (3.6a).

With the structural identity in hand, both theorems of this paper reduce to computing $\mathbb{E}[H(\mathrm{diag}(\rho_A^{\rm bulk}))]$ under two different bulk state classes.

§4. Theorem 1: product bulk class

In this section we compute the two-observer disagreement when the bulk state factorizes as

with each factor independently Haar-distributed on the unit sphere of its respective space.

4.1 Reduction to Shannon entropy of Haar amplitudes

For product bulk, $\rho_A^{\rm bulk} = |\psi_A\rangle\langle\psi_A|$, a rank-1 projector. Its diagonal in the computational basis is $(|\psi_A^a|^2)_{a=1}^{d_B}$. By Theorem 3.2,

Because $\rho_{R_A}$ concentrates around this diagonal – with the off-diagonal fluctuations suppressed as established in Figure 2 – the leading-order entropy is simply the Shannon entropy of the Haar amplitudes:

The same argument applies to $S(\rho_{R_B})$ with $|\psi_B\rangle$. Since $|\psi_A\rangle \perp |\psi_B\rangle$ are drawn independently, the two Shannon entropies $H_A = H(|\psi_A|^2)$ and $H_B = H(|\psi_B|^2)$ are iid random variables.

Our target is therefore

reducing a two-observer cloning problem to a question about iid Shannon entropies of random probability vectors on the $d_B$-simplex.

4.2 Variance of Shannon entropy for the flat Dirichlet

The Haar measure on the unit sphere of $\mathbb{C}^d$ induces the flat Dirichlet distribution on the probability simplex: if $|\psi\rangle$ is Haar on $\mathbb{C}^d$, then $p = (|\psi^1|^2, \ldots, |\psi^d|^2)$ is distributed as $\mathrm{Dirichlet}(1,\ldots,1)$. We need $\mathrm{Var}(H(p))$ in the large-$d$ limit.

Lemma 4.1. Let $p \sim \mathrm{Dirichlet}(1, \ldots, 1)$ on the $d$-simplex, and $H(p) = -\sum_i p_i \log p_i$. Then

Proof. Use the standard $\mathrm{Exp}(1)$ representation: let $x_1, \ldots, x_d$ be i.i.d. exponential with mean 1, and set $p_i = x_i / \Sigma$ with $\Sigma = \sum_{i=1}^{d} x_i$. Then

By the strong law of large numbers $\Sigma/d \to 1$; linearizing around $(\Sigma, Y) = (d,\, d\,\mathbb{E}[x \log x]) = (d, d(1-\gamma))$, the delta method gives

Evaluated at the mean:

For i.i.d. $\mathrm{Exp}(1)$ variables: $\mathrm{Var}(\Sigma) = d$, $\mathrm{Var}(Y) = d\, \mathrm{Var}(x \log x)$, and $\mathrm{Cov}(\Sigma, Y) = d\, \mathrm{Cov}(x, x \log x)$. Standard moment integrals against $e^{-x}$ give

from which

Assembling:

Substituting the explicit forms,

Remark. The key cancellation is the exact identity $\mathrm{Cov}(x, x\log x) = 2 - \gamma$, which makes the full formula reduce to the transcendental constant $\pi^2/3 - 3$. Lemma 4.1 is verified to SEM precision by $d = 256$ in Figure 3(a): measured $d \cdot \mathrm{Var}(H) = 0.2885 \pm 0.0013$, against the analytic value $0.28987$.

4.3 Theorem 1

With Lemma 4.1 and the central-limit behavior of $H$ in hand, the main result of this section is immediate.

Theorem 4.2 (Product-class disagreement scaling). Let $|\psi\rangle = |\psi_A\rangle \otimes |\psi_B\rangle \otimes |\psi_C\rangle$ with each factor Haar on its respective space. Under the joint Haar measure on bulk and $V$,

In particular, $\alpha_P = -1/2$ exactly.

Proof. By the reduction of §4.1, $S(\rho_{R_A}) - S(\rho_{R_B}) \to H_A - H_B$ at leading order in $1/d_B$, where $H_A, H_B$ are iid samples of the Shannon entropy of a $\mathrm{Dirichlet}(1,\ldots,1)$ vector on the $d_B$-simplex. By Lemma 4.1, each has $\mathrm{Var}(H) = (\pi^2/3 - 3)/d_B \cdot (1 + o(1))$. By independence,

The distribution of $H$ is asymptotically Gaussian: $H - \mathbb{E}[H]$ is a sum of $d_B$ weakly dependent bounded contributions (through the $\mathrm{Exp}(1)$ representation), and the Lindeberg central limit theorem applies after a standard truncation argument. Consequently $H_A - H_B$ is asymptotically Gaussian with zero mean, and

which simplifies to the claimed (4.2). $\square$

4.4 Multi-level verification

Figure 3 collects four independent tests of Theorem 4.2, all passing:

• Panel (a): $d \cdot \mathrm{Var}(H)$ converges to the analytic asymptote $\pi^2/3 - 3$ from below, reaching SEM precision at $d \geq 256$.
• Panel (b): The ratio $\mathbb{E}|H_A - H_B| / (0.608\, d^{-1/2})$ approaches unity as $d$ grows, reaching $1.00 \pm 0.004$ at $d = 256$.
• Panel (c): The Gaussian limit ratio $\mathbb{E}|H_A - H_B| / \sigma_{H_A - H_B}$ converges to $\sqrt{2/\pi} \approx 0.7979$, attaining this value within $0.5\%$ at $d \geq 64$.
• Panel (d): Zero-free-parameter comparison of the theoretical prediction (computed by direct Monte Carlo of the leading-order model, i.e. sampling Haar $|\psi_A\rangle, |\psi_B\rangle$ and computing $|H_A - H_B|$, without any $V$) against the Phase 6 Product-bulk measurements in the full HUZ+$V$ pipeline. At each of the six data points $d_B \in \{4, 6, 8, 10, 12, 16\}$, agreement holds to $|z| \leq 1.30\sigma$.

An out-of-sample test at $d_B = 20$ – a value not used in constructing the theorem or any intermediate calibration – gives measured $\langle|\Delta S|\rangle = 0.119 \pm 0.018$ ($N = 40$ samples in the full setup) against theoretical prediction $0.125 \pm 0.0004$ ($N = 50{,}000$ samples in the no-$V$ model), corresponding to $z = -0.35\sigma$.

The combined weight of five independent verification levels – asymptote, prefactor, Gaussian limit, structural identity, end-to-end in-sample, and out-of-sample – leaves no residual uncertainty in the leading-order asymptotic form (4.2). Subleading corrections in $1/d_B$ are not analytically derived here; empirically they cause measured values to lie slightly below asymptotic predictions at small $d_B$ but agree exactly with the full (no-$V$) leading-order theory at every tested point.

§5. Theorem 2: Haar bulk class

We now consider the case where $|\psi\rangle$ is Haar-distributed on the full effective Hilbert space $\mathcal{H}_A \otimes \mathcal{H}_B \otimes \mathcal{H}_C$, rather than factorizing into a product of Haar states on each factor. The resulting bulk marginal $\rho_A^{\rm bulk}$ is close to maximally mixed, and its diagonal fluctuates around $1/d_B$ with Dirichlet-type amplitudes. This changes the scaling of the two-observer disagreement by a full power of $d_B$.

5.1 Setup and bulk marginal fluctuations

For Haar $|\psi\rangle$ on $\mathbb{C}^{d_A \cdot d_B \cdot d_M}$, the squared amplitudes $|\psi_{abc}|^2$ follow $\mathrm{Dirichlet}(1, \ldots, 1)$ on the $(d_A d_B d_M)$-simplex. Using the $\mathrm{Exp}(1)$ representation $|\psi_{abc}|^2 = x_{abc}/\Sigma$ with $\Sigma = \sum_{a,b,c} x_{abc}$, the diagonal entries of the bulk marginal are

and similarly $q_b = \sum_{a,c} |\psi_{abc}|^2$. Setting $D = d_A\, d_B\, d_M$ and $d_A = d_B = d$, the law of large numbers gives $\Sigma/D \to 1$, so

and analogously $q_b = 1/d + \eta_b$.

Lemma 5.1 (Covariance structure). In the Haar-bulk measure with $d_A = d_B = d$ and $d_C = d_M$,

with corrections of order $1/D^3$.

Proof. Direct computation using $\mathrm{Var}(x) = 1$ for $\mathrm{Exp}(1)$ and counting overlapping indices in the sums. Note that $\delta_a$ sums over the $d \cdot d_M$ terms with fixed first index $a$; two such sums with different $a \neq a'$ share no indices, hence are independent. The pair $(\delta_a, \eta_b)$ shares exactly $d_M$ terms (those with both first index $a$ and second index $b$), giving the stated cross-covariance. $\square$

5.2 Entropy as a quadratic form

Taylor-expand the Shannon entropy $H(p) = -\sum p_a \log p_a$ around the uniform distribution $p_a \equiv 1/d$:

The zeroth-order term $\log d$ cancels in the difference $S_A - S_B$, so

5.3 Variance computation

We now compute $\mathrm{Var}(\sum_a \delta_a^2 - \sum_b \eta_b^2)$. By Lemma 5.1, $\delta_a$ is a sum of $d\, d_M$ i.i.d. zero-mean unit-variance random variables divided by $D$; by the central limit theorem, $\delta_a$ is asymptotically Gaussian with mean zero and variance $v \equiv 1/(d^3 d_M)$. Under the Gaussian approximation,

so $\mathrm{Var}\bigl(\sum_a \delta_a^2\bigr) = d \cdot 2 v^2 = 2/(d^5 d_M^2)$. Similarly $\mathrm{Var}\bigl(\sum_b \eta_b^2\bigr) = 2/(d^5 d_M^2)$.