EHyOut: Diverse Applications Across Disciplines

• EHyOut is a term used to denote a variety of constructs in fields such as quantum systems, many-body theory, data analysis, transformer architectures, hydrodynamics, and astrophysics.
• Its implementations range from output augmentation in IO-HEOM for tracking dynamical observables to robust functional outlier detection using area-based indices and refined attention in modern Hopfield models.
• Additional uses include a high-order ADER-DG hydrodynamics solver and analysis of extremely high-velocity protostellar outflows, underscoring its broad computational and empirical relevance.

EHyOut is not a single standardized research object. In current arXiv usage, the label is applied to several unrelated constructs spanning open quantum systems, nonequilibrium many-body theory, functional data analysis, modern Hopfield networks, high-order computational hydrodynamics, and protostellar outflows (Cirio et al., 2024, Foini et al., 2024, Mishra et al., 12 Dec 2025, Pulido et al., 8 Jul 2025, Hu et al., 2024, Mantilla et al., 16 May 2026, Matsushita et al., 2018). This plurality of meanings suggests that the term functions primarily as a local shorthand within specific papers rather than as a field-wide technical designation.

1. Terminological range

The documented uses of EHyOut are heterogeneous in both subject matter and formal content.

Domain	Meaning of EHyOut	Representative source
Open quantum systems	output-augmented hierarchy within IO-HEOM	(Cirio et al., 2024)
Nonequilibrium many-body theory	out-of-equilibrium ETH framework	(Foini et al., 2024)
OTO transport	effective field theory for out-of-time-ordered transport	(Mishra et al., 12 Dec 2025)
Functional data analysis	area-based functional outlier detector	(Pulido et al., 8 Jul 2025)
Transformer architectures	Outlier-Efficient Modern Hopfield Model	(Hu et al., 2024)
Numerical hydrodynamics	ExaHyPE ADER-DG hydrodynamics solver	(Mantilla et al., 16 May 2026)
Star formation	extremely high-velocity outflow toward MMS 5	(Matsushita et al., 2018)

These usages are not variants of a common formalism. Some are methods, some are physical phenomena, and some are broader theoretical programs. A concise treatment therefore requires disambiguation by domain rather than an attempt at unification.

2. EHyOut as output augmentation in IO-HEOM

In open-quantum-system theory, EHyOut denotes the output-augmented hierarchy inside the Input-Output Hierarchical Equations of Motion (IO-HEOM), a non-perturbative extension of standard HEOM for bosonic environments prepared in non-Gaussian input states and coupled linearly to the system through $H_{SB}=S\otimes B$ (Cirio et al., 2024). The starting point is the usual HEOM decomposition of the bath two-time correlator,

$$C(t)=\langle B(t)B(0)\rangle=\sum_k c_k e^{-\gamma_k t},$$

together with auxiliary density operators indexed by a multi-index $\mathbf n$. EHyOut extends this structure by adding output-tracking operators $\mathcal Y_t^{\mathrm{out};\alpha k}$ defined through spectrally decomposed cross-correlations between the output field and the bath-coupling superoperators,

$$\langle \phi^{\mathrm{out}}(t)\chi^\alpha_\tau\rangle=\sum_k c^{\alpha k}e^{-\gamma^{\alpha k}(t-\tau)}, \qquad \dot{\mathcal Y}_t^{\mathrm{out};\alpha k}=c^{\alpha k}\mathcal S_t^\alpha-\gamma^{\alpha k}\mathcal Y_t^{\mathrm{out};\alpha k}.$$

The resulting augmented ADOs carry both conventional HEOM indices and additional field indices that track output observables dynamically.

A central structural point is that the extra field order is bounded by construction. Static fields, including non-Gaussian inputs or outputs evaluated at a fixed time, enter with binary indices and explicit time-dependent source terms $\mathcal G_t$; dynamical output fields require a spectral ansatz and generate the output hierarchy proper. The general extended ADOs are written as

$$\rho^{(N^\phi,N)}_{\mathbf n^\phi,\mathbf n}(t) = \alpha_0^N\, \mathcal T_S \prod_\eta [\mathcal Y_t^\eta]^{n_\eta^\phi} \prod_\sigma [\Theta_t^\sigma]^{n_\sigma} \rho_S(t),$$

with an IO-HEOM evolution equation containing the standard HEOM kernel, decay terms for dynamical output indices, and source terms for static fields.

This formulation gives an exact finite-series mapping from reduced bath correlations to system-space objects built from free-field correlators and IO-HEOM ADOs. It thereby permits computation of dynamical bath-output observables such as field amplitudes, energy densities, higher-order moments, and normal-ordered correlators. In the Markovian white-noise limit, the framework reduces to an input-output Lindblad equation with explicit input-dependent driving terms and recovers canonical input-output relations such as

$$b_{\mathrm{out}}(t)=b_{\mathrm{in}}(t)+\sqrt{\kappa}\,a(t).$$

The computational workflow is explicit: choose spectral decompositions for $C(t)$ and, when required, for output cross-correlators; build the augmented hierarchy; initialize the input state via field superoperators or cumulants; integrate the resulting ODE system; reconstruct the desired output observables; and verify convergence by increasing hierarchy depth and the field-index coverage. The complexity scales as

$$\mathcal O(\text{HEOM-size}\times 2^{N_{\mathrm{tot}}}), \qquad N_{\mathrm{tot}}=2N_I^H N^\phi_{\mathrm{exp}} m_{\mathrm{dyn}}+m_{\mathrm{stat}}.$$

This makes EHyOut a controlled extension of regular HEOM rather than a separate dynamical theory.

3. EHyOut in nonequilibrium thermalization and out-of-time order

A second major usage places EHyOut in the theory of nonequilibrium many-body dynamics. In "Out-of-equilibrium Eigenstate Thermalization Hypothesis," EHyOut is an ETH-inspired statistical ansatz for the initial-state projector $$C(t)=\langle B(t)B(0)\rangle=\sum_k c_k e^{-\gamma_k t},$$0 in the energy eigenbasis, designed to describe relaxation from pure nonequilibrium states with extensive mean energy and sub-extensive fluctuations (Foini et al., 2024). The ansatz takes the form

$$C(t)=\langle B(t)B(0)\rangle=\sum_k c_k e^{-\gamma_k t},$$1

with large-deviation scaling $$C(t)=\langle B(t)B(0)\rangle=\sum_k c_k e^{-\gamma_k t},$$2. Its key dynamical ingredient is the cross-correlation

$$C(t)=\langle B(t)B(0)\rangle=\sum_k c_k e^{-\gamma_k t},$$3

whose normalized counterpart obeys

$$C(t)=\langle B(t)B(0)\rangle=\sum_k c_k e^{-\gamma_k t},$$4

This exponentially small cross-correlation controls transient relaxation through

$$C(t)=\langle B(t)B(0)\rangle=\sum_k c_k e^{-\gamma_k t},$$5

The framework was numerically tested in a tilted-field Ising chain with $$C(t)=\langle B(t)B(0)\rangle=\sum_k c_k e^{-\gamma_k t},$$6, $$C(t)=\langle B(t)B(0)\rangle=\sum_k c_k e^{-\gamma_k t},$$7, $$C(t)=\langle B(t)B(0)\rangle=\sum_k c_k e^{-\gamma_k t},$$8, using $$C(t)=\langle B(t)B(0)\rangle=\sum_k c_k e^{-\gamma_k t},$$9, $\mathbf n$0, and $\mathbf n$1 as observables.

A closely related but distinct line of work generalizes ETH itself to account for out-of-time-ordered correlators. "The Eigenstate Thermalization Hypothesis and Out of Time Order Correlators" shows that positive Lyapunov behavior in OTOCs requires nontrivial correlations among matrix elements that would vanish under an independent-random-variable reading of standard ETH (Foini et al., 2018). The generalized loop average is

$\mathbf n$2

and the $\mathbf n$3 case supplies the all-distinct-index contribution needed for OTOC growth. In this sense, EHyOut-style nonequilibrium ETH and generalized ETH address adjacent questions: the former concerns the statistics of the initial-state projector, whereas the latter concerns higher-order operator correlations required by chaotic dynamics.

A third development turns to hydrodynamic late-time OTO behavior. "Theory of Out-of-Time-Ordered Transport" constructs a Schwinger-Keldysh EFT on a 2-CTP contour, based on a strong-to-weak spontaneous symmetry breaking pattern $\mathbf n$4, with soft fields $\mathbf n$5 and density-like variable $\mathbf n$6 (Mishra et al., 12 Dec 2025). The quadratic theory is fixed entirely by conventional transport data $\mathbf n$7 and $\mathbf n$8, while quartic order introduces two genuinely OTO transport parameters, $\mathbf n$9 and $\mathcal Y_t^{\mathrm{out};\alpha k}$0, visible in $\mathcal Y_t^{\mathrm{out};\alpha k}$1 but absent from any collapsed 1-CTP correlator. The EFT predicts universal late-time tails such as $\mathcal Y_t^{\mathrm{out};\alpha k}$2, $\mathcal Y_t^{\mathrm{out};\alpha k}$3, and $\mathcal Y_t^{\mathrm{out};\alpha k}$4, thereby relating many OTOCs directly to ordinary transport coefficients while isolating a smaller set of genuinely new OTO observables.

At finite temperature near a quantum critical regime, OTO behavior has also been studied numerically in the one-dimensional Bose-Hubbard model. There, the normalized OTOC was fitted as

$\mathcal Y_t^{\mathrm{out};\alpha k}$5

and the extracted Lyapunov exponent exhibited a broad peak around $\mathcal Y_t^{\mathrm{out};\alpha k}$6 at $\mathcal Y_t^{\mathrm{out};\alpha k}$7, close to the chaos bound $\mathcal Y_t^{\mathrm{out};\alpha k}$8; by contrast, the peak disappeared at $\mathcal Y_t^{\mathrm{out};\alpha k}$9 and away from integer filling (Shen et al., 2016). The same work also extracted a butterfly velocity $$\langle \phi^{\mathrm{out}}(t)\chi^\alpha_\tau\rangle=\sum_k c^{\alpha k}e^{-\gamma^{\alpha k}(t-\tau)}, \qquad \dot{\mathcal Y}_t^{\mathrm{out};\alpha k}=c^{\alpha k}\mathcal S_t^\alpha-\gamma^{\alpha k}\mathcal Y_t^{\mathrm{out};\alpha k}.$$0 from the onset times of spatially separated OTOCs and proposed a two-copy echo-type measurement protocol without Hamiltonian inversion. Taken together, these results place one family of EHyOut usages squarely within the study of thermalization, scrambling, and transport beyond time order.

4. EHyOut as a functional outlier detector

In functional data analysis, EHyOut is a robust procedure that converts functional outlier detection into a six-dimensional multivariate problem built from area-based epigraph and hypograph indices (Pulido et al., 8 Jul 2025). For a sample of curves $$\langle \phi^{\mathrm{out}}(t)\chi^\alpha_\tau\rangle=\sum_k c^{\alpha k}e^{-\gamma^{\alpha k}(t-\tau)}, \qquad \dot{\mathcal Y}_t^{\mathrm{out};\alpha k}=c^{\alpha k}\mathcal S_t^\alpha-\gamma^{\alpha k}\mathcal Y_t^{\mathrm{out};\alpha k}.$$1, the paper defines

$$\langle \phi^{\mathrm{out}}(t)\chi^\alpha_\tau\rangle=\sum_k c^{\alpha k}e^{-\gamma^{\alpha k}(t-\tau)}, \qquad \dot{\mathcal Y}_t^{\mathrm{out};\alpha k}=c^{\alpha k}\mathcal S_t^\alpha-\gamma^{\alpha k}\mathcal Y_t^{\mathrm{out};\alpha k}.$$2

Unlike the Modified Epigraph Index and Modified Hypograph Index, these quantities are not normalized by $$\langle \phi^{\mathrm{out}}(t)\chi^\alpha_\tau\rangle=\sum_k c^{\alpha k}e^{-\gamma^{\alpha k}(t-\tau)}, \qquad \dot{\mathcal Y}_t^{\mathrm{out};\alpha k}=c^{\alpha k}\mathcal S_t^\alpha-\gamma^{\alpha k}\mathcal Y_t^{\mathrm{out};\alpha k}.$$3 or $$\langle \phi^{\mathrm{out}}(t)\chi^\alpha_\tau\rangle=\sum_k c^{\alpha k}e^{-\gamma^{\alpha k}(t-\tau)}, \qquad \dot{\mathcal Y}_t^{\mathrm{out};\alpha k}=c^{\alpha k}\mathcal S_t^\alpha-\gamma^{\alpha k}\mathcal Y_t^{\mathrm{out};\alpha k}.$$4. They therefore respond simultaneously to magnitude differences and localized shape deviations, and satisfy

$$\langle \phi^{\mathrm{out}}(t)\chi^\alpha_\tau\rangle=\sum_k c^{\alpha k}e^{-\gamma^{\alpha k}(t-\tau)}, \qquad \dot{\mathcal Y}_t^{\mathrm{out};\alpha k}=c^{\alpha k}\mathcal S_t^\alpha-\gamma^{\alpha k}\mathcal Y_t^{\mathrm{out};\alpha k}.$$5

EHyOut computes these indices არა only on the original curve but also on its first two derivatives. After representing each curve with a cubic spline interpolation, the feature vector for $$\langle \phi^{\mathrm{out}}(t)\chi^\alpha_\tau\rangle=\sum_k c^{\alpha k}e^{-\gamma^{\alpha k}(t-\tau)}, \qquad \dot{\mathcal Y}_t^{\mathrm{out};\alpha k}=c^{\alpha k}\mathcal S_t^\alpha-\gamma^{\alpha k}\mathcal Y_t^{\mathrm{out};\alpha k}.$$6 is

$$\langle \phi^{\mathrm{out}}(t)\chi^\alpha_\tau\rangle=\sum_k c^{\alpha k}e^{-\gamma^{\alpha k}(t-\tau)}, \qquad \dot{\mathcal Y}_t^{\mathrm{out};\alpha k}=c^{\alpha k}\mathcal S_t^\alpha-\gamma^{\alpha k}\mathcal Y_t^{\mathrm{out};\alpha k}.$$7

This construction emphasizes level, slope, and curvature anomalies in a unified way. The multivariate step then uses robust Mahalanobis distances based on the Comedian (COM) estimator of location and scatter,

$$\langle \phi^{\mathrm{out}}(t)\chi^\alpha_\tau\rangle=\sum_k c^{\alpha k}e^{-\gamma^{\alpha k}(t-\tau)}, \qquad \dot{\mathcal Y}_t^{\mathrm{out};\alpha k}=c^{\alpha k}\mathcal S_t^\alpha-\gamma^{\alpha k}\mathcal Y_t^{\mathrm{out};\alpha k}.$$8

with outliers flagged by the boxplot rule

$$\langle \phi^{\mathrm{out}}(t)\chi^\alpha_\tau\rangle=\sum_k c^{\alpha k}e^{-\gamma^{\alpha k}(t-\tau)}, \qquad \dot{\mathcal Y}_t^{\mathrm{out};\alpha k}=c^{\alpha k}\mathcal S_t^\alpha-\gamma^{\alpha k}\mathcal Y_t^{\mathrm{out};\alpha k}.$$9

The empirical evaluation covered 19 data-generation processes, each with 200 curves on $\mathcal G_t$0, contamination rates $\mathcal G_t$1, and performance metrics including MCC, AUC, and execution time. EHyOut was the only method whose per-DGP median MCC exceeded $\mathcal G_t$2 for all 19 designs, and it was also the second fastest competitor, with mean execution time approximately $\mathcal G_t$3 s. Real-data applications identified Canary Islands stations and Navacerrada as temperature or precipitation outliers in Spanish weather data, and flagged 38 countries in the United Nations world population dataset, including Saudi Arabia, Iraq, Afghanistan, Malaysia, Uganda, Cameroon, and Yemen. In this usage, EHyOut is therefore a concrete algorithmic pipeline rather than a theoretical principle.

5. EHyOut as OutEffHop in transformer architectures

In machine learning, EHyOut refers to the Outlier-Efficient Modern Hopfield Model, or OutEffHop, and the associated outlier-efficient Hopfield layers proposed as replacements for standard attention in large transformer-based models (Hu et al., 2024). The construction augments the associative memory with a no-op class and replaces the conventional log-sum-exp by a refined version containing an additional zero-energy point,

$\mathcal G_t$4

The corresponding retrieval rule is

$\mathcal G_t$5

where

$\mathcal G_t$6

Because the denominator contains $\mathcal G_t$7, the total probability mass can be strictly less than one, permitting abstention without forcing extreme logits elsewhere.

Applied once to query, key, and value embeddings, this yields an attention-like layer

$\mathcal G_t$8

The paper interprets this as a principled approximation of OutEffHop retrieval rather than an ad hoc attention variant. Theoretical claims include monotone energy descent and convergence of the retrieval dynamics, tighter one-step retrieval error bounds than the original modern Hopfield model, an exponential storage-capacity lower bound that exceeds the corresponding lower bound for the original modern Hopfield model, and a $\mathcal G_t$9-Lipschitz property of $$\rho^{(N^\phi,N)}_{\mathbf n^\phi,\mathbf n}(t) = \alpha_0^N\, \mathcal T_S \prod_\eta [\mathcal Y_t^\eta]^{n_\eta^\phi} \prod_\sigma [\Theta_t^\sigma]^{n_\sigma} \rho_S(t),$$0 from $$\rho^{(N^\phi,N)}_{\mathbf n^\phi,\mathbf n}(t) = \alpha_0^N\, \mathcal T_S \prod_\eta [\mathcal Y_t^\eta]^{n_\eta^\phi} \prod_\sigma [\Theta_t^\sigma]^{n_\sigma} \rho_S(t),$$1 to $$\rho^{(N^\phi,N)}_{\mathbf n^\phi,\mathbf n}(t) = \alpha_0^N\, \mathcal T_S \prod_\eta [\mathcal Y_t^\eta]^{n_\eta^\phi} \prod_\sigma [\Theta_t^\sigma]^{n_\sigma} \rho_S(t),$$2. A norm-based generalization bound is also derived, scaling as $$\rho^{(N^\phi,N)}_{\mathbf n^\phi,\mathbf n}(t) = \alpha_0^N\, \mathcal T_S \prod_\eta [\mathcal Y_t^\eta]^{n_\eta^\phi} \prod_\sigma [\Theta_t^\sigma]^{n_\sigma} \rho_S(t),$$3 up to logarithmic factors.

Empirically, the model was evaluated on BERT, OPT, ViT, and STanHop-Net, against vanilla attention and variants such as $$\rho^{(N^\phi,N)}_{\mathbf n^\phi,\mathbf n}(t) = \alpha_0^N\, \mathcal T_S \prod_\eta [\mathcal Y_t^\eta]^{n_\eta^\phi} \prod_\sigma [\Theta_t^\sigma]^{n_\sigma} \rho_S(t),$$4 and $$\rho^{(N^\phi,N)}_{\mathbf n^\phi,\mathbf n}(t) = \alpha_0^N\, \mathcal T_S \prod_\eta [\mathcal Y_t^\eta]^{n_\eta^\phi} \prod_\sigma [\Theta_t^\sigma]^{n_\sigma} \rho_S(t),$$5. Across four models, OutEffHop achieved an average reduction of $$\rho^{(N^\phi,N)}_{\mathbf n^\phi,\mathbf n}(t) = \alpha_0^N\, \mathcal T_S \prod_\eta [\mathcal Y_t^\eta]^{n_\eta^\phi} \prod_\sigma [\Theta_t^\sigma]^{n_\sigma} \rho_S(t),$$6 in average kurtosis and $$\rho^{(N^\phi,N)}_{\mathbf n^\phi,\mathbf n}(t) = \alpha_0^N\, \mathcal T_S \prod_\eta [\mathcal Y_t^\eta]^{n_\eta^\phi} \prod_\sigma [\Theta_t^\sigma]^{n_\sigma} \rho_S(t),$$7 in the maximum infinity norm of model outputs. For BERT, average kurtosis dropped from $$\rho^{(N^\phi,N)}_{\mathbf n^\phi,\mathbf n}(t) = \alpha_0^N\, \mathcal T_S \prod_\eta [\mathcal Y_t^\eta]^{n_\eta^\phi} \prod_\sigma [\Theta_t^\sigma]^{n_\sigma} \rho_S(t),$$8 to $$\rho^{(N^\phi,N)}_{\mathbf n^\phi,\mathbf n}(t) = \alpha_0^N\, \mathcal T_S \prod_\eta [\mathcal Y_t^\eta]^{n_\eta^\phi} \prod_\sigma [\Theta_t^\sigma]^{n_\sigma} \rho_S(t),$$9, max $$b_{\mathrm{out}}(t)=b_{\mathrm{in}}(t)+\sqrt{\kappa}\,a(t).$$0 from $$b_{\mathrm{out}}(t)=b_{\mathrm{in}}(t)+\sqrt{\kappa}\,a(t).$$1 to $$b_{\mathrm{out}}(t)=b_{\mathrm{in}}(t)+\sqrt{\kappa}\,a(t).$$2, FP16 perplexity from $$b_{\mathrm{out}}(t)=b_{\mathrm{in}}(t)+\sqrt{\kappa}\,a(t).$$3 to $$b_{\mathrm{out}}(t)=b_{\mathrm{in}}(t)+\sqrt{\kappa}\,a(t).$$4, and W8A8 perplexity from $$b_{\mathrm{out}}(t)=b_{\mathrm{in}}(t)+\sqrt{\kappa}\,a(t).$$5 to $$b_{\mathrm{out}}(t)=b_{\mathrm{in}}(t)+\sqrt{\kappa}\,a(t).$$6. For OPT, W8A8 perplexity improved from $$b_{\mathrm{out}}(t)=b_{\mathrm{in}}(t)+\sqrt{\kappa}\,a(t).$$7 to $$b_{\mathrm{out}}(t)=b_{\mathrm{in}}(t)+\sqrt{\kappa}\,a(t).$$8. In this context, EHyOut designates an associative-memory-based remedy for activation outliers and post-quantization degradation.

6. EHyOut as an ExaHyPE hydrodynamics solver

In computational fluid dynamics, EHyOut denotes ExaHyPE’s ADER-DG hydrodynamics solver for the compressible Euler equations, combining a high-order one-step discontinuous Galerkin update, a local space-time DG predictor, dynamic AMR, and an a posteriori subcell finite-volume limiter (Mantilla et al., 16 May 2026). The governing system is

$$b_{\mathrm{out}}(t)=b_{\mathrm{in}}(t)+\sqrt{\kappa}\,a(t).$$9

with ideal-gas closure

$C(t)$0

sound speed $C(t)$1, and default $C(t)$2. The solver uses degree-$C(t)$3 DG polynomials, Gauss-Legendre quadrature of order $C(t)$4, and a local predictor that solves the PDE on $C(t)$5 by a cell-local Galerkin method.

Interface coupling is provided by the Rusanov flux,

$C(t)$6

with $C(t)$7. Stability is controlled by

$C(t)$8

and the paper uses $C(t)$9. Troubled cells are identified by positivity of $$\mathcal O(\text{HEOM-size}\times 2^{N_{\mathrm{tot}}}), \qquad N_{\mathrm{tot}}=2N_I^H N^\phi_{\mathrm{exp}} m_{\mathrm{dyn}}+m_{\mathrm{stat}}.$$0 and $$\mathcal O(\text{HEOM-size}\times 2^{N_{\mathrm{tot}}}), \qquad N_{\mathrm{tot}}=2N_I^H N^\phi_{\mathrm{exp}} m_{\mathrm{dyn}}+m_{\mathrm{stat}}.$$1, an entropy-based discrete maximum principle using $$\mathcal O(\text{HEOM-size}\times 2^{N_{\mathrm{tot}}}), \qquad N_{\mathrm{tot}}=2N_I^H N^\phi_{\mathrm{exp}} m_{\mathrm{dyn}}+m_{\mathrm{stat}}.$$2, and finite-value checks. Rejected cells are evolved on a subcell grid with $$\mathcal O(\text{HEOM-size}\times 2^{N_{\mathrm{tot}}}), \qquad N_{\mathrm{tot}}=2N_I^H N^\phi_{\mathrm{exp}} m_{\mathrm{dyn}}+m_{\mathrm{stat}}.$$3 using a second-order TVD finite-volume scheme with Rusanov flux and minmod limiting.

The code reports conservative and primitive variables, sound speed, Mach number, entropy proxy $$\mathcal O(\text{HEOM-size}\times 2^{N_{\mathrm{tot}}}), \qquad N_{\mathrm{tot}}=2N_I^H N^\phi_{\mathrm{exp}} m_{\mathrm{dyn}}+m_{\mathrm{stat}}.$$4, vorticity $$\mathcal O(\text{HEOM-size}\times 2^{N_{\mathrm{tot}}}), \qquad N_{\mathrm{tot}}=2N_I^H N^\phi_{\mathrm{exp}} m_{\mathrm{dyn}}+m_{\mathrm{stat}}.$$5, baroclinic source term $$\mathcal O(\text{HEOM-size}\times 2^{N_{\mathrm{tot}}}), \qquad N_{\mathrm{tot}}=2N_I^H N^\phi_{\mathrm{exp}} m_{\mathrm{dyn}}+m_{\mathrm{stat}}.$$6, limiter masks, AMR statistics, conservation residuals, and error norms. Validation covered a strong-shock Sod-type problem, Shu-Osher shock-entropy interaction, the Woodward-Colella blast wave, a contact-driven vortex sheet, and a shock-interface interaction. In the Shu-Osher case, the phase-insensitive Shannon entropy of the post-shock density amplitude distribution at $$\mathcal O(\text{HEOM-size}\times 2^{N_{\mathrm{tot}}}), \qquad N_{\mathrm{tot}}=2N_I^H N^\phi_{\mathrm{exp}} m_{\mathrm{dyn}}+m_{\mathrm{stat}}.$$7 increased from $$\mathcal O(\text{HEOM-size}\times 2^{N_{\mathrm{tot}}}), \qquad N_{\mathrm{tot}}=2N_I^H N^\phi_{\mathrm{exp}} m_{\mathrm{dyn}}+m_{\mathrm{stat}}.$$8 bits for order 3 to $$\mathcal O(\text{HEOM-size}\times 2^{N_{\mathrm{tot}}}), \qquad N_{\mathrm{tot}}=2N_I^H N^\phi_{\mathrm{exp}} m_{\mathrm{dyn}}+m_{\mathrm{stat}}.$$9 bits for order 5 and $$C(t)=\langle B(t)B(0)\rangle=\sum_k c_k e^{-\gamma_k t},$$00 bits for order 7, versus $$C(t)=\langle B(t)B(0)\rangle=\sum_k c_k e^{-\gamma_k t},$$01 bits for the high-resolution reference. In the Woodward-Colella benchmark, a seventh-order ultra-fine reference run with $$C(t)=\langle B(t)B(0)\rangle=\sum_k c_k e^{-\gamma_k t},$$02 required more than $$C(t)=\langle B(t)B(0)\rangle=\sum_k c_k e^{-\gamma_k t},$$03 wall-clock hours, illustrating the practical motivation for combining high order with AMR and localized limiting.

7. EHyOut as an extremely high-velocity outflow

In star-formation studies, EHyOut refers to an Extremely High-Velocity outflow, specifically the compact EHV flow discovered toward MMS 5 (HOPS 88) in OMC-3 with ALMA (Matsushita et al., 2018). MMS 5 lies at $$C(t)=\langle B(t)B(0)\rangle=\sum_k c_k e^{-\gamma_k t},$$04 pc, is a Class 0 source with envelope mass $$C(t)=\langle B(t)B(0)\rangle=\sum_k c_k e^{-\gamma_k t},$$05–$$C(t)=\langle B(t)B(0)\rangle=\sum_k c_k e^{-\gamma_k t},$$06 and $$C(t)=\langle B(t)B(0)\rangle=\sum_k c_k e^{-\gamma_k t},$$07, and was previously known to drive an east-west CO outflow on $$C(t)=\langle B(t)B(0)\rangle=\sum_k c_k e^{-\gamma_k t},$$08 pc scales. The ALMA study combined ACA 7-m data with 12-m compact and extended arrays, and observed CO $$C(t)=\langle B(t)B(0)\rangle=\sum_k c_k e^{-\gamma_k t},$$09, SiO $$C(t)=\langle B(t)B(0)\rangle=\sum_k c_k e^{-\gamma_k t},$$10, C$$C(t)=\langle B(t)B(0)\rangle=\sum_k c_k e^{-\gamma_k t},$$11O $$C(t)=\langle B(t)B(0)\rangle=\sum_k c_k e^{-\gamma_k t},$$12, and N$$C(t)=\langle B(t)B(0)\rangle=\sum_k c_k e^{-\gamma_k t},$$13D$$C(t)=\langle B(t)B(0)\rangle=\sum_k c_k e^{-\gamma_k t},$$14 $$C(t)=\langle B(t)B(0)\rangle=\sum_k c_k e^{-\gamma_k t},$$15.

The systemic velocity is $$C(t)=\langle B(t)B(0)\rangle=\sum_k c_k e^{-\gamma_k t},$$16 km s$$C(t)=\langle B(t)B(0)\rangle=\sum_k c_k e^{-\gamma_k t},$$17. CO traces a lower-velocity outflow at $$C(t)=\langle B(t)B(0)\rangle=\sum_k c_k e^{-\gamma_k t},$$18–$$C(t)=\langle B(t)B(0)\rangle=\sum_k c_k e^{-\gamma_k t},$$19 km s$$C(t)=\langle B(t)B(0)\rangle=\sum_k c_k e^{-\gamma_k t},$$20 and an EHV component at $$C(t)=\langle B(t)B(0)\rangle=\sum_k c_k e^{-\gamma_k t},$$21–$$C(t)=\langle B(t)B(0)\rangle=\sum_k c_k e^{-\gamma_k t},$$22 km s$$C(t)=\langle B(t)B(0)\rangle=\sum_k c_k e^{-\gamma_k t},$$23, while SiO $$C(t)=\langle B(t)B(0)\rangle=\sum_k c_k e^{-\gamma_k t},$$24 selectively traces only the fastest, most collimated component. Morphologically, the CO outflow is V-shaped with position angle $$C(t)=\langle B(t)B(0)\rangle=\sum_k c_k e^{-\gamma_k t},$$25, opening angle $$C(t)=\langle B(t)B(0)\rangle=\sum_k c_k e^{-\gamma_k t},$$26, and deprojected length $$C(t)=\langle B(t)B(0)\rangle=\sum_k c_k e^{-\gamma_k t},$$27 AU. The red CO EHV jet has $$C(t)=\langle B(t)B(0)\rangle=\sum_k c_k e^{-\gamma_k t},$$28 AU, width $$C(t)=\langle B(t)B(0)\rangle=\sum_k c_k e^{-\gamma_k t},$$29 AU, deprojected length $$C(t)=\langle B(t)B(0)\rangle=\sum_k c_k e^{-\gamma_k t},$$30 AU, and position angle $$C(t)=\langle B(t)B(0)\rangle=\sum_k c_k e^{-\gamma_k t},$$31. The red SiO jet is smaller still, with $$C(t)=\langle B(t)B(0)\rangle=\sum_k c_k e^{-\gamma_k t},$$32 AU, width $$C(t)=\langle B(t)B(0)\rangle=\sum_k c_k e^{-\gamma_k t},$$33 AU, and deprojected length $$C(t)=\langle B(t)B(0)\rangle=\sum_k c_k e^{-\gamma_k t},$$34 AU. The jet-outflow axis offset on the red side is $$C(t)=\langle B(t)B(0)\rangle=\sum_k c_k e^{-\gamma_k t},$$35–$$C(t)=\langle B(t)B(0)\rangle=\sum_k c_k e^{-\gamma_k t},$$36.

Using an inclination $$C(t)=\langle B(t)B(0)\rangle=\sum_k c_k e^{-\gamma_k t},$$37, the inferred dynamical times are $$C(t)=\langle B(t)B(0)\rangle=\sum_k c_k e^{-\gamma_k t},$$38 yr for the CO outflow, $$C(t)=\langle B(t)B(0)\rangle=\sum_k c_k e^{-\gamma_k t},$$39 yr for the CO jet, and $$C(t)=\langle B(t)B(0)\rangle=\sum_k c_k e^{-\gamma_k t},$$40 yr for the SiO jet. The EHV component is knotty, with six knots separated on average by $$C(t)=\langle B(t)B(0)\rangle=\sum_k c_k e^{-\gamma_k t},$$41, corresponding to deprojected spacing $$C(t)=\langle B(t)B(0)\rangle=\sum_k c_k e^{-\gamma_k t},$$42 AU and ejection period $$C(t)=\langle B(t)B(0)\rangle=\sum_k c_k e^{-\gamma_k t},$$43–$$C(t)=\langle B(t)B(0)\rangle=\sum_k c_k e^{-\gamma_k t},$$44 yr. A sinusoidal fit to the knot wiggle suggests precession with period $$C(t)=\langle B(t)B(0)\rangle=\sum_k c_k e^{-\gamma_k t},$$45 yr. Four lines of evidence favor a nested wind scenario over pure jet entrainment: the outflow is larger than the jet, its dynamical time is longer by a factor of about $$C(t)=\langle B(t)B(0)\rangle=\sum_k c_k e^{-\gamma_k t},$$46, the axes are offset, and the knot periodicity indicates a time-variable inner engine. The paper nonetheless states that jet entrainment cannot be completely ruled out. In this astrophysical sense, EHyOut refers not to a computational method but to a compact, collimated, shock-traced molecular jet rooted at the base of a wider protostellar CO outflow.