---
schema_version: 1
id: PHYS-2026-08-06-01
date: 2026-08-06
updated_at: 2026-08-06
title: A complex pseudo-entropy does not establish emergent time
summary: "Separate the spectral branch of a reduced transition matrix, its short-time modular response, timelike CFT continuation, and the stronger holographic claim that complex entropy encodes Lorentzian geometry."
language: en
entry_kind: daily
status: published
level: research
user_difficulty: unrated
domains:
  - quantum-information
  - quantum-field-theory
  - general-relativity
  - cosmology
  - mathematical-physics
estimated_minutes: 90
---

## The claim under pressure

**A nonzero imaginary part of pseudo-entropy establishes that a reduced transition matrix carries phase information. It does not, without further structure, establish a Lorentzian direction in an emergent spacetime.**

Several mechanisms produce complex values:

- complex eigenvalues of a non-Hermitian reduced transition matrix;
- the branch chosen for the matrix logarithm;
- a real-time response controlled by correlations between the physical and modular Hamiltonians;
- analytic continuation of replica correlators to timelike separation;
- complex or mixed-signature saddles in a holographic calculation.

They can coexist. Their interpretations are different.

The useful task is therefore diagnostic: identify which mechanism produced the imaginary part, determine its invariant content, and state which additional dictionary permits a geometric reading.

## 1. The object is a transition amplitude, not a state

Let

$$
\mathcal H=\mathcal H_A\otimes\mathcal H_{\bar A}
$$

and choose two nonorthogonal pure states $\ket{\psi}$ and $\ket{\phi}$. The normalized transition operator is

$$
\tau^{\psi|\phi}
=\frac{\ket{\psi}\bra{\phi}}{\langle\phi|\psi\rangle},
\qquad
\Tr\tau^{\psi|\phi}=1.
$$

After tracing out $\bar A$,

$$
\tau_A^{\psi|\phi}
=\Tr_{\bar A}\tau^{\psi|\phi}.
$$

Pseudo-entropy is defined by

$$
S_A^{(p)}
=-\Tr_A\!\left(\tau_A^{\psi|\phi}
\log\tau_A^{\psi|\phi}\right).
$$

Nakata and collaborators introduced this quantity as a postselected generalization of entanglement entropy. It reduces to ordinary entanglement entropy when $\ket{\phi}=\ket{\psi}$.

For distinct states, $\tau_A$ generally fails to be Hermitian or positive. Its eigenvalues may be complex even though their sum is one. Standard entropy properties such as reality, positivity, concavity, and strong subadditivity do not follow from the definition.

Numerical studies have found violations of strong subadditivity for pseudo-entropy in field-theory and spin-system examples. Calling $S_A^{(p)}$ an entropy records its algebraic ancestry; it does not import every theorem about density matrices.

> [!margin: Normalization singularity]
> The definition requires $\langle\phi|\psi\rangle\neq0$. Near an overlap zero, the normalized transition operator and its pseudo-entropy can become singular. Such behavior may diagnose postselection or Loschmidt physics before it says anything about geometry.

### A finite-dimensional counterexample to the geometric inference

Take two qubits and the unnormalized vectors

$$
\ket{\psi}=\ket{00}+\sqrt2\ket{11},
\qquad
\ket{\phi}=\ket{00}-\ii\sqrt2\ket{11}.
$$

Their overlap is

$$
\langle\phi|\psi\rangle=1+2\ii.
$$

Tracing out the second qubit gives

$$
\tau_A
=\frac{1}{1+2\ii}
\begin{pmatrix}
1&0\\
0&2\ii
\end{pmatrix}.
$$

The eigenvalues are

$$
\lambda_0=\frac{1-2\ii}{5},
\qquad
\lambda_1=\frac{4+2\ii}{5},
\qquad
\lambda_0+\lambda_1=1.
$$

On the principal branch,

$$
S_A^{(p)}
=-\lambda_0\log\lambda_0
-\lambda_1\log\lambda_1
$$

is complex. The construction uses two qubits, postselection, and a partial trace. It contains no assumed spacetime dual.

This example is enough to disprove

$$
\Im S_A^{(p)}\neq0
\quad\Longrightarrow\quad
\text{emergent Lorentzian time}.
$$

It does not disprove more structured holographic proposals. Those proposals use additional input absent here.

## 2. The logarithm carries physical and conventional information

Suppose $\tau_A$ is diagonalizable:

$$
\tau_A=V\,\mathrm{diag}(\lambda_1,\ldots,\lambda_r)V^{-1}.
$$

For a specified holomorphic branch of the logarithm on a domain containing the spectrum,

$$
\log\tau_A
=V\,\mathrm{diag}(\log\lambda_1,\ldots,\log\lambda_r)V^{-1}.
$$

Then

$$
S_A^{(p)}=-\sum_j\lambda_j\log\lambda_j.
$$

Changing the branch for an eigenvalue by

$$
\log\lambda_j\mapsto\log\lambda_j+2\pi\ii n_j
$$

changes the pseudo-entropy by

$$
\Delta S_A^{(p)}
=-2\pi\ii\sum_jn_j\lambda_j.
$$

This shift need not be a pure imaginary constant because the $\lambda_j$ themselves can be complex. A statement about $\Im S_A^{(p)}$ is incomplete until the branch prescription is given.

A common real-time prescription starts at $t=0$, where $\tau_A(0)=\rho_A$ is positive, and continues the logarithm continuously. This works locally if the relevant eigenvalues remain inside a domain on which one branch is analytic.

Three obstructions can end that continuation:

1. an eigenvalue reaches zero, where the logarithm is singular;
2. an eigenvalue crosses the selected branch cut or winds around the origin;
3. eigenvectors coalesce at a defective degeneracy, so the spectral decomposition becomes singular.

The third event is the non-Hermitian analogue of an exceptional point. A matrix logarithm can still sometimes be defined through a contour integral, but continuity and branch assignment require a fresh analysis.

> [!margin: Track the spectrum]
> A plot of $S_A^{(p)}(t)$ alone can hide a branch jump. Track the complex eigenvalue trajectories of $\tau_A(t)$ and the overlap in the denominator.

For nondiagonalizable matrices, Jordan blocks add nilpotent terms to $\log\tau_A$. The trace formula remains meaningful when the spectrum avoids the branch cut and zero, but eigenvalues alone no longer describe every response of the logarithm.

This spectral bookkeeping separates a change of convention from a genuine continuous response. It also identifies points where the observable itself becomes nonanalytic.

## 3. Real-time pseudo-entropy has a modular response

Now specialize to unitary time evolution from an initial state:

$$
\ket{\Psi(t)}=\ee^{-\ii Ht}\ket{\Psi}.
$$

Define

$$
\tau(t,0)
=\frac{\ket{\Psi(t)}\bra{\Psi}}
{\langle\Psi|\Psi(t)\rangle},
\qquad
\tau_A(t,0)=\Tr_{\bar A}\tau(t,0).
$$

Assume the initial reduced state $\rho_A$ has full rank on its support and choose the logarithm by continuation from $t=0$. The initial modular Hamiltonian is

$$
K_A=-\log\rho_A.
$$

### Derivation of the first-order term

Expand the numerator:

$$
\ket{\Psi(t)}\bra{\Psi}
=\left(1-\ii tH+O(t^2)\right)
\ket{\Psi}\bra{\Psi}.
$$

The denominator is

$$
\langle\Psi|\Psi(t)\rangle
=1-\ii t\langle H\rangle+O(t^2),
$$

so its inverse contributes $1+\ii t\langle H\rangle$. Therefore

$$
\tau(t,0)
=\ket{\Psi}\bra{\Psi}
-\ii t\,(H-\langle H\rangle)
\ket{\Psi}\bra{\Psi}
+O(t^2).
$$

The normalization removes the component parallel to the original transition amplitude. After tracing out $\bar A$,

$$
\delta\tau_A
=-\ii t\,\Tr_{\bar A}
\left[
(H-\langle H\rangle)\ket{\Psi}\bra{\Psi}
\right]+O(t^2).
$$

For a trace-preserving first variation around the positive operator $\rho_A$,

$$
\delta S_A^{(p)}
=\Tr_A(\delta\tau_AK_A).
$$

Lifting $K_A$ to $K_A\otimes I_{\bar A}$ yields

$$
\boxed{
S_A^{(p)}(t,0)
=S_A
-\ii t\,
\langle K_A(H-\langle H\rangle)\rangle
+O(t^2)
}.
$$

Misumi derived this short-time response and tested it in finite-dimensional models and transverse-field Ising-chain quenches in a 2026 preprint.

Let

$$
\Delta H=H-\langle H\rangle,
\qquad
\Delta K_A=K_A-\langle K_A\rangle.
$$

Since $\langle\Delta H\rangle=0$,

$$
\langle K_A\Delta H\rangle
=\frac12\langle\{\Delta K_A,\Delta H\}\rangle
+\frac12\langle[K_A,H]\rangle.
$$

For Hermitian $H$ and $K_A$, the anticommutator expectation is real and the commutator expectation is imaginary. Hence

$$
\boxed{
\left.\frac{\dd}{\dd t}\Re S_A^{(p)}\right|_{t=0}
=\frac{1}{2\ii}\langle[K_A,H]\rangle
}
$$

and

$$
\boxed{
\left.\frac{\dd}{\dd t}\Im S_A^{(p)}\right|_{t=0}
=-\frac12
\langle\{\Delta K_A,\Delta H\}\rangle.
}
$$

The imaginary slope is a symmetrized modular covariance. It depends on the state, subsystem, and physical Hamiltonian. It can vanish even when the transition matrix becomes non-Hermitian, and it can remain nonzero when $[K_A,H]=0$.

This response contains temporal information in an operational sense: $H$ generates the evolution, and the sign reverses when the bra and ket in the transition matrix are exchanged. It still does not define a bulk proper time.

> [!margin: Modular flow is not laboratory time]
> $K_A$ generates modular flow for the reduced state. $H$ generates physical time evolution. Their covariance couples the two structures; it does not identify their parameters.

### What the linear response cannot tell you

The formula is local in time. At later times, higher connected correlations enter, the logarithmic branch can change, and the overlap $\langle\Psi|\Psi(t)\rangle$ may approach zero.

Near a critical point, the covariance can show finite-size scaling. That makes it a candidate susceptibility, not an established universal order parameter. One must specify the subsystem, ultraviolet regularization, normalization of $H$, and scaling limit.

The response also depends on which side of the transition matrix evolves. Evolving both ket and bra with the same unitary produces an ordinary density matrix and ordinary entanglement dynamics. Real-time pseudo-entropy keeps an amplitude-level comparison between two times.

## 4. Timelike CFT continuation produces a different imaginary term

For a vacuum interval in a two-dimensional CFT,

$$
S(\Delta x,\Delta t)
=\frac{c}{6}
\log\!\left(
\frac{\Delta x^2-\Delta t^2+\ii0}{\epsilon^2}
\right).
$$

Set $\Delta x=0$ and $\Delta t=T>0$. With

$$
\log(-T^2+\ii0)=\log T^2+\ii\pi,
$$

one obtains

$$
\boxed{
S_{\rm time}(T)
=\frac{c}{3}\log\frac{T}{\epsilon}
+\frac{\ii\pi c}{6}
}.
$$

The opposite $\ii0$ prescription reverses the sign of the imaginary constant. The term $\ii\pi c/6$ follows from causal ordering and the logarithmic branch.

It differs sharply from the modular-covariance term:

| Quantity | Origin | Typical dependence | What fixes the sign |
| --- | --- | --- | --- |
| $\pi c/6$ in vacuum timelike CFT | Analytic continuation of a twist correlator | Constant for one interval branch | Time ordering and $\ii0$ prescription |
| $-\tfrac12\langle\{\Delta K_A,\Delta H\}\rangle$ | Infinitesimal real-time transition matrix | State, subsystem, and Hamiltonian dependent | Exchange of forward and reverse transition matrices |
| $-2\pi\ii\sum_jn_j\lambda_j$ | Change of logarithm branch | Spectrum dependent | Chosen spectral continuation |
| Complex extremal area | Semiclassical holographic saddle | Geometry and saddle dependent | Boundary contour and saddle prescription |

The vacuum constant can survive when no state-dependent temporal response exists. Conversely, a finite spin chain can have a modular-covariance response without conformal symmetry or a bulk dual.

Recent work formulates timelike Rényi quantities directly through time-ordered twist correlators. This matters because analytic continuation of a closed-form entropy formula can obscure the replica contour and operator ordering that select the branch.

## 5. The replica route adds another continuation problem

For integer $n>1$, define the pseudo-Rényi quantity

$$
S_A^{(n)}
=\frac{1}{1-n}
\log\Tr\left(\tau_A^n\right).
$$

The von Neumann-like pseudo-entropy is formally obtained from

$$
S_A^{(p)}
=-\left.\partial_n
\log\Tr(\tau_A^n)\right|_{n=1}.
$$

In a path integral, integer $n$ specifies how replicas are sewn along the chosen region or contour. Reaching $n=1$ requires an analytic continuation away from the integers.

Integer data do not determine a unique analytic function without additional growth and regularity conditions. In practical CFT and holographic calculations, symmetry, operator product expansions, boundary conditions, and a chosen saddle family supply that extra input.

For timelike separation, one must also continue the operator positions and retain the ordering prescription. The continuations in $n$ and in spacetime kinematics are logically distinct. Performing them in different orders can expose branch and saddle ambiguities.

In a semiclassical bulk path integral,

$$
Z_n\sim\sum_\alpha
\exp\!\left[-I_\alpha(n)/G_N\right].
$$

Several real or complex saddles may satisfy the same boundary conditions. The smallest real part of the action often controls an asymptotic region, but the integration contour determines which saddles actually contribute.

As parameters vary, saddle dominance can change. Across a Stokes line, an asymptotic saddle decomposition can jump even when the exact boundary quantity remains analytic. A discontinuity in a leading semiclassical entropy may therefore reflect a change of approximation rather than a microscopic singularity.

> [!margin: A complex saddle is not selected by existence]
> Solving the complexified extremal equations finds candidates. Boundary conditions, homology, replica continuation, and the bulk integration contour decide whether a candidate contributes.

This is an active frontier. A 2025 proposal identifies boundary-anchored complex extremal surfaces as carriers of holographic timelike entropy and finds multiple candidate surfaces. A 2026 CFT analysis derives complex geodesics from time-ordered twist correlators and extends the construction to Rényi index $n>1$.

These developments strengthen the boundary-to-bulk calculation in specified AdS/CFT settings. They do not make every complex pseudo-entropy geometric.

## 6. What holography adds

The original holographic pseudo-entropy proposal relates reduced transition matrices in suitable CFT states to extremal areas in asymptotically AdS geometries. Timelike-entanglement constructions extend this idea to complex or mixed-signature extremal objects anchored on timelike boundary regions.

The geometric claim has content because several structures are matched at once:

1. a boundary replica or transition-matrix construction;
2. a large-$N$, semiclassical regime;
3. bulk boundary conditions and a homology prescription;
4. an extremal saddle with the correct analytic continuation;
5. agreement of variations, symmetries, and limiting cases between the two sides.

Only after these checks does an imaginary area acquire a controlled geometric interpretation.

Even then, “imaginary part equals proper time” is usually too strong. The regulated result can include counterterms, branch constants, several curve segments, and complex-coordinate contributions. Proper time is a diffeomorphism-invariant length along a specified real timelike curve; a complex extremal area is a different object unless a derivation identifies them.

### The de Sitter claim has an additional assumption

In proposed $\mathrm{dS}_3/\mathrm{CFT}_2$ constructions, the boundary theory is Euclidean and generally nonunitary, while the bulk contains Lorentzian de Sitter time. Pseudo-entropy is a natural candidate because the relevant boundary transition matrices and bulk extremal lengths can be complex.

Doi and collaborators argued that the imaginary part of timelike or de Sitter pseudo-entropy reflects emergent time in their examples. Takayanagi later presented pseudo-entropy and timelike entanglement as tools for investigating time emergence.

A 2025 preprint by Fujiki and collaborators gives a sharper conditional result: **assuming the proposed dS/CFT dictionary**, the first law of holographic pseudo-entropy for small perturbations matches the linearized Einstein equation in $\mathrm{dS}_3$ when complexified geodesics are included.

That is substantially stronger than observing a complex number. It matches a family of variations to a bulk dynamical equation.

It remains conditional on the dS/CFT proposal, the semiclassical approximation, the selected states, and the complex-geodesic prescription. It does not derive Lorentzian time from an arbitrary non-Hermitian quantum system.

## 7. An evidence ladder for “emergent time”

| Observation | Supported inference | Unsupported leap |
| --- | --- | --- |
| $\tau_A$ has complex eigenvalues | The reduced transition amplitude is non-Hermitian | A bulk spacetime exists |
| $S_A^{(p)}$ changes under a log branch | The quantity is branch sensitive | The shift measures elapsed time |
| $\Im\dot S_A^{(p)}(0)$ equals a modular covariance | The subsystem detects oriented unitary evolution at first order | Modular time and bulk proper time are identical |
| Timelike twist correlators yield $\ii\pi c/6$ | Causal continuation fixes a complex CFT observable | The constant alone reconstructs a metric |
| A complex extremal surface reproduces the CFT answer | A specified holographic dictionary has passed a nontrivial test | Every complex saddle is physical |
| Pseudo-entropy variations reproduce a bulk field equation | Strong conditional evidence for emergent bulk dynamics | A model-independent derivation of time |

The last two rows contain geometric evidence because they compare complete calculational structures, not merely the sign or existence of an imaginary part.

## False claim to diagnose

> In a holographic theory, every nonzero imaginary part of pseudo-entropy measures a bulk proper-time interval.

This fails for three independent reasons.

First, pseudo-entropy can acquire a branch-dependent imaginary part before any bulk limit is taken. Second, a holographic observable may receive contributions from several real, timelike, and complex curve segments. Third, a bulk saddle requires a contour and dominance prescription.

A defensible replacement is:

> In a specified holographic construction, a branch- and contour-fixed complex pseudo-entropy may encode timelike bulk information when a replica calculation, extremal saddle, and dynamical variation agree.

## Extended problem set

### Part A: branch constant

Starting from

$$
S=\frac{c}{6}
\log\frac{\Delta x^2-\Delta t^2+\ii0}{\epsilon^2},
$$

set $\Delta x=0$ and $\Delta t=T>0$. Derive the real and imaginary parts. Repeat with $-\ii0$ and explain the change.

### Part B: modular response

Starting from

$$
S_A^{(p)}(t)
=S_A-\ii t\langle K_A\Delta H\rangle+O(t^2),
$$

derive the commutator and covariance formulae. Analyze these cases:

1. $[K_A,H]=0$ but $\langle\Delta K_A\Delta H\rangle\neq0$;
2. the symmetrized covariance vanishes but $\langle[K_A,H]\rangle\neq0$;
3. both vanish at first order.

Does case 3 imply that $S_A^{(p)}(t)$ is constant?

### Part C: two-qubit spectrum

For the two-qubit example in Section 1:

1. verify $\Tr\tau_A=1$;
2. calculate $S_A^{(p)}$ numerically on the principal branch;
3. shift the branch of $\lambda_0$ by $2\pi\ii$ and calculate $\Delta S_A^{(p)}$;
4. explain why the result supplies no evidence for a geometric time direction.

### Part D: replica and saddle judgment

Suppose two complex bulk saddles have actions $I_1(n,eta)$ and $I_2(n,eta)$, where $\eta$ controls timelike separation. At $\eta=\eta_*$ their real parts cross.

List the information required before declaring a phase transition in the boundary pseudo-entropy. Your list should include the original integration contour, intersection numbers or an equivalent saddle-selection rule, the $n\to1$ continuation, finite-$G_N$ corrections, and the behavior of the exact boundary correlator.

<details>
<summary>Hints</summary>

- Use $\log(-a\pm\ii0)=\log a\pm\ii\pi$.
- For Hermitian $K_A$ and $H$, $\langle K_A\Delta H\rangle^*=\langle\Delta H K_A\rangle$.
- A vanishing first derivative leaves $O(t^2)$ and higher terms unconstrained.
- Equality of the real parts of two saddle actions does not prove that both saddles lie on the original integration contour.

</details>

<details class="solution">
<summary>Solution outline</summary>

For the timelike interval,

$$
S_{\rm time}^{(\pm)}
=\frac{c}{3}\log\frac{T}{\epsilon}
\pm\frac{\ii\pi c}{6}.
$$

The sign records the ordering prescription. The magnitude of the constant does not depend on the evolving many-body state in this vacuum formula.

Write

$$
z=\langle K_A\Delta H\rangle
=\frac12\langle\{\Delta K_A,\Delta H\}\rangle
+\frac12\langle[K_A,H]\rangle.
$$

Since $\dot S_A^{(p)}=-\ii z$,

$$
\Re\dot S_A^{(p)}(0)
=\frac{1}{2\ii}\langle[K_A,H]\rangle,
$$

$$
\Im\dot S_A^{(p)}(0)
=-\frac12\langle\{\Delta K_A,\Delta H\}\rangle.
$$

Commutation removes the real linear response, not the covariance term. Zero covariance removes the imaginary linear response, not the commutator term. If both vanish, the response begins at second or higher order unless another symmetry forces all coefficients to vanish.

For the two-qubit example,

$$
\lambda_0=0.2-0.4\ii,
\qquad
\lambda_1=0.8+0.4\ii.
$$

Using principal arguments gives approximately

$$
S_A^{(p)}\approx0.879-0.427\ii.
$$

Changing only the branch of $\lambda_0$ gives

$$
\Delta S_A^{(p)}
=-2\pi\ii\lambda_0
=-0.8\pi-0.4\pi\ii.
$$

The complex value is fully explained by a finite-dimensional reduced transition matrix and a logarithm branch.

At a saddle crossing, compare the candidate saddles with the integration cycle rather than selecting them by existence. Check whether the exact boundary object is singular, whether the replica continuation follows the same saddle family, and whether finite-$G_N$ terms smooth the leading transition. A Stokes jump in an asymptotic expansion need not be a phase transition of the exact theory.

</details>

## Research checks worth doing next

1. **Spectral tracking.** Follow every eigenvalue of $\tau_A(t)$ in the complex plane together with $\langle\Psi|\Psi(t)\rangle$. Mark branch-cut crossings and near-defective degeneracies.
2. **Orientation test.** Exchange the bra and ket transition states. Separate the odd imaginary response from branch constants that do not transform in the same way.
3. **Subsystem scaling.** Test how the modular covariance changes with interval size, ultraviolet cutoff, and system size near criticality.
4. **Replica consistency.** Compare direct spectral evaluation of $S_A^{(p)}$ with analytic continuation of $\Tr\tau_A^n$.
5. **Bulk saddle audit.** Derive the contributing complex saddles from the boundary correlator or a justified gravitational contour. Do not select a surface only because it produces the desired imaginary part.
6. **Dynamical matching.** Look for equations obeyed by variations of pseudo-entropy. A matched bulk field equation carries more information than a matched constant.

## Further Reading

- [Holographic Pseudo Entropy](https://arxiv.org/abs/2005.13801) — the original reduced-transition-matrix definition and holographic proposal.
- [Aspects of Pseudo Entropy in Field Theories](https://arxiv.org/abs/2106.03118) — field-theory properties, spin-system examples, and failures of ordinary entropy inequalities.
- [Real-time pseudo entropy and modular-Hamiltonian correlations](https://arxiv.org/abs/2606.14208) — the 2026 short-time covariance and commutator result used here.
- [Timelike entanglement entropy](https://arxiv.org/abs/2302.11695) — analytic continuation, mixed-signature holographic constructions, and the emergent-time proposal.
- [Imaginary part of timelike entanglement entropy](https://arxiv.org/abs/2410.22684) — twist-operator commutators and the field-theory origin of timelike imaginary terms.
- [Geometric interpretation of timelike entanglement entropy](https://arxiv.org/abs/2408.15752) — complex extremal surfaces and the problem of selecting among multiple saddles.
- [Temporal Entanglement from Twist Correlators in 2d Conformal Field Theory and Holography](https://arxiv.org/abs/2607.14012) — a 2026 preprint deriving timelike Rényi observables and complex geodesics from time-ordered twist correlators.
- [Entropic Interpretation of Einstein Equation in dS/CFT](https://arxiv.org/abs/2511.07915) — a conditional match between the pseudo-entropy first law and linearized de Sitter gravity.
- [Emergent Holographic Spacetime from Quantum Information](https://arxiv.org/abs/2506.06595) — Takayanagi's essay framing pseudo-entropy as a tool for the open problem of time emergence.

## What to retain

- A reduced transition matrix is normalized but need not be Hermitian or positive.
- The matrix logarithm requires a spectral branch and can fail at zeros or defective degeneracies.
- The first imaginary real-time response is a symmetrized covariance of $K_A$ and $H$.
- The vacuum timelike-CFT constant comes from causal analytic continuation, not that covariance.
- Replica continuation and spacetime continuation are separate operations.
- Complex bulk saddles require a contour and dominance prescription.
- Geometric evidence begins when a specified holographic dictionary matches families of observables or dynamical equations.
- A complex pseudo-entropy by itself proves none of that geometry.

Next: construct a two-qubit real-time example in which the logarithmic branch remains fixed, calculate the modular covariance exactly, and compare its orientation reversal with the branch-generated imaginary term.