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\rhead{\small\textit{\textcolor{gray}{The Toroidal Momentum Engine --- Austin (2026)}}}
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{\Huge\bfseries\color{headblue} The Toroidal Momentum Engine}

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{\large\itshape\textcolor{darkgray80}{A Framework for Cosmic Architecture, Topological Induction,\\
and a Workable Version of Infinity}}

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{\Large GK Austin II}\\[0.3cm]
{\textcolor{darkgray80}{Independent Researcher}}

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{\textcolor{darkgray80}{February 2026}}

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{\small\itshape\textcolor{darkgray80}{Methodology: Novel Heuristic Exploration (Stochastic Mashing)}}

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{\small\itshape\textcolor{darkgray80}{Format Note: This paper uses a dual-format structure. Each academic section is followed by a plain-language explainer (shown in a blue box). This is a deliberate design choice. As James Burke observed in \textbf{Connections} (1978), science and technology advance at speeds that exclude most of humanity from participation. This paper refuses that exclusion.}}

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% ═══════════ ABSTRACT ═══════════
\section*{Abstract}

This paper proposes a candidate geometric interpretation of cosmic architecture: a finite, circulating 3-Torus manifold in which cosmological inflation is reinterpreted as a topological merger event---the capture and assimilation of one toroidal manifold by another. We replace the singularity-dependent Big Bang with the ``Big Splash,'' a high-energy displacement event in a finite medium, and argue that the Hubble Tension is an observational artifact of Topological Refraction within a closed manifold. The 3-Torus provides what we term a ``Less Expensive Infinity'': boundless traversal within finite volume. We further propose that this infinity is not static but grows through toroidal merger events, each of which expands the manifold and injects energy indistinguishable from inflation. An experimental appendix outlines a terrestrial proof-of-concept: a toroidal recording medium capable of volumetric data storage at densities exceeding conventional optical media by orders of magnitude.

\begin{explainerbox}
Water is the closest example of a de facto model for the universe that we can observe on Earth. Drop a stone into a pool: you get rings---toroidal rings. Every time. When two sets of ripples meet, one absorbs the other and the combined pattern is bigger than either original. That is the core of this paper.

We propose the universe is shaped like a donut (a torus). It has no edges and no beginning or end---you can travel in any direction forever and loop back to where you started, even though the total space is finite. What we call the Big Bang wasn't an explosion from nothing. It was a collision: our donut-shaped universe swallowed another one, and that merger released the enormous energy we observe as inflation. The medium didn't need to travel faster than light. Like a pool that is already full of water, the quantum field was already everywhere---it just got displaced.

This paper lays out that argument, proposes it explains a major unsolved puzzle in physics (the Hubble Tension), and describes a physical experiment you could build on Earth to test whether toroidal geometry really does store information more efficiently.
\end{explainerbox}

% ═══════════ I. INTRODUCTION ═══════════
\section{I. Introduction: The Less Expensive Infinity}

Contemporary cosmology faces a paradox: the universe appears spatially flat to high precision (Planck 2018), yet the physical infinities implied by an open, flat, Euclidean manifold generate persistent theoretical pathologies---divergent integrals in quantum field theory, the cosmological constant problem, and the measure problem in eternal inflation. We propose a resolution grounded in topology rather than geometry: the universe is a 3-Torus ($T^3$), a manifold that is locally flat (consistent with all current observations of spatial flatness) but globally finite and closed.

A 3-Torus offers what we term a ``Less Expensive Infinity.'' It permits unbounded traversal---no observer will ever encounter an edge or boundary---while maintaining finite total volume. This is not infinity in extent, but infinity in experience. The distinction is consequential: a finite manifold eliminates the need for physical infinities, resolves certain divergence problems, and---critically---allows for observable topological signatures in the Cosmic Microwave Background (CMB), as explored by Cornish et al.\ (1998) and Luminet (2016).

Furthermore, we propose that this ``cheaper infinity'' is not static. The Block Universe, typically conceived as a frozen four-dimensional structure containing all of past, present, and future, is in our framework a circulating toroidal manifold that grows through merger events. When one toroid captures another, the total manifold expands. The closed loop becomes longer. The ``cheaper infinity'' becomes, in a precise topological sense, a bigger cheaper infinity.

\textbf{Scope and Status.} This paper presents a heuristic, semi-phenomenological framework. It proposes testable geometric and experimental signatures but does not yet constitute a full theory with derived field equations. The mechanisms described here are grounded in physical analogy, supported by laboratory and observational evidence, and formalized at the level of topological conjecture rather than mathematical proof. The Limitations section (IX) addresses each gap explicitly and outlines the formal program required to elevate the framework to a predictive theory.

\begin{explainerbox}
Physics has a problem with infinity. Many of our best equations produce nonsensical infinite answers because we assume the universe goes on forever in every direction. But what if it doesn't? What if space wraps around like the surface of a donut? You'd never hit a wall, so it would feel infinite, but the total amount of space would be finite. That's a `cheaper infinity.' And that donut can get bigger. If our donut-universe collides with and absorbs another one, the loop gets longer. That's what we call inflation.
\end{explainerbox}

% ═══════════ II. CORE THESIS ═══════════
\section{II. Core Thesis: Inflation as Topological Capture}

The standard inflationary model (Guth 1981, Linde 1982) posits a brief period of exponential expansion driven by a scalar inflaton field. While observationally successful, the model requires fine-tuning of initial conditions and introduces the inflaton---a particle with no independent experimental confirmation. We propose an alternative mechanism: inflation is the observational signature of a toroidal merger event.

\subsection{2.1 The Merger Mechanism}

Consider two 3-Torus manifolds existing in a higher-dimensional ambient space. When their boundaries come into contact, a topological surgery occurs: the two closed manifolds merge into a single, larger closed manifold. In formal terms, the merger is a cobordism---a higher-dimensional manifold whose boundary consists of the two pre-merger tori and the single post-merger torus (cf.\ Milnor 1965, \textit{Lectures on the h-Cobordism Theorem}). This process is analogous to galaxy mergers at cosmological scales---two self-gravitating systems coalesce into a single, more massive system, releasing gravitational energy in the process.

Critically, the merger is not merely a passive collision. The toroidal manifold actively warps the surrounding medium, creating an attractive gradient that draws nearby structures toward it. This is directly observable in fluid dynamics: underwater vortex rings distort the surrounding water and pull nearby objects inward. In footage analyzed by Caltech (Schr\"{o}der et al.\ 2019), a jellyfish was drawn into an underwater toroidal vortex ring---not by swimming toward it, but because the vortex reshaped the surrounding medium. The jellyfish underwent approximately 74 rotations within the ring before ejection, demonstrating the extraordinary energy contained within a stable toroidal structure.

In the toroidal case, the merger injects energy into the combined manifold through what we term Vortex Merging Acceleration. Caltech's computational simulations of vortex ring dynamics (Schr\"{o}der, SIGGRAPH 2019) demonstrated that when two toroidal vortex rings merge, the resulting structure is larger than either original---and the merger ejects a smaller daughter toroid, suggesting that toroidal mergers are generative: they reproduce. The mathematics governing these dynamics, formalized through the Biot-Savart law with terms for variable thickness, buoyancy, viscosity, and circulation, have been validated against physical footage to the point of visual indistinguishability.

The energy released during a toroidal merger event is indistinguishable, from an observer's perspective within the manifold, from the exponential expansion predicted by scalar-field inflation. The observer perceives rapid, isotropic expansion of their local space---because the manifold they inhabit has, in fact, grown.

\begin{explainerbox}
Standard physics says the universe expanded incredibly fast in its first moments because of a mysterious energy field (the `inflaton') that has never been directly detected. We're proposing something simpler: our donut-universe ran into another donut-universe and absorbed it.

But it's not just a random collision. The toroid actively warps the space around it and pulls things in---just like an underwater vortex ring pulls in a jellyfish. And when two toroids merge, Caltech's research shows the combined structure is bigger AND it ejects a smaller new toroid. The merger doesn't just grow---it reproduces.
\end{explainerbox}

\subsection{2.2 The Big Splash}

We reframe the initial conditions of the universe as a ``Big Splash'' rather than a Big Bang. In a finite, incompressible medium (the quantum vacuum), a high-energy displacement event does not produce an explosion radiating outward into empty space. It produces a toroidal vortex ring---as demonstrated by every drop of liquid falling into a pool. The Big Splash is the cosmological equivalent: an energetic event in a pre-existing medium that naturally generates toroidal geometry.

This framing eliminates the singularity and resolves a persistent conceptual difficulty: the apparent superluminal expansion of the early universe. In our model, the quantum field---like a pool already full of water---does not need to travel anywhere. It is already everywhere. The energy of the Big Splash displaces the pre-existing medium. This is analogous to the standard distinction in general relativity between recessional velocity (which may exceed $c$ because space itself is expanding) and local signal propagation (which remains subluminal): the medium is not transmitting information faster than light; it is responding to a disturbance that propagates through an already-filled volume at a finite characteristic speed. As a borrowed aphorism states: ``The speed of dark is faster than the speed of light, because it is already there.''

This principle received direct experimental confirmation in March 2026, when researchers at Technion--Israel Institute of Technology measured ``dark points''---optical phase singularities where wave amplitude drops to zero---moving faster than the speed of light within light fields (Bucher et al.\ 2026, \textit{Nature}). These vortex structures, which carry neither energy nor information, exceed the speed of light without violating relativity. The researchers explicitly compared the phenomenon to ``a vortex in a river overtaking the flow of water in which it exists.'' This is precisely the mechanism our model invokes: displacement patterns in a pre-existing medium propagate without the medium itself needing to travel.

This is observable in any rainstorm. Watching raindrops strike a pool, each drop creates a toroidal vortex ring. When two expanding ring patterns overlap, one absorbs the other. The drops in the shallow end and the drops in the deep end travel different distances through different depths of the same pool. An observer measuring both and assuming a uniform, flat medium would systematically misestimate distances---an exact analog of the Hubble Tension.

\begin{explainerbox}
Drop a stone into a pond. You don't get an explosion---you get rings. Every single time. We're proposing the Big Bang was the same kind of event: a `splash' in a medium that already existed.

This solves two problems at once. First, there's no singularity---no impossible point of infinite density. Second, the universe didn't need to expand faster than light. The medium was already there, like a pool full of water. The splash just displaced it. Nothing traveled. The pool responded.

And if you watch rain falling on a pool, the drops in the shallow end and the deep end create patterns that look different depending on where you're standing---exactly the kind of measurement bias we see in the Hubble Tension.
\end{explainerbox}

% ═══════════ III. HUBBLE TENSION ═══════════
\section{III. Reframing the Hubble Tension}

The Hubble Tension---the statistically significant disagreement between early-universe (CMB-derived, $H_0 \approx 67.4$ km/s/Mpc) and late-time (distance-ladder, $H_0 \approx 73.0$ km/s/Mpc) measurements of the Hubble constant---is among the most pressing unresolved problems in observational cosmology. We propose this tension is an artifact of Topological Refraction within a $T^3$ manifold.

\subsection{3.1 The Wrapping Number Effect}

In a 3-Torus, a photon emitted from a distant source may traverse the manifold with a wrapping number $n \geq 1$---that is, the photon's path includes one or more complete circuits of the torus before reaching the observer. If the observer assumes a simply-connected Euclidean geometry ($n = 0$), the inferred distance to the source is systematically overestimated.

We propose a simple toy model: the observed path length scales as $L_n \approx (1 + n)L_0$, where $L_0$ is the direct (unwrapped) path length. For a late-time source observed with $n = 1$, the distance is overestimated, which after accounting for the geometric distribution of wrapping paths, yields an apparent $\sim$8--9\% inflation in the inferred $H_0$. This is consistent with the observed magnitude of the Hubble Tension. We note that real geodesics in a $T^3$ FRW metric are more subtle than this additive scaling---photon paths in an expanding, multiply-connected manifold involve redshift accumulation and topological lensing effects that require full ray-tracing simulations to quantify. The toy model here serves as an intuition pump establishing the correct order of magnitude, not as a derived result.

Furthermore, if the quantum field medium is not uniform in depth---if it has topology analogous to a pool with a shallow end and a deep end---then the density of the field varies across the manifold. This depth variation would produce systematic biases in distance measurements that depend on the observer's position within the torus, potentially explaining not only the Hubble Tension but also the observed distribution of matter: galaxies cluster where the field is ``deep,'' voids form where it is ``shallow,'' and the large-scale cosmic web is a topographic map of the field's depth profile.

This model does not require dark energy as a distinct physical entity. The apparent acceleration of expansion is, in this framework, a geometric illusion produced by assuming Euclidean path lengths in a toroidal manifold.

\begin{explainerbox}
There's a major fight in physics right now. Two different ways of measuring how fast the universe is expanding give two different answers. This is called the Hubble Tension.

Our explanation: light wraps around the donut. If you assume the universe is flat and infinite, you calculate one distance. But if light actually goes around the donut before reaching you, the real distance is shorter than you thought. You'd overestimate how fast things are moving apart---exactly the discrepancy we see.

Think of rain on a pool: the drops in the shallow end meet the drops in the deep end, and the ones in the middle make the universe look like it was built faster than it was.
\end{explainerbox}

% ═══════════ IV. BLOCK UNIVERSE ═══════════
\section{IV. The Block Universe as a Growing Toroid}

In standard Block Universe theory (Minkowski 1908, Einstein 1915), all events---past, present, and future---coexist in a static four-dimensional structure. We propose a modification: the Block Universe is a 3-Torus that grows through topological merger events.

This introduces a distinction between internal time and external time. Internal time is the temporal dimension experienced by observers within the manifold. External time is the sequence in which merger events occur, expanding the manifold. The Block Universe has a history, but that history is not internal to the block. It is the history of the block's own topology changing through successive captures of other toroidal manifolds.

Each merger event extends the closed loop. The ``cheaper infinity'' becomes a larger cheaper infinity. The block accumulates. Past merger events are encoded in the topology of the current manifold, just as geological strata encode the history of the Earth.

\begin{explainerbox}
In standard physics, the Block Universe is frozen---past, present, and future all exist simultaneously, like frames on a film strip. We're adding something new: the film strip gets longer. Every time our toroid captures another toroid, the whole structure grows. The topology of our universe encodes the history of every merger it has undergone, like tree rings encoding the history of a tree.
\end{explainerbox}

% ═══════════ V. QUANTUM FIELDS ═══════════
\section{V. Quantum Fields as the Generator of Toroidal Geometry}

A critical question arises: what mechanism generates toroidal geometry? We propose that quantum fields are the active agent. These fields are not passive backgrounds; they are dynamic structures with inherent topological preferences.

The evidence for toroidal preference at multiple scales is suggestive: vortex rings form spontaneously in fluids (Kleckner and Irvine 2013), accretion disks around black holes adopt toroidal morphology (Circinus Galaxy), magnetic flux tubes in plasma physics are toroidal, and the tokamak---our most successful design for sustained fusion---is a torus. We propose this is Topological Induction: the process by which quantum fields, operating under energy-minimization principles, bias matter toward self-similar toroidal configurations at every accessible scale. Here `Topological Induction' is a phenomenological label for the observed cross-scale convergence toward toroidal structures under circulation and energy-minimization constraints, not yet a theorem derived from a specific field Lagrangian.

In this framework, a rotating Kerr black hole functions as a perpetual toroidal generator. Frame-dragging creates and sustains toroidal flow patterns in the surrounding spacetime. The black hole is a cosmological lathe, continuously producing toroidal structure from the quantum vacuum.

\begin{explainerbox}
Why does this shape keep showing up everywhere? Drop water: ring. Galaxy center: ring. Fusion reactor: ring. We're arguing quantum fields---the fundamental fabric of reality---naturally prefer toroidal shapes because these shapes minimize energy while maintaining circulation. Toroids are nature's preferred shape for anything that circulates, and at the deepest level, everything circulates.
\end{explainerbox}

% ═══════════ VI. PHYSICAL EVIDENCE ═══════════
\section{VI. Physical Evidence}

\begin{figure}[H]
\centering
\includegraphics[width=0.85\textwidth]{figure_plate.png}
\caption{\textit{Multi-scale evidence for toroidal geometry as a universal attractor.} \textbf{(a)} Water drop producing toroidal vortex ring. \textbf{(b)} Herschel Crater on Mimas---energy displacement recorded as toroidal geometry in rock. \textbf{(c)} Hydrogen atom electron probability distribution---quantum fields defaulting to toroidal geometry. \textbf{(d)} Double-slit interference and wave superposition diagrams.}
\end{figure}

\begin{figure}[H]
\centering
\includegraphics[width=0.55\textwidth]{panel_d_final.png}
\caption{Pond ripple interference (left) alongside the double-slit experiment (right). Three drop impact points create overlapping toroidal ring patterns identical in structure to quantum wave interference---the same phenomenon at different scales. The pond demonstrates the Triple-Drop Induction Model: three sources locking the system into a globally stable interference pattern.}
\end{figure}

\begin{figure}[H]
\centering
\includegraphics[width=0.85\textwidth]{panel_e_caltech.png}
\caption{\textit{Caltech vortex ring merger sequence with mathematical framework} (Schr\"{o}der, SIGGRAPH 2019). \textbf{Panels 1--3:} Two toroidal vortex rings approach, undergo violent topological surgery, and produce a larger combined structure plus an ejected daughter toroid. \textbf{Panel 4:} Traced filament path during merger. \textbf{Bottom rows:} Mathematical framework (Biot-Savart law) with terms for variable thickness, buoyancy, viscosity, and circulation---each validated against physical footage to visual indistinguishability. Source: Caltech/Peter Schr\"{o}der.}
\end{figure}

\subsection{6.1 The Caltech Experiment (Schr\"{o}der, 2019)}

Peter Schr\"{o}der's team at Caltech generated computer simulations of underwater bubble ring dynamics so accurate they are visually indistinguishable from physical footage. The research, presented at SIGGRAPH 2019, began as an attempt to understand solar flares---enormous loops of plasma erupting from the sun. Schr\"{o}der noted that solar flares, bubble rings, and ink chandeliers all share one thing in common: twisting, fluid doughnut-like shapes that rotate around a center line. The mathematical framework governing these phenomena---the Biot-Savart law applied to vortex filaments---was validated across all three contexts, demonstrating that toroidal dynamics are scale-invariant.

Critically, Schr\"{o}der's simulations captured the merger dynamics: when two vortex rings collide, they undergo topological surgery, producing a larger combined ring and ejecting smaller daughter structures. This is a direct laboratory analog of the cosmological merger mechanism proposed in this paper.

\subsection{6.2 The Chicago Experiment (Kleckner and Irvine, 2013)}

Researchers at the University of Chicago demonstrated that high-velocity impacts in a fluid medium robustly produce stable, knotted toroidal vortex structures. These structures persist, interact, and merge---exhibiting the exact dynamics our model predicts at cosmological scales.

\subsection{6.3 CMB Topology Searches}

Cornish, Spergel, and Starkman (1998) proposed searching the CMB for ``circles in the sky''---matched temperature patterns that would indicate a closed, finite topology. While current searches have not yielded definitive detections, our model is consistent with existing constraints. Furthermore, observed CMB anomalies---including the Cold Spot and the so-called Axis of Evil---which lack satisfactory explanation in simply-connected cosmologies, may represent direct observational signatures of toroidal flow structure and merger history. The CMB may function not merely as a snapshot of the early universe but as a flow map of the toroidal medium, encoding directional circulation patterns. Reprocessing existing Planck data using gradient vector field visualization---techniques standard in fluid dynamics---could reveal toroidal flow signatures and represents a concrete, achievable future research direction.

\subsection{6.4 The Circinus Galaxy}

Interferometric observations of the Circinus Galaxy reveal a dusty, clumpy torus surrounding its active galactic nucleus. In our framework, this morphology is a manifestation of Topological Induction---the galaxy is an ``iteration'' of the same blueprint that governs the global manifold.

\subsection{6.5 The Engine Argument: Observable Mechanisms Across All Media}

The standard Big Bang model posits a singularity---a dimensionless point of infinite density. The singularity has no physical analog anywhere in observable nature. No engine runs on a singularity. No process in any medium produces one. It is a mathematical abstraction that breaks all known physics at the moment it is most needed.

The toroidal model, by contrast, is a working engine observable in every medium: Water---drop impacts produce toroidal vortex rings with perfect reproducibility. Air---smoke rings, volcanic vortex rings (Mount Etna), and explosion shockwaves produce toroidal structures. Fire and Plasma---the tokamak, humanity's most successful fusion design, is a torus. Solid Matter---Herschel Crater on Mimas demonstrates impact energy producing toroidal structure frozen in rock. Space---accretion disks, Kerr black hole frame-dragging, galaxy morphology, and planetary ring systems exhibit toroidal geometry. Biology---jellyfish propulsion, the human heart's toroidal blood flow, and vortex ring locomotion demonstrate that evolution independently converges on toroidal mechanics.

The singularity model has zero working examples. The toroidal model has hundreds, across every scale and every medium. This does not constitute proof, but the burden of explanation shifts: why would nature prefer toroidal geometry everywhere except at the origin of the universe?

\begin{explainerbox}
Here's the simplest challenge to mainstream cosmology: show me a singularity. Show me one example, anywhere in nature, of a point of infinite density doing anything. You can't.

Now show me a toroid. They're everywhere. Drop water: toroid. Smoke ring: toroid. Fusion reactor: toroid. Galaxy center: toroid. Your heart pumps blood in a toroidal pattern. Volcanoes blow toroidal smoke rings.

One model has no working examples. The other is a working engine in every medium---water, air, fire, rock, space, and biology. That doesn't prove the universe is a toroid. But it's a striking asymmetry that deserves an answer.
\end{explainerbox}

\subsection{6.6 Contrast: The Infinite Nesting Problem}

The popular hypothesis that the universe exists inside a black hole suffers from an unresolvable infinite regression: if our universe is inside a black hole, and our universe contains black holes, each of those black holes presumably contains a universe, ad infinitum. The model has no exit condition.

The toroidal model does not suffer from this defect. A 3-Torus is self-contained. Its growth comes from combination---the merger of peer manifolds---not from infinite nesting.

\subsection{6.7 Water as a De Facto Cosmological Analog}

Of all terrestrial phenomena, fluid dynamics in an incompressible medium provides the most reliable analog for the mechanics of the universe as proposed in this framework. Water is incompressible, fills its container completely, transmits force at a finite speed, supports wave propagation, generates toroidal vortices spontaneously from displacement events, and preserves information about those events as persistent structures.

The quantum vacuum shares every one of these properties. Every major phenomenon described in this paper has a direct fluid dynamics analog: the Big Splash is a drop impact; inflation is vortex merger; the Hubble Tension is path-length distortion in a closed container; toroidal structure at every scale is vortex ring generation; and the CMB is the residual ripple pattern from the initial disturbance. Water has a speed limit for information propagation (the speed of sound), just as spacetime has the speed of light.

This structural correspondence suggests that fluid dynamics---rather than particle physics or string theory---may provide the most productive terrestrial framework for testing cosmological topology.

\begin{explainerbox}
Water is the closest thing we have on Earth to a working model of the universe. It's incompressible, it fills its container, it transmits force at a finite speed, and when you disturb it, it makes toroidal rings. Every major feature of our cosmological model has a direct water analog. This suggests that the most productive way to test theories about the universe might not be particle accelerators or space telescopes---it might be fluid dynamics.
\end{explainerbox}

\subsection{6.8 The Local Group Mass Distribution (Wempe et al.\ 2026)}

A January 2026 paper in \textit{Nature Astronomy} (Wempe, White, Helmi, Lavaux, and Jasche) provides striking observational evidence consistent with the toroidal framework. Using constrained $\Lambda$CDM simulations of the Local Group, the authors found that the mass distribution around the Milky Way and Andromeda is not spherical but is strongly concentrated in a flat sheet extending to at least 10 Mpc, with deep voids above and below. A spherical geometry was ``decisively rejected,'' with a minor-to-major axis ratio of approximately 0.24.

This configuration---a dense equatorial plane with empty regions on either side---is geometrically consistent with the cross-section of a toroidal manifold viewed from within. Additionally, the surface density of the sheet increases with distance from the Local Group, which is expected if the observer is situated inside a torus looking outward toward the thicker part of the ring. The velocity field is highly anisotropic, with material falling toward the sheet from the voids above and below---precisely the flow pattern expected inside a toroidal structure.

The authors interpret their findings within standard $\Lambda$CDM. We claim geometric compatibility, not uniqueness: a flattened sheet-like mass distribution is consistent with several structure-formation narratives within standard cosmology. What is noteworthy is that the specific configuration---equatorial concentration, polar voids, increasing radial density, and anisotropic infall---is predicted by our model on independent geometric grounds.

\begin{explainerbox}
A brand-new paper in \textit{Nature Astronomy} (January 2026) found something remarkable: the mass around our local neighborhood of galaxies isn't distributed in a sphere. It's concentrated in a flat sheet, like a pancake, with empty voids above and below.

A pancake is also what a toroid looks like from the side. And the density of the sheet increases as you look farther out---exactly what you'd expect if you're sitting inside a donut looking toward the thicker part of the ring. The researchers weren't trying to support our model. They were doing standard cosmology. But what they found is precisely what our model predicts.
\end{explainerbox}

% ═══════════ VII. VECTORS ═══════════
\section{VII. A Note on Dimensionality: Vectors, Not Dimensions}

Previous iterations of this framework referred to a ``6-Dimensional Tetroid.'' We clarify: the framework operates in standard 4D spacetime. The ``6'' refers to six directional vectors---3 spatial and 3 temporal---describing circulation patterns within the manifold, not additional dimensions. This is a descriptive framework operating within established 4D physics.

\begin{explainerbox}
An earlier version seemed to claim 6 dimensions. It doesn't. We're working in the same 4 dimensions everyone else uses. What we're adding is a description of how things flow within the donut: 3 directions in space and 3 directions in time (backward, present, forward). These are vectors---descriptions of flow---not extra dimensions.
\end{explainerbox}

% ═══════════ VIII. METHODOLOGY ═══════════
\section{VIII. Methodology: Stochastic Topological Mashing}

Over 500 unique parameter sets were explored within a 3D fractal engine, with topological stability assessed by measuring the persistence of the Euler Characteristic under $\pm$15\% randomized perturbation. Iterations yielded approximately 78\% convergence to a stable ``Attractor Triad'' (Torus: 42\%, Spiral: 24\%, Networked ``Glass Brain'': 12\%). The dominance of toroidal attractors across randomized parameter space is the primary empirical basis for the claim that toroidal geometry is a preferred topological configuration. Source code and mashing scripts are available upon request.

\begin{explainerbox}
We built a computer program that generates random 3D shapes, then tested whether those shapes stay stable when you add random noise. Over 500 runs, the donut shape dominated at 42\%. This suggests toroidal geometry isn't just one possibility among many---it's what nature defaults to.
\end{explainerbox}

% ═══════════ IX. LIMITATIONS ═══════════
\section{IX. Limitations and Future Work}

The Toroidal Momentum Engine is currently a heuristic framework. The principal limitations are: (1) the merger mechanism requires formalization in cobordism theory; (2) the Hubble Tension explanation requires ray-tracing simulations in $T^3$ cosmology; (3) the claim that quantum fields preferentially generate toroidal geometry requires first-principles derivation; (4) the stochastic mashing methodology does not constitute mathematical proof.

Future work aims to: formalize merger dynamics using geometric topology; develop ray-tracing simulations for quantitative Hubble Tension predictions; explore connections to flat-space holography (Pasterski, Shao, and Strominger 2017), where scattering amplitudes on a celestial sphere may be reinterpreted in a $T^3$ context; reprocess existing Planck CMB data using gradient vector field visualization to search for toroidal flow signatures; and conduct the toroidal data storage experiment described in Appendix A.

\begin{explainerbox}
We're honest about what this paper is and isn't. It's a framework---a set of ideas that hang together logically and are consistent with observations---but it's not a finished theory. The math needs tightening. The Hubble Tension explanation needs simulations. But the roadmap is clear, and the connections to existing physics provide multiple avenues for testing.
\end{explainerbox}

% ═══════════ APPENDIX A ═══════════
\newpage
\section*{Appendix A: Experimental Proposal --- Toroidal Volumetric Data Storage}

If the 3-Torus is nature's preferred geometry for information persistence, then a toroidal recording medium should exhibit superior storage characteristics compared to flat-topology media.

\subsection*{A.1 Experimental Design}

We propose fabricating a toroidal substrate of high-purity synthetic quartz with a major (ring) circumference of approximately 8 inches and a tube cross-section diameter of approximately 2 inches. Data encoding would be performed via femtosecond laser writing, with readout via phase-contrast microscopy or optical coherence tomography; error correction would employ Reed--Solomon codes adapted for circular topology. The proof-of-concept recording signal: ``Mary Had a Little Lamb.''

\subsection*{A.2 Storage Density}

A standard 12-inch vinyl record offers approximately 110 square inches of surface. The proposed toroid provides approximately 150 square inches of surface area, but the critical advantage is volumetric. Using CD-standard track pitch (1.6 micrometers), the 8-inch cross-section accommodates approximately 127,000 parallel ringlets. With Orbital Angular Momentum encoding, conservative estimates suggest storage densities exceeding conventional optical media by two to three orders of magnitude.

\subsection*{A.3 The Closed-Loop Advantage}

Unlike flat media with defined start and end points, a toroidal medium has no terminus. Data circulates. This property---information always present and always moving---is a direct analog of the Block Universe model.

\subsection*{A.4 Cosmological Implications}

This experiment does not prove the universe is a toroid. It demonstrates that if the universe is a toroid, certain observable properties---information density, persistence, recurrence---follow naturally from the geometry. And it is buildable with technology that exists today.

\begin{explainerbox}
Take a donut-shaped piece of high-quality glass. Use precision lasers to etch audio data into it---not just on the surface, but throughout the entire volume. Because the donut has no edges and more usable space than a flat disc, it should hold 100 to 1,000 times more data than a CD. This doesn't prove the universe is a donut. But if donut-shaped storage works better than flat storage, that's a measurable result consistent with the theory.
\end{explainerbox}

% ═══════════ APPENDIX B ═══════════
\section*{Appendix B: Supplementary Video Evidence}

\textbf{Video 1: Caltech Toroidal Vortex Ring Simulation.} Peter Schr\"{o}der's team demonstrates mathematically validated simulations of vortex ring merger dynamics, including topological surgery and daughter toroid ejection. Source: \url{https://youtu.be/lJfqgmbiqnQ}

\textbf{Video 2: Toroidal Vortex Ring Merger (Underwater Footage).} At the 16-second mark, two toroidal vortex rings merge, demonstrating energy release and structural coalescence. Source: \url{https://www.youtube.com/watch?v=pXsHmMMDFnA}

\textbf{Video 3: Jellyfish Captured by Underwater Vortex Ring.} A jellyfish is drawn into a toroidal vortex ring generated by a diver and undergoes approximately 74 rotations before ejection. The footage demonstrates two key properties: (1) the extraordinary rotational energy contained within a stable toroidal structure, and (2) the toroid's ability to warp the surrounding medium and attract nearby objects---a direct analog of the topological capture mechanism proposed for cosmological inflation. Source: \url{https://www.youtube.com/watch?v=JXkWSgU-CL0}

% ═══════════ BIBLIOGRAPHY ═══════════
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