Edge Breakout: Dynamics at Critical Boundaries

• Edge Breakout is a polysemous term describing threshold phenomena at material or conceptual boundaries where a slight change leads to significant shifts in transport, stability, or connectivity.
• It encompasses diverse applications such as shock breakouts in supernovae, turbulent state escape in boundary layers, anomalous skin effects in non-Hermitian quantum walks, and magnetic reconnection in coronal pseudostreamers.
• The concept underlines practical insights across disciplines, offering frameworks to predict early supernova emissions, control instability in complex fluids, and analyze dynamical transitions in theoretical models like the Breakout game.

Edge Breakout is a polysemous technical term used for several boundary-triggered dynamical phenomena. In the supplied literature it denotes, depending on field, shock breakout at the stellar surface or the outer edge of dense circumstellar material, state-space escape from the laminar basin through a localized edge state in boundary-layer transition, anomalous boundary-focused loss in non-Hermitian quantum walks, free-surface failure in sheared viscoelastic fluids, magnetic breakout reconnection at the separatrix of a coronal pseudostreamer, and edge-aligned periodic trajectories in an idealized mathematical Breakout model (Waxman et al., 2016, Khapko et al., 2016, Xue et al., 2021, Hemingway et al., 2017, Lynch et al., 2012, Jullian, 2019). Across these usages, the “edge” is the locus where a qualitative change in transport, stability, or connectivity occurs. This suggests a family resemblance rather than a single unified theory.

1. Terminological scope and shared structure

In astrophysical shock-breakout theory, the edge is a material boundary: either the stellar surface, where a radiation-mediated shock first satisfies $\tau \approx c/v_s$, or the sharp outer edge of dense circumstellar material (CSM), where photons begin to escape (Waxman et al., 2016, Margalit, 2021). In transitional shear flows, the edge is a codimension-one basin boundary in state space separating relaminarization from sustained turbulence, and “breakout” is the departure of a trajectory from the neighborhood of the edge state along its unstable manifold (Khapko et al., 2016). In non-Hermitian quantum dynamics, the edge is a physical lattice boundary under open boundary conditions, and breakout is the boundary loss anomaly produced by the cooperation of the non-Hermitian skin effect (NHSE) and imaginary-gap closing (Xue et al., 2021). In rheology, the edge is the fluid–air interface, whose instability under shear is called edge fracture or edge breakout (Hemingway et al., 2017). In solar magnetohydrodynamics, the edge is the separatrix/current-sheet layer above a pseudostreamer lobe where magnetic breakout reconnection begins (Lynch et al., 2012). In the plane-wide Breakout model, the edge is literal brick geometry, and breakout refers to recurrent edge-aligned collision patterns generated by special rational slopes (Jullian, 2019).

The common structural element is threshold behavior localized at a boundary. What varies is the relevant control variable: optical depth in shock breakout, phase-space distance to the edge state in bypass transition, complex spectral topology in non-Hermitian systems, stress balance at a free surface in complex fluids, magnetic topology in pseudostreamers, and Diophantine slope constraints in the mathematical game model.

2. Shock breakout at stellar and circumstellar edges

In canonical supernova breakout theory, the defining condition is that the optical depth ahead of the radiation-mediated shock drops to the point where diffusion competes with advection:

$\tau_{\rm bo} \simeq \frac{c}{v_{\rm sh}}.$

At the stellar surface, this “edge breakout” produces the earliest supernova emission, with a flash duration set by the larger of the planar breakout time and the light-crossing time $t_{\rm lc}=R_\star/c$ (Waxman et al., 2016). The planar breakout time is

$$t_{\rm bo} \simeq \frac{c}{\kappa \rho_{\rm bo} V_{\rm bo}},$$

and the breakout energy scales as

$$E_{\rm bo} \simeq 2.0 \times \frac{4\pi R_\star^2 c}{\kappa} V_{\rm bo}.$$

For red supergiants, blue supergiants, and compact Wolf–Rayet progenitors, the resulting flashes range from tens of eV to multi-keV, with durations from seconds to hundreds of seconds depending on $R_\star$, $V_{\rm bo}$, and $\rho_{\rm bo}$ (Waxman et al., 2016).

A distinct mechanism extends breakout in compact Wolf–Rayet stars close to the Eddington luminosity. When the local radiative luminosity approaches

$L_{\rm Edd}=\frac{4\pi G M c}{\kappa},$

the iron opacity bump at $T \simeq 200\ {\rm kK}$ can inflate a low-density envelope beneath a modest photospheric radius. In this case, the rise time is not set by $\tau_{\rm bo} \simeq \frac{c}{v_{\rm sh}}.$0 but by photon diffusion through the inflated layer:

$\tau_{\rm bo} \simeq \frac{c}{v_{\rm sh}}.$1

For the solar-metallicity $\tau_{\rm bo} \simeq \frac{c}{v_{\rm sh}}.$2 helium-star model, $\tau_{\rm bo} \simeq \frac{c}{v_{\rm sh}}.$3 gives $\tau_{\rm bo} \simeq \frac{c}{v_{\rm sh}}.$4, while $\tau_{\rm bo} \simeq \frac{c}{v_{\rm sh}}.$5 gives $\tau_{\rm bo} \simeq \frac{c}{v_{\rm sh}}.$6 for $\tau_{\rm bo} \simeq \frac{c}{v_{\rm sh}}.$7. The corresponding $\tau_{\rm bo} \simeq \frac{c}{v_{\rm sh}}.$8 solar-metallicity model gives $\tau_{\rm bo} \simeq \frac{c}{v_{\rm sh}}.$9 at $t_{\rm lc}=R_\star/c$0, whereas an uninflated $t_{\rm lc}=R_\star/c$1 polytrope with $t_{\rm lc}=R_\star/c$2 has $t_{\rm lc}=R_\star/c$3 (Moriya et al., 2015). The Type Ib event SN 2008D, with rise time $t_{\rm lc}=R_\star/c$4 and total duration $t_{\rm lc}=R_\star/c$5, is cited as a case naturally explained if breakout occurs within such an inflated envelope rather than at the bare surface (Moriya et al., 2015).

When the shock instead emerges through a thick stellar wind, the breakout radius is outside the star and the relevant condition is the wind optical depth

$t_{\rm lc}=R_\star/c$6

For a wind density $t_{\rm lc}=R_\star/c$7, the breakout duration is approximately $t_{\rm lc}=R_\star/c$8 and the peak luminosity obeys

$t_{\rm lc}=R_\star/c$9

For fast sub-relativistic shocks, $$t_{\rm bo} \simeq \frac{c}{\kappa \rho_{\rm bo} V_{\rm bo}},$$0, the time-integrated breakout spectrum is non-thermal and the time-resolved temperature is expected to drop by roughly an order of magnitude during the rising phase because photon escape steepens the shock, increases the compression ratio, and enhances photon production (Ioka et al., 2018). The framework yields a closure test linking $$t_{\rm bo} \simeq \frac{c}{\kappa \rho_{\rm bo} V_{\rm bo}},$$1, $$t_{\rm bo} \simeq \frac{c}{\kappa \rho_{\rm bo} V_{\rm bo}},$$2, and $$t_{\rm bo} \simeq \frac{c}{\kappa \rho_{\rm bo} V_{\rm bo}},$$3; XRT 080109/SN 2008D is discussed as broadly consistent with $$t_{\rm bo} \simeq \frac{c}{\kappa \rho_{\rm bo} V_{\rm bo}},$$4 and $$t_{\rm bo} \simeq \frac{c}{\kappa \rho_{\rm bo} V_{\rm bo}},$$5 in this thick-wind picture (Ioka et al., 2018).

Dense CSM with a sharp outer edge gives yet another edge-breakout regime. For a shell of mass $$t_{\rm bo} \simeq \frac{c}{\kappa \rho_{\rm bo} V_{\rm bo}},$$6 near radius $$t_{\rm bo} \simeq \frac{c}{\kappa \rho_{\rm bo} V_{\rm bo}},$$7,

$$t_{\rm bo} \simeq \frac{c}{\kappa \rho_{\rm bo} V_{\rm bo}},$$8

and breakout occurs near the outer edge when $$t_{\rm bo} \simeq \frac{c}{\kappa \rho_{\rm bo} V_{\rm bo}},$$9 (Margalit, 2021). Because the shocked shell is initially thin, the early expansion is planar rather than spherical. The analytic solution for the bolometric luminosity is a sum over planar diffusion eigenmodes, and in the optically thick limit the early planar decline approaches $$E_{\rm bo} \simeq 2.0 \times \frac{4\pi R_\star^2 c}{\kappa} V_{\rm bo}.$$0. If $$E_{\rm bo} \simeq 2.0 \times \frac{4\pi R_\star^2 c}{\kappa} V_{\rm bo}.$$1 is only marginally larger than $$E_{\rm bo} \simeq 2.0 \times \frac{4\pi R_\star^2 c}{\kappa} V_{\rm bo}.$$2, radiative losses in the planar phase steepen the early decline significantly beyond $$E_{\rm bo} \simeq 2.0 \times \frac{4\pi R_\star^2 c}{\kappa} V_{\rm bo}.$$3; at intermediate times, $$E_{\rm bo} \simeq 2.0 \times \frac{4\pi R_\star^2 c}{\kappa} V_{\rm bo}.$$4, the summed modes yield an approximate $$E_{\rm bo} \simeq 2.0 \times \frac{4\pi R_\star^2 c}{\kappa} V_{\rm bo}.$$5 decline (Margalit, 2021). This makes “edge breakout” in dense CSM analytically distinct from simpler one-zone spherical cooling models.

3. Edge states and turbulent breakout in boundary-layer flows

In the asymptotic suction boundary layer (ASBL), the edge is the laminar–turbulent boundary in state space: a codimension-one invariant set separating the basin of attraction of the laminar state from that of sustained turbulence. The edge state is the relative attractor on this boundary, dynamically a saddle with a high-dimensional stable manifold $$E_{\rm bo} \simeq 2.0 \times \frac{4\pi R_\star^2 c}{\kappa} V_{\rm bo}.$$6 and a low-dimensional unstable manifold $$E_{\rm bo} \simeq 2.0 \times \frac{4\pi R_\star^2 c}{\kappa} V_{\rm bo}.$$7 (Khapko et al., 2016).

The ASBL base flow is

$$E_{\rm bo} \simeq 2.0 \times \frac{4\pi R_\star^2 c}{\kappa} V_{\rm bo}.$$8

with displacement thickness $$E_{\rm bo} \simeq 2.0 \times \frac{4\pi R_\star^2 c}{\kappa} V_{\rm bo}.$$9 and Reynolds number

$R_\star$0

The laminar ASBL is linearly stable up to $R_\star$1, yet finite-amplitude perturbations can sustain turbulence for $R_\star$2; the simulations discussed were performed at $R_\star$3 (Khapko et al., 2016).

The localized edge state found by edge tracking is a three-dimensional structure recurrently characterized by a single low-speed streak flanked by staggered streamwise vortices. It is chaotic in time, fully localized in all three spatial directions, and exhibits calm phases alternating with bursts in the cross-flow energy

$R_\star$4

During calm phases, the active low-speed streak persists and meanders in the spanwise direction; during bursts, a sinuous instability develops, breaks the streak, and seeds a new localized core. Space–time diagnostics show erratic spanwise shifts over periods of $R_\star$5–$R_\star$6 time units (Khapko et al., 2016).

The paper’s “breakout” is the state-space escape that nucleates a turbulent spot. Noise-triggered trajectories first approach $R_\star$7, shadow the edge state during a calm phase, and then depart along $R_\star$8. In the $R_\star$9 projection, trajectories from diverse noisy initial conditions approach the compact region occupied by the chaotic edge state and then leave together along a single direction, consistent with a one-dimensional $V_{\rm bo}$0. The nucleation time $V_{\rm bo}$1 is defined by $V_{\rm bo}$2, and $V_{\rm bo}$3 is positively correlated with the minimal phase-space distance $V_{\rm bo}$4 to the calm phase of the edge state: closer approaches correspond to longer dwelling before breakout (Khapko et al., 2016).

Quantitatively, pre-nucleation streaks resemble edge-state calm phases. Across 13 edge calm-phase instances and pre-nucleation events, the reported streak widths are $V_{\rm bo}$5 and $V_{\rm bo}$6 in units of $V_{\rm bo}$7, the streak intensities are $V_{\rm bo}$8 and $V_{\rm bo}$9 in units of $\rho_{\rm bo}$0, and the sinuous wavelengths are $\rho_{\rm bo}$1 and $\rho_{\rm bo}$2 in units of $\rho_{\rm bo}$3 (Khapko et al., 2016). This establishes a dynamical-systems interpretation of bypass transition in which a localized edge state acts as a mediator of turbulent spot nucleation.

4. Non-Hermitian edge burst as boundary-focused loss

In non-Hermitian quantum dynamics, edge breakout is the “edge burst”: an anomalously large fraction of the total loss occurs at the system boundary under open boundary conditions (OBC). The effect is quantified through the site-resolved integrated loss probability

$\rho_{\rm bo}$4

with $\rho_{\rm bo}$5 for a normalized initial state, and through the edge burst ratio

$\rho_{\rm bo}$6

Its physical origin is the interplay of NHSE and imaginary-gap closing (Xue et al., 2021).

For the representative two-sublattice lossy walk, the Bloch Hamiltonian is

$\rho_{\rm bo}$7

Under OBC, the generalized Brillouin-zone radius is

$\rho_{\rm bo}$8

so $\rho_{\rm bo}$9 gives left-edge skin localization. The imaginary gap closes for $L_{\rm Edd}=\frac{4\pi G M c}{\kappa},$0, i.e.

$L_{\rm Edd}=\frac{4\pi G M c}{\kappa},$1

and in that regime the bulk loss profile becomes algebraic rather than exponential (Xue et al., 2021).

The universal bulk–edge scaling relation is

$L_{\rm Edd}=\frac{4\pi G M c}{\kappa},$2

Hence

$L_{\rm Edd}=\frac{4\pi G M c}{\kappa},$3

so the relative edge peak grows linearly with the distance of the initial state from the boundary. In the specific model, the algebraic bulk exponents are $L_{\rm Edd}=\frac{4\pi G M c}{\kappa},$4, $L_{\rm Edd}=\frac{4\pi G M c}{\kappa},$5, and $L_{\rm Edd}=\frac{4\pi G M c}{\kappa},$6 for $L_{\rm Edd}=\frac{4\pi G M c}{\kappa},$7, $L_{\rm Edd}=\frac{4\pi G M c}{\kappa},$8, and $L_{\rm Edd}=\frac{4\pi G M c}{\kappa},$9, respectively. If the NHSE is absent, for example at $T \simeq 200\ {\rm kK}$0, then $T \simeq 200\ {\rm kK}$1 and there is no edge burst; if the imaginary gap is open, $T \simeq 200\ {\rm kK}$2, the spatial decay is exponential and the edge peak is not anomalous (Xue et al., 2021).

Time-dependent perturbation theory supplies a real-space explanation. With the Hamiltonian split into onsite decay and hopping, and an initial state on the non-decaying $T \simeq 200\ {\rm kK}$3 sublattice, $T \simeq 200\ {\rm kK}$4 remains real and $T \simeq 200\ {\rm kK}$5 remains purely imaginary. The edge amplitude on the lossy site is dominated by only two non-decay feeders,

$T \simeq 200\ {\rm kK}$6

whereas a bulk lossy site has five distinct final-step quantum jumps feeding it (Wen et al., 2023). The boundary therefore has fewer destructive-interference channels, and NHSE-driven transport along the $T \simeq 200\ {\rm kK}$7 chain accumulates amplitude near $T \simeq 200\ {\rm kK}$8 and $T \simeq 200\ {\rm kK}$9. Numerical diagonalization further shows that the edge wave function is mainly propagated by eigenmodes with relatively large imaginary parts, i.e. modes that decay more slowly (Wen et al., 2023). OBC is essential; under periodic boundary conditions the boundary anomaly disappears (Xue et al., 2021).

5. Edge breakout as edge fracture in complex fluids

In rheology, edge breakout is edge fracture: the instability of the free surface where a sheared viscoelastic sample meets the outside medium, usually air. Above a threshold shear rate, the interface tilts and displaces, partially or fully breaking out and corrupting rheological measurements (Hemingway et al., 2017).

The analysis considers a planar-slab approximation to a cone-and-plate rheometer with flow in $\tau_{\rm bo} \simeq \frac{c}{v_{\rm sh}}.$00, gradient in $\tau_{\rm bo} \simeq \frac{c}{v_{\rm sh}}.$01, vorticity in $\tau_{\rm bo} \simeq \frac{c}{v_{\rm sh}}.$02, gap size $\tau_{\rm bo} \simeq \frac{c}{v_{\rm sh}}.$03, and a vertical fluid–air interface at $\tau_{\rm bo} \simeq \frac{c}{v_{\rm sh}}.$04. The key material functions are the shear stress $\tau_{\rm bo} \simeq \frac{c}{v_{\rm sh}}.$05 and the second normal stress difference

$\tau_{\rm bo} \simeq \frac{c}{v_{\rm sh}}.$06

The exact analytical onset criterion for the instability is

$\tau_{\rm bo} \simeq \frac{c}{v_{\rm sh}}.$07

where $\tau_{\rm bo} \simeq \frac{c}{v_{\rm sh}}.$08 is the shear-stress jump across the interface and $\tau_{\rm bo} \simeq \frac{c}{v_{\rm sh}}.$09 is the surface tension (Hemingway et al., 2017).

The corresponding linear growth rate for a perturbation of spanwise wavenumber $\tau_{\rm bo} \simeq \frac{c}{v_{\rm sh}}.$10 is

$\tau_{\rm bo} \simeq \frac{c}{v_{\rm sh}}.$11

with the most unstable mode at the smallest permissible wavenumber $\tau_{\rm bo} \simeq \frac{c}{v_{\rm sh}}.$12 (Hemingway et al., 2017). In the low-shear-rate limit, where $\tau_{\rm bo} \simeq \frac{c}{v_{\rm sh}}.$13 and $\tau_{\rm bo} \simeq \frac{c}{v_{\rm sh}}.$14, the threshold reduces to a Tanner-like form,

$\tau_{\rm bo} \simeq \frac{c}{v_{\rm sh}}.$15

The mechanistic sequence is explicit. A small interfacial tilt $\tau_{\rm bo} \simeq \frac{c}{v_{\rm sh}}.$16 exposes the shear-stress jump $\tau_{\rm bo} \simeq \frac{c}{v_{\rm sh}}.$17; tangential balance then requires a perturbation $\tau_{\rm bo} \simeq \frac{c}{v_{\rm sh}}.$18, which induces a shear-rate perturbation $\tau_{\rm bo} \simeq \frac{c}{v_{\rm sh}}.$19 through $\tau_{\rm bo} \simeq \frac{c}{v_{\rm sh}}.$20. This modifies $\tau_{\rm bo} \simeq \frac{c}{v_{\rm sh}}.$21 through $\tau_{\rm bo} \simeq \frac{c}{v_{\rm sh}}.$22, and the resulting normal-stress imbalance drives an extensional flow that advects the interface and amplifies the initial tilt. Surface tension opposes the curvature term and stabilizes short wavelengths (Hemingway et al., 2017). The centrality of $\tau_{\rm bo} \simeq \frac{c}{v_{\rm sh}}.$23 rather than $\tau_{\rm bo} \simeq \frac{c}{v_{\rm sh}}.$24 follows directly from the linearized $\tau_{\rm bo} \simeq \frac{c}{v_{\rm sh}}.$25–$\tau_{\rm bo} \simeq \frac{c}{v_{\rm sh}}.$26 interfacial stress balance.

The theory was validated in nonlinear simulations using both Johnson–Segalman and Giesekus constitutive models, with onset predictions insensitive to whether the $\tau_{\rm bo} \simeq \frac{c}{v_{\rm sh}}.$27 boundaries were hard-wall or Lees–Edwards. The same criterion also applies when the outside medium is an immiscible Newtonian bath of finite viscosity, provided $\tau_{\rm bo} \simeq \frac{c}{v_{\rm sh}}.$28 is replaced by the fluid–bath stress jump (Hemingway et al., 2017). This immediately yields mitigation strategies: reduce $\tau_{\rm bo} \simeq \frac{c}{v_{\rm sh}}.$29 by immersion in a higher-viscosity bath, increase $\tau_{\rm bo} \simeq \frac{c}{v_{\rm sh}}.$30, reduce $\tau_{\rm bo} \simeq \frac{c}{v_{\rm sh}}.$31, or operate where $\tau_{\rm bo} \simeq \frac{c}{v_{\rm sh}}.$32 is small because $\tau_{\rm bo} \simeq \frac{c}{v_{\rm sh}}.$33 has saturated.

6. Magnetic edge breakout in coronal pseudostreamers

In pseudostreamer CME models, edge breakout is magnetic breakout reconnection initiated at the boundary between a closed arcade and the overlying stalk field. The relevant “edge” is the overlying separatrix/current-sheet layer draped over the coronal null point above the pseudostreamer lobes (Lynch et al., 2012).

The simulations are 2.5-dimensional MHD calculations performed with the ARMS code in a Cartesian domain $\tau_{\rm bo} \simeq \frac{c}{v_{\rm sh}}.$34, $\tau_{\rm bo} \simeq \frac{c}{v_{\rm sh}}.$35, with periodic side boundaries, a line-tied lower boundary, and an open top. The initial field is a pseudostreamer embedded in a unipolar vertical background field built from a uniform field $\tau_{\rm bo} \simeq \frac{c}{v_{\rm sh}}.$36 plus an infinite series of line dipoles truncated at 25 terms with dipole moment $\tau_{\rm bo} \simeq \frac{c}{v_{\rm sh}}.$37. The topology contains two adjacent closed arcades and coronal null points at $\tau_{\rm bo} \simeq \frac{c}{v_{\rm sh}}.$38, $\tau_{\rm bo} \simeq \frac{c}{v_{\rm sh}}.$39, with characteristic length $\tau_{\rm bo} \simeq \frac{c}{v_{\rm sh}}.$40 (Lynch et al., 2012).

Ideal footpoint shearing parallel to the polarity inversion lines stores magnetic energy. When the left-lobe shear is reduced earlier than the right-lobe shear, the right arcade expands asymmetrically, distorting the coronal null and thinning the overlying separatrix into the first breakout current sheet, BCS1, above the right lobe. Between $\tau_{\rm bo} \simeq \frac{c}{v_{\rm sh}}.$41 and $\tau_{\rm bo} \simeq \frac{c}{v_{\rm sh}}.$42, this sheet reaches aspect ratios $\tau_{\rm bo} \simeq \frac{c}{v_{\rm sh}}.$43–$\tau_{\rm bo} \simeq \frac{c}{v_{\rm sh}}.$44, with $\tau_{\rm bo} \simeq \frac{c}{v_{\rm sh}}.$45 and $\tau_{\rm bo} \simeq \frac{c}{v_{\rm sh}}.$46–$\tau_{\rm bo} \simeq \frac{c}{v_{\rm sh}}.$47, exceeding tearing thresholds (Lynch et al., 2012).

Reconnection at BCS1 transfers restraining overlying flux from the right arcade into the left arcade and the open field, thereby enabling runaway expansion of the right lobe and the subsequent formation of a flare current sheet beneath it. During the impulsive flare phase, $\tau_{\rm bo} \simeq \frac{c}{v_{\rm sh}}.$48–$\tau_{\rm bo} \simeq \frac{c}{v_{\rm sh}}.$49, the first CME converts $\tau_{\rm bo} \simeq \frac{c}{v_{\rm sh}}.$50 into $\tau_{\rm bo} \simeq \frac{c}{v_{\rm sh}}.$51 over about $\tau_{\rm bo} \simeq \frac{c}{v_{\rm sh}}.$52, implying roughly $\tau_{\rm bo} \simeq \frac{c}{v_{\rm sh}}.$53 magnetic-to-kinetic conversion efficiency. Reconnection jets reach $\tau_{\rm bo} \simeq \frac{c}{v_{\rm sh}}.$54, with localized outflows approaching $\tau_{\rm bo} \simeq \frac{c}{v_{\rm sh}}.$55 and inflows near $\tau_{\rm bo} \simeq \frac{c}{v_{\rm sh}}.$56 in parts of the sheet (Lynch et al., 2012).

The same topology produces sympathetic eruption. In the gradual phase of the first CME, the first flare current sheet simultaneously acts as the overlying breakout sheet for the neighboring lobe, so the first flare sheet is also BCS2. This removes restraining flux from the left lobe, triggers a second flare current sheet, and launches a second CME. A key quantitative result is that flare reconnection processes flux about $\tau_{\rm bo} \simeq \frac{c}{v_{\rm sh}}.$57 times faster than pre-eruption breakout reconnection (Lynch et al., 2012).

Observable pre-eruption signatures are also edge-localized. Breakout reconnection jets produce bursty upflows along the pseudostreamer stalk and downflows in the adjacent arcade during the slow-rise phase. The simulations resolve plasmoid formation and ejection in both breakout and flare current sheets, and the breakout sheet creates a local minimum in magnetic energy density $\tau_{\rm bo} \simeq \frac{c}{v_{\rm sh}}.$58 that guides the flux-rope trajectory along a path of least resistance, producing early non-radial CME motion (Lynch et al., 2012).

7. Edge-aligned breakout in the plane-wide Breakout model

In the mathematical Breakout problem on $\tau_{\rm bo} \simeq \frac{c}{v_{\rm sh}}.$59, the plane is tiled by unit square bricks except for one missing square:

$\tau_{\rm bo} \simeq \frac{c}{v_{\rm sh}}.$60

The ball starts at $\tau_{\rm bo} \simeq \frac{c}{v_{\rm sh}}.$61 with $\tau_{\rm bo} \simeq \frac{c}{v_{\rm sh}}.$62, moves northeast with slope $\tau_{\rm bo} \simeq \frac{c}{v_{\rm sh}}.$63, reflects elastically from brick faces, and removes the brick it hits (Jullian, 2019).

The analysis uses a horizontal-edge unfolding convention, so the traveling ordinate is strictly increasing and satisfies

$\tau_{\rm bo} \simeq \frac{c}{v_{\rm sh}}.$64

Corner hits are excluded by requiring that the trajectory not pass through integer lattice points. The central combinatorial objects are ternary blocks

$\tau_{\rm bo} \simeq \frac{c}{v_{\rm sh}}.$65

which encode three successive horizontal-edge interactions (Jullian, 2019).

An elementary block is one for which the third encountered horizontal edge belongs to the already-broken top brick. This exact edge-alignment condition is given by parity-controlled equalities:

• if $\tau_{\rm bo} \simeq \frac{c}{v_{\rm sh}}.$66 and $\tau_{\rm bo} \simeq \frac{c}{v_{\rm sh}}.$67 have the same parity,

$\tau_{\rm bo} \simeq \frac{c}{v_{\rm sh}}.$68

• if $\tau_{\rm bo} \simeq \frac{c}{v_{\rm sh}}.$69 and $\tau_{\rm bo} \simeq \frac{c}{v_{\rm sh}}.$70 have different parity,

$\tau_{\rm bo} \simeq \frac{c}{v_{\rm sh}}.$71

A necessary constraint is $\tau_{\rm bo} \simeq \frac{c}{v_{\rm sh}}.$72 (Jullian, 2019). These conditions rigorously encode recurrent travel along aligned edges; in the terminology of the paper, they are the signal of elementary edge-breakout behavior.

Because bricks are destroyed, ordinary geometric periodicity is replaced by relative periodicity: in finitely many half-strips, the increment sequences of destroyed-brick coordinates are periodic or eventually periodic. The paper proves that if an irreducible rational slope $\tau_{\rm bo} \simeq \frac{c}{v_{\rm sh}}.$73 is elementary, then

$\tau_{\rm bo} \simeq \frac{c}{v_{\rm sh}}.$74

and if $\tau_{\rm bo} \simeq \frac{c}{v_{\rm sh}}.$75, then

$\tau_{\rm bo} \simeq \frac{c}{v_{\rm sh}}.$76

Moreover, any elementary sequence of blocks is periodic, and its minimal block period is either one of $\tau_{\rm bo} \simeq \frac{c}{v_{\rm sh}}.$77 or, up to circular permutation, has the form

$\tau_{\rm bo} \simeq \frac{c}{v_{\rm sh}}.$78

(Jullian, 2019).

The constructive families are equally explicit. When the relevant block conditions hold, the slopes

$\tau_{\rm bo} \simeq \frac{c}{v_{\rm sh}}.$79

yield elementary periodic block sequences $\tau_{\rm bo} \simeq \frac{c}{v_{\rm sh}}.$80, $\tau_{\rm bo} \simeq \frac{c}{v_{\rm sh}}.$81, and $\tau_{\rm bo} \simeq \frac{c}{v_{\rm sh}}.$82, respectively (Jullian, 2019). The resulting theory is a precise arithmetic classification of when edge-following periodic destruction patterns occur in an infinite Breakout geometry.

The broader significance is methodological. Here “edge breakout” is neither instability nor transport anomaly, but a rigid combinatorial recurrence enforced by parity, Sturmian slicing structure, and stackability constraints. That usage is formally unrelated to the physical meanings surveyed above, yet it preserves the same core intuition: a boundary interaction becomes dynamically organizing rather than incidental.