Error Geometry and Causal Robustness

“Error is not noise, but geometric boundary; robustness is not luck, but geometric invariant.”

🎯 Core Ideas

In the previous chapter, we learned how spacetime geometry serves as minimal lossless compression of causal constraints. Now we face a practical question:

When causal structure has uncertainty (measurement errors, quantum fluctuations, finite samples), are our conclusions still robust?

Traditional method: “Point estimate + error bars”

GLS New Perspective:

$$\boxed{\text{Error = Geometric Region in Parameter Space (Credible Region)}}$$

$$\boxed{\text{Robustness = Conclusion Holds Over Entire Credible Region}}$$

Analogy:

Traditional method is like describing a route to a friend:

• “Go straight 500 meters from here, then turn left”

What if your GPS has ±50 meter error? You might end up in a completely different place!

Geometric Method:

• “Within a circle centered on me with radius 50 meters, no matter which point you’re at, turning left will get you to the destination”

This is true robustness!

📖 From Point Estimation to Region Estimation

Limitations of Traditional Statistical Inference

Classical Procedure:

1. Collect data $\{X_1,\dots ,X_n\}$
2. Estimate parameter ${\hat{\theta }}_n$
3. Calculate standard error $\text{SE}({\hat{\theta }}_n)$
4. Give confidence interval $[{\hat{\theta }}_n-2\text{SE},{\hat{\theta }}_n+2\text{SE}]$

Problems:

• Confidence interval is one-dimensional (for scalar parameters)
• For multi-dimensional parameters, separate confidence intervals ignore correlations
• Causal conclusions often depend on complex combinations of parameters

Example (Linear Regression):

$$Y_i={\beta }_0+{\beta }_1{X}_i+{ϵ}_i$$

We might care about conditional effect:

$$\psi (\beta )={\beta }_1+{\beta }_0/\bar{X}$$

From confidence intervals of ${\beta }_0,{\beta }_1$ alone, we cannot accurately infer confidence interval of $\psi (\beta )$!

Geometric Perspective: Confidence Ellipsoid

New Method: Treat error as geometric region in parameter space $\Theta$

Fisher Information Metric:

At each parameter point $\theta$, define local metric:

$$g_{\theta }(u,v):=u^{\top }I(\theta )v$$

where $I(\theta )$ is Fisher information matrix.

Intuition:

• Large eigenvalue of $I(\theta )$ → Parameter “easily identifiable” in that direction
• Small eigenvalue of $I(\theta )$ → Parameter “hard to identify” in that direction

Confidence Ellipsoid:

$${\mathcal{R}}_n(\alpha ):=\left\{\theta \in \Theta :n(\theta -{\hat{\theta }}_n{)}^{\top }I({\hat{\theta }}_n)(\theta -{\hat{\theta }}_n)\le {\chi }_{d,1-\alpha }^2\right\}$$

graph TB
    subgraph "Parameter Space Θ"
        CENTER["Point Estimate<br/>θ̂ₙ"]
        ELLIPSE["Confidence Ellipsoid<br/>ℛₙ(α)"]
        TRUE["True Value θ₀"]

        CENTER -.->|"Semi-axis Directions Determined by I⁻¹"| ELLIPSE
        ELLIPSE -->|"Contains with Probability 1-α"| TRUE
    end

    INFO["Fisher Information I(θ)"] -->|"Determines Shape"| ELLIPSE

    style ELLIPSE fill:#fff4e1
    style TRUE fill:#ffe1e1

Key Property:

Asymptotic Coverage Theorem:

$$P_{{\theta }_0}({\theta }_0\in {\mathcal{R}}_n(\alpha ))\to 1-\alpha ,n\to \infty$$

That is: True parameter falls inside ellipsoid with probability $1-\alpha$!

🎨 Geometric Operations on Credible Regions

Projection of Linear Functions

Suppose causal effect we care about is linear function of parameters:

$$\psi (\theta )=c^{\top }\theta$$

Question: Range of $\psi$ on credible region ${\mathcal{R}}_n(\alpha )$?

Geometric Answer:

$${\Psi }_n(\alpha ):=\{\psi (\theta ):\theta \in {\mathcal{R}}_n(\alpha )\}$$

This is projection of ellipsoid in direction $c$!

Analytic Solution:

$${\psi }_{\max }=c^{\top }{\hat{\theta }}_n+\sqrt{\frac{{\chi }_{d,1-\alpha }^2}{n}{c}^{\top }I({\hat{\theta }}_n{)}^{-1}c}$$

$${\psi }_{\min }=c^{\top }{\hat{\theta }}_n-\sqrt{\frac{{\chi }_{d,1-\alpha }^2}{n}{c}^{\top }I({\hat{\theta }}_n{)}^{-1}c}$$

Analogy:

Imagine ellipsoid is a watermelon, $c$ is direction of cutting knife:

• After cutting, cross-section (projection) is an ellipse
• Major and minor axes of ellipse determined jointly by watermelon shape and knife angle

graph LR
    ELLIPSOID["3D Confidence Ellipsoid<br/>ℛₙ(α)"] -->|"Project Along Direction c"| INTERVAL["1D Confidence Interval<br/>[ψ_min, ψ_max]"]

    DIRECTION["Projection Direction c"] -.->|"Determines"| INTERVAL

    style ELLIPSOID fill:#e1f5ff
    style INTERVAL fill:#ffe1e1

Local Linearization of Nonlinear Functions

If causal effect is nonlinear function $\psi :\Theta \to {\mathbb{R}}^k$, what to do?

Delta Method (first-order approximation):

Near ${\hat{\theta }}_n$:

$$\psi (\theta )\approx \psi ({\hat{\theta }}_n)+D\psi ({\hat{\theta }}_n)\cdot (\theta -{\hat{\theta }}_n)$$

where $D\psi$ is Jacobian matrix.

Projected Ellipsoid:

$${\mathcal{S}}_n(\alpha ):=\left\{y\in {\mathbb{R}}^k:n(y-\psi ({\hat{\theta }}_n){)}^{\top }{\Sigma }^{-1}(y-\psi ({\hat{\theta }}_n))\le {\chi }_{k,1-\alpha }^2\right\}$$

where

$$\Sigma :=D\psi ({\hat{\theta }}_n)\cdot I({\hat{\theta }}_n{)}^{-1}\cdot D\psi ({\hat{\theta }}_n{)}^{\top }$$

Physical Meaning: Uncertainty ellipsoid of nonlinear effect!

🔍 Identifiable Sets in Causal Inference

What Is Identifiable Set?

In many causal problems, even with infinite data, we cannot uniquely determine certain parameters.

Definition (Identifiable Set):

$$\mathcal{I}:=\{\theta \in \Theta :\theta \text{ compatible with observed distribution and causal assumptions}\}$$

Example 1 (Omitted Variable Bias):

True model: $Y={\beta }_{1}X+{\beta }_{2}Z+ϵ$

But $Z$ is unobservable!

Identifiable set:

$$\mathcal{I}=\{({\beta }_1,{\beta }_2):{\beta }_1^{\text{obs}}={\beta }_1+{\beta }_2{\gamma }_{ZX}\}$$

where ${\beta }_1^{\text{obs}}$ is observed regression coefficient, ${\gamma }_{ZX}$ is regression coefficient of $Z$ on $X$.

Geometry: This is a line, not a single point!

Example 2 (Weak Identification of Instrumental Variables):

When instrumental variable is “weak”, identifiable set of structural parameters may be unbounded or very “flat” region.

Intersection of Credible Region and Identifiable Set

GLS Core Insight:

Causal conclusions should be based on:

$$\boxed{{\mathcal{R}}_n(\alpha )\cap {\mathcal{I}}_n}$$

not just point estimate ${\hat{\theta }}_n$!

where ${\mathcal{I}}_n$ is data-driven estimate of identifiable set.

graph TB
    subgraph "Parameter Space Θ"
        TRUST["Credible Region<br/>ℛₙ(α)<br/>(Statistical Uncertainty)"]
        IDENT["Identifiable Set<br/>ℐₙ<br/>(Causal Constraints)"]
        INTER["Intersection<br/>ℛₙ∩ℐₙ"]

        TRUST -.->|"Intersection"| INTER
        IDENT -.->|"Intersection"| INTER
    end

    INTER --> ROBUST["Robust Causal Conclusion<br/>(Holds Over Entire Intersection)"]

    style TRUST fill:#e1f5ff
    style IDENT fill:#ffe1e1
    style INTER fill:#fff4e1,stroke:#ff6b6b,stroke-width:3px

Definition (Geometric Robustness):

Let $\psi :\Theta \to \mathbb{R}$ be causal effect. If there exists interval $[L,U]$ such that

$$\{\psi (\theta ):\theta \in {\mathcal{R}}_n(\alpha )\cap {\mathcal{I}}_n\}\subset [L,U]$$

then we say “causal conclusion $\psi (\theta )\in [L,U]$ is geometrically robust at level $\alpha$”.

In Particular: If $L>0$ (or $U<0$), we can robustly assert direction of effect!

Convex Optimization for Linear Identifiable Sets

Common Case: Identifiable set can be represented as linear inequalities:

$$\mathcal{I}=\{\theta :A\theta \le b\}$$

Then ${\mathcal{R}}_n(\alpha )\cap \mathcal{I}$ is intersection of ellipsoid and polyhedron (convex set).

Extrema of Causal Effect:

For linear effect $\psi (\theta )=c^{\top }\theta$:

$${\psi }_{\max }^{\ast }:=\sup \{c^{\top }\theta :\theta \in {\mathcal{R}}_n(\alpha ),A\theta \le b\}$$

$${\psi }_{\min }^{\ast }:=\inf \{c^{\top }\theta :\theta \in {\mathcal{R}}_n(\alpha ),A\theta \le b\}$$

This is a Quadratic Programming problem, can be solved efficiently!

Geometric Robustness Criterion Theorem:

If ${\psi }_{\min }^{\ast }\ge \delta >0$, then we can robustly assert:

$$\text{Causal effect is positive, and at least }\delta$$

And this conclusion holds for all $\theta \in {\mathcal{R}}_n(\alpha )\cap \mathcal{I}$!

🌐 Multi-Experiment Aggregation: Intersection and Union of Regions

Problem Scenario

In reality, we often have multiple data sources:

• Experiment 1: Randomized controlled trial (RCT), sample $n_1=500$
• Experiment 2: Observational study, sample $n_2=5000$
• Experiment 3: RCT from another region, sample $n_3=300$

Traditional Meta-Analysis:

Calculate point estimates ${\hat{\theta }}_1,{\hat{\theta }}_2,{\hat{\theta }}_3$ of each study, then weighted average.

Problems:

• How to judge whether studies are truly consistent?
• How to systematically identify conflicts?
• Point estimate differences may come from sampling error, not real effect differences!

Geometric Meta-Analysis

Idea: Each study gives a credible region ${\mathcal{R}}_k({\alpha }_k)$, $k=1,2,3$

Consensus Region (intersection):

$${\mathcal{R}}_{\text{cons}}:=\bigcap _{k=1}^K{\mathcal{R}}_k({\alpha }_k)$$

Physical Meaning: Parameter range simultaneously supported by all studies

Permissible Region (union):

$${\mathcal{R}}_{\text{perm}}:=\bigcup _{k=1}^K{\mathcal{R}}_k({\alpha }_k)$$

Physical Meaning: Parameter range supported by at least one study

Conflict Region (symmetric difference):

$${\mathcal{R}}_{\text{conflict}}:={\mathcal{R}}_{\text{perm}}\setminus {\mathcal{R}}_{\text{cons}}$$

Physical Meaning: Parameter range supported by only some studies, where controversy exists

graph TB
    subgraph "Credible Regions of Three Studies"
        R1["Study 1<br/>ℛ₁(α₁)"]
        R2["Study 2<br/>ℛ₂(α₂)"]
        R3["Study 3<br/>ℛ₃(α₃)"]
    end

    R1 -.->|"Intersection"| CONS["Consensus Region<br/>ℛ_cons"]
    R2 -.->|"Intersection"| CONS
    R3 -.->|"Intersection"| CONS

    R1 -.->|"Union"| PERM["Permissible Region<br/>ℛ_perm"]
    R2 -.->|"Union"| PERM
    R3 -.->|"Union"| PERM

    PERM -.-> CONFLICT["Conflict Region<br/>ℛ_conflict = ℛ_perm \ ℛ_cons"]
    CONS -.-> CONFLICT

    style CONS fill:#e1ffe1,stroke:#00aa00,stroke-width:3px
    style CONFLICT fill:#ffe1e1,stroke:#ff0000,stroke-width:2px

Consistency Judgment

Strong Consistency: ${\mathcal{R}}_{\text{cons}}\ne \varnothing$ (consensus region non-empty)

Weak Consistency: “Volume” of ${\mathcal{R}}_{\text{conflict}}$ relatively small

Significant Conflict: ${\mathcal{R}}_{\text{cons}}=\varnothing$ (consensus region empty!)

Then we can clearly assert: Studies have fundamental contradiction, not vaguely say “results somewhat different”.

Consensus Interval for Causal Effect

For effect of interest $\psi (\theta )$:

$${\Psi }_{\text{cons}}:=\{\psi (\theta ):\theta \in {\mathcal{R}}_{\text{cons}}\}$$

Robust Conclusion: Only when $\psi (\theta )\in {\Psi }_{\text{cons}}$ can we say this effect value is supported by all studies.

Examples:

• ${\Psi }_{\text{cons}}=[0.2,0.5]$: All studies consistently support effect between 0.2 and 0.5
• ${\Psi }_{\text{cons}}=[-0.1,0.3]$: Contains 0, cannot robustly assert direction!
• ${\Psi }_{\text{cons}}=\varnothing$: Studies conflict, no consensus

⚙️ Experimental Design: Shaping Future Credible Regions

New Perspective

Traditional experimental design goal: Minimize variance

GLS Geometric Perspective:

$$\boxed{\text{Experimental Design = Shaping Geometric Shape of Future Credible Region}}$$

Key Insight: By choosing experimental scheme (sample allocation, covariate design, etc.), we can actively shape Fisher information matrix $I(\theta ;\xi )$, thus shaping shape of credible ellipsoid!

Fisher Information and Region Volume

Volume of credible ellipsoid:

$$\text{Vol}({\mathcal{R}}_n(\alpha ;\xi ))=C_{d,\alpha }\cdot \det (I_n({\theta }_0;\xi ){)}^{-1/2}$$

where:

• $\xi$ is design variable (e.g., sample allocation scheme)
• $C_{d,\alpha }$ is constant

D-Optimal Design:

$${\xi }^{\ast }=\arg \underset{\xi }{\max }\det I_n({\theta }_0;\xi )$$

Geometric Meaning: Minimize volume of credible ellipsoid, make parameter estimation “tightest overall”!

graph LR
    DESIGN["Experimental Design ξ"] -->|"Determines"| FISHER["Fisher Information<br/>I(θ;ξ)"]
    FISHER -->|"Determines Shape"| ELLIPSE["Credible Ellipsoid<br/>ℛₙ(α;ξ)"]

    OPT["Optimization<br/>max det I"] -.->|"Shrinks"| ELLIPSE

    style ELLIPSE fill:#fff4e1
    style OPT fill:#e1ffe1

Directional Distinguishability: c-Optimal Design

If we only care about specific causal effect $\psi (\theta )=c^{\top }\theta$, don’t need overall optimal!

c-Optimal Design:

$${\xi }^{\ast }=\arg \underset{\xi }{\min }{c}^{\top }{I}_n({\theta }_0;\xi {)}^{-1}c$$

Geometric Meaning:

• Don’t pursue minimum overall ellipsoid volume
• Specifically compress semi-axis in direction $c$
• Concentrate resources to improve resolution of this specific causal effect

Analogy:

• D-optimal = All-round development (all subjects must be good)
• c-optimal = Specialization (only need math good, for math department application)

Example: Sample Allocation in Linear Regression

Model:

$$Y_i={\beta }_0+{\beta }_1{X}_i+{ϵ}_i,{ϵ}_i\sim \mathcal{N}(0,{\sigma }^2)$$

Design Problem: How to allocate $n$ samples at two levels $X\in \{x_1,x_2\}$?

Let allocation proportion be $\xi =(p,1-p)$, i.e., $np$ samples at $x_1$, $n(1-p)$ samples at $x_2$.

Fisher Information Matrix:

$$I(\beta ;\xi )=\frac{n}{{\sigma }^2}\left(\begin{array}{cc}1& \bar{x}(\xi )\\ \bar{x}(\xi )& \overline{x^2}(\xi )\end{array}\right)$$

where $\bar{x}(\xi )=p{x}_1+(1-p)x_2$.

D-Optimal Design (minimize inverse of determinant):

$$\det I(\beta ;\xi )\propto \overline{x^2}(\xi )-\bar{x}(\xi {)}^2=\text{Var}(X)$$

Maximize variance $\Rightarrow$ Equal allocation at extremes: $p^{\ast }=1/2$

c-Optimal Design (only care about slope ${\beta }_1$):

$$c=(0,1{)}^{\top },c^{\top }{I}^{-1}c\propto \frac{1}{\text{Var}(X)}$$

Also get $p^{\ast }=1/2$ (equal allocation at extremes maximizes variance of $X$)

🔗 Connections with GLS Theory

Connection with Causal Diamond

In GLS theory, boundary $\partial D=E^{+}\cup E^{-}$ of causal diamond $D(p,q)$ encodes complete information.

Analogy:

• Causal Diamond ↔ Credible Region ${\mathcal{R}}_n(\alpha )$
• Boundary $\partial D$ ↔ Ellipsoid Boundary
• Bulk Reconstruction ↔ Inferring Interior Parameters from Boundary

All reflect idea that boundary encodes complete information!

Connection with Time Scale

Uncertainty of unified time scale $\kappa (\omega )$ can be geometrized:

$$\kappa (\omega )\in {\mathcal{R}}_{\kappa }:=[{\kappa }_{\min }(\omega ),{\kappa }_{\max }(\omega )]$$

Robust Causal Conclusion: Only when conclusions based on all $\kappa$ values in ${\mathcal{R}}_{\kappa }$ are consistent, are they robust!

Connection with Null-Modular Double Cover

Estimation error of modulation function $g_{\sigma }(\lambda ,x_{\perp })$ can be represented as confidence region:

$$g_{\sigma }\in {\mathcal{R}}_g$$

Robustness: Range of modular Hamiltonian $K_D$ over entire ${\mathcal{R}}_g$:

$$K_D[{\mathcal{R}}_g]=\left\{2\pi \sum _{\sigma }{\int }_{E^{\sigma }}{g}_{\sigma }{T}_{\sigma \sigma }:g_{\sigma }\in {\mathcal{R}}_g\right\}$$

🌟 Core Formula Summary

Fisher Information Metric

$$\boxed{g_{\theta }(u,v):=u^{\top }I(\theta )v}$$

Confidence Ellipsoid (Credible Region)

$$\boxed{{\mathcal{R}}_n(\alpha ):=\left\{\theta :n(\theta -{\hat{\theta }}_n{)}^{\top }I({\hat{\theta }}_n)(\theta -{\hat{\theta }}_n)\le {\chi }_{d,1-\alpha }^2\right\}}$$

Projection Interval for Linear Effect

$$\boxed{{\Psi }_n(\alpha )=\left[c^{\top }{\hat{\theta }}_n-\sqrt{\frac{{\chi }_{d,1-\alpha }^2}{n}{c}^{\top }{I}^{-1}c},c^{\top }{\hat{\theta }}_n+\sqrt{\frac{{\chi }_{d,1-\alpha }^2}{n}{c}^{\top }{I}^{-1}c}\right]}$$

Geometric Robustness

$$\boxed{{\mathcal{C}}_n(\alpha ):=\{\psi (\theta ):\theta \in {\mathcal{R}}_n(\alpha )\cap {\mathcal{I}}_n\}}$$

Multi-Experiment Consensus Region

$$\boxed{{\mathcal{R}}_{\text{cons}}:=\bigcap _{k=1}^K{\mathcal{R}}_k({\alpha }_k)}$$

D-Optimal Design

$$\boxed{{\xi }^{\ast }=\arg \underset{\xi }{\max }\det I_n({\theta }_0;\xi )}$$

💭 Thinking Questions

Question 1: Why Ellipsoid Instead of Sphere?

Hint: Consider correlations between parameters

Answer:

If parameters are completely independent ($I(\theta )$ diagonal), confidence region is a sphere (same uncertainty in all directions).

But in practice, parameters are often correlated:

• Intercept ${\beta }_0$ and slope ${\beta }_1$ usually negatively correlated (seesaw effect)
• Fisher information matrix $I(\theta )$ non-diagonal
• Confidence region is ellipsoid (different uncertainty in different directions)

Geometric Intuition:

• Long axis of ellipsoid → Direction where parameters “hard to identify”
• Short axis of ellipsoid → Direction where parameters “easy to identify”

Question 2: What Does Empty Consensus Region Mean?

Hint: Think of quantum uncertainty principle

Answer:

$${\mathcal{R}}_{\text{cons}}=\bigcap _{k=1}^K{\mathcal{R}}_k=\varnothing$$

Physical Meaning:

1. Significant Conflict: Studies have fundamental contradiction, cannot simultaneously satisfy all constraints
2. Model Mismatch: Assumptions of some studies may be wrong
3. Heterogeneity: Different studies may measure different parameters (e.g., effects in different populations)

Quantum Analogy:

Like simultaneously precisely measuring position and momentum → Uncertainty principle forbids!

Intersection of credible regions of multiple studies empty → “Geometric incompatibility” in parameter space!

Question 3: How to Define Error Geometry in Quantum Gravity?

Hint: Recall quantum fluctuations of causal diamond

Answer:

In quantum gravity, spacetime geometry itself has quantum fluctuations!

Classical GLS:

$${\mathcal{R}}_{\text{classical}}=\{g_{\mu \nu }:\text{satisfies causal constraints}\}$$

Quantum GLS (path integral):

$${\mathcal{R}}_{\text{quantum}}=\int \mathcal{D}{g}_{\mu \nu }{e}^{iS[g]}\delta (\text{causal constraints})$$

Geometric Uncertainty:

• At Planck scale ${\ell }_P$, spacetime has “foam” fluctuations
• Credible region becomes measure in function space
• Robustness → Topological invariance under quantum fluctuations

Example:

Correction terms to Bekenstein-Hawking entropy:

$$S=\frac{A}{4G}+\alpha \log \left(\frac{A}{{\ell }_P^2}\right)+\cdots$$

Credible region ${\mathcal{R}}_{\alpha }$ of $\alpha$ determines robust predictions of quantum gravitational corrections!

🎯 Core Insights

1. Error = Geometric Boundary

   Traditional: Error = Supplementary information

   GLS: Error = Geometric region in parameter space (credible region)

2. Robustness = Geometric Invariance

   Traditional: Robustness = “Results similar”

   GLS: Robustness = Conclusion holds over entire credible region

3. Causal Inference = Credible Region ∩ Identifiable Set

   $$\text{Robust Causal Conclusion}⟷{\mathcal{R}}_n(\alpha )\cap {\mathcal{I}}_n$$

4. Meta-Analysis = Intersection and Union of Regions

   • Consensus = Intersection $\bigcap {\mathcal{R}}_k$
   • Conflict = Symmetric difference $\bigcup {\mathcal{R}}_k\setminus \bigcap {\mathcal{R}}_k$

5. Experimental Design = Shaping Geometric Shape

   Actively shape credible ellipsoid through Fisher information $I(\theta ;\xi )$

📚 Connections with Other Chapters

With Causal Geometrization (Chapter 8)

• Chapter 8: Spacetime geometry = Compression of causal constraints
• This chapter: Parameter geometry = Compression of statistical constraints

Unified Perspective:

$$\text{Geometry}=\text{Optimal Encoding of Constraints}+\text{Error Boundary}$$

With Boundary Theory (Chapter 6)

• Boundary encodes bulk information ↔ Ellipsoid boundary encodes parameter uncertainty
• Uncertainty of Brown-York energy ↔ Ellipsoid shape determined by Fisher information

With Unified Time (Chapter 5)

Uncertainty of time scale $\kappa (\omega )$:

$$\kappa \in [{\kappa }_{\min },{\kappa }_{\max }]$$

Robust Causal Arrow: Only when all $\kappa \in [{\kappa }_{\min },{\kappa }_{\max }]$ give same causal direction, is arrow robust!

📖 Further Reading

Classical Statistics:

• van der Vaart (1998): Asymptotic Statistics (asymptotic theory)
• Bickel & Doksum (2015): Mathematical Statistics (confidence regions)

Causal Inference:

• Manski (2003): Partial Identification (identifiable sets)
• Imbens & Rubin (2015): Causal Inference (robustness analysis)

Experimental Design:

• Pukelsheim (2006): Optimal Design of Experiments (Fisher information and design)

GLS Theory Source Documents:

• error-geometry-causal-robustness.md (source of this chapter)

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