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%\shorttitle{Magnetic Breakdown and Reconnection}
%\shortauthors{Martin}
\newcommand{\keywords}[1]{\begin{center}\textbf{Keywords:} #1\end{center}}

\begin{document}

\title{Magnetic Breakdown as the Incipient Phase of Magnetic Reconnection: \\ Energy Conservation in Colliding Magnetic Flux Tubes}

\author{Fred Lane Martin}
\affil{University of Tennessee}
\maketitle
\begin{abstract}
Magnetic Reconnection has long been recognized as the dominant mechanism responsible for solar flare energy release. However, standard models fail to adequately account for the timing, intensity, and spatial structure observed in flare events. This study introduces Magnetic Breakdown as the incipient physical mechanism underlying Magnetic Reconnection, framed through a conservation of energy analysis between interacting magnetic flux tubes. By modeling field lines as mechanical photon-based chains in Photony Theory, we show that the breakdown of these structures initiates energy release preceding reconnection. Observational support is provided through early-phase flare data and analytical treatment of chain fragmentation energetics. 
\end{abstract}

\keywords{magnetic breakdown --- magnetic reconnection --- solar flares --- magnetic fields --- plasma physics --- flux tubes --- flux ropes --- Photony Theory --- magnetic fields --- charge --- charge photons --- magnetic chains --- transduction --- free electrons}

\section{Introduction}
The release of magnetic energy during solar flares has traditionally been explained through Magnetic Reconnection (E.R. Priest et al., 2002), in which magnetic field lines of opposite polarity realign in a resistive layer, releasing energy in the form of heat, particle acceleration, and electromagnetic radiation. However, this paradigm is incomplete when considering the rapid onset and high-energy content of flares, as well as their detailed temporal and spatial structure. The present work introduces the concept of Magnetic Breakdown as the initiating process that triggers reconnection and accounts for the earliest observable flare signatures. This model is based on Photony Theory, where magnetic fields consist of quantized mechanical chains of electromagnetic-infrared (EM-IR) photon chainlets. We investigated the energetic dynamics of these chains during flux tube collisions and established a conservation-based framework for energy transformation.

\section{\textcolor{black}{Photony Theory Core Principles}}
\begingroup
\color{black}

\paragraph{Planck Scale Foundation: Reality and spacetime have been built from the ground up, starting at the Planck Length = Lp, $ \ 1.616 \times 10^{-35}$ scale, with energy filaments as the fundamental building blocks.}

\subparagraph{1. Filament Spin Dynamics: The behavior of energy filaments (spinning, orientation, linking) determines the properties of particles and phenomena observed.}

\subparagraph{2. Manhattan Metric: An orthogonal guiding principle in Planck Space, literally guiding energy filaments.}

\subparagraph{3. Photony Theory offers a mechanical, bottom-up explanation for the Universe, starting from the Planck scale.}

\subparagraph{4. Photony Theory emphasizes the dynamic behavior of energy filaments as the basis for particle properties, bonding, and interactions.}

\subparagraph{5. Photony theory seeks to align with and underpin the Standard Model while providing a fresh perspective on fundamental physics and extending to astrophysics.}


\paragraph{The core principles of Photony pertain to magnetic and electric chains as physical models of magnetic and electric field lines. Free surface electrons transduce single-ended electric chains from accumulating charge (static elemental charge photons). Free electrons within a conductive body transduce magnetic chain loops from moving charge (current flow, comprised of dynamic elemental charge photons).}

\paragraph{Single-ended electric chains in the context of solar phenomena are transduced by free electrons, which are coupled to the flux tube and rope surfaces and are responsible for the formation of electric field structures. These free electrons generate electric photon chains in the presence of charge accumulation and potential gradients along the flux tube surfaces. These electric chains are modeled to replace the electric field “lines” convention in solar and other phenomena pertaining to potential. (Martin,  2023)}

\paragraph{The internal free electrons transduce magnetic chains in and around flux tubes and ropes, which bind the internal plasma and atoms described by a frozen-in state. This magnetic transduction occurs within the conductive plasma as free electrons generate quantized, structured magnetic chains that comprise the magnetic fields. These magnetic chain loops are modeled to replace the magnetic field “lines” convention as seen in various magnetic configurations, including the Larmor radius.}

\paragraph{In magnetic structures such as solar flux tubes and flux ropes, the Larmor radius serves as a geometric indicator of how particles interact with the local magnetic field configuration. Defined classically as the radius of curvature of a charged particle’s motion around magnetic field lines, the Larmor radius in Photony Theory is reinterpreted as a marker of spatial coherence between particle motion and quantized field structure. Magnetic fields are not continuous media but are composed of structured photon chains formed via transduction by internal free electrons. Electrons do not orbit in abstract field lines; rather, they move toroidally along these quantized chains, as illustrated in a later figure showing the mechanical application of Larmor motion. This toroidal behavior reflects not true gyromotion but a mechanically guided trajectory imposed by the geometry of the photon chains. When the Larmor radius is small relative to the flux tube or rope dimensions, electron motion remains confined within the geometry of the field structure. However, as the Larmor radius grows—due to local field weakening or kinetic perturbation—electrons may begin to deviate from the curvature of the chains. This is not a transition to magnetization loss but a sign of increasing mechanical stress on the field structure. In Photony Theory, such stress accumulates until the photon chain ruptures in an event called Magnetic Breakdown, releasing stored magnetic energy through a coherent, directional mechanism. The result is a burst of particle acceleration and energy redistribution that precedes classical reconnection, rooted not in resistivity or fluid diffusion but in structural failure at the quantized level.}

\paragraph{The chains are comprised of interlinked transduced EM-IR photons, which are pairs of elemental charge (virtual) photons, as transduced by the free electrons. The transduction or interlinking of charge photons (charge) into photons further comprising chains is the core principle of Photony, providing physical models as opposed to fictitious field “lines.” These distinctions are central to Photony Theory’s redefinition of electromagnetic field structure as composed of real, quantized chains built by distinct electron populations through different transduction mechanisms.}

\paragraph{Photony Theory reconceives magnetic fields as quantized, mechanical structures, composed of discrete "photon chains" that behave mechanically, twisting, storing tension, and transmitting energy through elastic dynamics. Contrasting classical Maxwellian models, which treat fields as continuous vector entities, this framework considers fields as tangible, quantized linkages of photon units that can rupture, a process termed Magnetic Breakdown, when mechanical strain exceeds a critical rupture threshold. This rupture, predicted as a distinct, quantized precursor, triggers the abrupt release of magnetic energy into electromagnetic emission and the acceleration of plasma electrons.}

\paragraph{A compelling body of external observational evidence supports this mechanism. Photony Theory proposes that magnetic photon chains rupture under mechanical stress, releasing stored tension energy in a sudden, directional manner. This energy release occurs not through gradual dissipation but as a structural failure that delivers momentum preferentially to mobile, low-mass particles, especially electrons, resulting in non-thermal acceleration. Because the magnetic rupture process in solar flux tubes is coherent (energy release follows the internal structure of the photon chain) and directional (magnetic rupture energy propagates along the path of greatest mechanical tension of the magnetic structure), it does not immediately distribute energy isotropically to the bulk plasma. Thus, thermal heating emerges as a secondary effect, occurring after the initial energy release, rather than as its primary product (Martin, 2025). Warmuth \& Mann (Warmuth, 2020) found that in larger flares, the nonthermal electron energy often equals or exceeds the thermal energy by up to an order of magnitude. This aligns with Photony Theory’s implication that Magnetic Breakdown events lead to coherent energy release, preferentially accelerating electrons. Furthermore, Warmuth \& Mann (Warmuth, 2020) emphasize the importance of conductive and radiative losses, particularly in small events, suggesting that magnetic stress release mechanisms must operate efficiently in nonthermal channels.}

\paragraph{Additional emphasis regarding nonthermal electrons arises from  (Fleishman, et al., 2022), who report that in the 10 September 2017 flare, nearly all electrons in a large coronal volume became nonthermal, with minimal thermal contribution, stating that a “solar flare accelerates nearly all electrons in a large coronal volume.” … “Solar flares, driven by prompt release of free magnetic energy in the solar corona…” This confirms Photony Theory’s description of volume-wide chain rupture, leading to widespread, coherent acceleration across magnetic structures.}

\paragraph{rom the Global Energetics of Solar Flares series, Aschwanden et al.  (Aschwanden, 2016) estimate a warm-target low-energy cutoff for nonthermal electrons around 6.2 ± 1.6 keV, considerably lower than classical estimates (about 21 keV). They find the nonthermal-to-magnetic energy ratio to be approximately 0.41 and the thermal-to-magnetic ratio to be around 0.08, indicating that magnetic energy is primarily converted into nonthermal electron populations, consistent with Magnetic Breakdown/rupture dynamics.}

\paragraph{In consideration of quantitative energy partitioning and conservation, early electron acceleration, and volume-wide nonthermal populations, the Photony Theory model posits that flux tube and rope magnetic containment fields behave as structural, quantized photon chains, whose mechanical failure under collision-based strain efficiently channels stored mechanical magnetic energy into primarily nonthermal electrons, as the incipient phase of classical reconnection phenomena.}
\endgroup


\section{Background}
The concept of flexible tubes of force has its origins in Michael Faraday’s pioneering work on fields. Faraday introduced the idea of lines of force, not merely as abstract mathematical constructs but as physical entities capable of exerting tension and pressure. He envisioned these lines, and more explicitly tubes of force, as real structures that extended through space, even across a vacuum. This marked a shift in thinking about electromagnetism, providing a physical model rather than relying solely on action-at-a-distance. Though groundbreaking, Faraday’s tubes of force were not fully appreciated at the time. (Faraday, 1852)
\paragraph{James Clerk Maxwell later built upon Faraday’s vision, offering a more rigorous theoretical framework. He integrated Faraday’s tubes into his field equations, proposing that these tubes behaved like rotating systems within a hypothetical ether. Maxwell imagined that the centrifugal force of these rotations could explain phenomena such as attraction and repulsion between charges. However, the mechanics of tubes rotating in the same direction posed problems at contact points. To address this, Maxwell suggested that intermediate layers of counter-rotating particles—similar to idler wheels—might exist between the tubes to maintain mechanical feasibility. This visualization helped bridge field theory with mechanical intuition and was foundational in uniting electricity, magnetism, and light.}
\paragraph{S.G. Starling, writing in 1941, provided a pedagogical refinement of Maxwell's ideas by describing magnetic lines as tubes of force. He highlighted the existence of pressure between these tubes and showed how current and magnetic rotation could be conceptualized through causal motion, reinforcing the idea of field structures having mechanical properties. Starling’s account further legitimized the physical reality of field tubes in magnetic interactions.}
\paragraph{J.J. Thomson, best known for discovering the electron, also contributed to this conceptual lineage. He supported Faraday’s view of lines of force as real entities and referred to them as chains of polarized particles in a dielectric medium. According to Thomson, these lines (which he called tubes of electrostatic inductance) were not merely geometrical abstractions but carried physical meaning, capable of sustaining tension and transmitting force across space. (Thomson, 1893)}
\paragraph{Together, Faraday, Maxwell, Starling, and Thomson laid the intellectual and theoretical groundwork for understanding fields as composed of dynamic, flexible, and mechanically meaningful tubes of force—a concept that continues to influence contemporary field models, including Photony Theory and modern plasma physics.} 
\paragraph{The mechanical tension Faraday attributed to tubes, manifested as magnetic pressure and elasticity, is explained in Photony Theory by the elastic properties of magnetic chains under stress. In this way, Photony Theory provides a quantized, mechanical foundation that modernizes and substantiates Faraday’s and Thomson’s classical field concepts, connecting nineteenth-century intuition with twenty-first-century field physics.}
\paragraph{Scientific figures such as Heinrich Hertz, Einstein, and others eventually agreed with the ramifications of Faraday's theory.  From these pioneers’ concepts, particularly regarding Thomson’s chains of polarized particles, and notwithstanding a dielectric medium, Photony Theory builds directly on this legacy, providing a modern, quantized model of field lines as interlinked chains of elemental charge photons.} 
\paragraph{These "Photony chains" serve the same role as tubes of force and polarized particles, but with defined quantum structure and mechanical behavior. These chains store energy, convey spin, and can rupture under stress, especially during solar flux tube collisions, releasing energy in a process known as magnetic breakdown. In essence, Photony chains are the structural and energetic realization of the classical tubes of force, transforming a 19th-century conceptual model into a 21st-century physical theory.}

\section{Flux Tube and Rope Composition}
Sunspots are surface manifestations of strong magnetic flux tubes emerging from the solar interior. While they are often visualized as roughly circular features, sunspots frequently exhibit irregular shapes or cluster into complex active regions. This irregularity arises because sunspots form where bundles of fibrils (small magnetic flux threads) coalesce at the photosphere. (Parker, 1979) (Fan, 2009)  A single large-scale flux tube is often composed of multiple fibril groups, and as these groups rise through the convection zone, slight variations in buoyancy, magnetic tension, and convective flows can cause multiple emergence points at the surface. This process naturally leads to the formation of multiple sunspots or sunspot groups linked by a common underlying magnetic structure, rather than a single monolithic spot.
\paragraph{Active regions near sunspots provide ideal environments for magnetic breakdown. Subsurface convection and rotational shear generate twisted flux tubes that carry significant magnetic helicity. (Fan, 2009) (E.R. Priest, 1979) As these twisted flux systems emerge into the solar atmosphere, they interact with preexisting fields or with other emerging tubes, creating sites of strong magnetic stress accumulation. When two flux tubes collide or press against one another in the low corona, the embedded magnetic chains—conceptualized in Photony Theory as mechanical structures—are stretched, compressed, and entangled, leading to an increase in local field stress. }
\paragraph{Once a critical stress threshold is exceeded, magnetic breakdown can initiate, leading to localized energy release and particle acceleration. Observational evidence supporting this model includes measurements of magnetic shear near polarity inversion lines, pre-flare brightenings in extreme ultraviolet (EUV) and X-ray wavelengths, and evolving topological changes captured by vector magnetograms. (S. Toriumi, 2019)    (Schrijver, 2009) These phenomena are consistent with the early stages of flare initiation, where magnetic breakdown serves as the precursor to large-scale reconnection and eruption. Along with rising flux tubes, subsurface convection and shear generate twisted flux tubes (ropes), which rise and interact in the corona. As they rise and fall, they may collide, magnetic chains are stretched, compressed, and entangled, increasing local field stress until the critical threshold is met. Observations of magnetic shear, pre-flare brightenings, and evolving topologies from vector magnetograms support this model.}
\paragraph{Magnetic flux tubes in the solar atmosphere are not singular, monolithic structures but are instead composed of multiple extended, smaller magnetic fibrils (shown in Figure 1). Fibrils are fine-scale, thread-like magnetic features that trace the field line structures observed particularly in the chromosphere and photosphere. High-resolution solar observations and magnetohydrodynamic (MHD) simulations have demonstrated that large-scale flux tubes, such as those associated with sunspots, prominences, and coronal loops, are actually bundles of these finer fibril structures. (Parker, 1979) (Priest, 2014)  As fibrils cluster and interact, they form coherent larger tubes that can collectively trap and guide plasma and hydrogen across various layers of the solar atmosphere. This fibril composition persists even into the corona, where fine filamentation within magnetic loops remains visible despite large-scale expansion. Thus, solar flux tubes should be understood as composite structures, consisting of many dynamic, interacting fibrils that together give rise to the observed large-scale magnetic behavior.}

\begin{figure}
    \centering
    \includegraphics[width=1.0\linewidth]{image1.png}
    \caption{Flux Tubes/Ropes, and Fibrils Structure}
    \label{fig:enter-label}
\end{figure}

\section{Solar Flux Tubes and Ropes}
Solar flux tubes and ropes (twisted tubes) are central structures in understanding solar magnetism and flare activity. These are concentrated bundles of magnetic field lines (chains) that confine plasma within the solar atmosphere. They often emerge from the solar interior and extend through the photosphere, chromosphere, and into the corona (shown in Figure 2). Sunspots mark the surface footprint of large flux tubes, where intense magnetic fields inhibit convection, causing localized cooling and the dark appearance of the spots. These sunspots commonly form part of active regions, areas on the Sun associated with heightened magnetic complexity and flare potential.
\paragraph{Flux tubes can rise through the solar atmosphere due to a combination of buoyancy, thermal gradients, and gravitational stratification. In the deeper convection zone, plasma motion and rotational shear may twist magnetic field lines into coiled flux ropes. When the magnetic pressure within a flux tube exceeds the external gas pressure, the tube becomes buoyant and begins to ascend. The external density and pressure drop as it rises, while temperature gradients across the photosphere, chromosphere, and corona influence its expansion and curvature. In the lower atmosphere, the rising flux is moderated by gas pressure and gravity, but in the upper layers—especially the transition region and corona—flux tubes can expand more freely and reconnect with overlying field lines (magnetic chains) or other emerging structures.}
\paragraph{This dynamic environment, especially within complex active regions, creates ideal conditions for flux tube collision, twisting, and entanglement, setting the stage for energy buildup and release via magnetic breakdown and reconnection. Understanding how flux tubes rise and evolve across solar layers is essential for modeling solar storms and predicting their onset.}

\section{Magnetic Chain Convergence and Flux Tube Interaction}

\paragraph{Gravity and buoyancy are causal factors in Magnetic Reconnection theory and provide natural mechanisms for enhancing reconnection efficiency. In solar environments, gravitational forces significantly influence plasma motion. During the early phase of the Reconnection Process, plasma from inside magnetic flux tubes is heated, and buoyancy causes it to rise, thinning the current sheet and reducing its length, which can speed up the reconnection process. This effect helps the Sweet-Parker model better align with observational data from solar flares, coronal mass ejections, and magnetospheric substorms. In addition, the instability of the interchange provides another pathway for improved reconnection. (J. Birn, 2007), (Yokoyama, 1995)}

\begin{figure}
    \centering
    \includegraphics[width=1\linewidth]{Flux Tube Rising.png}
    \caption{Magnetic Convergence in the Magnetic Reconnection Zone}
    \label{fig:enter-label}
\end{figure}

\section{Frozen-in Plasma}
In Photony Theory, magnetic fields are structured as quantized chains composed of elemental charge photons. When assembled into longer structures, these mechanical entities form elongated interlinked (classical) photon chainlets (as continuous chains that define and model the magnetic field lines encompassing and containing “frozen-in” solar plasma as shown in Figure 3. When solar magnetic flux tubes—curled Larmor cylindrical regions of magnetic fields of chains confining frozen-in plasma—collide, their embedded magnetic chains interact. Stress builds in the region of interaction, especially where opposing spin orientations lead to topological incompatibility. When a critical density and stress threshold is reached, the chains fragment—a process defined as Magnetic Breakdown.

\begin{figure}
    \centering
    \includegraphics[width=1\linewidth]{larmor.png}
    \caption{Larmor Radius and Magnetic Chains Containment}
    \label{fig:enter-label}
\end{figure}

\paragraph{\textbf{}The Larmor radius is also relevant to the dynamics of Magnetic Breakdown as described by Photony Theory. When the Larmor radius becomes comparable to the scale length of magnetic gradients, such as those near current sheets or within flux ropes, electrons may experience local instabilities that contribute to magnetic fragmentation. In this regime, high-energy electrons confined to small spiral orbits may be trapped within collapsing magnetic chain structures, further enhancing their acceleration. These electrons then become primary sources of Gyrosynchrotron emission and also influence the evolution of the Magnetic Breakdown process.}

\section{Sweet-Parker Model and Extended Eruption}
The Sweet-Parker model is the classical resistive reconnection framework in magnetohydrodynamics (MHD), describing a steady-state configuration where opposing magnetic fields converge in a long, thin diffusion region. Within this model, the reconnection rate is limited by plasma resistivity, leading to a relatively slow energy release. Although useful in idealized cases, the Sweet-Parker model underpredicts the explosive flare dynamics observed in the corona. 
\paragraph{The Magnetic Breakdown framework posits that before a resistive diffusion region forms, localized fragmentation of magnetic chains initiates rapid energy liberation. Magnetic breakdown catalyzes reconnection by pre-conditioning the field geometry, generating plasma outflows, and reducing magnetic tension.}
\paragraph{Figure 4 shows the generalized Sweet-Parker Model. The magnetic ﬁelds, B, have opposite directions above and below a reconnection layer. Plasma is frozen into the magnetic ﬁelds, enters the Reconnection Layer at the velocity Vz, and moves out at the velocity Vx in opposing +/- directions. (Goldstein, 2013) The plasma outflow is in the x direction (x-axis) and its thickness 2δ in the inflow direction (z-axis).}
\paragraph{The Sweet-Parker model defines a diffusion layer between converging magnetic fields. However, its energy release rate is too slow.}

\begin{figure}
    \centering
    \includegraphics[width=1\linewidth]{image3.png}
    \caption{The Generalized Sweet–Parker Model}
    \label{fig:enter-label}
\end{figure}

\paragraph{Photony-enhanced models introduce Magnetic Breakdown first, accelerating reconnection. Key equations include:}
\begin{equation}
\rho V_z L = \rho V_x \delta
\end{equation}
\begin{equation}
V_x = \frac{B}{\sqrt{4\pi \rho}}
\end{equation}
\begin{equation}
V_z = V_A \frac{\delta}{L}
\end{equation}

Here, VA is the Alfvén speed, derived from energy conservation. Assuming that the magnetic energy is converted into kinetic energy in the outflow, equate:

\begin{equation}
1/2 \cdot \rho \cdot (Vx)^2 = B^2/(8 \cdot \pi)
\end{equation}

\begin{equation}
\rho \cdot (Vx)^2 = B^2/(4 \cdot \pi)
\end{equation}

4π is the Photony Compression Factor (Martin, 2023). The product, 4π, represents the interaction between energy and quasi-static Planck space, which appears in many fundamental physics equations and relates to how energy filaments compress Planck space.

Solving for Vx:

\begin{equation}
Vx = B/sqrt(4 \cdot \pi \cdot \rho)
\end{equation}

\begin{center}
\[
Note: V_{x} = V_{A}
\]
\end{center}




This relation shows that the reconnection inflow speed Vz is constrained by the geometry of the reconnection layer. Because δ / L is typically very small in high-conductivity plasmas, the reconnection rate predicted by this model is very slow. 

This geometric bottleneck is a key limitation of the classical Sweet–Parker Model when applied to fast energy-release phenomena such as solar flares, which often require enhanced or alternative mechanisms to explain observed reconnection rates.

\section{The Photony Rendition of the Sweet-Parker Model}
Figure 4 shows an expanded view of single layers of oppositely spinning elemental charge photon chains. The fields of magnetic chains are generated by opposing current sheets, each converging on the Reconnection Zone at velocity Vz from opposite directions.  
The Sweet–Parker reconnection geometry (Figure 4) and the Photony-based magnetic chain model shown in Figure 5 align well:
\paragraph{The length, L, is aligned with the Horizontal x-axis: }
This represents the length of the current sheet (sometimes also called the reconnection layer extent or arcade span), running along the polarity inversion line. It is where oppositely directed magnetic field lines (chains) run nearly parallel in opposing directions on either side of the sheet.
Note that the dimensions of x and y are relatively small compared to the solar radius.
\paragraph{δ (thickness) aligned with the Vertical z-axis: }
This marks the thickness across which the magnetic field reverses, and it's the critical dimension for calculating the current via $I = B \cdot \delta / \mu_0$. In this narrow vertical dimension, the two magnetic domains press toward each other, bringing oppositely directed field lines (chains) into contact.
\paragraph{Depth of x-y plane aligned with the y-axis: }
This represents the depth of the current sheet, δ, which doesn’t directly affect 2δ current sheet dynamics but is important for total volumetric energy and 3D modeling of reconnection. is also relevant to compute the total current as $J \cdot A$, where the cross-sectional area, A, includes the current sheet depth ($A = L\cdot\delta$). 

\begin{figure}
    \centering
    \includegraphics[width=1\linewidth]{image5.png}
    \caption{Expanded view of single layers of oppositely spinning elemental charge photon chains}
    \label{fig:enter-label}
\end{figure}

The expanded view of single layers shows oppositely spinning elemental charge photon chains generated by opposing current sheets while converging on the Reconnection Zone at velocity Vz from opposite directions. The magnetic ﬁelds, BU and BL, are modeled as magnetic chains with opposite spin directions above and below the reconnection layer.
\paragraph{Magnetic chains (loops) around the current sheet:  These represent the Photony model’s visualization of magnetic fields as structured photon chains, looping from one side of the current sheet to the other. This is conceptually compatible with magnetic field chains threading around and into the reconnection region.}
\paragraph{Reconnection zone between the opposing current sheets: This central region is where Magnetic Breakdown initiates and reconnection proceeds. Field lines (or chains) from above and below are forced into this narrow layer due to plasma inflow and magnetic tension. The combined thickness of the two current sheets is assigned initially to the thickness of the connection zone.}
\paragraph{Approaching magnetic chains under compression and tension: }
In standard MHD, mechanical pressure and shearing that build up before reconnection are often described as magnetic tension or stress. The pre-breakdown phase in magnetic flux tube interactions involves both magnetic strain and stress, each playing a distinct but interconnected role in the lead-up to energy release. Magnetic strain refers to the geometric deformation of the field structure, manifesting as twisting, stretching, and compression of photonic chains within interacting flux tubes. This deformation arises from shear flows, rotational motion near sunspots, and the converging dynamics of neighboring magnetic systems. As strain accumulates, internal forces develop, manifested as magnetic stress, including magnetic tension, pressure gradients, and Lorentz forces that act to restore equilibrium. These stresses are concentrated particularly in regions of high curvature or entanglement within the chain structure. When the magnetic stress exceeds the mechanical strength of the photonic chains, fragmentation occurs, initiating magnetic breakdown. This process releases stored magnetic energy through chain rupture, marking the transition from a strained but stable configuration to one of rapid reconfiguration and energy dissipation. Thus, magnetic breakdown is precipitated by a sequence of escalating strain-induced stress, aligning with both physical intuition and observational characteristics of early flare dynamics.
\paragraph{The Photony chain-based Sweet-Parker model is visualized more concretely as physical compression and misalignment of adjacent chains, which initiates collisions, structural collapse, and chain-reaction-based Magnetic Breakdown. (Martin, 2023)}


\section{Energy Release Timeline}
This chain-based model is physically consistent and adds explanatory depth that standard field-line diagrams lack. It captures the geometry, directional flows, field structure, and the onset mechanism for reconnection—all in line with Sweet–Parker theory but adapted to the elemental charge photon chain framework.

\paragraph{Energy release occurs in five stages:}
\begin{enumerate}
    \item Magnetic Tension Buildup (0--2 min)
    \item Breakdown Trigger (\textasciitilde2 min)
    \item Chain Breakdown Acceleration (2--25 min)
    \item Peak Output (\textasciitilde25 min)
    \item Declining Energy: Breakdown + Reconnection (25--90 min)
\end{enumerate}

Stage 1- Magnetic Tension Buildup
Time Range: 0 – 2 Minutes
This corresponds directly with the local Sweet-Parker Model wherein elemental charge photon chains begin moving into the interaction zone. Magnetic shear and tension increase between flux tubes. This phase also represents the gradual accumulation of magnetic stress in the flux tube system due to the movement, twisting, and compression of magnetic flux tubes. Energy rises slowly.
\paragraph{Stage 2- Extended Eruption Magnetic Breakdown Trigger }
Beginning at approximately 2 minutes, this marks the onset of structural failure in the large-scale magnetic configuration. Magnetic field lines (chains) lose coherence, and the internal magnetic chain structure collapses in a cascading fashion. This is the incipient, nonlinear trigger phase leading to full eruption.
$P = P_{trigger}$ = 0.5 × 10²² W
\paragraph{Stage 3- Rising Chain Breakdown}
Time Range: 2–25 min
Chain collapse accelerates energy release nonlinearly. This model represents exponential chain failure, not reconnection.
Equation: 
$P_{2}(t) = P_{trigger} + (P_{peak} - P_{trigger})(1 - e^{-(t - 2)/\tau})$ 
where τ = 5 min
\paragraph{Stage 4- Peak Energy Output}
Around 25 minutes, the system reaches its maximum rate of magnetic energy release. The stored magnetic energy has largely been converted into kinetic, thermal, and radiative outputs. This marks the culmination of the exponential rise in energy seen during breakdown and early reconnection.
Ppeak = 1.44 × 10²² W
Stage 5- Declining Energy: Breakdown + Reconnection
Rime range: 25 – 90 minutes

\section{Sweet–Parker Model: Theoretical Model and Energy Timeline}
In Figure 6, the X-class Flare Energy Timeline for Local Magnetic Reconnection (Sweet–Parker) provides an energy release rate (power) view and a cumulative view of how magnetic energy is cumulated in a localized flare region over time. This plot focuses on a modeled total energy budget that begins near zero at two minutes, rises to a peak of approximately 3.5 × 10²¹ joules around 3.5 minutes, and then gradually tapers to about 2.3 × 10²¹ joules by twelve minutes. This curve reflects a realistic energy envelope for a confined reconnection site in the lower corona during an X-class flare based on observationally motivated timelines and the expected evolution of energy storage and dissipation. The plot represents energy released and is expressed in joules. It represents how much energy is released cumulatively during a specific, event-related timescale.
\paragraph{The local Sweet-Parker plot shown in Figure 6 models the energy release timeline of a strong (high-intensity) solar flare (X-class) by modeling the stages of magnetic activity with a continuum of realistic energy values.}

\begin{figure}
    \centering
    \includegraphics[width=1\linewidth]{image6.png}
    \caption{X-class Flare Energy Timeline- Local Magnetic Reconnection (Sweet-Parker)}
    \label{fig:enter-label}
\end{figure}

\paragraph{Figure 6 shows the energy release rate and cumulative energy plot from a physically derived, time-dependent energy release function using the Sweet–Parker model. The release rate is plotted in red and has units of joules per second (J/s). It peaks at around 3.5 minutes, consistent with the expected reconnection dynamics, and is modeled using an exponential rising profile and a falling Gaussian profile to simulate the natural rise and fall of flare activity. }
\paragraph{The blue curve in the same plot shows the cumulative energy, obtained by integrating the energy release rate over time. This blue cumulative energy curve closely follows the original Sweet–Parker timeline plot profile, rising to roughly the same peak value and gradually tapering toward a final state that matches earlier models.}

\section{Extended Eruption Model and Energy Timeline}
The extended eruption plot shown in Figure 7 models the energy release timeline of a strong (high intensity) solar flare (model) by combining theoretical stages of magnetic activity with realistic energy values. It follows a five-stage structure. 
\paragraph{The labeled stages on the Extended Eruption Timeline plot represent the key phases of an X-class flare and CME event as modeled in the eruption energy curve. The extended eruption flare model presents a time-resolved profile of magnetic energy release during a large-scale X-class solar event. }
\paragraph{Unlike the shorter, more impulsive Sweet–Parker scenario, this case models a sustained, high-energy flare spanning 90 minutes and incorporates both Magnetic Breakdown and reconnection dynamics.}
\paragraph{The plot displays two key outputs:}
\begin{enumerate}
    \item The energy release rate (blue curve), power expressed in watts (J/s)
    \item The cumulative energy released (red curve), expressed in joules (J)
   \end{enumerate}

\begin{figure}
    \centering
    \includegraphics[width=1\linewidth]{image7.png}
    \caption{X-class Flare Extended Eruption Energy Timeline}
    \label{fig:enter-label}
\end{figure}
\paragraph{The blue curve rises slowly at first as magnetic tension builds, then accelerates through a nonlinear Magnetic Breakdown phase. It reaches a peak power output of 1.44 × 10²² W around 25 minutes, after which it gradually declines following a Gaussian profile, representing the transition into reconnection and magnetic relaxation. The cumulative energy curve, shown in orange, is derived by integrating the release rate over time and is constrained to a total energy budget of 3.0 × 10²⁵ J, consistent with observational estimates for extended X-class eruptions.}
\paragraph{The expression for power, or energy per unit time, is defined as $P = U_{B} × V_{in} × A$, where $U_{B}$ is the magnetic energy density, $V_{in}$ is the reconnection inflow speed, and A is the cross-sectional area of inflow. This equation links the instantaneous power output to the magnetic field structure and reconnection environment.}

\paragraph{The model curve itself is composed of:}

\begin{enumerate}
    \item A quasi-linear ramp-up phase (0–2 min)
    \item An exponential rise from breakdown (2–25 min)
    \item A Gaussian decline (25–90 min)
   \end{enumerate}


\paragraph{Together, these stages capture the distinct phases of flare evolution—from tension buildup and cascading magnetic failure to reconnection and large-scale field reconfiguration. The resulting profile closely matches known flare behaviors and provides a physically interpretable basis for the modeled energy release.}
\paragraph{X-class Flare Extended Eruption Timeline Parameters:}

\paragraph{All the parameters, equations, and modeling structures required to replicate the final combined energy release rate and cumulative energy plot for an X-class extended eruption flare are provided below. All values are directly verified against the final curve data.}


\section{Breakdown as a Conservative Energy Process}
The breakdown follows conservation of energy:

\paragraph{Magnetic breakdown is modeled as a physically conservative energy release process that occurs when interacting flux tubes reach a critical mechanical instability. At the core of this process is a chain-based structure defined by Photony Theory—quantized photon chains that are represented by magnetic field lines within solar plasma.} 

\paragraph{These chains, embedded in colliding magnetic flux tubes, store tension and energy due to mutual compression and topological stress. Breakdown initiates once the mechanical integrity of these chains fails under excess tension or compression.}

\begin{equation}
E_{\text{critical}} = \frac{1}{2} \mu_0 \frac{B_{\text{chain}}^2}{\rho_{\text{chain}}}
\end{equation}
\text{where:}

\begin{equation}
\mu_{0}\ \text{is the permeability of free space,}
\end{equation}

\begin{equation}
\text  B_{chain} \ \text{is the B-field strength, and }
\end{equation}

\begin{equation} \rho_{chain} \ \text{is the density of the flux tube. }
\end{equation}


\paragraph{This transformation results in kinetic plasma energy, radiation, and thermal heat—all magnetically derived.}

\paragraph{Breakdown occurs when the compression and elastic tension in these chains exceeds their mechanical integrity, causing a chain reaction.}

\paragraph{At this threshold, magnetic chains rupture and fragment, converting stored magnetic energy into broad-spectrum electromagnetic and photon (chainlet) energy release collectively as electromagnetic radiation, followed by plasma kinetic energy and localized thermal heating. This transformational Magnetic Breakdown process strictly follows the law of energy conservation.}

\paragraph{This model's Magnetic Breakdown process accounts for the early and intense energy release preceding full reconnection. The energy release path reflects the findings of Aschwanden and others, where \textbf{flare onset and reconnection are best explained by a magnetic energy source, not plasma pressure or coronal thermal dynamics}. This fact cannot be overstated with regard to the energy release mechanisms during the Magnetic Breakdown incipient phase of Magnetic Reconnection Theory}

\paragraph{The mechanical failure of magnetic chains effectively front-loads the flare energy budget, setting the stage for rapid reconnection and subsequent eruption.}


\section{The Sequential Stages and Pathway of Magnetic Chain Fragmentation}

\paragraph{In Magnetic Breakdown, the observed partitioning of flare energy into different channels maps well onto the sequential stages of magnetic chain fragmentation as shown in Figure 8.} 

\begin{figure}
    \centering
    \includegraphics[width=1\linewidth]{energy flow chart.png}
    \caption{Magnetic Breakdown Stages from Magnetic Chain Compression to Magnetic Breakdown and Flare Onset to an Extended Eruption.}
    \label{fig:enter-label}
\end{figure}

\paragraph{The breakdown process initiates energy release before classical reconnection, producing a distinct fragmentation spectrum of electromagnetic radiation as chains rupture under mechanical stress.}

In Magnetic Breakdown, the observed partitioning of flare energy into different channels maps well onto the sequential stages of photonic chain fragmentation as shown in Figure 8. The breakdown process initiates energy release before classical reconnection, producing a distinct fragmentation spectrum of electromagnetic radiation as chains rupture under mechanical stress. 
\paragraph{These early releases—especially in the extreme ultraviolet (EUV) and X-ray bands—correspond to the initial photon emissions that precede substantial plasma motion or large-scale restructuring. This timing distinction is critical: photon emissions occur nearly instantaneously with chain rupture, whereas the secondary nonthermal particles and CME launch follow the onset of reconnection and magnetic restructuring.}
\paragraph{Using the benchmark energy ratios from Aschwanden et al. (2017), we see that approximately 51 percent of dissipated magnetic energy is channeled into nonthermal electrons, 17 percent into nonthermal ions, and 7 percent into the kinetic energy of coronal mass ejections (CMEs). Another 7 percent contributes to thermal heating of flare plasma, and the remaining 18 percent is likely distributed across alternative mechanisms or represents modeling uncertainties. These values, derived from a statistically complete sample of GOES M1.0 flares, align with the breakdown model in that the bulk of energy is stored and released via magnetic mechanisms, before and independent of plasma-driven dynamics. (Aschwanden, 2016)}

\paragraph{In this context, it is essential to distinguish thermal vs. nonthermal particles. Thermal electrons and ions follow a Maxwellian velocity distribution, characteristic of particles heated to equilibrium by collisions, and are responsible for the gradual heating phase and soft X-ray emissions. In contrast, nonthermal electrons and ions are accelerated to higher energies through magnetic processes such as electric fields at current sheets or magnetic breakdown sites. They exhibit a power-law distribution and are associated with impulsive bursts, hard X-ray emission, and particle precipitation into the chromosphere.}

\paragraph{Therefore, the timing and composition of energy release during a flare strongly support a model where Magnetic Breakdown precedes Magnetic Reconnection, with photonic chain fragmentation initiating early, high-frequency radiation and conditioning the field structure for subsequent particle acceleration and CME launch.}


\section{\textcolor{black}{Quantized Chain Energetics and Energy Accumulation}}

\begingroup
\color{black}

To quantify magnetic energy buildup within the framework of Photony Theory, we model magnetic fields as composed of discrete, spin-aligned photon chains. Each chain is formed by interlinked \emph{classical photons}, where each classical photon is a composite of two elemental charge photons. These classical photons possess a characteristic spin frequency associated with the infrared/microwave boundary at $\nu = 3.76 \times 10^{13}$ Hz. This represents the minimum energy required for photon production at approximately an 8 micron wavelength.

The energy of each classical photon is given by Planck's relation:
\begin{equation}
E_{\gamma} = h \cdot \nu,
\end{equation}

where $h = 6.626 \times 10^{-34} \, \text{J·s}$ is Planck’s constant. Substituting:
\[
E_{\gamma} = (6.626 \times 10^{-34}) \cdot (3.76 \times 10^{13}) \approx 2.49 \times 10^{-20} \, \text{J}.
\]


Assuming that each photon occupies a spatial length approximately equal to its wavelength, $\lambda = 8 \, \mu\text{m} = 8 \times 10^{-6} \, \text{m}$, and that photons are transduced continuously over a duration of $t = 600$ seconds, the number of classical photons (as chainlets, further interlinked during transduction) in a single chain is:
\[
N_{\gamma/\text{chain}} = \frac{c \cdot t}{\lambda} = \frac{(3.0 \times 10^8)(600)}{8 \times 10^{-6}} \approx 2.25 \times 10^{16}.
\]


The total energy stored in a single chain is then:


\begin{align*}
E_{\text{chain}} &= N_{\gamma/\text{chain}} \cdot E_{\gamma} \\
                 &= (2.25 \times 10^{16}) \cdot (2.49 \times 10^{-20}) \\
                 &\approx 5.60 \times 10^{-4} \, \text{J}.
\end{align*}



To model flare-scale energy release, we consider a flux rope comprising $N_{\text{chains}}$ magnetic chains. If a rupture event affects a transverse bundle of $10^{22}$ chains, the total released energy becomes:

\begin{equation}
E_{\text{rupture}} = N_{\text{chains}} \cdot E_{\text{chain}}
\end{equation}
\begin{align*}
E_{\text{rupture}} 
& =  10^{22} \cdot 5.60 \times 10^{-4} \\
                   & \approx 5.6 \times 10^{18} \, \text{J}
\end{align*}

This value represents a localized magnetic breakdown, demonstratable by the Sweet-Parker Model, and when extended across larger volumes or multiple reconnection sites, the cumulative energy can approach or exceed $10^{25}$ J—consistent with the observed energetics of X-class solar flares. This approach provides a quantized, mechanical explanation for magnetic energy storage and release, linking the photon-scale architecture of the magnetic field to macroscopic flare dynamics.

\endgroup

\section{\textcolor{black}{Chain Fragmentation Energetics in Photony Theory- A Mathematical Framework}}

\begingroup
\color{black}

\paragraph{The following mathematical framework assumes an X-class flare resulting from collisions of flux tubes, each having a magnetic flux density of 0.3 T. To strengthen the quantitative foundation of Photony Theory, we examine the energetics of magnetic chain rupture in the context of an X-class solar flare, assuming a magnetic field strength of $B = 0.3\,\text{T}$.}


\subsection*{Magnetic Energy Density}

The magnetic energy density is given by:
\begin{equation}
u_B = \frac{B^2}{2\mu_0}
\end{equation}

where:

$\mu_0 = 4\pi \times 10^{-7}\,\text{H/m}$ 

Substituting $B = 0.3\,\text{T}$:

\[
u_B = \frac{(0.3)^2}{2 \cdot 4\pi \times 10^{-7}} \approx 3.6 \times 10^4\,\text{J/m}^3
\]

\subsection*{Volume of Fragmentation}
\textcolor{black}{
Assuming the rupture affects a cylindrical region (corresponding to a flux tube section) of the magnetic field with radius $r = 5 \times 10^6\,\text{m}$ and length $l = 10^7\,\text{m}$, the volume is:}
\[
V = \pi r^2 l = \pi (5 \times 10^6)^2 \cdot 10^7 \approx 7.85 \times 10^{20}\,\text{m}^3
\]

\subsection*{Total Stored Magnetic Energy}


The total magnetic energy stored in the affected volume is then:

\begin{align*}
E_{\text{stored}} 
&= u_B \cdot V = 3.6 \times 10^4 \cdot 7.85 \times 10^{20} \\
&\approx 2.83 \times 10^{25} \, \text{J}
\end{align*}



This magnetic energy stored is consistent with energy releases observed in large X-class solar flares and the Extended Eruption Energy Timeline (Figure 7).

\subsection*{Mechanical Basis of Chain Rupture}

In Photony Theory, magnetic fields are composed of a multiplicity of quantized photon chains that store additional mechanical energy (beyond that of plasma containment) via mechanical torsion and curvature. Rupture occurs when accumulated stress exceeds a critical threshold:

\begin{equation}
\tau_{\text{crit}} = \frac{F_{\text{rupture}}}{A_{\text{chain}}}
\end{equation}

Each photon chain segment can be modeled as an elastic element with torsional energy:
\begin{equation}
E_{\text{chain}} = \frac{1}{2} k_{\theta} \theta^2
\end{equation}

where:

$k_{\theta}$ is the torsional stiffness and $\theta$ is the angular displacement. 

While $k_{\theta}$ must be empirically calibrated, this formulation enables parametric estimation of localized energy release per rupture event, forming a bridge between structural mechanics and flare-scale energetics.

\endgroup

\section{Flux Tube Twisting and the Transition to Flux Ropes (and Instability)}
Twisting contributes to flux rope formation and potential instability. When twist exceeds 1.25--2 turns, systems may erupt. Kink instability and reconnection drive energy release in these scenarios.

\paragraph{Twisting of magnetic flux tubes is a fundamental aspect of solar magnetic structure and plays a significant role in the onset of magnetic breakdown. As flux tubes emerge from the solar interior, they often exhibit intrinsic twist due to subsurface rotational shear and convective motion, resulting in magnetic helicity—a measure of the winding of magnetic field lines. (Fan, 2009) This twist contributes to the accumulation of magnetic stress and energy within the tube. When two flux tubes interact or collide in the solar corona, their internal twist influences whether the system remains stable, reconnects, or undergoes magnetic breakdown.} 
\paragraph{Regions of concentrated twist are particularly prone to instabilities such as the kink instability, which can localize magnetic tension and trigger chain fragmentation in models like Photony Theory. In this framework, twisted photonic chains experience increased rotational tension, making them more susceptible to rupture under stress, which initiates energy release via magnetic breakdown.}

\paragraph{Importantly, as the internal twist increases and becomes more coherent, a twisting flux tube can transition into a flux rope—a magnetic configuration in which field lines wrap around a central axis in a helical fashion. This transformation typically occurs when the twist angle exceeds approximately 1.25 to 2 full turns, a threshold beyond which the structure acquires a distinct topological identity. (T. Torok, 2005) Photospheric shearing flows, sunspot rotation, and continued helicity injection are common drivers of this evolution. (Fan, 2009) In some cases, a flux rope may emerge fully formed from below the photosphere, while in others, it develops through magnetic reconnection processes, such as tether-cutting reconnection, that reconfigure the magnetic field into a coherent rope structure. (G. Aulanier, 2010) }

\paragraph{During Magnetic Breakdown, the twist within such ropes may either intensify locally or begin to redistribute. The rupture of magnetic chains leads to a rapid release of stored magnetic tension, and this may result in either partial unwinding of field line twist or its reconfiguration into newly formed magnetic structures. In eruptive events, magnetic twist is often observed to transfer into reconnected field lines or expelled with a coronal mass ejection (CME), supporting a transition from magnetic breakdown to large-scale reconnection dynamics. (S. Inoue, 2018) Observational and numerical studies confirm that highly twisted magnetic configurations are more susceptible to destabilization and eruptive outcomes, further supporting the role of twist as a precursor and catalyst in magnetic breakdown and flare onset. (Aschwanden, 2016) (T. Torok, 2004)}


\section{Observational Support}
Instruments like RHESSI and SDO show early radiation spikes consistent with magnetic breakdown before reconnection. Observations support Magnetic Breakdown as the incipient phase of Magnetic Reconnection. These show initial bursts of radiation in EUV and hard X-ray spectra prior to the classic reconnection phase. This timing is consistent with the energy release rate model, where Magnetic Breakdown (as shown in Figure 6) begins energy release prior to the formation of a Magnetic Reconnection site. Quantitative estimates showing magnetic energy dominates the flare energy budget, further supporting the Magnetic Breakdown hypothesis, are provided. (A.G. Emslie, 2005)  (Aschwanden, 2016) 

\paragraph{Recent observational studies, particularly those from (Aschwanden, 2016), support the magnetic nature of this energy origin. They demonstrate energy closure—where the total of kinetic, thermal, and CME energies nearly equals the dissipated magnetic energy—with a mean ratio of Esum/Emag = 0.87 ± 0.18. Moreover, the dominant portion of energy in flares resides in non-thermal electrons and early magnetic dissipation, not in thermalized plasma energy, which represents a smaller fraction (typically less than 15 percent of total dissipated energy).}


\paragraph{
\textcolor{black}{The total energy released by a large solar flare is on the order of $3\times 10^{25}$ joules, predominantly in the form of kinetic energy and electromagnetic radiation. Among all plausible sources of this energy, only the magnetic field offers a sufficient energy density to account for such output. Priest, E.R., \& Forbes, T. 2002}}

\section{Magnetic Breakdown Fragmentation of Magnetic Chains}
The breakdown process resembles brittle fracture propagation in materials under stress. Once local chains fragment, cascading failure propagates through the magnetic skeleton at near-relativistic speeds. This chain reaction (shown in Figure 9) explains the impulsive phase of flare emissions, characterized by rapid increases in intensity across multiple wavelengths and particle fluxes.

\begin{figure}
    \centering
    \includegraphics[width=1\linewidth]{image9.png}
    \caption{Chain Fragmentation During Magnetic Breakdown}
    \label{fig:enter-label}
\end{figure}

In the Photony model, magnetic fields are supported by coherent chains of elemental charge photons transduced by current-carrying electrons. These chains maintain alignment, electromagnetic curvature, and Lorentz tension across the flux tube or twisted magnetic rope structure. 

When shear, compression, or overextension is applied, these chains undergo rapid destabilization. The resulting Magnetic Breakdown fragments of magnetic chains including smaller chainlets, qualify as classical photons.

\paragraph{The distribution of fragments in terms of Spectral Band and Observations is shown in Table 1. The Spectral Bands range from radio waves (including microwaves) to photons with various observational designations and characteristics based on energy and effects.}

% Requires: \usepackage{graphicx}
\begin{table}[h]
    \centering
    \begin{tabular}{|l|l|}
    \hline
    \textbf{Spectral Band} & \textbf{Observations} \\ \hline
    Radio (kHz--MHz)       & Type II \& III bursts \\ \hline
    Microwaves (GHz)       & Gyrosynchrotron       \\ \hline
    Infrared               & Continuum rise        \\ \hline
    Visible                & White-light flares    \\ \hline
    UV                     & Line/continuum brightenings \\ \hline
    EUV                    & Broadline emission    \\ \hline
    Soft X-ray             & Flare classification  \\ \hline
    Hard X-ray             & Bremsstrahlung        \\ \hline
    Gamma-ray              & Nuclear lines, pions  \\ \hline
    \end{tabular}
    \caption{Spectral Bands and Their Associated Observations}
    \label{tab:spectral_bands}
\end{table}

\paragraph{Type II and Type III Bursts: Chain Sections and Long Chainlet Fragmentation:}
\subparagraph{In the context of Type II and Type III solar radio wave emissions, a burst refers to a sudden, transient enhancement of radio emission at specific frequencies. These bursts are direct signatures of energetic processes in the solar atmosphere, specifically related to accelerated electrons, and are observed as rapidly evolving features in solar dynamic spectra.}

\subparagraph{Type II bursts (20–150 MHz): Produced by shock waves propagating through the corona, often CME-associated. These shocks accelerate ambient electrons that generate Langmuir waves and convert them to radio emissions near the local plasma frequency and its harmonic.}
\subparagraph{Type III bursts (0.5–20 MHz): Caused by fast electron beams (10–100 keV) streaming along open field lines, generating downward-drifting radio emission as they move into decreasing plasma density.}

\subparagraph{These bursts are not photon-driven processes. They occur below the quantum threshold (~37.6 THz) and are modeled via collective plasma wave interactions, not quantized photons. However, they remain essential diagnostics of early particle acceleration, electric field development, and topological magnetic restructuring triggered by Magnetic Breakdown.}

\paragraph{Type II and III bursts serve as remote sensing markers for distinct chainlet-driven processes: Type III reflects beam acceleration along open field lines (chains) at flare onset, while Type II indicates macroscopic energy release via shock formation. In Photony Theory, their timing and spectral drift profiles are signatures of the magnetic chain fragmentation cascade (chain reaction), which drives fast particle ejection and large-scale coronal reconfiguration. Magnetic Breakdown in coronal regions produces sharp electron beams that excite plasma frequencies matching observed Type II/III bursts.}

\paragraph{Gyrosynchrotron Emission:}
\subparagraph{Gyrosynchrotron emission (from Table 1) is nonthermal radiation produced when high-energy electrons spiral around magnetic field lines (chains) at relativistic speeds, that is, speeds close to the speed of light. This process is common in solar flares and active regions with strong magnetic fields and energetic particles. }
\subparagraph{As electrons accelerate in the solar atmosphere, especially during flares, they follow helical paths along magnetic field lines. When their velocities are high enough (typically in the range of tens of keV to MeV), the resulting radiation is referred to as Gyrosynchrotron, which is a combination of two processes: cyclotron emission, which occurs at nonrelativistic speeds, and synchrotron emission, which dominates at highly relativistic speeds. Gyrosynchrotron sits in between and describes emission from relativistic but not ultra-relativistic electrons.}
\subparagraph{This emission is typically observed in the microwave range (hundreds of MHz to tens of GHz) and has a broad, smooth spectrum. Its intensity and polarization depend on magnetic field strength, electron energy distribution, and viewing angle. Gyrosynchrotron radiation is a key diagnostic tool in solar radio astronomy as it helps determine the properties of accelerated electrons and magnetic field structures in flare regions. Instruments like EOVSA and Nobeyama Radioheliograph routinely detect this emission during solar flares, offering insights into the dynamics of flare particle acceleration and Magnetic Reconnection.}

\paragraph{Continuum Rise- Infrared:}
\subparagraph{Continuum rise (from Table 1) in solar flares refers to a broadband increase in radiation across a wide range of wavelengths rather than a spike in a specific spectral line. This emission type occurs when a flare produces excess light over a continuous spectrum, often observed in the visible, infrared, or ultraviolet regions. One of the most well-known manifestations is the white-light flare, where visible light intensity rises significantly, typically near the footpoints of flare loops.}
\subparagraph{The continuum emission is thought to arise from the lower chromosphere or photosphere through direct heating or thick-target Bremsstrahlung, where high-energy electrons decelerate in denser layers and release energy. Hydrogen recombination continua, such as the Balmer and Paschen continua, may also contribute.}


\paragraph{Visible, White-Light Flares:}
\subparagraph{Physical Process: H⁻ opacity enhancement, hydrogen recombination, and chromospheric compression. Visible continuum emission during solar flares arises from high-energy chainlet fragments depositing energy into the deep chromosphere and upper photosphere.}
\subparagraph{The visible portion of the electromagnetic spectrum, ranging approximately from 430 THz to 770 THz (corresponding to wavelengths of 700 to 390 nanometers), represents a narrow but observationally rich window through which solar activity can be directly detected. Unlike emissions in the microwave, infrared, or X-ray bands, visible light from the Sun can be observed both with ground-based and space-based instruments, offering high temporal and spatial resolution.}

\paragraph{Continuum Brightenings- UV:}
\subparagraph{Continuum brightenings (from Table 1) represent a broad increase in radiation across a range of wavelengths rather than at discrete lines. This typically results from the recombination of ionized hydrogen (e.g., Balmer or Paschen continua) or free–free emission (Bremsstrahlung), and it often occurs deeper in the solar atmosphere. Continuum brightening is especially prominent in white-light and ultraviolet flares, reflecting intense heating and energy deposition in the chromosphere or even the upper photosphere.}

\subparagraph{The ultraviolet (UV) portion of the electromagnetic spectrum, covering wavelengths from approximately 10 to 400 nanometers (or photon energies from approximately 3 to 124 electron volts), serves as a critical diagnostic window for examining solar flare activity in the transition region and upper chromosphere. }

\subparagraph{Emissions in this spectral range are characterized primarily by two radiative behaviors: brightening in discrete spectral lines associated with atomic transitions and enhancements in broadband continuum emission due to recombination and free-bound processes.}


\subparagraph{Together, line and continuum brightenings provide a detailed picture of how energy from a flare is distributed across layers of the solar atmosphere. Instruments like IRIS, SDO/EVE, and RHESSI capture both types of emission and help trace the flare’s thermal and non-thermal evolution. Their combined presence is a hallmark of major flares, showing localized heating in specific plasma states and broad energy deposition across multiple wavelengths}

\paragraph{Soft and Hard X-ray Emissions: }

\subparagraph{Physical Process- Soft x-rays: Thermal bremsstrahlung (free-free emission thermal radiation) from high-temperature coronal plasma heated to 10⁷ K.}
\subparagraph{Soft X-ray emission in solar flares is primarily produced by thermal bremsstrahlung—radiation emitted when ions deflect high-energy electrons in a hot plasma. This process dominates in the coronal source region heated by magnetic chain collapse and associated compression, typically during the early-to-mid phase of flare development.} \subparagraph{The electron temperatures (10–30 MK) produce soft X-ray photons, consistent with the energy from chain fragmentation-driven heating in the upper corona.} 
\paragraph{Physical Process- Hard X-rays:. Nonthermal Bremsstrahlung from photon chainlet-accelerated electrons impacting the chromosphere.}

\subparagraph{Hard X-ray (HXR) emission in solar flares originates from nonthermal electrons, typically in the 10–300 keV range, accelerated during the incipient phase of Magnetic Reconnection, when the earliest magnetic field collapse and chain breakdown occur. In this phase, magnetic field structures lose stability and fragment into chain-like configurations.}
\subparagraph{As these magnetic chains collapse, they generate localized electric fields capable of accelerating electrons to high energies. These electrons are directed downward along evolving magnetic field lines (chains) and collide with dense chromospheric plasma, where they lose energy via Bremsstrahlung radiation in the hard X-ray range.}

\paragraph{Gamma Ray Emissions: Gamma Ray Emissions – Nuclear and Pion Contributions}
\subparagraph{Physical Process- Nuclear and Pion Emission and Pair Production: Ion acceleration and nuclear interactions in the chromosphere and low corona. Chain breakdown may accelerate protons and alpha particles to energies exceeding 10 MeV, enabling inelastic collisions with ambient nuclei. These collisions produce gamma-ray emissions through prompt nuclear de-excitation, neutron capture, and neutral pion (π0) decay. Pion production requires proton energies above ~300 MeV, resulting in broadband gamma continua above ~70 MeV.}
\subparagraph{Insight: In Photony Theory, these signatures arise from the upper tail of the photon chainlet energy distribution. As magnetic chains fragment into denser chromospheric layers, their concentrated magnetic energy is converted into proton acceleration. The resulting gamma-ray lines and continua validate Photony Theory’s prediction that Magnetic Breakdown can initiate both electron and ion acceleration, with gamma rays serving as the terminal observational signatures of the chainlet energy cascade.}




\section{\textcolor{black}{Comparative Framework for Solar Flare Initiation Mechanisms}}

\begingroup
\color{black}

A robust understanding of solar flare initiation necessitates engagement with the diverse array of models proposed across theoretical, observational, and computational domains. While Photony Theory introduces a mechanistic framework based on magnetic chain fragmentation and photon-based energy transduction, the astrophysical literature has long advanced alternative paradigms grounded in plasma physics, magnetohydrodynamics (MHD), and statistical mechanics. 

This section presents a comparative survey of the principal flare initiation mechanisms currently recognized in the field, highlighting their conceptual foundations, operational strengths, and inherent limitations.

Classical reconnection models, including the Sweet–Parker and Petschek formulations, remain foundational to flare theory. These approaches rely on the formation of narrow current sheets where magnetic energy is dissipated resistively. However, the rapid energy conversion required in solar flares often exceeds the rates predicted by idealized reconnection alone, leading to augmented models that incorporate turbulent or plasmoid-mediated enhancements. 

Variants such as the breakout model or tether-cutting reconnection further refine the initiation sequence by invoking topological changes that precede and facilitate the main reconnection event. These mechanisms are particularly effective in explaining flare triggers in complex magnetic arcades, but they presuppose specific structural preconditions and field configurations.

Flux rope instability models offer an alternative pathway to eruption, grounded in magnetostatic thresholds. The kink instability, triggered when magnetic twist surpasses a critical helical threshold, and the torus instability, activated by a decaying overlying field, both represent coherent, well-defined eruption criteria. 

Torus instability is a magnetohydrodynamic mechanism governing the eruption of magnetized plasma structures, such as solar flux ropes, when the confining magnetic field above them decreases too rapidly with height. It occurs when the decay index n declines with height h, exceeds a critical threshold, typically around 1.5. In this regime, the upward-directed hoop force generated by the internal current of the flux rope surpasses the downward magnetic tension from the overlying field, initiating an eruption. This phenomenon is distinct from kink instability, which is triggered by excessive twist within the flux rope itself and leads to helical deformation without necessarily causing an eruption. While kink instability is governed by internal magnetic structure and the critical twist threshold, torus instability is driven by the external field environment and its gradient. Both instabilities are central to understanding solar flare and coronal mass ejection dynamics, with torus instability offering a robust criterion for the loss of equilibrium in pre-eruptive magnetic configurations.





Statistical and phenomenological approaches, such as avalanche or self-organized criticality (SOC) models, characterize the solar corona as a system evolving near marginal stability. In these frameworks, energy release events (flares) are emergent phenomena resulting from a cascade of small-scale instabilities. Such models are adept at reproducing the observed power-law distribution of flare energies and sizes, yet they typically lack predictive power at the level of individual events and are limited in describing specific physical onset conditions.



\begin{table}[ht]
\centering
\scriptsize
\renewcommand{\arraystretch}{1.05}
\begin{tabular}{|p{3.4cm}|p{3.4cm}|}
\hline
\textcolor{black}{\textbf{Mechanism}} & \textcolor{black}{\textbf{Challenge}} \\
\hline
\textcolor{black}{\textbf{Sweet–Parker/Petschek}: Reconnection via current sheets.} & \textcolor{black}{Too slow unless enhanced by turbulence or plasmoids.} \\
\hline
\textcolor{black}{\textbf{Breakout/Tether-Cut}: Topology change triggers flare.} & \textcolor{black}{Needs precise geometry and field weakening.} \\
\hline
\textcolor{black}{\textbf{Flux Rope Instability}: Eruption from twist or weak restraint.} & \textcolor{black}{Requires accurate twist or decay index.} \\
\hline
\textcolor{black}{\textbf{SOC Models}: Critical state yields avalanche flaring.} & \textcolor{black}{Lacks event-level onset physics.} \\
\hline
\textcolor{black}{\textbf{Loop/Beam}: Models post-flare flows and heating.} & \textcolor{black}{Assumes reconnection as cause.} \\
\hline
\textcolor{black}{\textbf{3D MHD}: Simulates eruption evolution.} & \textcolor{black}{Very compute-intensive; low detail scale.} \\
\hline
\end{tabular}
\caption{\textcolor{black}{Flare initiation mechanisms (models) and their challenges (limitations).}}
\label{tab:flare_models_compact10}
\end{table}

Post-eruptive observational models, which focus on chromospheric evaporation, mass flows, and beam-driven heating, address the evolution of flare emission following energy release. These approaches, supported by high-resolution data from instruments such as IRIS and SDO/AIA, elucidate the consequences of flare initiation but are not independently sufficient to define the triggering process itself. 

Similarly, modern 3D MHD simulations represent an integrative class of models capable of combining reconnection, flux emergence, shear motion, and eruption dynamics within a single numerical framework. While they offer the most comprehensive simulations of flare evolution, their computational complexity and resolution constraints limit the granularity of their physical insight into early flare onset.

The key characteristics and limitations of these models are summarized in Table 2, which presents a comparative overview in compact form. This table highlights both the breadth of theoretical perspectives and the recurring challenges that complicate flare initiation research.

This comparative review underscores both the diversity and the limitations of current flare initiation models. Each provides valuable insight into specific aspects of flare physics but often falls short of offering a unified mechanism that encompasses early energy buildup, discrete triggering, and predictive utility. By contrast, Photony Theory introduces a quantized and mechanically grounded mechanism of energy accumulation and release, which may offer a complementary or alternative interpretation capable of resolving initiation timing and early flare dynamics.


\endgroup



\section*{\textcolor{black}{Predictive Modeling of Flares with Photony Theory: Early Classification of M- and X-Class Solar Flares}}




\begingroup
\color{black}

Photony Theory, grounded in a quantized, mechanical framework for magnetic energy storage and release, enables not only descriptive modeling of flare dynamics but also predictive capabilities that bridge theoretical physics and real-time space weather forecasting. A notable application is the algorithmic classification of solar flares during their early energy release phase, distinguishing between M-class and X-class events based solely on initial magnetic energy release rates. This predictive model provides a rigorous, data-driven demonstration of Photony Theory’s operational viability.

\subsection*{Background: Flare Class Definitions and Energy Metrics}

The classification of solar flares, as per NOAA GOES instrumentation, is based on peak soft X-ray flux in the 1–8 Å band. Each letter class (A, B, C, M, X) denotes a tenfold increase in intensity, and empirical studies show that energy release rates (power output) scale accordingly. Table 4.21 of the dissertation outlines typical energy release rates for each class. For instance:

\begin{itemize}
\item M-class flares release on the order of $\sim 10^{22}$ watts.
\item X-class flares release $\sim 10^{23} - 10^{24}$ watts.
\end{itemize}

These real-time detectable emissions—EUV, X-ray flux—serve as the input for early-stage predictive algorithms.

\subsection*{Method: Polynomial-Based Predictive Classification Algorithm}

The proposed classifier distinguishes flare classes by analyzing early energy release rate data and extrapolating its trajectory over time. Key features of the algorithm include:

\begin{enumerate}
\item \textbf{Input sampling}: The model uses 10 early-time power samples taken over the interval 2.0 to 2.9 minutes post-breakdown (0.1-minute resolution), simulating near-instantaneous detection.

\item \textbf{Polynomial fitting}: These samples are fitted to a 7th-degree polynomial:

\begin{equation}
P(t) = a_0 + a_1 t + a_2 t^2 + \ldots + a_7 t^7
\end{equation}

capturing the nonlinear rise of energy release in early flare dynamics.

\item \textbf{Reference models}: Two standardized flare templates are constructed, representing typical M-class and X-class flares using empirically validated 7th-degree polynomials derived from historical energy profiles and theoretical modeling.

\textbf{M-Class Flare Power Polynomial}

\begin{align*}
P_M(t) &= -1.035 \times 10^{14} + 3.524 \times 10^{14} t \\
       &\quad - 4.614 \times 10^{13} t^2 + 2.991 \times 10^{12} t^3 \\
       &\quad - 1.071 \times 10^{11} t^4 + 1.992 \times 10^9 t^5 \\
        &\quad - 1.730 \times 10^7 t^6 + 5.379 \times 10^4 t^7
\end{align*}


\textbf{X-Class Flare Power Polynomial}
\begin{align*}
P_X(t) &= -2.356 \times 10^{21} + 1.132 \times 10^{21} t \\
       &\quad - 2.271 \times 10^{20} t^2 + 2.426 \times 10^{19} t^3 \\
       &\quad - 1.483 \times 10^{18} t^4 + 5.054 \times 10^{16} t^5 \\
       &\quad - 8.452 \times 10^{14} t^6 + 5.362 \times 10^{12} t^7
\end{align*}

\item \textbf{Energy constraints}: The cumulative energy release modeled over a flare's lifetime is integrated to meet observed constraints:
\begin{itemize}
\item M-class flares: $E_{\text{total}} = 1.0 \times 10^{23} \, \text{J}$
\item X-class flares: $E_{\text{total}} = 3.0 \times 10^{25} \, \text{J}$
\end{itemize}

\item \textbf{Extrapolation and comparison}: The fitted early polynomial $P_{\text{fit}}(t)$ is extrapolated from $t = 0$ to $80$ minutes and compared against both reference models via Root Mean Square Error (RMSE):


\begin{align*}
\text{RMSE}_M &= \sqrt{ \frac{1}{n} \sum_{i=1}^{n} (P_{\text{fit}}(t_i) - P_M(t_i))^2 } \\
\text{RMSE}_X &= \sqrt{ \frac{1}{n} \sum_{i=1}^{n} (P_{\text{fit}}(t_i) - P_X(t_i))^2 }
\end{align*}


\end{enumerate}

The flare class corresponding to the lower RMSE is assigned as the predicted outcome.

\subsection*{Synthetic Testing and Quantitative Results}

To validate the predictive model, synthetic input curves were generated using interpolation between $P_M(t)$ and $P_X(t)$ with weighting factors $f = 0.4, 0.6, 0.8$. This created flare profiles partially resembling either class:

\[
P_{\text{synthetic}}(t) = P_M(t) + f \cdot (P_X(t) - P_M(t))
\]

Each profile was tested against both reference models using RMSE to determine classification accuracy.



\textbf{Classification Outcomes:}

\vspace{0.5em}
\begin{center}
\resizebox{0.935\linewidth}{!}{%
\begin{tabular}{|c|c|c|c|}
\hline
\textbf{Case} & \textbf{RMSE to M (W)} & \textbf{RMSE to X (W)} & \textbf{Prediction} \\
\hline
40\% & $3.22 \times 10^{21}$ & $3.80 \times 10^{21}$ & M-class \\
60\% & $3.57 \times 10^{21}$ & $2.96 \times 10^{21}$ & X-class \\
80\% & $5.41 \times 10^{21}$ & $1.77 \times 10^{21}$ & X-class \\
\hline
\end{tabular}
}
\end{center}





The algorithm successfully predicted the correct flare class in all cases, including borderline interpolations. Moreover, predictions were made using just the first 0.9 minutes of post-breakdown data—highlighting the model’s speed and sensitivity.

A three-minute advance warning was demonstrated based on synthetic flare data modeled from standardized M-class and X-class energy release rate curves. These curves were designed using cumulative energy constraints and empirically validated timelines consistent with real observational data from instruments such as GOES and SDO.


\subsection*{Physical Interpretation and Operational Utility}

This algorithm embodies a central assertion of Photony Theory: that quantized magnetic chain rupture produces not only rapid and class-distinct energy release but also early divergences in power profiles that are predictive of total flare evolution. The polynomial fits reflect the physical mechanisms of magnetic stress accumulation (pre-flare ramp), breakdown-triggered acceleration (exponential rise), and reconnection-driven dissipation (Gaussian decay)—all observable in satellite X-ray and EUV records.

By connecting these modeled dynamics with real-time data, the method enables forward-looking flare classification suitable for early warning systems and mission-critical space weather forecasting. This also exemplifies how Photony Theory can move from theoretical physics to operational science, integrating chain-level energetics into predictive analytics.

\endgroup







\section{Conclusions}
\textcolor{black}{A mechanically grounded model of flare initiation is presented, rooted in the principles of Photony Theory. Magnetic Breakdown is identified as the incipient phase of Magnetic Reconnection and serves as the physical mechanism responsible for initiating flare energy release. This model aligns with both theoretical formulations and observational data, providing an enhanced framework for predictive space weather forecasting. Magnetic fields are conceptualized as coherent chains of transduced photon energy generated by the directional chain linkage activity of free electrons. These photon chains store magnetic energy in quantized, structured forms that can accumulate stress under photospheric motion. Energy release occurs through a process of chain fragmentation, in which coherent sequences rupture and convert stored mechanical energy into directed particle acceleration and wave emission. This mechanism favors early nonthermal electron acceleration, with thermalization following as a secondary effect through collisions and plasma wave interactions. The energetic structure of each chain, determined by the spin frequency and energy of constituent photons, allows for precise calculation of energy density and cumulative release during fragmentation. A real-time classification algorithm has been developed to exploit this structure, distinguishing M-class from X-class flares by analyzing the initial power curve using polynomial fitting and root-mean-square error comparisons to standard flare profiles. The algorithm provides a predictive window of at least three minutes, utilizing early energy release data recorded within the first minute of flare evolution. While initially validated using synthetic data derived from empirically constrained flare models, the reference trajectories reflect well-established flare energetics recorded by missions such as GOES and SDO. These findings affirm that Photony-based modeling not only offers a quantized and directional theory of magnetic energy storage and release, but also enables the development of actionable prediction tools grounded in fragmentation energetics.}


\bibliographystyle{aasjournal}
\begin{thebibliography}{}
\bibitem[Aschwanden(2017)]{Aschwanden2017} Aschwanden, M.J. 2016, GLOBAL ENERGETICS OF SOLAR FLARES. III. NONTHERMAL ENERGIES ApJ
\bibitem[Aulanier(2010)]{Aulanier2010} Aulanier, G., et al. 2010, FORMATION OF TORUS-UNSTABLE FLUX ROPES AND ELECTRIC CURRENTS IN ERUPTING SIGMOIDS, ApJ
\bibitem[Birn(2007)]{Birn2007} Birn, J., et al. 2007, Reconnection of Magnetic Fields: Magnetohydrodynamics and Collisionless Theory and Observations. Cambridge Univ. Press
\bibitem[Emslie et al.(2005)]{Emslie2005} Emslie, A.G., et al. 2005, Energy partition in two solar flare/CME events, Journal of Geophysical Research: Space Physics, JGR: Space Physics
\bibitem[Fan(2009)]{Fan2009} Fan, Y. 2009, Magnetic Fields in the Solar Convection Zone, Living Reviews in Solar Physics
\bibitem[Faraday(1852)]{Faraday1852} Faraday, M. 1852, ON THE PHYSICAL CHARACTER OF THE LINES OF MAGNETIC FORCE, Philosophical Magazine
\bibitem[Fleishman, et al., 2022]{Fleishman,2022} Solar flare accelerates nearly all electrons in a large coronal volume, Nature Astronomy
\bibitem[Goldstein \& Lyatsky(2013)]{Goldstein2013} Goldstein, W. L. 2013, Effect of interchange instability on magnetic reconnection, Nonlinear Processes in Geophysics, 365–377.
\bibitem[Inoue(2018)]{Inoue2018} Inoue, S., et al. 2018, Formation and dynamics of a solar eruptive flux tube, Nature Communications
\bibitem[Martin(2023)]{Martin2023} Martin, F. 2023, Photony Foundations
\bibitem[Martin(2025)]{Martin2025} Causes and Presitors of Solar Storms- a dissertation (in print)
\bibitem[Parker(1979)]{Parker1979} Parker, E. 1979, Cosmical Magnetic Fields: Their Origin and Their Activity, Oxford University Press
\bibitem[Priest \& Forbes(2002)]{Priest2002} Priest, E.R., \& Forbes, T. 2002, The magnetic nature of solar flares, A\&A
\bibitem[Priest(2014)]{Priest2014} Priest, E.R. 2014, Magnetohydrodynamics of the Sun. Cambridge University Press.
\bibitem[Schrijver(2009)]{Schrijver2009} Schrijver, C. 2009, Driving major solar flares and eruptions: A review, Advances in Space Research
\bibitem[Thomson(1893)]{Thomson1893} Thomson, J. 1893, Notes on Recent Researches in Electricity and Magnetism. Clarendon Press.
\bibitem[Toriumi(2019)]{Toriumi2019} Toriumi, S., et al. 2019, Flare-productive active regions, Living Reviews in Solar Physics
\bibitem[Török(2004)]{Torok2004} Török, T., et al. 2004, THE KINK INSTABILITY IN SOLAR ERUPTIONS, ApJ Letters
\bibitem[Warmuth \& Mann(2020)]{Warmuth2020} Warmuth, A. \& Mann, G. 2020, Thermal-nonthermal energy partition in solar flares derived from X-ray, EUV, and bolometric observations
\bibitem[Yokoyama(1995)]{Yokoyma1995} Yokoyama, T. 1995, Magnetic reconnection as the origin of X‑ray jets and Hα surges on the Sun Nature, 42-44.
\end{thebibliography}

\end{document}
