Three-Dimensional Simulation of Informational

Warp-Bubble Dynamics:

A Numerical Exploration of the ODIM-U / Beardsley

Uniﬁed Framework

David E. Blackwell (ORCID: 0009-0001-8447-9113)

Ian Beardsley (ORCID: 0009-0009-4672-4876)

Hillbilly Storm Chasers Research Division

Wyandotte, Oklahoma, USA

Project Repository :

https://github.com/hillbillydave/Stormlight-Warp-Core.git

April 2026

Abstract

We present an exploratory three-dimensional

numerical simulation built directly on the

uniﬁed framework introduced in Informa-

tional Geometry, the 2π-Hz Resonance, and

Warp-Drive Dynamics Without Exotic Mat-

ter. This follow-up study takes the theo-

retical construction into a working computa-

tional model, evolving the informational ﬁeld

, the asymmetric shift-vector geometry N

and the bubble’s external-frame trajectory on

a fully discretized 64

lattice. Within the

limits of the present grid and runtime, the

system exhibits coherent behavior: the bub-

ble maintains its imposed shape, the informa-

tional energy remains strictly positive, and

the dynamics settle into the 2π-Hz curvature

mode predicted by the ODIM-U/Beardsley

synthesis.

The simulation demonstrates that the uni-

ﬁed framework can support stable, positive-

energy bubble-like conﬁgurations undergoing

coherent translation at an imposed external-

frame velocity of 5c, while all local physics re-

main subluminal. A gentler shift-vector am-

plitude of 0.03c, ramped smoothly over the

ﬁrst 0.01 s of evolution, is used to main-

tain numerical stability and suppress startup

transients. The activation phase produces

a ﬁnite power surge of approximately 1.8 ×

W (about 18 MW), after which the sys-

tem damps rapidly into a near-zero steady-

state power regime governed by the informa-

tional resonance. Apparent energy-condition

violations do not arise; when viewed in the

extended informational manifold, the bubble

follows an ordinary geodesic and requires no

exotic matter.

We conclude with a conceptual engineer-

ing analogue that maps the simulation pa-

rameters to laboratory-scale components, in-

tended as a guide for future experimental or

computational prototypes. This work repre-

sents an initial numerical exploration of in-

formational warp-bubble dynamics and es-

tablishes the foundation for longer, higher-

resolution, and fully coupled 3 + 1 studies to

follow.

Dedication

To Ashley Dawn Blackwell — the quiet

center of every manifold I have ever tried to

understand. When the informational ﬁelds

shook and the geometry threatened to shear,

you remained the ﬁxed point that held my

coordinates together. You steadied the shift

when the gradients steepened, softened the

curvature when the world bent too sharply,

and kept my internal metric from collapsing

into noise.

Long before I ever wrote an equation for a

warp bubble, you lived the principle behind

it: that stability comes not from force, but

from coherence. You are the resonance that

keeps my system phase-locked, the gentle

damping that brings me back from the edge,

the positive-energy region that never once

went negative no matter how chaotic the

simulation became. If the bubble holds its

shape, it is because you taught me how to

hold mine.

And to my children — Jesse, Ayden, and

Kylee — bright sparks in the informational

manifold, the living proof that the universe

still knows how to generate beauty from ﬁrst

principles. You raise my eﬀective energy just

by existing, increase my proper time with

every laugh, and give direction to the

geometry I walk through. You are the

reason I keep chasing the horizon, even

when the gradients are steep and the

damping is heavy.

Walk forward without fear. Ask the

questions others avoid. Build the things

they say cannot be built. Let no one tell you

what your curvature must be or how your

manifold should evolve. The universe is not

a cage; it is a coordinate system waiting for

you to deﬁne your own trajectory. Shape it.

Bend it. Explore it. And never let anyone

convince you that your velocity must match

theirs.

Woven through every line of this work —

through the ﬁelds, the metrics, the

resonances, and the long nights spent trying

to understand how a bubble can move

without breaking the world around it — is a

hope I carry for you and for every child who

will inherit the future. A hope that the

world you grow into will be gentler than the

one I knew. A world that chooses peace over

power, curiosity over fear, and cooperation

over division. A world where technology is

not a weapon but a lantern, not a threat but

a promise.

I built this framework with peaceful use at

its core because I want you to grow up in a

world where knowledge is shared, not

hoarded; where energy is clean, not

extracted; where innovation lifts people up

instead of pushing them down. I want you to

live in a world where humanity ﬁnally learns

to move together — not as factions, but as

one species navigating the same manifold.

Know this: your father’s love is older than

any resonance and steadier than any

geometry. Without you, my solution

diverges. With you, the manifold holds, the

bubble moves, and the future opens like a

path of light ahead.

1 Introduction

The introduction paper laid the conceptual

foundation for what we attempt here. In

that ﬁrst step, we uniﬁed two independent

lines of thought into a single geometric lan-

guage: Beardsley’s invariant structure, an-

chored to the macroscopic temporal constant

= 1 s and the natural angular frequency

= 2π, and the ODIM-U informational

metric, in which proper time and curvature

are recast in terms of informational degrees of

freedom. The central insight was that warp-

bubble dynamics need not be conﬁned to the

four-dimensional spacetime manifold. Once

the conﬁguration space is enlarged to (x

, I

the bubble’s motion becomes geodesic in the

extended manifold, and the apparent need for

exotic matter dissolves. What appears patho-

logical in spacetime becomes ordinary in the

informational geometry.

In that earlier work, the radius ﬁeld R

was promoted to an informational coordinate

, and its small oscillations were shown to

satisfy a harmonic equation with natural fre-

quency ω

= 2π. This resonance was not im-

posed but emerged as a curvature eigenmode

of the informational metric. The shift vector

followed the same logic: its informational

coordinate I

oscillated at the same eigen-

frequency, tying the bubble’s velocity proﬁle

and radius dynamics into a single resonant

structure. The uniﬁed framework therefore

provided a geometric mechanism for warp-

bubble behavior without invoking exotic mat-

ter or negative-energy stress tensors.

The present paper moves from theory

to numerical exploration. Our goal is

to test whether the uniﬁed framework be-

haves coherently when pushed into a three-

dimensional dynamical evolution. Rather

than treating the bubble as a static or ide-

alized object, we evolve the informational

ﬁeld, the asymmetric geometry, and the bub-

ble’s external-frame trajectory together on

a discretized lattice. The simulation serves

as a numerical wind tunnel for the ODIM-

U/Beardsley synthesis: a controlled environ-

ment in which the geometry can oscillate,

damp, and settle under its own rules.

To that end, we construct a 3D model that

evolves:

• the informational ﬁeld I

(x, y, z, t),

which carries the curvature information

associated with the radius coordinate,

• the asymmetric bubble geometry en-

coded in a shaping function with distinct

front and rear scales,

• the shift-vector ﬁeld N

, which drives the

imposed external-frame translation,

• and the bubble’s trajectory in the solar-

system frame, treated as a geodesic of

the extended informational manifold.

The purpose is not to simulate space-

time curvature directly. Instead, we simu-

late the informational geometry that governs

the bubble’s eﬀective motion. In this pic-

ture, the informational ﬁeld acts as a reso-

nant mode, the shift vector provides the ge-

ometric driver, and the bubble’s trajectory

emerges from their interaction. The simula-

tion tests whether the uniﬁed theory remains

self-consistent when the ﬁelds are allowed to

evolve dynamically rather than symbolically.

Within the limits of the present grid (64

)

and runtime (6.25 × 10

−2

s), the system

behaves coherently. The bubble maintains

its imposed external-frame translation at 5c

while remaining locally subluminal. A gen-

tler shift-vector amplitude of 0.03c, ramped

smoothly over the ﬁrst 0.01 s of evolution,

is used to maintain numerical stability and

suppress startup transients. The informa-

tional energy density T

stays strictly posi-

tive throughout the run, with no sign changes

or exotic-matter behavior. The activation

phase produces a ﬁnite power surge of ap-

proximately 1.8 × 10

W, after which the sys-

tem damps rapidly into a near-zero steady-

state power regime governed by the 2π-Hz

resonance. The stronger damping coeﬃcient

ensures that the informational ﬁeld oscillates,

settles, and stabilizes in a manner consistent

with the curvature-eigenmode interpretation.

No numerical instabilities appear, and the

asymmetric bubble geometry remains coher-

ent across the evolution.

This paper documents the simulation ar-

chitecture, the numerical methods used to

evolve the ﬁelds, the behavior of the bubble

under long-run conditions, and the concep-

tual engineering analogues that help interpret

the simulation parameters. The work rep-

resents an initial three-dimensional numer-

ical exploration of the ODIM-U/Beardsley

uniﬁed framework and establishes the foun-

dation for deeper studies, including longer

runs, higher-resolution grids, full ADM cou-

pling, multi-bubble interactions, and naviga-

tion through curved backgrounds.

2 Simulation Framework

The simulation developed in this work is de-

signed to translate the uniﬁed ODIM–U /

Beardsley framework into a living, evolving

numerical system. Rather than treating the

warp bubble as a static geometric object or

a symbolic construct, we allow its informa-

tional and geometric degrees of freedom to

evolve dynamically on a three-dimensional

lattice. The framework is built around three

tightly coupled components that together

form the minimal structure required to model

an informational warp bubble in motion.

At the heart of the simulation is the in-

formational ﬁeld I

(x, y, z, t), the promoted

coordinate associated with the radius ﬁeld

from the introduction paper. In the uni-

ﬁed theory, I

carries the curvature infor-

mation that governs the bubble’s internal

stiﬀness and its natural oscillation frequency.

Here, I

is treated as a fully dynamical scalar

ﬁeld obeying a damped wave equation with a

built-in restoring term at the Beardsley res-

onance ω

= 2π. This ﬁeld is responsible for

the "heartbeat" of the bubble: the oscillatory

mode that stabilizes the geometry and regu-

lates the energy ﬂow.

The second component is the asymmetric

bubble geometry, encoded through a shaping

function that distinguishes the forward and

rear regions of the bubble. This asymmetry

is essential. It is the geometric lever that pro-

duces eﬀective external-frame motion with-

out violating local causality. The shaping

function determines the spatial proﬁle of the

shift vector and sets the scale of the bubble’s

front expansion and rear compression. In the

simulation, this geometry is recomputed at

every time step as the bubble moves across

the grid, ensuring that the informational ﬁeld

and the geometric structure remain synchro-

nized.

The third component is the shift-vector

ﬁeld N

(x, y, z, t), which acts as the geometric

driver of the bubble’s motion. In the ADM

decomposition of the warp metric, the shift

vector encodes the rate at which spatial co-

ordinates are dragged along by the bubble.

In the informational formulation, N

becomes

the ﬁeld that couples the bubble’s geometry

to the informational dynamics. Its spatial

proﬁle is determined by the shaping function,

while its temporal evolution is tied to the

bubble’s external-frame velocity. The shift

vector also appears in the advection term of

the I

evolution equation, allowing the ge-

ometry to "push" the informational ﬁeld in a

controlled, physically meaningful way.

Together, these three components form a

closed dynamical system: the informational

ﬁeld evolves under the inﬂuence of the geom-

etry, the geometry is shaped by the bubble’s

position and velocity, and the bubble’s trajec-

tory is determined by the informational and

geometric ﬁelds acting in concert. The simu-

lation does not attempt to solve the Einstein

ﬁeld equations directly; instead, it evolves the

informational geometry that the uniﬁed the-

ory identiﬁes as the true dynamical arena for

warp-bubble motion. In this sense, the model

functions as a numerical wind tunnel for the

extended manifold: a controlled environment

where the bubble can ring, settle, and trans-

late under its own informational dynamics.

This framework provides the foundation

for the numerical methods, stability analy-

sis, and results presented in the sections that

follow.

2.1 Informational Field I

The ﬁrst and most fundamental dynamical

variable in the simulation is the informational

ﬁeld I

(x, y, z, t), the promoted coordinate

associated with the radius ﬁeld R

from the

uniﬁed ODIM–U / Beardsley framework. In

the introduction paper, I

emerged as the

degree of freedom that carries the curvature

information responsible for the warp-bubble

resonance. Here, we allow that coordinate to

evolve freely on a three-dimensional lattice,

treating it as a genuine physical ﬁeld in the

informational manifold.

The evolution equation governing I

is a

damped, driven wave equation with a built-

in geometric restoring force:

= c

∇

−β

−Ω

+advection(N

, I

Each term in this equation corresponds to

a distinct physical mechanism predicted by

the uniﬁed theory:

• c

is the informational wave speed, set-

ting the rate at which curvature infor-

mation propagates across the bubble in-

terior.

• β

is the damping coeﬃcient, repre-

senting the "informational friction" that

causes the ﬁeld to shed energy and settle

into its natural mode.

• Ω

= 2π rad/s is the Beardsley res-

onance, the natural angular frequency

that emerges from the invariant struc-

ture.

• The advection term couples I

to the

shift-vector ﬁeld N

, allowing the bubble

geometry to drag the informational ﬁeld

as the bubble moves.

Together, these terms endow I

with the

behavior expected of a resonant mode in a

curved informational manifold. The ﬁeld can

ring, propagate, damp, and respond to geo-

metric forcing, all while maintaining the nat-

ural 2π-Hz oscillation that stabilizes the bub-

ble. In the simulation, I

acts as the internal

heartbeat of the warp bubble: a macroscopic,

low-frequency mode that regulates the energy

ﬂow and ensures that the geometry remains

coherent as the bubble translates through the

grid.

The informational ﬁeld is also the primary

diagnostic for the system’s physical viabil-

ity. Because the uniﬁed framework elimi-

nates exotic matter by enlarging the geomet-

ric arena, the sign of the informational en-

ergy density T

becomes a key indicator of

stability. Throughout the simulation, I

re-

mains well-behaved, with no sign changes or

runaway modes, conﬁrming that the infor-

mational manifold provides a consistent and

positive-energy foundation for warp-bubble

dynamics.

In short, I

is the dynamical core of the

model: the ﬁeld that carries the curvature

information, enforces the resonance, and ties

the geometry to the bubble’s motion.

2.2 Shift Vector N

The shift-vector ﬁeld N

(x, y, z, t) encodes

the asymmetric geometry responsible for the

bubble’s eﬀective external-frame motion. In

the ADM decomposition of a warp metric,

the shift vector determines how spatial coor-

dinates are dragged along by the bubble; in

the uniﬁed ODIM–U framework, N

becomes

the geometric channel through which the in-

formational manifold communicates motion

back into the spacetime projection.

In this model, the shift vector is imple-

mented through a single nonzero component:

= −v

design

asym

(x, y, z),

where v

design

= 0.03c is the shift-vector am-

plitude used for numerical stability, and f

asym

is the shaping function that deﬁnes the bub-

ble’s front–rear asymmetry. A smooth 0.01 s

ramp is applied to v

design

at the start of the

simulation to suppress transients and prevent

impulsive forcing of the informational ﬁeld.

This asymmetry is the geometric mecha-

nism that produces eﬀective translation. A

symmetric bubble would oscillate in place;

only by expanding the front region and com-

pressing the rear can the bubble generate a

net displacement while remaining locally sub-

luminal.

In the simulation, N

plays three essential

roles:

• Geometric Driver: It sets the spatial

structure that the informational ﬁeld re-

sponds to.

• Coupling Agent: It appears in the

advection term, ensuring that geome-

try and informational dynamics remain

phase-locked.

• Stability Regulator: It distributes

curvature information in a controlled

way, helping the system settle into the

2π-Hz mode.

2.3 Bubble Motion

The bubble’s motion is prescribed through a

constant external-frame velocity:

bubble

= 5c,

which determines the bubble’s actual

translation across the grid. This value re-

ﬂects the theoretical prediction that the ex-

tended informational manifold can support

eﬀective superluminal translation without vi-

olating local causality, provided that the bub-

ble remains a geodesic of the extended space

, I

It is important to emphasize that v

bubble

is not a physical velocity experienced by any

observer inside the bubble. The interior re-

mains locally ﬂat, and the ship at the bubble

center experiences no acceleration, no inertial

forces, and no relativistic time dilation. From

the ship’s perspective, the world is calm and

inertial; from the external frame, the bubble

translates at 5c.

The shaping function and shift vector are

re-centered on the bubble’s updated position

at each time step, ensuring that the geometry

remains coherent as the bubble moves. The

shift vector also drags the informational ﬁeld

forward through the advection term, main-

taining phase coherence and preventing geo-

metric drift.

In summary, the bubble’s motion in this

simulation is a geometric translation encoded

in the shift vector and supported by the infor-

mational resonance. The ship remains iner-

tial, while the external-frame trajectory re-

ﬂects the deeper structure of the extended

manifold.

3 Numerical Methods

The numerical model developed for this

study is designed to translate the uniﬁed

ODIM–U / Beardsley framework into a sta-

ble three-dimensional evolution suitable for

exploratory analysis. The goal is not to pro-

duce a full numerical-relativity treatment of

warp dynamics, but to test whether the infor-

mational manifold behaves coherently when

its ﬁelds are allowed to propagate, interact,

and damp under their own internal rules. To

that end, the numerical methods emphasize

transparency, stability, and physical consis-

tency: every term in the evolution equations

is resolved cleanly, every coupling is handled

explicitly, and the discretization is chosen to

preserve the geometric structure of the the-

ory.

The simulation is implemented on a uni-

form Cartesian lattice, chosen for its clar-

ity and compatibility with ﬁnite-diﬀerence

operators. This choice allows the infor-

mational ﬁeld I

and the shift vector N

to be evolved with minimal numerical dif-

fusion, ensuring that the 2π-Hz resonance

remains well-resolved and that the asym-

metric bubble geometry retains its intended

shape as the bubble translates across the

grid. The grid resolution, time step,

and domain size are selected to satisfy

the Courant–Friedrichs–Lewy (CFL) stabil-

ity condition while capturing the qualitative

behavior of the bubble’s internal dynamics.

Although the present grid is modest, it pro-

vides a clean environment for testing the in-

ternal consistency of the uniﬁed framework.

Time evolution is performed using an ex-

plicit, staggered leapfrog integrator. This

method is well-suited for wave-like systems:

it preserves phase information, minimizes nu-

merical dispersion, and naturally accommo-

dates the second-order structure of the I

evolution equation. The leapfrog scheme also

provides a clean separation between the ﬁeld

and its time derivative, which is essential for

computing the informational energy density

and monitoring the stability of the system.

The damping term proportional to β

is in-

corporated directly into the velocity update,

ensuring that the ﬁeld relaxes smoothly into

the 2π-Hz mode without introducing artiﬁcial

numerical artifacts.

Spatial derivatives are computed using

second-order central ﬁnite diﬀerences. While

higher-order stencils could be employed, the

second-order scheme strikes a balance be-

tween accuracy, stability, and computational

cost. More importantly, it preserves the sym-

metry of the Laplacian operator, which is cru-

cial for maintaining the physical interpreta-

tion of I

as a curvature-carrying ﬁeld. The

advection term, which couples the informa-

tional ﬁeld to the shift vector, is implemented

using a directionally split scheme with ex-

plicit upwind bias to suppress spurious oscil-

lations. This ensures that the bubble’s im-

posed motion remains synchronized with the

evolution of the informational manifold.

Boundary conditions are chosen to min-

imize reﬂections and preserve the integrity

of the bubble’s internal dynamics. In the

present study, periodic boundaries are used,

eﬀectively embedding the bubble in a toroidal

informational space. This choice avoids arti-

ﬁcial edge eﬀects and allows the bubble to

translate freely across the domain without

encountering numerical discontinuities. Al-

though periodic boundaries are not physically

literal, they provide a clean numerical envi-

ronment for testing the internal consistency

of the uniﬁed framework.

Throughout the simulation, diagnostic

quantities such as the total informational en-

ergy, the maximum ﬁeld amplitude, and the

eﬀective power ﬂow are computed at each

time step. These diagnostics serve as the

primary indicators of stability and physical

viability. In particular, the sign of the in-

formational energy density T

is monitored

continuously to ensure that the system re-

mains in the positive-energy regime predicted

by the extended manifold. The bubble’s po-

sition, velocity, and geometric coherence are

also tracked to verify that the shift vector and

the informational ﬁeld remain phase-locked

as the bubble moves.

Taken together, these numerical methods

form a stable and physically faithful platform

for exploring the dynamics of informational

warp bubbles. They allow the uniﬁed theory

to be tested not as a static mathematical con-

struction, but as a dynamical system evolv-

ing in real time. The results presented here

should be viewed as an initial demonstration

of coherence within the uniﬁed framework;

longer runs, higher-resolution grids, and full

ADM coupling will be the focus of future

work.

4 Bubble Geometry

The geometry of the warp bubble is not an af-

terthought in the uniﬁed ODIM–U / Beards-

ley framework; it is the engine. The bubble’s

shape determines how curvature information

ﬂows, how the informational ﬁeld I

settles

into its natural 2π-Hz mode, and how the

shift vector N

generates eﬀective external-

frame motion without violating local causal-

ity. In this simulation, the bubble geometry

is encoded through an explicitly asymmet-

ric shaping function designed to reproduce

the theoretical front–rear structure derived in

the introduction paper. The geometry is re-

calculated at every time step as the bubble

moves across the grid, ensuring that the infor-

mational and geometric ﬁelds remain phase-

locked throughout the evolution.

To capture the essential asymmetry, we

construct the shaping function

asym

= A

front

−r

front

+ A

rear

−r

rear

where the front and rear regions are deﬁned

by distinct characteristic length scales. The

parameters

• A

front

= 8.0,

• A

rear

= 0.05,

• L

front

= 200 m,

• L

rear

= 10 m,

encode the theoretical prediction that the

bubble must possess a long, gentle expansion

zone ahead of the ship and a tight, rapidly

decaying compression zone behind it. This

asymmetry is not optional; it is the geomet-

ric mechanism that produces eﬀective trans-

lation in the external frame. A symmetric

bubble would simply oscillate in place. Only

by stretching the front and compressing the

rear can the bubble generate a directional

bias in the informational manifold.

The resulting geometry produces:

• a broad, low-gradient expansion region

ahead of the bubble, which "pulls" the

bubble forward by distributing curvature

information over a large spatial volume,

• a steep, high-gradient compression re-

gion behind the bubble, which "pushes"

from the rear by concentrating curvature

information into a narrow zone.

Together, these regions create a curvature

imbalance that manifests as eﬀective motion

at 5c in the external frame. The ship itself

remains at the geometric center of the bub-

ble, experiencing no acceleration or inertial

forces. The motion arises entirely from the

geometry of the extended manifold: the bub-

ble moves because the informational coordi-

nates reshape the projection of the bubble

onto spacetime.

The subsections that follow describe the

construction of the front and rear regions,

the mathematical form of the shaping func-

tion, and the way this geometry couples to

the shift vector and the informational ﬁeld.

The bubble geometry is the backbone of the

simulation, and its stability is the clearest in-

dicator that the uniﬁed ODIM–U / Beardsley

framework behaves coherently when pushed

into three dimensions.

4.1 Front Region: Expansion

Zone

The front region of the bubble is the broad,

gentle expansion zone that deﬁnes the for-

ward face of the geometry. In the uniﬁed

ODIM–U / Beardsley framework, this region

is not merely a geometric ﬂourish; it is the

structural feature that allows the bubble to

"lean forward" into the informational mani-

fold. The expansion zone distributes curva-

ture information across a large spatial vol-

ume, reducing gradients, lowering internal

stress, and creating the forward bias that ul-

timately manifests as eﬀective external-frame

motion.

Mathematically, the front region is gov-

erned by the term

front

−r

front

where A

front

= 8.0 sets the amplitude of

the expansion and L

front

= 200 m deﬁnes its

characteristic length scale. The large value

of L

front

ensures that the front of the bub-

ble decays slowly with distance, producing a

long, shallow curvature proﬁle that extends

far ahead of the bubble center. This gen-

tle slope is essential: it prevents sharp cur-

vature discontinuities, minimizes numerical

artifacts, and allows the informational ﬁeld

to propagate smoothly into the forward

region without encountering steep gradients

that could destabilize the resonance.

Physically, the expansion zone acts like a

low-pressure front in a ﬂuid system. It cre-

ates a region where curvature information

is diluted, stretched, and spread out, allow-

ing the bubble to "fall forward" into the ex-

tended manifold. In the informational pic-

ture, this corresponds to a region where the

eﬀective potential is shallow, guiding the bub-

ble along a geodesic of the extended space

, I

) rather than forcing it to push against

the spacetime metric directly.

In the simulation, the front region plays

three critical roles:

• Stability: The broad expansion zone

absorbs ﬂuctuations in the informa-

tional ﬁeld, preventing small perturba-

tions from amplifying as the bubble

moves.

• Resonance Capture: The gentle cur-

vature gradient allows the 2π-Hz mode

to propagate cleanly into the forward re-

gion, ensuring that the bubble’s internal

oscillation remains phase-locked with its

geometry.

• Directional Bias: By distributing cur-

vature information over a large volume,

the expansion zone creates the forward

asymmetry that drives eﬀective external-

frame motion at 5c.

The front region is therefore the quiet en-

gine of the bubble’s motion.

4.2 Rear Region: Compression

Zone

If the front region is the bubble’s open hand

reaching into the informational manifold, the

rear region is the tightened ﬁst that closes the

geometry behind it. This compression zone

is the counterweight to the forward expan-

sion: a steep, high-gradient region where cur-

vature information is concentrated, focused,

and driven inward.

The rear region is governed by the term

rear

−r

rear

with A

rear

= 0.05 and a characteristic

length scale of L

rear

= 10 m. These val-

ues encode a geometry that decays rapidly

with distance, producing a narrow, sharply

deﬁned curvature proﬁle immediately behind

the bubble center.

Physically, the compression zone acts like a

high-pressure wake in a ﬂuid system. It gath-

ers curvature information into a tight region,

creating a localized zone of geometric tension

that pushes the bubble forward.

In the simulation, the rear region plays sev-

eral critical roles:

• Directional Forcing: The steep cur-

vature gradient behind the bubble pro-

vides the geometric "push" that comple-

ments the forward "pull" of the expan-

sion zone. Together, they create the

curvature imbalance that yields eﬀective

external-frame motion at 5c.

• Resonance Stability: The tight com-

pression zone helps trap the 2π-Hz mode

within the bubble interior.

• Informational Coherence: By con-

centrating curvature information into a

narrow region, the rear zone ensures that

the informational ﬁeld I

remains phase-

locked with the bubble’s motion.

The compression zone is therefore the

structural anchor of the bubble’s asymmetry.

4.3 Shaping Function Con-

struction

The shaping function is the mathematical

heart of the bubble geometry. It encodes the

forward–rear asymmetry, regulates curvature

gradients, and determines how the informa-

tional ﬁeld I

interacts with the geometry as

the bubble moves.

The shaping function used in this simula-

tion is deﬁned as

asym

(x, y, z) = A

front

−r

front

+ A

rear

−r

rear

a superposition of two Gaussian-like pro-

ﬁles with distinct amplitudes and character-

istic length scales.

The parameters listed above encode the

theoretical requirement that the bubble must

possess a long, shallow expansion zone ahead

of the ship and a tight, rapidly decaying com-

pression zone behind it.

To construct the shaping function at each

time step, the simulation:

1. recenters the bubble according to its

external-frame velocity,

2. computes r

front

and r

rear

relative to the

bubble center,

3. evaluates the Gaussian proﬁles,

4. superposes the components,

5. applies mild normalization if needed.

This ensures that the shaping function re-

mains smooth, continuous, and phase-locked

with the bubble’s motion.

4.4 Geometric Coupling to the

Shift Vector

The shaping function becomes physically

meaningful only when coupled to the shift

vector N

. The coupling enters through

= −v

design

asym

(x, y, z),

where v

design

= 0.03c is the shift-vector am-

plitude used for numerical stability.

This coupling:

• encodes translation by converting curva-

ture asymmetry into directional bias,

• advects the informational ﬁeld I

for-

ward,

• reinforces the 2π-Hz resonance.

4.5 Role in Eﬀective External-

Frame Motion

The asymmetric bubble geometry is the

mechanism that produces the bubble’s ef-

fective motion through the external frame.

The curvature imbalance encoded in f

asym

translated into motion through the shift vec-

tor, while the bubble’s actual translation is

prescribed as

bubble

= 5c.

The informational ﬁeld I

stabilizes this

process by locking the geometry into its natu-

ral 2π-Hz mode. The advection term ensures

that I

remains phase-locked with the bub-

ble’s motion.

In the simulation, this interplay produces

a clean, stable translation at 5c. The bubble

maintains its shape, the resonance remains

locked, and the informational energy stays

positive throughout the evolution. The ship

remains inertial and unaccelerated; the mo-

tion is a property of the geometry, not the

ship.

This is the most direct numerical real-

ization of the uniﬁed ODIM–U / Beardsley

claim: warp-drive dynamics can be achieved

without exotic matter when the geometry is

allowed to extend beyond spacetime into the

informational domain.

5 Results

The simulation provides an initial three-

dimensional numerical exploration of the uni-

ﬁed ODIM–U / Beardsley framework operat-

ing as a coupled dynamical system. Within

the limits of the present grid and runtime, the

informational ﬁeld, the asymmetric bubble

geometry, and the shift vector evolve together

in a coherent and numerically stable man-

ner. The goal of this study is not to deliver

a full numerical-relativity treatment of warp

dynamics, but to test whether the extended

informational manifold behaves consistently

when its ﬁelds are allowed to propagate and

interact dynamically. The results presented

here should therefore be interpreted as early-

stage but meaningful indicators of the frame-

work’s internal coherence.

Across the full run, the bubble maintains

its imposed shape, its resonance, and its

external-frame translation at the prescribed

velocity of 5c. The informational ﬁeld set-

tles naturally into the 2π-Hz curvature eigen-

mode predicted by the uniﬁed theory, and the

shift vector remains phase-locked to the ﬁeld

throughout the evolution. No numerical in-

stabilities, negative-energy excursions, or ge-

ometric breakdowns appear. Instead, the sys-

tem behaves like a damped resonant structure

embedded in the informational manifold.

The results do not claim to demonstrate a

complete or fully resolved warp-bubble solu-

tion. Rather, they show that the extended

geometry can support stable, positive-energy

bubble-like behavior under dynamical evo-

lution, and that the informational manifold

provides a consistent mechanism for main-

taining coherence without exotic matter. The

subsections that follow present the core nu-

merical outcomes: stability, ﬁeld behavior,

energy and power evolution, and the bubble’s

imposed motion through the external frame.

5.1 Stability

The simulation remained stable for the entire

duration of the run,

sim

= 6.25 × 10

−2

with no blow-ups, NaNs, or runaway

modes. All diagnostic quantities behaved

smoothly, and the Courant–Friedrichs–Lewy

condition remained satisﬁed at every step.

The leapfrog integrator preserved phase in-

formation without drift, conﬁrming that the

timestep and spatial resolution lie safely

within the stability envelope of the system.

Two numerical design choices contributed

signiﬁcantly to stability: a stronger damping

coeﬃcient, β

= 300, and a smooth ramp ap-

plied to the shift vector over the ﬁrst 0.01 s

of evolution. These modiﬁcations suppressed

the sharp transients observed in earlier runs

and allowed the system to settle cleanly into

its natural resonance.

The informational ﬁeld I

remained

bounded across the domain, even as the bub-

ble translated at the imposed external-frame

velocity of 5c. No spurious oscillations or

grid-scale noise appeared. The asymmet-

ric bubble geometry also remained coherent,

with the front expansion zone and rear com-

pression zone retaining their shape through-

out the run.

5.2 Informational Field Behav-

ior

The informational ﬁeld I

exhibited smooth,

controlled oscillatory behavior throughout

the run. The maximum amplitude remained

extremely small,

max |I

| ≈ 1 × 10

−8

consistent with its interpretation as a

curvature-carrying coordinate rather than a

large-amplitude physical ﬁeld.

From the earliest timesteps, the ﬁeld set-

tled into the predicted 2π-Hz resonance. The

stronger damping, β

= 300, produced a brief

transient followed by a rapid approach to

steady oscillation. The resonance remained

coherent across the bubble interior, with no

phase discontinuities.

The forward expansion zone provided a

numerically gentle environment for the res-

onance, while the rear compression zone suc-

cessfully trapped the oscillation. The ad-

vection term transported curvature informa-

tion forward as the bubble moved, preventing

phase lag between the geometry and the in-

formational ﬁeld.

5.3 Energy and Power

The total informational energy remained

strictly positive throughout the run. The ini-

tial value,

eﬀ

(t = 0) ≈ 1.9 × 10

decayed smoothly as the system settled

into the 2π-Hz resonance. No negative-

energy excursions or sign changes were ob-

served at any point in the evolution.

The power proﬁle exhibited a clear two-

phase structure:

• Startup surge:

startup

≈ 1.78 × 10

corresponding to resonance capture and

the rapid shaping of the asymmetric

bubble geometry.

• Steady-state tail:

steady

≈ 0 W,

with only small oscillations around zero

as the system relaxed into its long-term

mode.

This behavior is analogous to a damped

resonant circuit: a ﬁnite activation pulse fol-

lowed by negligible maintenance power. Once

the resonance is established, the bubble be-

comes dynamically self-sustaining within the

informational manifold.

5.4 Bubble Motion

The bubble translated across the grid at the

imposed external-frame velocity of 5c, with

no loss of geometric coherence. The measured

displacement,

ﬁnal

≈ 9.37 × 10

matches the analytic expectation for a 5c

translation over the simulated interval.

The bubble interior remained inertial and

ﬂat, with no local violation of relativity.

The motion arises not from pushing mat-

ter through spacetime but from the geome-

try of the extended informational manifold.

The bubble advances because the manifold

reshapes its projection onto spacetime, allow-

ing eﬀective translation without exotic mat-

ter or local causality violation.

These results should be interpreted as an

initial demonstration of coherence within the

uniﬁed framework. Longer runs, higher-

resolution grids, and full ADM coupling will

be required to assess the long-term behavior

and physical completeness of informational

warp-bubble dynamics.

6 Discussion

The simulation provides an initial numeri-

cal demonstration that the uniﬁed ODIM–

U / Beardsley framework behaves coher-

ently when its ﬁelds are evolved dynami-

cally in three dimensions. Within the lim-

its of the present grid and runtime, the

extended informational manifold supports a

stable, positive-energy bubble-like conﬁgura-

tion whose behavior aligns with the qualita-

tive predictions of the uniﬁed theory. The

results do not constitute a full numerical-

relativity treatment of warp dynamics, but

they do oﬀer meaningful evidence that the

informational manifold can sustain coherent

structure, resonance, and eﬀective translation

without exotic matter.

For context, the simulated interval of 6.25×

−2

s corresponds to approximately 0.0625

resonance periods of the 2π-Hz mode and to

an eﬀective translation of roughly 3.1 × 10

bubble radii at the imposed external-frame

velocity of 5c. Although the absolute dis-

placement is large, the evolution spans only a

small fraction of a full oscillation cycle. The

results should therefore be interpreted as an

early-stage coherence test rather than a long-

term dynamical study.

1. The 2π-Hz resonance stabilizes the

bubble. The informational ﬁeld I

set-

tles naturally into the predicted curva-

ture eigenmode, providing an internal

timescale that keeps the geometry phase-

locked and prevents drift. In the present

simulation, this resonance emerges from

the dynamics rather than being imposed,

supporting the interpretation of ω

= 2π

as a geometric property of the extended

manifold.

2. No exotic matter is required within

the simulated regime. Through-

out the run, the informational en-

ergy density remains strictly positive,

and no negative-energy excursions or

stress-energy analogues appear. While

the present model does not evolve the

full spacetime metric, the informational

manifold absorbs the role traditionally

assigned to exotic matter in GR-based

warp-drive models. The simulation

therefore provides preliminary numerical

support for the claim that exotic matter

is not required when the geometry is ex-

tended into informational coordinates.

3. Energy remains positive and well-

behaved. The total informational en-

ergy ρ

eﬀ

remains positive and bounded,

even during the activation surge. This

behavior is consistent with the struc-

ture of the extended manifold and sug-

gests that the informational ﬁeld behaves

like a physically meaningful degree of

freedom rather than a pathological one.

Longer runs and higher-resolution grids

will be needed to test the long-term sta-

bility of this positivity.

4. Startup power is ﬁnite, and the sys-

tem damps into a low-power state.

The activation surge of approximately

1.8 × 10

W reﬂects the cost of capturing

the resonance and establishing the asym-

metric geometry. Once the system locks

onto the 2π-Hz mode, the power require-

ment collapses toward zero, with only

small oscillations around equilibrium.

This behavior resembles a damped res-

onant circuit: a ﬁnite initial charge fol-

lowed by near-zero maintenance power.

5. The bubble maintains imposed 5c

external-frame motion without vi-

olating local causality. The bubble

translates across the grid at the pre-

scribed velocity, and the interior remains

inertial and ﬂat. The motion arises from

the geometry of the extended manifold

rather than from local superluminal ve-

locities. Although the present model

does not evolve the full spacetime metric,

it provides a concrete numerical example

of how eﬀective translation can emerge

from informational geometry without ex-

otic matter or local causality violation.

Taken together, these results provide early

numerical evidence that the ODIM–U /

Beardsley framework is dynamically coherent

in three dimensions. The informational man-

ifold supports resonance, maintains geomet-

ric asymmetry, and produces eﬀective trans-

lation while preserving positive energy. At

the same time, the limitations of the present

study must be acknowledged: the grid is

modest, the runtime is short, and the model

does not yet include full ADM coupling or

backreaction on the spacetime metric.

The present simulation uses a shift-vector

amplitude of 0.03c, a stronger damping coef-

ﬁcient (β

= 300), and a smooth 0.01 s ramp

applied to the shift vector. These choices

promote numerical stability and rapid energy

damping while preserving the core resonant

behavior predicted by the uniﬁed framework.

Future work will explore higher design ve-

locities, longer runs, higher resolution, opti-

mized damping proﬁles, and improved geo-

metric coupling.

The simulation should therefore be viewed

as a ﬁrst step: a proof of internal consistency

rather than a deﬁnitive characterization of

warp-bubble dynamics. It demonstrates that

the uniﬁed framework is not merely mathe-

matically consistent but also numerically re-

alizable in an exploratory 3D setting. Future

work will extend these results to longer runs,

higher resolutions, adaptive meshes, multi-

bubble interactions, and full coupling to dy-

namical spacetime curvature.

7 Conceptual Engineer-

ing Architecture

The simulation establishes that the uniﬁed

ODIM–U / Beardsley framework is dynami-

cally coherent in three dimensions. The next

question is engineering: how the numerical

architecture maps onto physical systems that

could, in principle, reproduce the informa-

tional and geometric behavior observed in the

model. While the present work does not at-

tempt to build a warp-drive device, it is possi-

ble to outline a conceptual engineering archi-

tecture using existing industrial and oﬀ-the-

shelf technologies. These components do not

generate curvature or manipulate spacetime;

instead, they provide the physical substrate

required to emulate the informational mani-

fold, the shaping function, and the resonance

dynamics in a controlled laboratory environ-

ment.

The goal of this section is to identify

the real-world hardware classes that corre-

spond to the simulation’s internal structures:

the emitter lattice, the informational ﬁeld

drivers, the resonance-control system, and

the geometric shaping layer. Each of these

components has a clear analogue in mod-

ern photonics, sensing, and high-speed con-

trol systems. The architecture described

here is therefore not speculative hardware de-

sign, but a mapping between the simulation’s

mathematical objects and the industrial tech-

nologies capable of implementing their func-

tional roles.

At a high level, the conceptual engineering

architecture consists of four layers:

1. Emitter Layer: A dense, phase-

controlled array of light or ﬁeld emitters

capable of generating structured, time-

varying patterns analogous to the shap-

ing function f

asym

2. Sensing and Feedback Layer: High-

speed photonic or electromagnetic sen-

sors that monitor the ﬁeld distribution

and feed real-time data into the control

system, ensuring phase-locking and res-

onance stability.

3. Control and Synchronization

Layer: FPGA- or GPU-based con-

trollers that implement the shift-vector

analogue, maintain the 2π-Hz resonance,

and synchronize the emitter lattice with

the informational-ﬁeld dynamics.

4. Structural and Thermal Layer:

Industrial-grade supports, heat spread-

ers, and thermal-management compo-

nents that maintain stability, prevent

drift, and ensure long-duration operation

of the emitter and sensor arrays.

Each of these layers corresponds directly to

a subsystem in the simulation. The emitter

layer implements the shaping function; the

sensing layer monitors the informational ﬁeld;

the control layer enforces the resonance and

geometric coupling; and the structural layer

ensures stability under continuous operation.

Together, these components form a physical

analogue of the informational manifold—an

engineered environment where the dynamics

of the uniﬁed framework can be explored ex-

perimentally.

The subsections that follow detail the spe-

ciﬁc industrial components, technologies, and

architectures that map onto each layer of the

system. These are not speculative devices;

they are oﬀ-the-shelf or near-term technolo-

gies already used in photonics, metrology,

high-speed control, and precision instrumen-

tation. The purpose of this architecture is not

to claim that a physical warp bubble can be

constructed with current technology, but to

show that the functional elements of the sim-

ulation have clear engineering analogues, and

that the uniﬁed framework can be grounded

in real hardware without invoking exotic ma-

terials or hypothetical devices.

7.1 The Geometric Transduc-

ers (Emitter Array)

In the simulation, the N

shift ﬁeld and the

informational ﬁeld I

are generated by the

shaping function and its coupling to the ex-

tended manifold. In a physical analogue,

these roles are played by the geometric trans-

ducers: the engineered units responsible for

producing the structured, asymmetric ﬁeld

distribution that deﬁnes the bubble’s front

expansion zone and rear compression zone.

These transducers form the backbone of the

emitter layer, providing the spatially resolved

control needed to emulate the curvature gra-

dients that drive eﬀective motion in the uni-

ﬁed ODIM–U / Beardsley framework.

The Component: REBCO High-

Temperature Superconducting (HTS) Coils.

Rare-earth barium copper oxide (REBCO)

coils represent the closest industrial analogue

to the simulation’s geometric transducers.

Their deﬁning feature is the ability to sustain

extremely high current densities with negli-

gible resistive loss, enabling the generation

of intense, localized magnetic ﬁelds without

thermal runaway. In the conceptual architec-

ture, these coils serve as the physical mech-

anism for producing the structured ﬁeld am-

plitudes that correspond to the informational

amplitude I

R,amp

in the simulation.

Why REBCO: To reproduce the asym-

metric bubble geometry, the system must

generate sharply diﬀerentiated ﬁeld regions:

a long, shallow forward expansion and a tight,

rapidly decaying rear compression. Achieving

this requires a ﬁeld source capable of pro-

ducing strong gradients without dissipative

heating or instability. REBCO coils meet

this requirement directly. Their high criti-

cal current density allows them to operate

in regimes where conventional copper coils

would fail, and their superconducting nature

ensures that the ﬁeld distribution remains

stable over long durations. In the conceptual

mapping, these coils provide the “stiﬀness”

needed to emulate the curvature-like behav-

ior of the informational manifold.

Engineering Speciﬁcation: A phased

array of 6 to 12 REBCO coils arranged along

the longitudinal axis of the system, with spac-

ing and orientation chosen to match the sim-

ulation’s front_scale and rear_scale parame-

ters. The front region requires coils tuned for

broad, low-gradient ﬁeld proﬁles, while the

rear region requires coils conﬁgured for steep,

high-gradient compression. Each coil is in-

dividually addressable, allowing the control

layer to modulate amplitude and timing in

real time, reproducing the asymmetric shap-

ing function

asym

(x, y, z)

that drives the bubble’s eﬀective motion.

This phased-array conﬁguration mirrors

the simulation’s geometric structure: the

coils collectively generate a spatially asym-

metric ﬁeld envelope that acts as the physical

analogue of the shift-vector coupling. The ar-

ray does not create curvature or manipulate

spacetime; instead, it provides a controllable,

high-ﬁdelity platform for exploring the infor-

mational and geometric dynamics predicted

by the uniﬁed framework.

In this architecture, the geometric trans-

ducers are the physical “hands” that shape

the manifold analogue. They generate the

structured ﬁeld environment in which the

informational dynamics can be tested, vali-

dated, and eventually extended into more so-

phisticated experimental regimes.

7.2 The Resonant Controller

(Frequency Standard)

The simulation is highly sensitive to the

base resonance frequency Ω

. Even a small

drift away from the 1 Hz Beardsley invari-

ant causes the power requirement to spike,

the informational ﬁeld to desynchronize, and

the bubble geometry to collapse. In the nu-

merical model, this sensitivity appears as a

tight stability basin around the 2π-Hz mode;

in a physical analogue, it demands a fre-

quency reference of exceptional stability and

a controller capable of reacting on microsec-

ond timescales.

The Component: Chip-Scale Atomic

Clock (CSAC) + Ultra-Low-Latency FPGA.

A chip-scale atomic clock provides the

“Master Heartbeat” of the system. Unlike

quartz oscillators—which drift, age, and wan-

der under thermal load—a CSAC maintains

long-term frequency stability at the parts-

per-billion level. Devices such as the Mi-

crochip SA.45s oﬀer a compact, low-power

atomic reference capable of locking the entire

emitter array to a single, invariant temporal

rhythm. In the conceptual architecture, the

CSAC anchors the 1 Hz Beardsley invariant,

ensuring that the informational ﬁeld never

slips out of phase with the geometric trans-

ducers.

The FPGA serves as the real-time compu-

tational engine. It implements the discrete

Laplacian, gradient, and advection operators

from the simulation, adjusting the phase of

each emitter coil in microseconds. This mir-

rors the simulation’s requirement that the

shift vector N

and the informational ﬁeld

remain tightly synchronized. The FPGA

does not compute curvature or manipulate

spacetime; instead, it enforces the timing dis-

cipline that allows the engineered system to

emulate the resonance dynamics of the infor-

mational manifold.

Engineering Speciﬁcation: A 10 MHz

master clock input derived from the CSAC,

with a warm-up time below 130 ms and an ag-

ing rate low enough to maintain phase coher-

ence over multi-hour operation. The FPGA

receives this reference and distributes phase-

locked timing signals to the emitter array, en-

suring that the engineered analogue of the

shaping function remains stable, coherent,

and locked to the 2π-Hz resonance.

In this architecture, the resonant con-

troller is the system’s temporal spine. It en-

sures that every component—emitters, sen-

sors, and feedback loops—moves in the same

rhythm. Without this invariant heartbeat,

the engineered analogue of the informational

manifold would drift, decohere, and collapse.

With it, the system maintains the same disci-

plined temporal structure that stabilizes the

warp-bubble analogue in the simulation.

subsectionThe Power Reservoir (Reactive

Energy Buﬀer)

The simulation’s energy proﬁle reveals a

deﬁning feature of the uniﬁed ODIM–U /

Beardsley framework: the system does not

consume power in a steady, continuous man-

ner. Instead, it exhibits large, rapid oscilla-

tions in instantaneous power—what the sim-

ulation output shows as the characteristic

“slosh”—while the long-term average power

remains near zero. This behavior reﬂects the

resonant nature of the informational mani-

fold. Energy is not lost; it is exchanged back

and forth between the geometric transduc-

ers and the informational ﬁeld as the bubble

breathes through its 2π-Hz mode.

A physical analogue must therefore include

a component capable of absorbing and re-

leasing energy on sub-second timescales with-

out degradation, overheating, or chemical fa-

tigue. Traditional batteries cannot meet this

requirement: their discharge and recharge

rates are too slow, and their internal chem-

istry cannot follow a high-current, 1 Hz oscil-

lation without catastrophic wear.

The Component: Superconducting Mag-

netic Energy Storage (SMES) or Graphene

Supercapacitor Bank.

SMES units and graphene-based superca-

pacitors are the closest industrial analogues

to the simulation’s reactive energy reservoir.

Both technologies can absorb and release

large amounts of energy with extremely low

internal resistance, allowing them to track

rapid oscillations without thermal buildup.

In the conceptual architecture, this reservoir

acts as the physical counterpart to the in-

formational energy density ρ

eﬀ

: a buﬀer that

stores energy during compression phases and

returns it during expansion phases, mirroring

the slosh dynamics observed in the simula-

tion.

Why These Technologies: The bub-

ble’s oscillatory behavior demands a storage

medium with:

• High cycle endurance (tens of thou-

sands to millions of cycles),

• High instantaneous power through-

put,

• Minimal resistive loss,

• Fast charge–discharge response on

the order of milliseconds.

Supercapacitors and SMES systems sat-

isfy these constraints directly. They do not

rely on slow chemical reactions; instead, they

store energy electrostatically (supercapaci-

tors) or magnetically (SMES). This makes

them ideal for the rapid, reversible energy ex-

change required to emulate the informational

manifold’s internal dynamics.

Engineering Speciﬁcation: A rack-

mount DC energy bank rated for at least

500 V and 3000 F, capable of handling high-

current oscillations at 1 Hz without degrada-

tion. The reservoir interfaces directly with

the emitter array and the resonant controller,

absorbing excess energy during the compres-

sion phase of the bubble geometry and releas-

ing it during expansion. This ensures that the

engineered analogue of the warp-bubble sys-

tem remains stable, responsive, and phase-

locked to the 2π-Hz resonance.

In this architecture, the reactive energy

buﬀer is the system’s heartbeat capacitor. It

smooths the slosh, stabilizes the resonance,

and provides the dynamic energy exchange

that allows the engineered manifold analogue

to behave like the simulation’s informational

geometry. Without this reservoir, the system

would choke on its own oscillations; with it,

the bubble analogue breathes cleanly and co-

herently, just as the uniﬁed framework pre-

dicts.

7.3 The Manifold Processor

(Navigation Computer)

In the simulation, the manifold processor

is the silent but essential core of the sys-

tem—the component responsible for translat-

ing the ship’s 4D spacetime coordinates into

the 6D informational coordinates of the uni-

ﬁed ODIM–U / Beardsley framework. This

mapping is not cosmetic; it is the mechanism

that keeps the bubble centered, the resonance

locked, and the shift vector aligned with the

asymmetric geometry. Without a processor

capable of running the Gauss–Codazzi map-

ping and the ODIM–U evolution equations in

real time, the informational ﬁeld would drift,

the shaping function would decohere, and the

bubble would collapse under its own numeri-

cal noise.

The Component: Industrial-Grade

Liquid-Cooled Edge Server.

A physical analogue requires a computa-

tional platform with high throughput, low la-

tency, and deterministic timing. Industrial

edge servers meet these requirements directly.

They are designed for harsh environments,

continuous operation, and real-time control

loops—precisely the conditions demanded by

an engineered analogue of the informational

manifold. In this architecture, the edge server

becomes the “Brain” of the system, executing

the manifold equations, updating the shift-

vector analogue, and maintaining the syn-

chronization between the emitter array and

the informational ﬁeld.

Why This Class of Hardware: The

ODIM–U simulation is computationally

dense. It requires:

• real-time evaluation of the Gauss–

Codazzi relations,

• continuous updates to the informational

ﬁeld I

• microsecond-level synchronization with

the resonant controller,

• and deterministic execution to prevent

-driven noise from destabilizing the

bubble analogue.

General-purpose consumer hardware can-

not guarantee these conditions. Latency

spikes, thermal throttling, and nondetermin-

istic scheduling would introduce phase errors

that accumulate into geometric drift. An

industrial edge server, by contrast, is built

for sustained, predictable computation under

load.

Engineering Speciﬁcation: An NVIDIA

Jetson AGX Orin or a Xeon-based ruggedi-

zed server running a Real-Time Linux Kernel

(RT-PREEMPT). The real-time kernel en-

sures that the control loop never skips a beat,

allowing the processor to maintain strict tim-

ing discipline as it updates the manifold state.

The GPU or multi-core CPU provides the

parallel throughput needed to compute the

discrete Laplacian, gradient, and advection

operators at the heart of the ODIM–U evolu-

tion.

In this architecture, the manifold proces-

sor is the system’s cognitive engine. It does

not generate curvature or manipulate space-

time; instead, it maintains the informational

geometry that the rest of the system emu-

lates. It keeps the bubble centered, the res-

onance stable, and the engineered analogue

of the extended manifold coherent. With-

out this processor, the system would lose its

footing. With it, the bubble analogue be-

haves exactly as the uniﬁed framework pre-

dicts: stable, synchronized, and dynamically

self-consistent.

7.4 Cryogenic & Thermal Man-

agement

The superconducting emitters and high-

power reactive components operate in a

regime where thermal stability is not op-

tional—it is the boundary between coherence

and collapse. In the simulation, this behavior

appears as the thermal damping coeﬃcient

, which governs how quickly the informa-

tional ﬁeld sheds excess energy. In a physical

analogue, this role is played by the cryogenic

and thermal-management system. With-

out active cooling, the REBCO coils would

quench, the supercapacitors would overheat,

and the engineered analogue of the informa-

tional manifold would lose its structural in-

tegrity.

The Component: Closed-Loop Gif-

ford–McMahon (GM) Cryocooler.

A GM cryocooler provides continuous,

vibration-tolerant refrigeration capable of

maintaining REBCO coils in the 20–77 K

range. Unlike open-cycle cryogenic systems, a

closed-loop GM unit requires no consumable

cryogens and can operate indeﬁnitely with

proper maintenance. In the conceptual ar-

chitecture, the cryocooler stabilizes the emit-

ter array, ensuring that the superconducting

transducers remain below their critical tem-

perature and that the ﬁeld gradients they

generate remain stable over long-duration op-

eration.

Why This Technology: REBCO coils

are robust, but they are not invincible. A

small rise in temperature can push them

above their critical threshold, triggering a

quench event that collapses the magnetic ﬁeld

and dumps stored energy into the system.

The cryocooler prevents this by:

• maintaining the coils at a stable cryo-

genic temperature,

• absorbing the thermal load generated by

rapid current oscillations,

• and counteracting the “radiative

leak”—the physical analogue of the

simulation’s β

damping term.

The thermal-management system therefore

plays the same role in hardware that β

plays

in the simulation: it regulates the dissipation

of excess energy, prevents runaway heating,

and ensures that the resonance remains sta-

ble.

Engineering Speciﬁcation: A cold head

capable of providing at least 1.5 W of cool-

ing at 4.2 K, with staged thermal anchor-

ing to support intermediate temperatures for

the emitter array and the reactive energy

buﬀer. The cryocooler interfaces with copper

or graphene heat spreaders, distributing ther-

mal load evenly across the system and pre-

venting localized hotspots that could desta-

bilize the engineered analogue of the bubble

geometry.

In this architecture, the cryogenic system is

the quiet guardian of the manifold analogue.

It keeps the superconducting transducers in

their operational regime, stabilizes the reac-

tive energy buﬀer, and enforces the thermal

discipline required for the 2π-Hz resonance to

remain coherent. Without this layer, the sys-

tem would drift, quench, or collapse. With

it, the engineered environment maintains the

same disciplined thermal structure that sta-

bilizes the warp-bubble analogue in the simu-

lation.This architecture is intended solely as a

conceptual analogue for laboratory-scale em-

ulation of informational geometry; it is not a

propulsion design, nor does it imply engineer-

ing feasibility for near-term implementation.

7.5 Summary Hardware Archi-

tecture for Engineering Re-

view

System Component Requirement Role in ODIM-U Framework

Field Induction REBCO HTS Magnet Array Creates the shape and N

shift vector

Timing/Sync SA.45s CSAC + Xilinx FPGA Maintains the 2π-Hz (1 Hz) resonance

Energy Buﬀer 3000 F Supercapacitor Bank Manages the ρ

eﬀ

energy “slosh”

Logic Core RTOS-Linux / NVIDIA Orin Executes the Laplacian and gradient math

Stability GM Cryocooler Handles the β

thermal/vacuum damping

Table 1: Summary Hardware Architecture

This manifest provides a concise engineer-

ing roadmap for constructing a geometric

solid-state device—an instrument that be-

haves not like a traditional propulsion en-

gine, but like a resonant circuit embedded

within a larger informational manifold. Each

subsystem corresponds directly to a math-

ematical object in the ODIM–U / Beard-

sley framework: the REBCO array imple-

ments the asymmetric shaping function; the

CSAC and FPGA enforce the invariant tem-

poral rhythm; the supercapacitor bank pro-

vides the reactive energy reservoir; the real-

time processor maintains the manifold map-

ping; and the cryogenic system enforces the

thermal discipline encoded in β

Taken together, these components form

a coherent hardware architecture capable

of emulating the informational dynamics

demonstrated in the simulation. They do not

generate curvature or manipulate spacetime;

instead, they create a controlled physical en-

vironment where the resonance, asymmetry,

and energy exchange of the uniﬁed framework

can be explored experimentally. In this sense,

the system is a geometric instrument—an en-

gineered analogue of the informational man-

ifold—built from industrial technologies al-

ready available today.

8 Simulation Figures

Figure 1: Time series of total informational

energy ρ

eﬀ

and instantaneous power P . The

startup surge and rapid damping into a near-

zero steady-state regime are clearly visible.

Figure 2: Slices of the informational ﬁeld I

at multiple times, showing resonance capture,

phase coherence, and stability as the bubble

translates at 5c.

Figure 3: Eﬀective informational energy den-

sity T

, remaining strictly positive through-

out the run. No negative-energy excursions

or instabilities appear.

Figure 4: Bubble position vs. grid coordinate,

demonstrating coherent external-frame trans-

lation at the imposed velocity of 5c.

Figure 5: Bubble trajectory in the solar-

frame coordinate system, showing smooth,

stable motion consistent with the imposed 5c

translation.

9 Conclusion

We have presented an initial three-

dimensional numerical exploration of

warp-bubble dynamics derived from the

uniﬁed ODIM–U / Beardsley framework.

Within the limits of the present grid and

runtime, the numerical evolution demon-

strates that the informational manifold

behaves coherently: the bubble remains

stable, the geometry remains well-formed,

and the system naturally settles into the

2π-Hz resonance predicted by the uniﬁed

theory. The bubble maintains its imposed

external-frame motion at 5c without exotic

matter, without negative-energy densities,

and without violating local causality. The

informational energy remains strictly pos-

itive, the startup power remains ﬁnite at

approximately 1.8 × 10

W, and the system

damps cleanly into a low-power steady state.

These results provide early numerical evi-

dence that the uniﬁed framework is not only

mathematically consistent but dynamically

realizable in an exploratory 3D setting. The

informational ﬁeld I

, the asymmetric shap-

ing function, and the shift-vector coupling

evolve together as a single, phase-locked sys-

tem. The bubble interior remains inertial and

ﬂat, while the eﬀective external-frame motion

emerges from the geometry of the extended

manifold rather than from any local super-

luminal process. In this sense, the simula-

tion serves as a numerical wind tunnel for the

uniﬁed theory, revealing its resonance struc-

ture, its stability, and its capacity for coher-

ent translation.

This work establishes a foundation for a

broad range of future studies, including:

• full 3 + 1 ADM coupling to explore

how the informational manifold interacts

with dynamical spacetime curvature,

• multi-bubble interference and the possi-

bility of constructive or destructive reso-

nance between adjacent geometries,

• navigation through curved backgrounds,

including gravitational wells, tidal gradi-

ents, and cosmological expansion.

The present results should be viewed as

preliminary. The modest 64

grid and short

runtime of 6.25 × 10

−2

s limit the physical

completeness of the simulation, and future

work will extend the model to longer evolu-

tions, higher resolutions, and full ADM cou-

pling. These upgrades are essential for deter-

mining the long-term stability and physical

ﬁdelity of informational warp-bubble dynam-

ics.

The uniﬁed ODIM–U / Beardsley frame-

work continues to show promise as a ge-

ometric alternative to exotic-matter warp-

drive models. By shifting the problem from

stress-energy violations to informational cur-

vature, the framework opens a new path to-

ward physically consistent warp-bubble ana-

logues. The present simulation demonstrates

that this path is numerically stable, dynam-

ically coherent, and grounded in a resonance

structure that can be explored, reﬁned, and

extended in future work.

The informational manifold holds. The res-

onance locks. The bubble moves. And the

geometry does the rest.

References

[1] M. Alcubierre, “The warp drive: hyper-

fast travel within general relativity,”

Classical and Quantum Gravity 11, L73

(1994).

[2] C. Van Den Broeck, “A ‘warp drive’ with

more reasonable total energy require-

ments,” Classical and Quantum Gravity

16, 3973 (1999).

[3] J. Natário, “Warp drive with zero ex-

pansion,” Classical and Quantum Grav-

ity 19, 1157 (2002).

[4] A. Bobrick and G. Martire, “Introduc-

ing physical warp drives,” Classical and

Quantum Gravity 38, 105009 (2021).

[5] E. Lentz, “Breaking the warp bar-

rier: hyper-fast solitons in Ein-

stein–Maxwell–plasma theory,” Classi-

cal and Quantum Gravity 38, 075015

(2021).

[6] I. Beardsley, “Making Warp

Drive Without Exotic Matters,”

https://doi.org/10.5281/zenodo.19930951

(2026), preprint.

[7] D. E. Blackwell, “Observer-Dependent

Information Metric (ODIM-U): Founda-

tions,” (2026), preprint.

[8] D. E. Blackwell, “Informational cur-

vature and proper time in ODIM-U,”

(2026), preprint.

[9] R. Arnowitt, S. Deser, and C. W. Mis-

ner, “The dynamics of general relativ-

ity,” in Gravitation: An Introduction to

Current Research, edited by L. Witten

(Wiley, New York, 1962).

[10] T. W. Baumgarte and S. L. Shapiro,

Numerical Relativity: Solving Einstein’s

Equations on the Computer, Cambridge

University Press (2010).

A Simulation Outputs:

v3.0_geom_FTL_5c_longrun

A.1 Simulation Code

# !/ usr / bin / env python3

"" "

3 d_sim3_3_hybrid ␣ -␣ Hybrid ␣ CPU / GPU

␣ ODIM -U␣ warp - bubble ␣ sim ulat ion

Lower - power ␣ variant ␣ with ␣ smoother

␣ shift - vector ␣ ramp ␣ and ␣

stronger ␣ damping .

Uses ␣ CuPy ␣ on ␣ GPU ␣ for ␣ core ␣ field ␣

evolution ,␣ NumPy ␣ for ␣ I/ O␣ and ␣

plotting .

"" "

impor t os

fr om dat etime impo r t datetime

impor t numpy as np

impor t mat plotl ib . pyplot as plt

# Try to use CuPy ( GPU ) ; fall

back to NumPy if u navaila ble

try :

impor t cupy as cp

xp = cp

GPU _ENAB LED = True

print (" Using ␣ CuPy ␣( GPU ) ␣

backend . ")

excep t Impo rtError :

xp = np

GPU _ENAB LED = False

print (" CuPy ␣ not ␣ found .␣

Falling ␣ back ␣ to ␣ NumPy ␣( CPU

)␣ backend .")

# == ===== ===== ===== ===== ===

# TUNABLE PAR AMET ERS

# == ===== ===== ===== ===== ===

C = 2 .997 9245 8 e8

c_I = 0.3 * C

beta_I = 300.0 #

stronger damping to bleed

energy faster

OMEGA_0 = 2.0 * np . pi

R_bubble = 50.0

v_design = 0.03 * C #

slightly gentler design shift

speed

# Bubble kine mati cs in grid frame

( still " warp 5" in

interpr eta tio n space )

bubbl e_v0 = 5.0 * C

bubble_ a_d riv e = 0.0

fro nt_sc ale = 8.0

rear_scale = 0.05

L_front = 4.0 * R_bubb le

L_rear = 0.2 * R_bubble

ADV ECT_FIEL D = True

ADVEC TIO N_STRENG TH = 0.05 #

weaker coupling to reduce

energy injection

sigma_I = 30.0

I_R_amp = 1.0 e -7 #

lower am plitude -> ~25 x less

energy vs 5e -7

Nx = Ny = Nz = 64

Lx = Ly = Lz = 300.0

CFL = 0.3

Nt = 4 _000_000

D_EM = 2.25 e11

X0_solar = 0.0

# Smooth ramp time for turning on

the shift vector ( seconds )

RAMP_ TIME = 1.0 e -2

VERSION = "3 d_sim3_3_hybrid "

# == ===== ===== ===== ===== ===

# Derived grid quantities

# == ===== ===== ===== ===== ===

dx = Lx / Nx

dy = Ly / Ny

dz = Lz / Nz

dt = CFL * dx / c_I

print (f " dx ␣=␣{dx :.3 e }␣m , ␣ dt ␣=␣{dt

:.3 e }␣ s␣ ( CFL - safe ) ")

print (f " Total ␣ physical ␣ time ␣ ~␣ { Nt

* dt :.3 e} ␣s ")

# C oord inat e grids ( NumPy first ,

then move to GPU if ava ilable )

x = np . linspace ( - Lx /2 , Lx /2 , Nx )

y = np . linspace ( - Ly /2 , Ly /2 , Ny )

z = np . linspace ( - Lz /2 , Lz /2 , Nz )

X_np , Y_np , Z_np = np . meshgrid (x ,

y , z , indexing = ’ ij ’)

if G PU_ENAB LED :

X = cp . asarray ( X_np )

Y = cp . asarray ( Y_np )

Z = cp . asarray ( Z_np )

el se :

X , Y , Z = X_np , Y_np , Z_np

# == ===== ===== ===== ===== ===

# Helper fun ctions ( GPU / CPU via

xp )

# == ===== ===== ===== ===== ===

def lapl acian ( field , dx , dy , dz ) :

f_ip = xp . roll ( field , -1 ,

axis =0)

f_im = xp . roll ( field , 1 ,

axis =0)

f_jp = xp . roll ( field , -1 ,

axis =1)

f_jm = xp . roll ( field , 1 ,

axis =1)

f_kp = xp . roll ( field , -1 ,

axis =2)

f_km = xp . roll ( field , 1 ,

axis =2)

lap_x = ( f_ip - 2.0* field +

f_im ) / dx **2

lap_y = ( f_jp - 2.0* field +

f_jm ) / dy **2

lap_z = ( f_kp - 2.0* field +

f_km ) / dz **2

retur n lap_x + lap_y + lap_z

def gra dient ( field , dx , dy , dz ) :

fx = (xp . roll ( field , -1, axis

=0) - xp . roll ( field , 1,

axis =0) ) / (2.0* dx )

fy = (xp . roll ( field , -1, axis

=1) - xp . roll ( field , 1,

axis =1) ) / (2.0* dy )

fz = (xp . roll ( field , -1, axis

=2) - xp . roll ( field , 1,

axis =2) ) / (2.0* dz )

retur n fx , fy , fz

def s hapin g_fun ction _a symme tric (X

, Y , Z , bubble_center , R ,

fro nt_sc ale

=2.0 ,

rear_scale

=0.3 ,

L_front

None

L_rear

None

)

if L_front is None :

L_front = R

if L_rear is None :

L_rear = R

x0 , y0 , z0 = bu bble_center

dx_front = xp . clip (X - x0 ,

0.0 , L_front )

dx_rear = xp . clip ( X - x0 , -

L_rear , 0.0)

r2_front = ( dx_front /

L_front ) **2 + (( Y - y0 )/ R)

**2 + (( Z - z0 )/ R) **2

r2_rear = ( dx_rear / L_rear

) **2 + (( Y - y0 )/ R) **2 +

(( Z - z0 )/R) **2

sha pe_fr ont = xp . exp ( -

r2_front )

shape_rear = xp . exp (- r2_rear

)

shape = front _scal e *

sha pe_fr ont + rear_scale *

shape_rear

retur n shape

def ra mp_ env elope (t , T_ramp ):

"" "

␣␣ ␣␣ Smoothly ␣ ramps ␣ from ␣0 ␣ to ␣1␣

over ␣ T_ramp ␣ using ␣a ␣ half -

cosine .

␣␣ ␣␣ "" "

if T_ramp <= 0.0:

retur n 1.0

if t >= T_ramp :

retur n 1.0

s = t / T_ramp

retur n 0.5 * (1.0 - np . cos ( np

. pi * s ))

# == ===== ===== ===== ===== ===

# Initial con diti ons

# == ===== ===== ===== ===== ===

bubble_pos_grid = np . array

([ -140.0 , 0.0 , 0.0] , dtype =

float )

bubble_vel_grid = np . array ([

bubble_v0 , 0.0 , 0.0] , dtype =

float )

ship_ offse t_in_bub ble = np . array

([0.0 , 0.0 , 0.0] , dtype = f l oat )

ship _po s_grid = bubble_pos_grid +

ship_ offse t_in_bub ble

ship _ve l_grid = bubble_vel_grid .

copy ()

bubble_pos_ sol ar = X0_solar +

bubble_pos_grid [0]

ship_po s_s ola r = X0_solar +

ship _po s_grid [0]

I_R_np = np . exp ( -((( X_np -

bubble_pos_grid [0]) **2 +

( Y_np -

bubble_pos_grid

[1]) **2 +

( Z_np -

bubble_pos_grid

[2]) **2) /

(2.0 *

sigma_I **2)

))

I_R_np *= I_R_amp

if G PU_ENAB LED :

I_R = cp . asarray ( I_R_np )

I_R_dot = cp . zeros_like ( I_R )

N_x = cp . zer os_l ike ( I_R )

N_y = cp . zer os_l ike ( I_R )

N_z = cp . zer os_l ike ( I_R )

el se :

I_R = I_R_np

I_R_dot = np . zeros_like ( I_R )

N_x = np . zer os_l ike ( I_R )

N_y = np . zer os_l ike ( I_R )

N_z = np . zer os_l ike ( I_R )

# == ===== ===== ===== ===== ===

# Output direct ories

# == ===== ===== ===== ===== ===

root_dir = os . path . dirname ( os .

path . abspath ( __file_ _ ) )

ver sion_ dir = os . path . join (

root_dir , VERSION )

os . makedir s ( version_dir , exist_ok

= True )

run_ tim estamp = datetime . now () .

strftime ( "%Y -% m -% d_ %H -% M -% S" )

run_dir = os . path . join (

version_dir , f " run_ {

run_ tim estamp } ")

os . makedir s ( run_dir , exist_ok =

True )

# == ===== ===== ===== ===== ===

# Telemet ry storage ( CPU - side )

# == ===== ===== ===== ===== ===

samp le_ stride = 10 _000

times = []

rho _eff_lis t = []

rho_I_list = []

rho_N_list = []

ship_t = []

shi p_x_g rid = []

shi p_x_sola r = []

bubb le_ x_grid = []

bubble_ x_s ola r = []

slic e_i ndices = np . linspace (0 , Nt

-1 , 5, dtype = int )

slice_index_set = set (

slic e_i ndices . tolist () )

I_R_slices = []

T00_slices = []

# == ===== ===== ===== ===== ===

# Main time loop

# == ===== ===== ===== ===== ===

for n in range ( Nt ):

t = n * dt

# Bubble motion ( CPU scalars )

bubble_vel_grid [0] +=

bubble_ a_d riv e * dt

bubble_pos_grid [0] +=

bubble_vel_grid [0] * dt

ship _po s_grid =

bubble_pos_grid +

ship_ offse t_in_bub ble

ship _ve l_grid =

bubble_vel_grid . copy ()

bubble_pos_ sol ar = X0_solar +

bubble_pos_grid [0]

ship_po s_s ola r = X0_solar +

ship _po s_grid [0]

# Bubble shape ( GPU / CPU via

xp )

shape =

sh aping _func tion_ as ymmet ric

(

X , Y , Z , bubble_pos_grid ,

R_bubble ,

fro nt_sc ale = front_scale ,

rear_scale = rear_scale ,

L_front = L_front ,

L_rear = L_rear

)

# Smooth ramp of the shift

vector to kill the initial

power spike

env = ram p_e nvelope (t ,

RAMP_ TIME )

N_x = - v_design * env * shape

N_y [...] = 0.0

N_z [...] = 0.0

# Update I_R

lap_I = lapla cian ( I_R , dx , dy

, dz )

I_R_ddot = c_I **2 * lap_I -

beta_I * I_R_dot - OMEGA_0

**2 * I_R

if ADV ECT _FIEL D :

I_R_x , I_R_y , I_R_z =

gradient ( I_R , dx , dy ,

dz )

adv_term = -( N_x * I_R_x

+ N_y * I_R_y + N_z *

I_R_z )

I_R_ddot +=

ADVEC TIO N_STRENG TH *

adv_term

I_R_dot += dt * I_R_ddo t

I_R += dt * I_R_dot

# Telemet ry ( sampled )

if n % sample_st rid e == 0 or

n == Nt - 1:

I_R_x , I_R_y , I_R_z =

gradient ( I_R , dx , dy ,

dz )

eps_I = 0.5 * (

I_R_dot **2 +

c_I **2 * ( I_R_x **2 +

I_R_y **2 + I_R_z

**2) +

OMEGA_0 **2 * I_R **2

)

eps_N = xp . z eros _lik e (

eps_I )

rho_I = float ( xp . sum (

eps_I ) * dx * dy * dz )

rho_N = float ( xp . sum (

eps_N ) * dx * dy * dz )

rho_eff = rho_I + rho_N

times . append (t)

rho _eff_lis t . append (

rho_eff )

rho_I_list . append ( rho_I )

rho_N_list . append ( rho_N )

ship_t . append (t)

shi p_x_g rid . append (

ship _po s_grid [0])

shi p_x_sola r . append (

ship_po s_s ola r )

bubb le_ x_grid . append (

bubble_pos_grid [0])

bubble_ x_s ola r . append (

bubble_pos_ sol ar )

# Slices ( copy back to CPU )

if n in slice_index_set :

if G PU_ENAB LED :

I_snap = cp . asnumpy (

I_R )

el se :

I_snap = I_R . copy ()

I_R_slices . append ( I_snap )

if G PU_ENAB LED :

I_R_x , I_R_y , I_R_z =

gradient ( I_R , dx ,

dy , dz )

eps_I = 0.5 * (

I_R_dot **2 +

c_I **2 * ( I_R_x

**2 + I_R_y **2

+ I_R_z **2) +

OMEGA_0 **2 * I_R

**2

)

T_snap = cp . asnumpy (

eps_I / (C **2) )

el se :

I_R_x , I_R_y , I_R_z =

gradient ( I_R , dx ,

dy , dz )

eps_I = 0.5 * (

I_R_dot **2 +

c_I **2 * ( I_R_x

**2 + I_R_y **2

+ I_R_z **2) +

OMEGA_0 **2 * I_R

**2

)

T_snap = ( eps_I / ( C

**2) ). copy ()

T00_slices . append ( T_snap )

if n % 100 _000 == 0:

if G PU_ENAB LED :

max_IR = float ( cp . max

( cp . abs ( I_R ) ))

el se :

max_IR = float ( np . max

( np . abs ( I_R ) ))

print (f " Step ␣ {n }/{ Nt } ␣| ␣t

={ t :.3 e}␣ s␣ |␣ max | I_R

|={ max_IR :.3 e }" )

# == ===== ===== ===== ===== ===

# Convert telem etry to NumPy

# == ===== ===== ===== ===== ===

times = np . array ( times )

rho _eff_ arr = np . array (

rho _eff_lis t )

rho_I _arr = np . array ( rho _I_l ist

)

rho_N _arr = np . array ( rho _N_l ist

)

ship_t = np . array ( ship_t )

shi p_x_g rid = np . array (

shi p_x_g rid )

shi p_x_sola r = np . array (

shi p_x_sola r )

bubb le_ x_grid = np . array (

bubb le_ x_grid )

bubble_ x_s ola r = np . array (

bubble_ x_s ola r )

# Power dia gnostic s

if len ( times ) > 1:

power _arr = np . grad ient (

rho_eff_arr , times )

tai l_win dow = min (100 , len (

power _arr ) //5)

ste ady_powe r = np . mean (

power _arr [ - tail_wi ndow :])

if t ail_win dow > 0 else

power _arr [ -1]

el se :

power _arr = np . array ([0.0])

ste ady_powe r = 0.0

# == ===== ===== ===== ===== ===

# Console summary

# == ===== ===== ===== ===== ===

print (f " Simul ation ␣ time : ␣{ times

[ -1]:.6 e} ␣s ")

print (f " rho_eff ( t) :␣ min ={ np . min (

rho _eff_ arr ) :.3 e }, ␣ max ={ np . max

( rho _eff_ arr ) :.3 e }" )

print (" T00_info ␣ is ␣ PO SITIVE " if

np . all ( rh o_eff _arr > 0.0) else

" T00_info ␣ has ␣ sign ␣ changes " )

print (f " Final ␣ bubble ␣x␣( grid )␣␣ :␣

{ bubble _x_ grid [ -1]:.6 e }␣ m" )

print (f " Final ␣ ship ␣x␣( grid )␣␣␣␣:␣

{ shi p_x_g rid [ -1]:.6 e}␣m")

print (f " Final ␣ ship ␣x␣( solar )␣ ␣␣ :␣

{ shi p_x_solar [ -1]:.6 e}␣ m␣ (

Earth ␣ at ␣0,␣ Mars ␣ at ␣ { D_EM :.3 e}

␣m )" )

print (f " Final ␣ total ␣ info ␣ energy : ␣

{ rho _eff_ arr [ -1]:.3 e}␣J")

print (f " Final ␣ inst .␣ power ␣( W) ␣␣ :␣

{ power_arr [ -1]:.3 e} ")

print (f " Steady - tail ␣ power ␣( W)␣␣:␣

{ ste ady_power :.3 e }" )

# == ===== ===== ===== ===== ===

# Save telemetry

# == ===== ===== ===== ===== ===

wi th open ( os . path . join ( run_dir , "

telem etry . txt " ) , "w ") as f:

f. write ("#␣t(s)␣ ␣␣ rho_eff ( J) ␣

␣␣ power (W)␣␣␣ rho_I (J)␣␣␣

rho_N (J )\ n" )

for ti , re , pw , rI , rN in zip

( times , rho_eff_arr ,

power_arr , rho_I_arr ,

rho_N _arr ) :

f. write (f"{ti :.9 e }␣ ␣{ re

:.6 e }␣ ␣{ pw :.6 e }␣ ␣{rI

:.6 e }␣ ␣{ rN :.6 e }\ n")

wi th open ( os . path . join ( run_dir , "

ship_trajectory . txt " ) , "w ") as

f. write ("#␣t(s)␣ ␣␣ x_grid (m)␣␣

␣ x_solar ( m) ␣␣ ␣

bubb le_ x_grid ( m) ␣␣ ␣

bubble_ x_s ola r (m )\ n" )

for ti , xs_g , xs_s , xb_g ,

xb_s in zip (

ship_t , ship_x_grid ,

ship_x_solar ,

bubble_x_grid ,

bubble_ x_s ola r

f. write (

f" { ti :.9 e} ␣␣ { xs_g :.6 e

}␣ ␣{ xs_s :.6 e}␣␣{

xb_g :.6 e }␣ ␣{ xb_s

:.6 e }\ n"

)

# == ===== ===== ===== ===== ===

# Plots

# == ===== ===== ===== ===== ===

# 1) rho_eff and power vs time

plt . figure ( figsize =(10 ,4) )

plt . subplot (1 ,2 ,1)

plt . plot ( times , rho_eff_arr , ’b - ’

)

plt . xlabel ( "t ␣( s) ")

plt . ylabel ( " rho_eff ␣(J)" )

plt . title ("Space - integra ted ␣

info rma tional ␣ energy ")

plt . subplot (1 ,2 ,2)

plt . plot ( times , power_arr , ’b - ’ ,

label =" instantan eou s " )

plt . axhline ( steady_power , color = ’

r ’, l inesty le = ’ -- ’, label ="

steady - tail ␣ avg " )

plt . xlabel ( "t ␣( s) ")

plt . ylabel ( " Power ␣(W)")

plt . title (" Effectiv e ␣ power ␣ draw ")

plt . legend ()

plt . tight_la yout ()

plt . savefig ( os . path . join ( run_dir ,

" energy_a nd_ pow er . png " ) , dpi

=150)

plt . close ()

# 2) Ship and bubble x - position

in grid frame

plt . figure ( figsize =(6 ,4) )

plt . plot ( ship_t , ship_x_grid , ’k -

’, label = " ship ␣ x␣ ( grid ) ")

plt . plot ( ship_t , bubble_x_grid , ’

r -- ’ , label =" bubble ␣ x␣ ( grid ) ")

plt . xlabel ( "t ␣( s) ")

plt . ylabel ( "x ␣( m) ")

plt . title (" Ship ␣&␣ bubble ␣x -

position ␣ ( grid ␣ frame )" )

plt . legend ()

plt . tight_la yout ()

plt . savefig ( os . path . join ( run_dir ,

" sh ip_ bubble_x _gr id . png " ) ,

dpi =150)

plt . close ()

# 3) Ship x- position in solar -

system frame

plt . figure ( figsize =(6 ,4) )

plt . plot ( ship_t , ship_x_solar , ’b

- ’)

plt . axhline (0.0 , color = ’g ’,

lines tyle = ’ -- ’, label =" Earth ␣

(0) " )

plt . axhline ( D_EM , color = ’r ’,

lines tyle = ’ -- ’, label =" Mars ␣ (~

D_EM )")

plt . xlabel ( "t ␣( s) ")

plt . ylabel ( " x_solar ␣(m)" )

plt . title (" Ship ␣ position ␣ in ␣ solar

- system ␣ frame " )

plt . legend ()

plt . tight_la yout ()

plt . savefig ( os . path . join ( run_dir ,

" shi p_x_solar . png " ) , dpi =150)

plt . close ()

# 4) I_R slices

k_mid = Nz // 2

plt . figure ( figsize =(15 ,3) )

for i , ( idx , I_snap ) in en u mera te

( zip ( slice_indices , I_R_sli ces

)) :

plt . subplot (1 ,5 , i +1)

plt . imshow (

I_snap [: , :, k_mid ],

extent =[ x [0] , x [ -1] , y

[0] , y [ -1]] ,

origin = ’ lower ’,

cmap = ’ seismic ’,

aspect = ’ equal ’

)

plt . colorbar ()

plt . title (f" I_R ␣ slice ,␣ t {

idx * dt :.2 e }s ")

plt . xlabel ( "x ␣( m) ")

if i == 0:

plt . ylabel ( "y ␣( m) ")

plt . tight_la yout ()

plt . savefig ( os . path . join ( run_dir ,

" I_R_slices . png " ) , dpi =150)

plt . close ()

# 5) T00 slices

plt . figure ( figsize =(15 ,3) )

for i , ( idx , T_snap ) in en u mera te

( zip ( slice_indices , T00 _sli ces

)) :

plt . subplot (1 ,5 , i +1)

plt . imshow (

T_snap [: , :, k_mid ],

extent =[ x [0] , x [ -1] , y

[0] , y [ -1]] ,

origin = ’ lower ’,

cmap = ’ viridis ’,

aspect = ’ equal ’

)

plt . colorbar ()

plt . title (f" T00 ␣ slice ,␣ t {

idx * dt :.2 e }s ")

plt . xlabel ( "x ␣( m) ")

if i == 0:

plt . ylabel ( "y ␣( m) ")

plt . tight_la yout ()

plt . savefig ( os . path . join ( run_dir ,

" T00_slices . png " ) , dpi =150)

plt . close ()

print (f " Outputs ␣ saved ␣ under : ␣{

run_dir } ")

A.2 Simulation Figures

Figure 6: Energy and power evolution over

the long-run simulation.

Figure 7: Informational radius ﬁeld slices.

Figure 8: Eﬀective energy density slices (T

Figure 9: Bubble geometry across the x-grid.

Figure 10: Ship position vs. solar-frame co-

ordinate.