3 Quantum-Gravity-Inspired Instability Hamiltonian for Fusion Reactor Design In this section we outline a fusion-reactor design framework inspired by the Hamiltonian and spectral analysis used in quantum cosmology. The key idea is to treat plasma instabilities as collective degrees of freedom governed by an effective instability Hamiltonian, whose spectrum encodes stability, marginality, and disruption. By engineering the "potential" and "geometry" of this Hamiltonian, we obtain a principled way to design coils, walls, and control systems such that all relevant instability modes are stabilized with a finite spectral gap. We deliberately mirror the structure of the quantum-gravity construction in order to transfer both its conceptual tools (Hamiltonian stability analysis, critical couplings, boundary-condition engineering) and its discrete, covariant numerical methodology into the fusion context. 3.1 Instability coordinates and effective Hamiltonian We begin by introducing a set of collective instability coordinates, denoted by = ( _1, _2, ..., _N), which parameterize the amplitudes of a chosen set of important modes. These can be, for example, global kink, tearing, ballooning, and edge-localized mode (ELM) perturbations, or reduced representations of displacement fields obtained from the MHD energy principle. The central object is an effective instability Hamiltonian H_plasma acting on functions ( ): H_plasma ( ) = [ T( ; P) + U( ; P) ] ( ), (3.1) where - P is a vector of engineering and control parameters (coil currents and geometry, pressure and current profiles, wall properties, feedback gains, etc.); - T( ; P) is a kinetic-like operator that encodes how gradients and currents drive the instability; - U( ; P) is an effective potential that encodes stabilizing effects such as magnetic line bending, curvature and shear, wall response, and active control fields. In the simplest one-dimensional prototype, with a single coordinate representing the amplitude of a dominant mode, we may write H_plasma = -D(P) d²/d ² + U( ; P), (3.2) where D(P) > 0 is an effective "mobility" for the mode. This directly mirrors the one-dimensional minisuperspace Hamiltonian of the cosmological model, with the role of the scale factor a replaced by the instability amplitude 3.2 Quadratic stability and the stabilizing mass term To connect more directly to the quantum-gravity analysis, we decompose the potential U into a destabilizing contribution and a stabilizing "mass" term. For a single mode, a convenient parametrization is U( ; P) = -1/2 (P) ² + 1/2 _stab(P)² ² + 1/4 (P) , (3.3) where - (P) > 0 quantifies the effective drive due to bad curvature, pressure gradients, and current profiles; - _stab(P)² is a positive stabilizing parameter, directly analogous to the cosmological mass term _2 in the matter potential; - (P) 0 encodes nonlinear saturation that bounds the mode amplitude at large | |. The quadratic part of the potential is then U_quad( ; P) = 1/2 [ _stab(P)² - (P) ] ². (3.4) If _stab(P)² < (P), the curvature is negative around = 0 and the mode is linearly unstable; if _stab(P)² > (P), the curvature is positive and = 0 is a linearly stable equilibrium point. This mimics the role of the matter coupling _2 in the universe Hamiltonian: sufficiently large stabilizing coupling drives the system into a bounded-below, gapped spectral regime. In the multi-mode case, we write W = 1/2 _{i,j} G_ij(P) _i _j + higher-order terms, (3.5) where W is the MHD energy functional and G_ij(P) is the metric on the instability configuration space. The instability Hamiltonian can then be written in a Laplace Beltrami form H_plasma = - _eff² _LB( ; P) + U( ; P), (3.6) where _LB is the Laplace Beltrami operator associated with the metric G_ij and _eff is a scaling constant (there is no claim of literal quantization here; we use the spectral analogy as a design tool). 3.3 Critical coupling and the stability phase transition The key structural feature imported from the quantum-gravity model is the existence of a critical coupling _stab* at which the spectrum of H_plasma undergoes a transition from unstable (unbounded below) to stable (bounded below with a positive gap). Let _min(P) denote the lowest eigenvalue of H_plasma for a given parameter set P. We define _stab* implicitly by _min( _stab) < 0 for _stab < _stab*, (3.7a) _min( _stab) = 0 for _stab = _stab* (marginal stability), (3.7b) _min( _stab) > 0 for _stab > _stab*, (3.7c) where we suppress the dependence on other parameters for clarity. In a practical fusion device, _stab is not a single scalar but an effective parameter that can be realized by several engineering knobs, including: - Magnetic well depth and curvature (particularly in stellarators); - Strength and spectral content of resonant magnetic perturbations; - Shear and safety-factor profile shaping via current drive and pressure profiles; - Wall properties (conducting, resistive, or actively controlled); - Feedback control gains and bandwidth for specific modes. The fusion design goal is to choose a region of parameter space P in which all relevant instability Hamiltonians H_plasma(P) have _min(P) > 0 with a finite gap, and in which admissible operational variations do not cross the critical hypersurface _min = 0. It is often convenient to phrase this in terms of a Euclidean trace Z_plasma( ; P) = Tr [ exp( - H_plasma(P) ) ], (3.8) in direct analogy with the Euclidean partition function used to diagnose instability in the cosmological setting. If H_plasma has a negative eigenvalue, Z_plasma( ; P) grows rapidly as increases. If H_plasma is positive definite, Z_plasma( ; P) decays monotonically with . Thus, the transition from growth to decay in Z_plasma as a function of _stab and other parameters provides another practical diagnostic of the critical stability surface. 3.4 Discrete, covariant formulation and geometric control In the quantum-gravity construction, the Laplace Beltrami operator on configuration space is discretized on a lattice or graph in a way that respects the underlying measure and ensures self-adjointness. Inspired by this, we discretize the instability configuration space for fusion as follows. First, we choose a finite set of representative values for each _i, defining a grid or more general graph in -space: _i { _i^(1), _i^(2), ..., _i^(n_i) }. (3.9) We then construct a weighted graph Laplacian L_LB that approximates the continuous _LB and is symmetric with respect to the discrete measure induced by the metric G_ij. The discrete instability Hamiltonian is H_plasma^disc(P) = - _eff² L_LB + U_disc(P), (3.10) where U_disc(P) is a diagonal matrix that stores the values of the potential U at each node in the discrete configuration space. This finite-dimensional matrix has a well-defined spectrum that can be computed directly for any given parameter set P. The dependence of _min(P) and higher eigenvalues on coil geometry, wall configuration, and feedback settings provides a spectral objective for design optimization. Geometric control enters through the way engineering elements deform U and the metric G_ij. Each coil set C_k, sensor, and actuator is associated with a deformation term U_k( ), such that U( ; P) = U_0( ) + _k g_k(P) U_k( ), (3.11) where g_k(P) are control parameters (coil currents, displacements, etc.). By choosing coil shapes and placements so that their induced U_k( ) are aligned with the geometry encoded in G_ij and the discrete Laplacian, we obtain a kind of "LB-covariant" control: stabilizing deformations that do not introduce artificial instabilities or unphysical distortions elsewhere in configuration space. 3.5 Boundary-condition engineering: walls and divertor A central role in the cosmological stability analysis is played by the boundary conditions imposed on the wavefunction at the endpoints of configuration space. Dirichlet, Neumann, and Robin families each correspond to different physical assumptions about the early universe and lead to different spectral properties of the Hamiltonian. In the fusion setting, the reactor wall, blanket, and divertor play an analogous role. Different material and control choices correspond to different effective boundary conditions on the plasma perturbations: - A perfectly conducting wall imposes constraints on magnetic perturbations that resemble Dirichlet-type conditions on certain components of the displacement field. - A resistive wall leads to mixed (Robin-like) conditions, where both the value and derivative of the perturbation appear, with coefficients set by wall resistivity, thickness, and geometry. - An actively controlled wall, equipped with sensors and feedback coils, effectively promotes the boundary condition to a dynamical degree of freedom coupled to the plasma. We model these possibilities via a boundary parameter _BC that enters the boundary relation schematically as '( _edge) + _BC ( _edge) = 0, (3.12) where _edge denotes a boundary point in configuration space associated with large amplitude displacements or interaction with the wall. The parameter _BC can then be varied to represent different wall materials, passive stabilization schemes, and active control laws. In the discrete framework, _BC becomes an explicit part of P, and its influence on _min(P) and Z_plasma( ; P) can be systematically explored. This elevates the wall and divertor from passive hardware to active elements in Hamiltonian engineering: by tuning _BC we effectively move along a family of boundary conditions and search for regimes where the instability Hamiltonian is spectrally gapped. 3.6 Non-perturbative whole-device simulation The instability Hamiltonian framework is not restricted to small linear perturbations around a fixed equilibrium. In a non-perturbative, whole-device picture, we represent the plasma, wall, and coil system as a three-dimensional graph whose nodes correspond to finite-volume cells in the plasma and structural components, and whose edges represent flux tubes, transport paths, and current connections. Let X(t) denote the vector of all state variables attached to the nodes (densities, temperatures, magnetic fields, velocities, and reduced instability amplitudes). We define a covariant discrete time update X(t + t) = U_disc[ X(t); P ], (3.13) where U_disc is built from discrete differential operators that respect conservation laws (energy, momentum, and magnetic flux) up to controlled dissipative terms. At selected times, or whenever the system configuration changes appreciably, we linearize the dynamics around X(t), project onto the reduced instability coordinates , and reconstruct an instantaneous H_plasma^disc(P; t). The lowest eigenvalues and the corresponding Euclidean trace Z_plasma( ; P; t) then provide real-time diagnostics of proximity to the critical stability surface _min = 0. A control system can be designed to adjust fast-acting parameters in P (such as RMP currents and feedback gains) to keep the evolving device trajectory inside a region of configuration space where the instability Hamiltonians remain spectrally gapped. In the next section we will show how this framework can be specialized to particular fusion concepts, including stellarator coil optimization, tokamak disruption avoidance, and the design of magnetic geometries with topologically protected stability properties. Figure 3.1: Schematic depiction of the instability configuration space and the effective potential U( ; P), showing a transition from an unstable negative-curvature regime to a stable, gapped regime as _stab is increased. Figure 3.2: Illustration of the discrete -space graph, the associated graph Laplacian L_LB, and the deformation of the potential U_disc(P) by coil currents and wall parameters. Figure 3.3: Example parameter scan showing _min(P) and Z_plasma( ; P) across a two-dimensional slice of engineering parameters, with the critical stability curve _min = 0 highlighted. 4 Illustrative Toy Model for a Single Instability Mode In this section we instantiate the general framework of Sec. 3 in the simplest nontrivial setting: a single instability coordinate governed by a 1D instability Hamiltonian. Although this model is deliberately minimal, it already exhibits the key phenomena of interest: 1. an unstable regime with a negative lowest eigenvalue; 2. a critical stabilizing coupling _stab* at which the spectrum becomes marginal; 3. a stable, gapped regime for _stab > _stab*. The purpose of this section is to demonstrate, in a fully explicit numerical example, how the critical coupling picture and the spectral gap can be computed and visualized for a concrete Hamiltonian of the form introduced in Sec. 3. This provides a template for more realistic multi-mode applications in later sections. 4.1 Model definition We consider a single real coordinate that represents the amplitude of a dominant instability mode (for example, an n=1 kink or tearing perturbation in a reduced description). The effective instability Hamiltonian is taken to be of the form H_plasma = -D d²/d ² - 1/2 ² + 1/2 _stab² ² + 1/4 , (4.1) where D > 0 is a kinetic prefactor, > 0 quantifies the destabilizing drive, _stab is the stabilizing coupling, and 0 controls the nonlinear saturation of large-| | fluctuations. The potential term can be written as U( ; _stab) = -1/2 ² + 1/2 _stab² ² + 1/4 = 1/2 ( _stab² - ) ² + 1/4 (4.2) For fixed and , it is clear at the level of the quadratic approximation that the curvature at =0 changes sign when _stab² = (4.3) For _stab² < , the origin is a local maximum and the system is linearly unstable; for _stab² > , the origin becomes a local minimum. The quartic term ensures that U( ; _stab) grows at large | |, so the Hamiltonian is bounded below whenever the quadratic part is not too negative. In practice, the combined effect of the kinetic term and the quartic potential leads to a well-defined critical value _stab* at which the lowest eigenvalue of H_plasma crosses zero. In the numerical example below we adopt dimensionless units and fix D = 1, = 1, = 0.2, (4.4) leaving _stab as the control parameter to be scanned. 4.2 Discretization and numerical method To obtain the spectrum of H_plasma we discretize the coordinate on a finite interval [-L, L], with L = 5, (4.5) using a uniform grid of N points, _j = -L + (j-1) , j = 1,...,N, = 2L / (N-1). (4.6) The second derivative is approximated by the standard three-point finite difference stencil with Dirichlet boundary conditions, d² /d ²( _j) ( _{j+1} - 2 _j + _{j-1}) / ², (4.7) which yields a tridiagonal Laplacian matrix ( _ )_{jj} = -2/ ², ( _ )_{j,j+1} = ( _ )_{j+1,j} = 1/ ². (4.8) The kinetic term is then represented by T = -D _ , (4.9) while the potential term becomes a diagonal matrix U_{jj}( _stab) = U( _j; _stab). (4.10) The discrete Hamiltonian is thus H_disc( _stab) = T + U( _stab), (4.11) a real symmetric N × N matrix whose eigenvalues can be obtained by standard linear algebra routines. In the numerical results presented here we use N = 400, which is sufficient to resolve the low-lying spectrum and the qualitative structure of the stability transition for the chosen parameters. For each value of _stab we compute the full set of eigenvalues and record the lowest eigenvalue _min( _stab). 4.3 Spectral results and critical coupling Figure 4.1 illustrates the shape of the potential U( ; _stab) for several representative values of _stab. For small _stab the negative quadratic term dominates near =0, producing a locally inverted potential that drives instability. As _stab increases, the curvature at the origin becomes less negative, then vanishes, and finally positive, while the quartic term ensures confinement at large | |. Figure 4.2 shows the numerically computed ground-state eigenvalue _min( _stab) as a function of _stab for the parameter choices in Eq. (4.4). For small _stab the lowest eigenvalue is negative, reflecting the presence of an energetically favored unstable direction in configuration space. As _stab is increased, _min( _stab) rises monotonically, crossing zero at a critical value _stab* and becoming positive thereafter. Define _min( _stab*) = 0. (4.12) Numerically, for the parameter values adopted here, _stab* is found to be close to the quadratic threshold _stab² = , with small quantitative shifts due to the finite domain and quartic term. For _stab > _stab* the Hamiltonian is positive definite and the gap ( _stab) = _1( _stab) - _min( _stab) (4.13) between the ground state and the first excited state is strictly positive. The existence and size of this gap provide a natural quantitative measure of how robustly the instability has been suppressed by the stabilizing coupling. In parallel with Sec. 3, one may also form the Euclidean trace Z_plasma( ; _stab) = Tr [ exp( - H_disc( _stab) ) ], (4.14) which in this toy model can be computed by summing over the discrete eigenvalues. For _stab < _stab* the contribution of the negative ground-state eigenvalue causes Z_plasma to grow rapidly with , whereas for _stab > _stab* the trace decays monotonically, as expected for a positive-definite Hamiltonian. 4.4 Interpretation and limitations Although the model considered here is highly simplified, it already captures the structure needed for fusion design: 1. Unstable regime: A negative lowest eigenvalue represents an energetically favored unstable mode; in a reactor, this corresponds to a configuration in which the chosen instability can grow spontaneously. 2. Critical coupling: The crossing _min( _stab*) = 0 marks a sharp transition between unstable and stable regimes in the space of control parameters. In a physical device, the effective _stab would be realized through a combination of coil currents, profile shaping, and wall response. 3. Stable, gapped regime: For _stab > _stab* the positive gap quantifies how far the system is from the onset of instability. Designing an operational window with a large minimum gap provides a quantitative margin of safety against parameter drifts and transient perturbations. At the same time, this toy model has clear limitations: - It describes only a single degree of freedom, whereas real plasmas exhibit a high-dimensional spectrum of coupled modes. - The potential parameters ( , ) are chosen phenomenologically rather than derived from a specific MHD equilibrium. - Spatial structure and geometry (e.g. toroidal vs helical configurations) are not yet represented; these enter more naturally in the multi-mode, Laplace Beltrami framework of Sec. 3. Nonetheless, the toy model serves three essential roles: 1. It provides a fully explicit example of the critical-coupling phenomenon in a setting that is numerically transparent. 2. It demonstrates how the instability Hamiltonian approach can be implemented concretely as a matrix eigenvalue problem. 3. It furnishes a template for constructing more realistic reduced models associated with specific classes of instabilities in tokamaks and stellarators. In subsequent sections, we generalize this construction to multi-mode Hamiltonians derived from MHD energy principles, and we discuss how to embed such reduced spectral analyses into whole-device optimization loops for stellarator coil design and tokamak disruption avoidance. Figure 4.1: Effective potential U( ; _stab) from Eq. (4.2) for several values of _stab, illustrating the transition from a locally inverted potential (unstable) to a confining, single-well potential (stable) as the stabilizing coupling increases. Figure 4.2: Numerically computed lowest eigenvalue _min( _stab) of H_plasma vs _stab. The curve crosses zero at the critical value _stab*, separating an unstable regime ( _min < 0) from a stable, gapped regime ( _min > 0). 5 Synergy with the MAS Landau Fluid Eigenvalue Code To connect the quantum gravity-inspired fusion (QG Fusion) framework to existing tools, we compare it to the MAS code: a five-field Landau fluid eigenvalue solver for plasma stability in general tokamak geometry. This section outlines how MAS can act both as (i) a physical instantiation of our instability Hamiltonian programme and (ii) a testbed for extending the framework from toy models to realistic reactor configurations. 5.1 Conceptual bridge: Hamiltonian formalism and spectral analysis MAS solves a global, five-field Landau fluid eigenvalue problem, A X = B X, in realistic tokamak geometry, where is complex and the eigenmodes include Alfvén eigenmodes, kink modes, and drift-wave instabilities. QG Fusion introduces an effective instability Hamiltonian H_plasma(P) = , whose spectrum { } encodes stability (all > 0), marginality ( _min = 0), or disruption ( _min < 0) for a reduced set of instability coordinates The Toy Model in Sec. 4 explicitly constructs and diagonalizes such a one-dimensional H_plasma, showing how critical couplings and spectral gaps arise in practice. Synergy. MAS s eigenvalue problem can be viewed as a high-dimensional, physics-complete realization of the spectral analysis proposed in the QG Fusion framework. In particular, MAS provides: - physically grounded spectra (including kinetic and geometric effects) against which reduced Hamiltonians H_plasma can be calibrated; - a route to validate that the instability Hamiltonian formalism reproduces known stability properties in realistic geometries. 5.2 Stability margins and critical transitions MAS identifies stability boundaries in parameter space (e.g. thresholds for kinetic ballooning modes, RSAE frequency sweeps, onset of drift-wave instabilities) and resolves singularities via kinetic effects and Landau closures. QG Fusion formalizes these transitions as the crossing of a critical stabilizing coupling _stab* defined by _min( _stab*) = 0, where _min is the lowest eigenvalue of H_plasma. The Toy Model numerically demonstrates this transition: for a 1D Hamiltonian, _min( _stab) crosses from negative to positive at a well-defined _stab*, beyond which a finite spectral gap opens. Synergy. MAS simulations provide a physical mapping from reactor parameters (pressure gradients, current profiles, equilibrium shape) to effective stability margins. These can be encoded as an effective _stab (or a small set of such couplings) in the QG Fusion framework. Conversely, the QG formalism offers a unified mathematical language in which MAS stability boundaries appear as critical surfaces _min = 0 in a reduced Hamiltonian space. 5.3 Geometry and boundary-condition engineering MAS is formulated in realistic flux coordinates (Boozer coordinates), retaining geometric effects such as elongation, triangularity, and shaping that influence stability limits. QG Fusion emphasizes boundary conditions as an explicit control knob: the reactor wall, blanket, and divertor are treated analogously to cosmological boundary conditions, with parameters (e.g. _BC) encoding conducting, resistive, or actively controlled boundaries. The Toy Model implements a simple Dirichlet boundary condition on a finite interval in , serving as a minimal example of how boundary choices influence the spectrum of H_plasma. Synergy. MAS can be used to explore boundary-condition families e.g. perfectly conducting vs. resistive walls, different pedestal and scrape-off-layer models and to track how these choices deform the spectrum. This directly instantiates the QG Fusion idea that changing wall/divertor physics corresponds to moving along a boundary-condition manifold in the space of instability Hamiltonians. 5.4 Discretization and covariant formulation MAS discretizes the governing equations using finite differences radially and Fourier decomposition poloidally/toroidally, assembling large sparse matrices for eigenvalue solves. QG Fusion proposes a graph Laplacian / discrete Laplace Beltrami formulation in the reduced instability configuration space, ensuring that the discrete operators respect the underlying geometry and measure. The Toy Model uses a one-dimensional, finite-difference Laplacian to approximate -d²/d ², which is the simplest instance of this covariant discretization idea. Synergy. The numerical machinery in MAS well-tested sparse eigenvalue solvers, coordinate mapping, and matrix construction already embodies many of the discretization principles advocated in the QG Fusion framework. MAS thus becomes a natural platform for: - introducing more geometric, graph-based discretizations of selected reduced subspaces; - testing whether LB-like discrete operators in reduced -spaces can be consistently embedded into a full Landau fluid eigenvalue code. 5.5 From linear eigenvalue problems to control MAS currently focuses on linear stability: it computes growth rates and mode structures for a broad spectrum of instabilities, including MHD, EP-driven Alfvén eigenmodes, and drift-wave turbulence precursors. QG Fusion proposes to turn spectral data into control diagnostics, e.g. by monitoring the lowest eigenvalue _min(t) or Euclidean traces Z_plasma( ; t) in real time and actively steering the system away from the _min = 0 surface. The Toy Model shows concretely how _min( _stab) can be tracked as a function of a control parameter, demonstrating a simple instance of spectral feedback . Synergy. MAS s eigenvalue solver can be embedded in higher-level control workflows: for example, given a slowly evolving equilibrium sequence (or predicted evolution), MAS can: - compute _min(t) for key modes; - feed this into a control algorithm that adjusts coil currents, profile actuators, or RMP amplitudes to keep _min bounded away from zero. This is precisely the QG Fusion paradigm: treat the instability spectrum as the primary control observable, rather than individual macroscopic quantities alone. 5.6 Multi-mode and whole-device modeling MAS already supports multi-field, multi-harmonic modeling: it evolves five coupled fields, with multiple poloidal harmonics and realistic toroidal coupling, capturing complex mode structures and kinetic effects. QG Fusion generalizes from a single -coordinate to a multi-mode instability configuration space = ( _1,..., _N), and further to whole-device graphs where nodes represent spatial cells and edges encode fluxes and couplings. The Toy Model provides the simplest case (N = 1), serving as an analytically transparent starting point before moving to a multi-mode Laplace Beltrami operator. Synergy. MAS is a natural candidate for implementing the multi-mode instability Hamiltonians envisioned in the QG Fusion framework: - One can define reduced instability coordinates _i as projections of MAS eigenmodes (e.g. dominant kinks, ballooning branches, AE families). - The MAS matrices then induce an effective metric G_ij and potential U( ) in this reduced space, concretely realizing the multi-mode H_plasma = - _eff² _LB + U( ) structure. - This is particularly promising for stellarator optimization and tokamak disruption avoidance, where one wants to design geometry and operating scenarios such that all relevant modes lie in a spectrally gapped regime. 5 Case Study: Reversed-Shear Alfvén Eigenmode in DIII-D #159243 as an Instability Hamiltonian In this section we show how the instability Hamiltonian framework can be instantiated in a realistic fusion scenario by mapping the MAS Landau-fluid eigenvalue problem for the reversed-shear Alfvén eigenmode (RSAE) in DIII-D discharge #159243 into an effective one-mode Hamiltonian H_plasma. This provides a concrete bridge between: 1. MAS: a five-field Landau-fluid eigenvalue solver in realistic geometry, producing the RSAE spectrum, damping rates, and mode structures. 2. QG Fusion: an abstract instability Hamiltonian H_plasma = T( ) + U( ) whose spectral gap encodes stability. 3. Toy model: a 1D Schrödinger-type Hamiltonian with tunable potential and critical coupling. We will work with the n = 4 RSAE at t = 805 ms in DIII-D #159243, where MAS has already been used as a verification and validation benchmark. 5.1 MAS eigenvalue problem and physical setup The MAS Landau-fluid model evolves five fields ( , A_ , P_i, u_ i, n_i) in Boozer coordinates ( , , ), retaining FLR, diamagnetic drifts, finite E_ , and ion/electron Landau damping. After linearization, Fourier decomposition in the toroidal angle, and discretization in the radial coordinate, MAS solves a generalized eigenvalue problem of the form A( ) X = B( ) X, (5.1) where - = _r + i is the complex eigenfrequency, - X is the stacked vector of radial profiles of all poloidal harmonics of the five fields, - parameterizes the included physics (from reduced-MHD up to the full Landau-fluid model), - A, B are sparse matrices encoding geometry, equilibrium, and kinetic terms. For the RSAE application, the code uses DIII-D #159243 equilibrium at t = 805 ms, with reversed magnetic shear and q_min 2.945. MAS computes the n = 4 Alfvén and acoustic continua, as well as discrete RSAE/TAE branches. In particular, MAS calculates the n = 4 RSAE with four hierarchical physics models: - (a) ideal reduced-MHD, - (b) ideal full-MHD, - (c) Landau-fluid without ion-FLR, - (d) full Landau-fluid with ion-FLR. The comprehensive Landau-fluid model yields for the dominant n = 4 RSAE f_RSAE 62.8 kHz, _d / _r -1.67 %, (5.2) where _d < 0 encodes net damping from Landau, continuum and radiative mechanisms. 5.2 From MAS eigenvalues to a reduced instability coordinate The MAS eigenvector X_RSAE defines the full multidimensional structure of the mode in configuration space. To connect to the QG Fusion Hamiltonian, we introduce a single instability coordinate _RSAE as the amplitude of this eigenmode: Y( , ; t) _RSAE(t) X_RSAE( , ) + (orthogonal modes), (5.3) where Y collects all fluctuating fields. The orthogonal components are treated as stable bath modes in this first toy reduction. The linear MAS dynamics then induces an effective equation for _RSAE(t), d _RSAE/dt = -i ( ) _RSAE = -i _r( ) _RSAE + ( ) _RSAE, (5.4) with ( ) < 0 for a damped RSAE in the pure thermal-plasma MAS model. To embed this in an instability Hamiltonian with a real spectrum, we rotate to Euclidean time = it and define an effective spectral parameter _RSAE( ) -2 ( ), (5.5) so that - ( ) < 0 _RSAE( ) > 0 (stable / damped mode), - ( ) = 0 _RSAE( ) = 0 (marginal), - ( ) > 0 _RSAE( ) < 0 (growing instability). In the QG Fusion language, _RSAE( ) becomes the ground-state eigenvalue of an effective Hamiltonian H_plasma^(RSAE)( ) ( ) = _RSAE( ) ( ), (5.6) defined on the single coordinate = _RSAE. Figure 5.1: Schematic mapping from the MAS eigenvalue ( ) to the reduced Hamiltonian eigenvalue _RSAE( ), showing ( ) from different physics levels (reduced-MHD, full-MHD, Landau w/o FLR, Landau + FLR) and the sign of _RSAE. 5.3 Effective potential for the RSAE instability coordinate We now posit a minimal instability Hamiltonian for RSAE of the form H_plasma^(RSAE)( ) = -( _eff² / 2M_eff) d²/d ² + U_RSAE( ; ), (5.7) with an effective potential U_RSAE( ; ) = 1/2 ( ) ² + 1/4 _4 + ... (5.8) Here - is the RSAE mode amplitude, - M_eff and _eff set an overall scale (they can be fixed by matching the MAS-derived energy norm), - ( ) is a curvature that encodes linear stability, - _4 > 0 stabilizes the system nonlinearly at large | |. We identify ( ) with the MAS-derived spectral parameter, ( ) = c_ _RSAE( ) = -2 c_ ( ), (5.9) for some positive scale factor c_ > 0. With this convention: - Damped RSAE (MAS Landau-fluid): < 0 ( ) > 0: a convex potential with a single minimum at =0. - Marginal RSAE: = 0 ( ) = 0: flat quadratic term, marginal confinement. - Unstable RSAE (with EP drive): > 0 ( ) < 0: double-well structure with =0 as a local maximum. Figure 5.2: Effective RSAE potential U_RSAE( ; ) below, at, and above the critical point: (a) > 0 (stable, MAS Landau-fluid point); (b) = 0 (critical); (c) < 0 (EP-driven unstable RSAE). For DIII-D #159243 in the thermal plasma limit modeled by MAS (no EP drive in the code), we are clearly in the > 0 regime: the RSAE is linearly damped with _d / _r -1.7 %. 5.4 Defining a critical coupling for RSAE stability Let _stab be a scalar stabilizing coupling aggregating bulk kinetic effects (Landau damping, continuum damping, FLR, etc.), and _EP be an EP drive parameter (proportional to fast-ion pressure gradient, NBI power, etc.). Then we can model the curvature as ( _stab, _EP) = _0 ( _stab / _stab^(0) - _EP / _EP^(0) ), (5.10) where - _stab^(0) is the value corresponding to the comprehensive MAS Landau-fluid model (with FLR and Landau damping), - _EP^(0) is the EP drive level that would exactly cancel the MAS-predicted damping. The critical stability surface is then ( _stab, _EP) = 0 _stab / _stab^(0) = _EP / _EP^(0). (5.11) This plays the role of the critical coupling line in the quantum-gravity analogy. - For _EP _EP^(0), the MAS point ( _stab^(0), 0) lies deep in the stable regime. - As _EP increases at fixed _stab, the system approaches the critical surface. - Beyond the surface, < 0 and the RSAE amplitude is driven into one of the nonlinear wells: an EP-driven RSAE burst. Figure 5.3: Stability phase diagram in the ( _stab, _EP) plane, showing the MAS Landau-fluid point, the critical line = 0, and the EP-driven unstable region. MAS data enters in three ways: - The slope of with respect to _stab can be calibrated by comparing the four physics levels (reduced-MHD full-MHD Landau w/o FLR Landau + FLR) and their effect on the damping of the RSAE. - The MAS value _d / _r -1.7 % fixes the vertical offset: it tells us how far below marginality the thermal plasma is at the experimental equilibrium. - EP physics, not yet included self-consistently in MAS for this shot, would move us horizontally in _EP and could be taken from a hybrid-MHD or gyrokinetic EP module. Thus MAS supplies the thermal baseline curve _RSAE( _stab), while EP physics supplies an additive drive that shifts it toward zero, as envisioned in the QG Fusion framework. 5.5 Toward real-time spectral control for RSAEs In the QG Fusion paradigm, the key diagnostic quantity is the minimum eigenvalue of H_plasma, _min(p) min Spec H_plasma(p), (5.12) where p denotes a vector of control parameters (profiles, coil currents, heating power, etc.). For the RSAE example, we can define an operational spectral stability indicator Z_RSAE(t) _RSAE( _stab(t), _EP(t)) / _ref, (5.13) with _ref > 0 a normalization constant chosen so that Z_RSAE = 1 corresponds to the MAS Landau-fluid baseline for DIII-D #159243. - Z_RSAE(t) > 0: net damped RSAE (spectrum gapped above zero). - Z_RSAE(t) = 0: marginal RSAE; the system lies on the critical surface. - Z_RSAE(t) < 0: EP-driven unstable RSAE. In a whole-device control loop, MAS (or a reduced-order surrogate trained on MAS) would be used to: 1. Reconstruct or forecast ( _stab(t), _EP(t)) from diagnostics and equilibrium reconstructions. 2. Map it into _RSAE(t) via Eq. (5.5). 3. Compute Z_RSAE(t) and enforce a constraint Z_RSAE(t) Z_target > 0 (5.14) by adjusting: - EP sources (NBI power, geometry), - bulk kinetic stabilization ( profile shaping, q_min control, differential heating), - possibly targeted RMP fields that alter the effective boundary conditions experienced by the RSAE. This is exactly the QG-inspired picture: we steer not just local gradients, but the entire spectrum of an effective Hamiltonian H_plasma, using MAS to evaluate it in realistic geometry. Figure 5.4: Conceptual real-time loop where equilibrium reconstruction MAS (or surrogate) _RSAE control actuator updates (NBI, coils, profile control) to enforce Z_RSAE Z_target. 6 Fitted RSAE Toy Hamiltonian We now construct an explicit, dimensionless toy Hamiltonian for the n = 4 RSAE in DIII-D #159243, whose ground-state eigenvalue ( _stab, _EP) is fitted to the MAS result that the thermal-plasma RSAE is damped with _d / _r -1.67 % at t = 805 ms. The goal is a simple, closed-form mapping ( _stab, _EP) ( _stab, _EP) with: - > 0: net-damped, spectrally gapped RSAE, - = 0: marginal RSAE, - < 0: EP-driven unstable RSAE. We work in dimensionless units, choosing the normalization so the MAS thermal-plasma point sits at a convenient value. 6.1 Normalization from MAS RSAE damping From MAS, for the n = 4 RSAE in #159243 we have (comprehensive Landau fluid model): _d / _r -1.67 %, f_RSAE 62.8 kHz. (6.1) Define the reference time scale _A 1 / _r, (6.2) and the corresponding dimensionless damping rate _th _d _A = _d / _r -0.0167. (6.3) We define a dimensionless spectral parameter by _RSAE -2 / | _d|. (6.4) Then: - At the MAS thermal baseline (no EP drive, full Landau fluid physics, = _d): _RSAE^(MAS) = -2 _d / | _d| = 2. (6.5) - At marginality ( = 0): _RSAE = 0. (6.6) - At an EP-driven RSAE with = +| _d|: _RSAE = -2. (6.7) Thus the MAS point sits at = 2 in these units, fixing the vertical normalization of the toy Hamiltonian s spectrum. 6.2 Dimensionless control parameters Introduce dimensionless stabilizing and driving parameters _stab / _stab^(0), _EP / _EP^(0). (6.8) Here _stab^(0) is the stabilizing bulk kinetic coupling (Landau damping, continuum damping, FLR, etc.) corresponding to the full MAS Landau fluid model, and _EP^(0) is the EP drive level that would exactly cancel the MAS damping (reach = 0 at _stab = _stab^(0)). By construction: - MAS thermal baseline: ( , ) = (1, 0), - Critical EP-driven marginal point: ( , ) = (1, 1). 6.3 Linear fitted model for ( _stab, _EP) We choose a minimal linear model for the net growth rate ( , ) = _d + _EP^(0) , (6.9) where _d < 0 is the MAS thermal damping and _EP^(0) > 0 is an EP-driven growth coefficient. Imposing criticality at ( , ) = (1, 1): (1, 1) = 0 _d + _EP^(0) = 0 _EP^(0) = - _d. (6.10) Thus ( , ) = _d ( - ). (6.11) Using the definition (6.4), we obtain ( , ) = -2 ( , ) / | _d| = 2 ( - ). (6.12) In terms of the original dimensional parameters, ( _stab, _EP) = 2 ( _stab / _stab^(0) - _EP / _EP^(0) ). (6.13) This fitted RSAE toy Hamiltonian spectrum gives: - MAS thermal point ( _stab = _stab^(0), _EP = 0): = 2(1 - 0) = 2 (damped, gapped), (6.14) - Critical line ( = 0): _stab / _stab^(0) = _EP / _EP^(0), (6.15) - EP-unstable region: _EP > _EP^(0) at _stab = _stab^(0) < 0. (6.16) 6.4 RSAE toy Hamiltonian in -space We now embed ( , ) into a 1D instability Hamiltonian for the RSAE amplitude : H_RSAE( , ) = -( _eff² / 2M_eff) d²/d ² + U_RSAE( ; , ), (6.17) with U_RSAE( ; , ) = 1/2 ( , ) ² + 1/4 _4 (6.18) Using (6.12), the quadratic curvature is proportional to ( - ). The regimes are: - Deeply stable RSAE (MAS thermal baseline): ( , ) = (1, 0) = 2, harmonic well + quartic term; = 0 is a robust minimum. - Marginal RSAE: ( , ) = (1, 1) or any point on = = 0, potential purely quartic near the origin: marginal confinement. - EP-driven RSAE burst: ( , ) with > < 0, potential with a local maximum at = 0 and two minima at finite | |, representing finite-amplitude RSAE saturation. 6.5 Example parameter choices Examples: 1. Stable but closer to marginality ( = 1, = 0.5): = 2(1 - 0.5) = 1 > 0. (6.19) The spectral gap is half the MAS thermal value. 2. Exactly marginal ( = 0.9, = 0.9): = 2(0.9 - 0.9) = 0. (6.20) 3. Unstable ( = 1, = 1.2): = 2(1 - 1.2) = -0.4 < 0. (6.21) The toy Hamiltonian thus provides a fully explicit spectral map in terms of the control parameters, suitable for numerical experiments and control-oriented studies. Appendix A. Clarifying the RSAE Toy Hamiltonian and Spectral Normalization In this appendix we summarize and clarify the conceptual choices underlying the fitted RSAE toy Hamiltonian, ensuring that the mapping between the control parameters ( _stab, _EP), the dimensionless variables ( , ), the spectral parameter , and the potential curvature is fully consistent. A.1 Sign structure and unstable regimes We work with normalized control parameters _stab / _stab^(0), _EP / _EP^(0), (A.1) where _stab^(0) is the stabilizing bulk kinetic coupling corresponding to the full MAS Landau fluid model (Landau damping, continuum damping, FLR), and _EP^(0) is the EP drive level that would bring the RSAE to marginality at _stab = _stab^(0). The fitted RSAE eigenvalue is d

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