Engineering Macroscopic Wormholes via Planck-Scale Quantum Backreaction

1. Introduction

The modern study of traversable wormholes sits at the interface of geometry, quantum matter, and causal structure. Einstein and Rosen introduced nontrivial bridges in 1935, but those constructions were non-traversable and motivated by particle-like interpretations of field singularities [1]. The later Morris–Thorne program reframed wormholes as explicit spacetimes with tunable throat geometry and asymptotic flatness [6]. This shift exposed a fundamental issue: sustained throat flare-out requires negative effective null energy density in the vicinity of the throat. Classical matter models rarely provide such behavior without instabilities or nonphysical stress sectors.

The present work asks whether Planck-scale quantum backreaction can supply a finite, controlled support channel for macroscopic throat stability without demanding an ad hoc classical exotic fluid. Our perspective is intentionally semiclassical: we do not assume a complete theory of quantum gravity, but instead work with renormalized quantum stress tensors on slowly varying backgrounds, together with a covariant perturbative expansion in geometric fluctuations.

The chronology problem appears immediately. Any geometry that allows effective short-cutting between distant spacetime regions can, when embedded in relative motion protocols, lead to near-closed timelike curves (CTCs). Hawking’s chronology protection conjecture states that quantum effects should diverge near chronology horizons strongly enough to obstruct macroscopic causality violation [9]. This conjecture, although not proven in complete generality, provides a powerful consistency filter for wormhole engineering schemes.

Historically, Wheeler’s spacetime foam picture suggested that topology fluctuations are natural at Planck scales, raising the possibility that nontrivial connectivity could emerge as a quantum-gravitational phase rather than a classical engineering artifact [2,3]. Subsequent developments in quantum field theory in curved spacetime, especially renormalization of $\langle T_{\mu \nu }\rangle$, supplied calculational tools for examining backreaction near horizons and compact regions [14,15,17]. Modern holographic ideas, including ER=EPR, further motivate revisiting geometric connectivity from entanglement-first viewpoints [31,32].

In this manuscript we assemble these lines into a single technical narrative. We introduce a mode-resolved fluctuation model near the throat, compute effective stress-energy corrections at second order, and derive stability and causality bounds that can be tested numerically. We emphasize where assumptions enter. 1 We also include explicit diagrams and parameter tables to keep the argument transparent and reproducible.

Contributions and Paper Roadmap

Our main contributions are:

• A detailed historical-to-technical bridge from Einstein–Rosen geometry to semiclassical throat stabilization.
• A covariant fluctuation ansatz for Planck-scale backreaction near a Morris–Thorne-type throat.
• Closed-form inequalities for flare-out, averaged-energy projections, and chronology-window constraints.
• Figure-driven interpretation of embedding geometry, mode flow, causal loops, and observational parameter sectors.
• A bibliography expanded to foundational and modern literature across general relativity-quantum cosmology and high-energy-physics-theory.

2. Historical Background of Wormhole Physics

2.1. From Einstein–Rosen Bridges to Geometric Topology Change

Einstein and Rosen’s original bridge identified two exterior Schwarzschild regions across a coordinate patch, but the wormhole pinches off too quickly for traversability [1]. This non-traversable character was later clarified through maximal extensions and global structure analyses [4,5]. Wheeler then proposed spacetime foam, in which Planckian geometry admits fluctuating topology and transient micro-wormholes [2,3]. Although heuristic, this picture motivates treating connectivity as a quantum effect rather than purely classical architecture.

2.2. Traversability and Exotic Support

Morris and Thorne formalized traversable wormholes with humanly passable tidal constraints and no event horizons [6]. The key geometric condition is throat flare-out, encoded by $b\left(r_0\right)=r_0$ and $b^{\prime }\left(r_0\right)<1$. Einstein equations then imply NEC violation near $r_0$, often interpreted as “exotic matter.” Subsequent work by Visser and others broadened constructions via cut-and-paste junctions, thin shells, and alternative equations of state [10,11,12,13].

2.3. Chronology Programs and Self-Consistency

The Morris–Thorne–Yurtsever protocol showed that relative motion of wormhole mouths can produce time shifts and CTC risk [7]. Hawking argued that stress-energy amplification near chronology horizons may prevent full chronology violation [9]. Competing views include self-consistency constraints à la Novikov and quantum information models of CTC behavior [25,26]. Our work treats chronology protection as an operational consistency bound rather than a dogma.

3. Energy Conditions in General Relativity

For a static, spherically symmetric ansatz,

$$\mathrm{d}{s}^2=-e^{2\mathrm{\Phi }\left(r\right)}\mathrm{d}{t}^2+{\displaystyle \frac{\mathrm{d}{r}^2}{1-b\left(r\right)/r}}+r^2(\mathrm{d}{\theta }^2+{sin}^2\theta \mathrm{d}{\phi }^2),$$

(1)

we define throat radius $r_0$ by $b\left(r_0\right)=r_0$. Radial null vectors yield

$$8\pi G{T}_{\mu \nu }{k}^{\mu }{k}^{\nu }={\displaystyle \frac{b^{\prime }\left(r\right)r-b\left(r\right)}{r^3}}+2\left(1-{\displaystyle \frac{b\left(r\right)}{r}}\right){\displaystyle \frac{{\mathrm{\Phi }}^{\prime }\left(r\right)}{r}}.$$

(2)

At the throat, regular $\mathrm{\Phi }$ gives

$${\left.T_{\mu \nu }{k}^{\mu }{k}^{\nu }\right|}_{r_0}={\displaystyle \frac{b^{\prime }\left(r_0\right)-1}{8\pi G{r}_0^2}}<0,$$

(3)

confirming NEC violation for flare-out. 2

3.1. Averaged Conditions and Quantum Inequalities

Ford and Roman derived bounds on negative energy duration and magnitude in quantum field theory [18,19]. A schematic worldline bound reads

$${\int }_{-\infty }^{\infty }\rho \left(\tau \right)f\left(\tau \right)\mathrm{d}\tau \ge -{\displaystyle \frac{C}{{\tau }_0^4}},$$

(4)

for smooth sampling function f of width ${\tau }_0$. Wormhole support sectors must therefore satisfy an optimization problem: sufficient negative effective energy near $r_0$, but not so persistent as to violate quantum inequality bounds.

3.2. Comparative Energy-Condition Taxonomy

Table 1. Energy-condition behavior across representative exotic spacetimes.

Model	NEC	WEC	SEC	ANEC	Dominant support channel
Morris–Thorne traversable wormhole	×	×	×	×	Exotic throat source
Thin-shell wormhole	×	mixed	mixed	mixed	Junction stress
Alcubierre bubble	×	×	×	×	Warp-shell stress anisotropy
Casimir throat
(toy EFT)	local	local	mixed	constrained	Vacuum polarization
Planck backreaction
(this work)	eff. local	mixed	mixed	constrained	Renormalized quantum stress

Table 1. Energy-condition behavior across representative exotic spacetimes.

Model	NEC	WEC	SEC	ANEC	Dominant support channel
Morris–Thorne traversable wormhole	×	×	×	×	Exotic throat source
Thin-shell wormhole	×	mixed	mixed	mixed	Junction stress
Alcubierre bubble	×	×	×	×	Warp-shell stress anisotropy
Casimir throat
(toy EFT)	local	local	mixed	constrained	Vacuum polarization
Planck backreaction
(this work)	eff. local	mixed	mixed	constrained	Renormalized quantum stress

4. Quantum Field Theory in Curved Spacetime

4.1. Semiclassical Einstein Equation

We adopt

$$G_{\mu \nu }+\mathrm{\Lambda }{g}_{\mu \nu }=8\pi G\left(T_{\mu \nu }^{\mathrm{cl}}+{\langle T_{\mu \nu }\rangle }_{\mathrm{ren}}\right),$$

(5)

with ${\nabla }^{\mu }{\langle T_{\mu \nu }\rangle }_{\mathrm{ren}}=0$. Renormalization introduces geometric counterterms proportional to $R^2$, $R_{\mu \nu }{R}^{\mu \nu }$, and $R_{\mu \nu \alpha \beta }{R}^{\mu \nu \alpha \beta }$ in effective action language [14,17].

4.2. Vacuum Polarization Near Horizons and Throats

Near nearly trapped regions, mode redshifting amplifies state dependence of ${\langle T_{\mu \nu }\rangle }_{\mathrm{ren}}$. 3 For static sectors, point-splitting gives

$${\langle T_{\mu \nu }\rangle }_{\mathrm{ren}}=\underset{x^{\prime }\to x}{lim}{\mathcal{D}}_{\mu \nu }\left[G^{\left(1\right)}(x,x^{\prime })-G_{\mathrm{Had}}^{\left(1\right)}(x,x^{\prime })\right],$$

(6)

where $G_{\mathrm{Had}}^{\left(1\right)}$ subtracts the universal short-distance singularity.

4.3. Fluctuation Expansion and Stochastic Gravity Motivation

Write

$$g_{\mu \nu }={\overline{g}}_{\mu \nu }+h_{\mu \nu },\langle h_{\mu \nu }\rangle =0,$$

(7)

with correlator

$$\langle h_{\mu \nu }\left(x\right)h_{\alpha \beta }\left(y\right)\rangle ={\ell }_{\mathrm{P}}^2{\mathcal{N}}_{\mu \nu \alpha \beta }(x,y).$$

(8)

Then

$$\langle G_{\mu \nu }[\overline{g}+h]\rangle =G_{\mu \nu }\left[\overline{g}\right]+\langle G_{\mu \nu }^{\left(2\right)}\left[h\right]\rangle +\mathcal{O}\left(h^3\right),$$

(9)

which motivates an effective source

$$T_{\mu \nu }^{\mathrm{br}}\equiv {\displaystyle \frac{1}{8\pi G}}\langle G_{\mu \nu }^{\left(2\right)}\left[h\right]\rangle .$$

(10)

5. Semiclassical Backreaction Framework

5.1. Mode-Resolved Throat Ansatz

We parameterize the near-throat fluctuation packet as

$$\delta g_{\mu \nu }^{\left(n\right)}\left(r\right)={\epsilon }_{n}exp\left[-{\displaystyle \frac{{(r-r_0)}^2}{2{\sigma }_n^2}}\right]cos\left({\displaystyle \frac{n\pi (r-r_0)}{L_n}}\right){\mathrm{\Pi }}_{\mu \nu }^{\left(n\right)},$$

(11)

with $|{\epsilon }_n|\ll 1$ and projector tensors ${\mathrm{\Pi }}_{\mu \nu }^{\left(n\right)}$. A finite set $n\le N_{*}$ captures EFT-resolved modes.

Define aggregated amplitude

$$\mathcal{A}\left(r\right)=\sum _{n=1}^{N_{*}}{\epsilon }_n{e}^{-{(r-r_0)}^2/\left(2{\sigma }_n^2\right)}.$$

(12)

Backreaction-corrected shape and redshift functions are modeled as

$$\begin{array}{cc}\hfill b_{\mathrm{e}ff}\left(r\right)& =b_0\left(r\right)+{\alpha }_b{r}_0\mathcal{A}\left(r\right)+{\beta }_b{r}_0{\mathcal{A}}^2\left(r\right),\hfill \end{array}$$

(13)

$$\begin{array}{cc}\hfill {\mathrm{\Phi }}_{\mathrm{e}ff}\left(r\right)& ={\mathrm{\Phi }}_0\left(r\right)+{\alpha }_{\mathrm{\Phi }}\mathcal{A}\left(r\right)+{\beta }_{\mathrm{\Phi }}{\mathcal{A}}^{\prime }\left(r\right).\hfill \end{array}$$

(14)

5.2. Stability and Perturbative Control Window

Linear radial perturbations $r\left(t\right)=r_0+\delta r\left(t\right)$ satisfy

$$\ddot{\delta r}+{\mathrm{\Omega }}_r^2\delta r=0,$$

(15)

with

$${\mathrm{\Omega }}_r^2\propto {\left.{\displaystyle \frac{{\partial }^2{V}_{\mathrm{e}ff}}{\partial r^2}}\right|}_{r_0}.$$

(16)

A viable semiclassical window requires simultaneously:

$$\begin{array}{cc}\hfill & b_{\mathrm{e}ff}\left(r_0\right)=r_0,b_{\mathrm{e}ff}^{\prime }\left(r_0\right)<1,\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{\langle G^{\left(2\right)}\rangle }{G\left[\overline{g}\right]}}\right|\lesssim {\eta }_{\mathrm{m}ax}\ll 1,\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill & {\mathrm{\Omega }}_r^2>0,\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill & {\mathcal{Q}}_{\mathrm{Q}I}\equiv {\tau }_0^4\left|\int \rho \left(\tau \right)f\left(\tau \right)\mathrm{d}\tau \right|\le C_{\mathrm{Q}I}.\hfill \end{array}$$

(20)

Equation (20) encodes quantum inequality compliance.

5.3. Representative Parameter Datasets

Table 2. Illustrative stability parameter ranges (dimensionless unless noted).

Parameter	Symbol	Min	Max	Nominal	Physical interpretation
Throat radius (km)	$r_0$	${10}^{-3}$	${10}^2$	1	Macroscopic bridge scale
Mode amplitude	${\epsilon }_n$	${10}^{-9}$	${10}^{-2}$	${10}^{-5}$	Planck-seeded geometric fluctuation
Coherence width ($r_0$ units)	${\sigma }_n/r_0$	${10}^{-3}$	${10}^{-1}$	${10}^{-2}$	Shell localization near throat
Backreaction ratio	$\eta$	${10}^{-6}$	${10}^{-1}$	${10}^{-3}$	EFT validity diagnostic
Chronology indicator	${\chi }_{\mathrm{C}TC}$	0	1	$0.25$	Proximity to causal-loop onset

Table 2. Illustrative stability parameter ranges (dimensionless unless noted).

Parameter	Symbol	Min	Max	Nominal	Physical interpretation
Throat radius (km)	$r_0$	${10}^{-3}$	${10}^2$	1	Macroscopic bridge scale
Mode amplitude	${\epsilon }_n$	${10}^{-9}$	${10}^{-2}$	${10}^{-5}$	Planck-seeded geometric fluctuation
Coherence width ($r_0$ units)	${\sigma }_n/r_0$	${10}^{-3}$	${10}^{-1}$	${10}^{-2}$	Shell localization near throat
Backreaction ratio	$\eta$	${10}^{-6}$	${10}^{-1}$	${10}^{-3}$	EFT validity diagnostic
Chronology indicator	${\chi }_{\mathrm{C}TC}$	0	1	$0.25$	Proximity to causal-loop onset

Table 3. Planck-scale fluctuation amplitudes for selected mode families.

Mode index n	${\epsilon }_n$	${\sigma }_n/r_0$	$L_n/r_0$	Comment
1	$1.0\times {10}^{-5}$	$2.0\times {10}^{-2}$	$0.9$	Dominant low-frequency support mode
2	$6.5\times {10}^{-6}$	$1.8\times {10}^{-2}$	$0.6$	Corrective mode suppressing over-flare
3	$4.0\times {10}^{-6}$	$1.2\times {10}^{-2}$	$0.45$	Localized compensation near $r_0^{+}$
4	$2.2\times {10}^{-6}$	$1.0\times {10}^{-2}$	$0.35$	Higher mode, small chronology sensitivity
5	$1.5\times {10}^{-6}$	$8.0\times {10}^{-3}$	$0.30$	Ultraviolet tail within EFT cutoff

Table 3. Planck-scale fluctuation amplitudes for selected mode families.

Mode index n	${\epsilon }_n$	${\sigma }_n/r_0$	$L_n/r_0$	Comment
1	$1.0\times {10}^{-5}$	$2.0\times {10}^{-2}$	$0.9$	Dominant low-frequency support mode
2	$6.5\times {10}^{-6}$	$1.8\times {10}^{-2}$	$0.6$	Corrective mode suppressing over-flare
3	$4.0\times {10}^{-6}$	$1.2\times {10}^{-2}$	$0.45$	Localized compensation near $r_0^{+}$
4	$2.2\times {10}^{-6}$	$1.0\times {10}^{-2}$	$0.35$	Higher mode, small chronology sensitivity
5	$1.5\times {10}^{-6}$	$8.0\times {10}^{-3}$	$0.30$	Ultraviolet tail within EFT cutoff

6. Wormhole Geometry and Stability Analysis

6.1. Flare-out with Backreaction Corrections

Expanding

$$\Delta \left(r\right)\equiv r-b_{\mathrm{e}ff}\left(r\right)={\Delta }_1(r-r_0)+{\displaystyle \frac{1}{2}}{\Delta }_2{(r-r_0)}^2+\cdots ,$$

(21)

flare-out demands ${\Delta }_1>0$. In terms of coefficients in (13),

$${\Delta }_1=1-b_0^{\prime }\left(r_0\right)-{\alpha }_b{r}_0{\mathcal{A}}^{\prime }\left(r_0\right)-2{\beta }_b{r}_0\mathcal{A}\left(r_0\right){\mathcal{A}}^{\prime }\left(r_0\right)>0.$$

(22)

Hence a positive classical margin $1-b_0^{\prime }\left(r_0\right)$ can be reduced or enhanced depending on mode phasing. 4

6.2. Embedding Geometry (TikZ)

Figure 1. Embedding cross-section of an effective traversable throat.

Figure 1. Embedding cross-section of an effective traversable throat.

6.3. Embedding Surface (pgfplots)

Figure 2. Canonical embedding surface for the effective throat geometry.

Figure 2. Canonical embedding surface for the effective throat geometry.

6.4. Backreaction Process Diagram

Figure 3. Backreaction pipeline used in the present framework.

Figure 3. Backreaction pipeline used in the present framework.

7. Chronology Protection and CTC Constraints

7.1. Causal-Loop Onset and Horizon Diagnostics

Let $\Delta T$ denote induced proper-time shift between mouths under motion protocol. A CTC-accessible sector emerges when $\Delta T>L_{AB}/c$ for spatial separation $L_{AB}$. We define chronology margin

$$\mathrm{\Gamma }\equiv 1-{\displaystyle \frac{\Delta Tc}{L_{AB}}}.$$

(23)

$\mathrm{\Gamma }<0$ indicates formal CTC access; $\mathrm{\Gamma }\approx 0^{+}$ indicates chronology-horizon approach.

Near $\mathrm{\Gamma }\to 0^{+}$, renormalized stress can scale as

$$\parallel {\langle T_{\mu \nu }\rangle }_{\mathrm{ren}}\parallel \sim {\mathrm{\Gamma }}^{-p},p>0,$$

(24)

with model- and state-dependent exponent. 5

7.2. Causal Diagram with CTC Loop

Figure 4. Schematic causal loop in mouth-displaced wormhole protocol.

Figure 4. Schematic causal loop in mouth-displaced wormhole protocol.

7.3. Chronology Protection Inequality Set

A conservative consistency region can be phrased as

$$\begin{array}{cc}\hfill \mathrm{\Gamma }& \ge {\mathrm{\Gamma }}_{\mathrm{m}in}>0,\hfill \end{array}$$

(25)

$$\begin{array}{cc}\hfill max\parallel {\langle T_{\mu \nu }\rangle }_{\mathrm{ren}}\parallel & \le T_{\mathrm{U}V},\hfill \end{array}$$

(26)

$$\begin{array}{cc}\hfill {\displaystyle \frac{{\ell }_{\mathrm{P}}^2\mathcal{R}}{1+{\chi }_{\mathrm{C}TC}}}& \ll 1,\hfill \end{array}$$

(27)

where $\mathcal{R}$ is characteristic curvature scalar. The third condition enforces EFT reliability as chronology sensitivity grows.

8. Observational Signatures and Experimental Proposals

8.1. Cosmological Signatures and Spectral Modulations

In inflation-adjacent parametrizations, coherence-scale backreaction can induce tiny oscillatory imprints in the scalar power spectrum:

$${\mathcal{P}}_{\mathcal{R}}\left(k\right)=A_s{\left({\displaystyle \frac{k}{k_{*}}}\right)}^{n_s-1}\left[1+{\delta }_{0}sin\left(\omega ln{\displaystyle \frac{k}{k_{*}}}+\varphi \right)\right],$$

(28)

with ${\delta }_0\ll 1$.

Figure 5. Illustrative primordial spectrum modulation induced by coherence-scale corrections.

Figure 5. Illustrative primordial spectrum modulation induced by coherence-scale corrections.

8.2. Energy-Condition Histogram Diagnostics

Figure 6. Comparative NEC diagnostic histogram for representative models.

Figure 6. Comparative NEC diagnostic histogram for representative models.

8.3. Analog Gravity Platforms

Acoustic black holes and BEC analogs provide controlled settings for horizon-like mode conversion and correlation measurements. While they do not realize full Einstein dynamics, they can test vacuum amplification channels and effective causal response under engineered backgrounds [44,45,46].

Table 4. Predicted signatures and indicative experimental channels.

Signature class	Preferred probe	Notes
Oscillatory primordial tilt correction	CMB + LSS joint fits	Sensitive to small ${\delta }_0$ with broad priors
Ringdown phase drift in compact mergers	3G GW detectors	Degenerate with beyond-GR damped mode sectors
Near-horizon correlation asymmetry	BEC analog experiments	Tests mode-conversion analogs of vacuum polarization
Effective light-cone deformation	Precision atom interferometry	Requires control of environmental decoherence

Table 4. Predicted signatures and indicative experimental channels.

Signature class	Preferred probe	Notes
Oscillatory primordial tilt correction	CMB + LSS joint fits	Sensitive to small ${\delta }_0$ with broad priors
Ringdown phase drift in compact mergers	3G GW detectors	Degenerate with beyond-GR damped mode sectors
Near-horizon correlation asymmetry	BEC analog experiments	Tests mode-conversion analogs of vacuum polarization
Effective light-cone deformation	Precision atom interferometry	Requires control of environmental decoherence

9. Discussion

9.1. Limitations of Semiclassical Gravity

The semiclassical Einstein equation assumes a classical metric sourced by quantum expectation values. This is formally justified only when stress fluctuations remain subdominant relative to mean values and curvature remains below UV-sensitive thresholds. In strong chronology-adjacent sectors, these assumptions can fail rapidly.

9.2. Comparison with Alternative Models

Classical exotic-fluid models often provide direct flare-out but pay a large stability or microphysical consistency cost. Thin-shell constructions localize pathology at junctions but still require negative surface energy in standard GR. Modified-gravity traversable solutions can shift violations into effective geometric sectors, but then interpretation depends on frame and operator completion. Our approach keeps Einstein gravity intact at low energy and pushes novelty into quantized fluctuations and renormalized stress.

9.3. Consistency Constraints and Falsifiability

A useful outcome of the present program is not “proof of traversability,” but a compact falsifiability map:

• If chronology diagnostics force $\mathrm{\Gamma }\to 0$ before stable support develops, the model fails operationally.
• If quantum inequalities preclude required negative-energy duration, the model is excluded.
• If backreaction ratio $\eta$ exceeds perturbative bounds, EFT control is lost.
• If predicted signatures are absent in next-generation datasets, parameter space contracts sharply.

10. Conclusions and Future Work

We constructed a detailed semiclassical framework for Planck-scale backreaction support of macroscopic wormhole throats and analyzed its compatibility with energy constraints and chronology protection. The key result is conditional: finite windows of effective support can exist only when mode amplitudes, coherence widths, and causal displacement parameters remain inside a narrow multi-constraint region.

Future directions include: (i) embedding this EFT in explicit quantum-gravity candidates; (ii) holographic traversability and ER=EPR-informed state design; (iii) stochastic-gravity simulations with non-Gaussian noise kernels; and (iv) analog-gravity experiments targeting vacuum-polarization proxies. 6

Appendix E.1. Near-Throat Series Expansion

Let $x=(r-r_0)/r_0$ with $\left|x\right|\ll 1$. Expand

$$\begin{array}{cc}\hfill \mathrm{\Phi }\left(r\right)& ={\mathrm{\Phi }}_0+{\mathrm{\Phi }}_{1}x+{\displaystyle \frac{1}{2}}{\mathrm{\Phi }}_2{x}^2+{\displaystyle \frac{1}{6}}{\mathrm{\Phi }}_3{x}^3+\mathcal{O}\left(x^4\right),\hfill \end{array}$$

(A7)

$$\begin{array}{cc}\hfill b\left(r\right)& =r_0\left[1+b_{1}x+{\displaystyle \frac{1}{2}}{b}_2{x}^2+{\displaystyle \frac{1}{6}}{b}_3{x}^3+\mathcal{O}\left(x^4\right)\right].\hfill \end{array}$$

(A8)

Then

$$1-{\displaystyle \frac{b\left(r\right)}{r}}={\displaystyle \frac{(1-b_1)x+\frac{1}{2}(1-b_2)x^2+\mathcal{O}\left(x^3\right)}{1+x}},$$

(A9)

and flare-out requires $b_1<1$.

Using effective source terms $({\rho }_{\mathrm{e}ff},p_{r,\mathrm{e}ff},p_{t,\mathrm{e}ff})$, one obtains

$$\begin{array}{cc}\hfill b_1& =8\pi G{r}_0^2{\rho }_0,\hfill \end{array}$$

(A10)

$$\begin{array}{cc}\hfill b_2& =2b_1+8\pi G{r}_0^2{\rho }_1,\hfill \end{array}$$

(A11)

$$\begin{array}{cc}\hfill {\mathrm{\Phi }}_1& ={\displaystyle \frac{1+8\pi G{r}_0^2{p}_{r,0}}{2(1-b_1)}}.\hfill \end{array}$$

(A12)

The radial conservation equation gives

$$p_{r,1}=2(p_{t,0}-p_{r,0})-({\rho }_0+p_{r,0}){\mathrm{\Phi }}_1.$$

(A13)

Appendix E.2. Effective Exoticity Budget

Define

$${\mathcal{E}}_0\equiv -8\pi G{r}_0^2({\rho }_0+p_{r,0})=1-b_1,$$

(A14)

with decomposition

$${\mathcal{E}}_0={\mathcal{E}}_0^{\mathrm{c}l}+{\mathcal{E}}_0^{\mathrm{r}en}+{\mathcal{E}}_0^{\mathrm{\Xi }}.$$

(A15)

A practical target is reduced classical burden, i.e., ${\mathcal{E}}_0^{\mathrm{c}l}\ll {\mathcal{E}}_0$, while retaining QI compatibility.

Appendix E.3. Stability Potential Expansion

For radial shell variable $a\left(\tau \right)$,

$${\dot{a}}^2+V\left(a\right)=0,$$

(A16)

with

$$V\left(a\right)=V_0+V_1(a-a_0)+{\displaystyle \frac{1}{2}}{V}_2{(a-a_0)}^2+{\displaystyle \frac{1}{6}}{V}_3{(a-a_0)}^3+\cdots .$$

(A17)

Static equilibrium requires $V_0=V_1=0$, linear stability requires $V_2>0$, and the sign of $V_3$ controls finite-amplitude asymmetry.