CHIHIRO: shape, equilibrium and distinct instability diagnostics Bee Rosa Davis Research note · Version 0.3 · Updated 8 October 2026 CHIHIRO investigates how plasma geometry enters a stability calculation. The work spans shaped ballooning comparisons, equilibrium calculations, external-kink analysis and a Rust implementation that keeps different instability channels explicit. TL;DR The question: Within a stated tokamak stability model, how do geometric and spectral quantities relate to where pressure-driven instabilities set in as plasma pressure rises? What the work does: Saved sweeps of a Miller-shaped ballooning model and an ITER-like graph-spectrum calculation across normalized pressure, plus an interactive shear–pressure model across several cross-section shapes. What is still open: Whether any tracked diagnostic converges under refinement and predicts a measured limit on independent discharges, rather than recovering the model’s own or textbook thresholds. THE RESEARCH IN CONTEXT A tokamak confines hot plasma in a doughnut-shaped magnetic field. Raising plasma pressure improves fusion output, but beyond some point the plasma becomes unstable and confinement can be lost. Designers summarize pressure with normalized beta, β_N; the empirical Troyon limit puts the usual ceiling near β_N ≈ 2.8. Ballooning modes, driven by the pressure gradient on the outer side of the torus, are opposed by magnetic shear, the twist of field lines between surfaces. Cross-section shape—elongation κ and triangularity δ—changes this balance, and relating shape to stability is a central problem in reactor design. The saved records come from two separate calculations. The first is a ballooning model using Miller shaping (κ = 1.7, δ = 0.33) swept over 18 values of β_N from 0.5 to 5, recording at each step the sign of a spectral gap, the most unstable radius and whether the local Suydam test fails. The second builds a graph from an ITER-like equilibrium (major radius 6.2 m, minor radius 2 m, 5.3 T, 15 MA) at two resolutions: a 48 × 64 grid of 2,313 interior nodes for the baseline spectrum at β_N = 1.8, and a coarser 40 × 48 grid of 1,447 nodes for the sweep of Laplacian eigenvalues and Cheeger-constant bounds across β_N from 1 to 3.5; the two spectra are not directly comparable. The two records behave differently, which is why they cannot be read as one result. In the shaped ballooning sweep the spectral gap is positive only at β_N = 0.5 and negative from 0.8 upward. In the graph calculation the first nonzero eigenvalue stays positive throughout—a graph Laplacian cannot change sign—while falling from about 2.9 × 10⁻³ to 3.5 × 10⁻⁴ as β_N rises. The deposited manuscript treats the Troyon limit separately, through an external-kink energy calculation it reports across 20 current profiles. AVAILABLE RECORD Deposited preprint, computational implementation and saved model results; experimental predictive accuracy remains to be established. TECHNICAL FIGURE saved model calculations Saved model calculations with κ = 1.7 and δ = 0.33. Cases are selected, not interpolated into a claimed confinement transition. Playback frames: https://www.davisgeometric.com/research/figure-data/player-plasma.json The saved shaped-plasma sweep Model output βN | Signed spectral diagnostic | Worst radial position x | all_stable flag | Suydam violated fraction 0.5 | 0.009823 | 0.0500 | true | 1 0.8 | -0.009491 | 0.8346 | false | 1 1 | -0.198168 | 0.8577 | false | 1 1.2 | -0.398591 | 0.8577 | false | 1 1.4 | -0.608436 | 0.8808 | false | 1 1.6 | -0.769188 | 0.8808 | false | 1 1.8 | -0.897392 | 0.8808 | false | 1 2 | -1.006743 | 0.8808 | false | 1 2.2 | -1.093224 | 0.8808 | false | 1 2.4 | -1.162348 | 0.9038 | false | 1 2.6 | -1.205199 | 0.9038 | false | 1 2.8 | -1.241508 | 0.9269 | false | 1 3 | -1.251243 | 0.9269 | false | 1 3.2 | -1.279691 | 0.9500 | false | 1 3.5 | -1.280360 | 0.9500 | false | 1 4 | -1.142735 | 0.9500 | false | 1 4.5 | -0.881999 | 0.8808 | false | 1 5 | -1.163572 | 0.8808 | false | 1 The flags are reproduced independently, including disagreement between diagnostics. The source calls the signed value spectral_gap; it is not a universally nonnegative graph-Laplacian gap. This is the shaped model, not a plasma discharge or the separate graph experiment. plasma.json — Saved project data; values read directly How shape changes the ballooning boundary The saved Experiment 12 compares circular, elongated, D-shaped and negative-triangularity configurations. At ten magnetic-shear values per geometry, it records the first pressure-gradient stability boundary and the ratio of shear to that boundary. The resulting curves flatten at different levels over part of the sampled range. A separate calculation searches for a peak in this ratio and checks the tangent condition at the stationary point. The numerical plateau, the calculus identity and any theorem about uniqueness are different claims. The expanded illustrated study gives the definitions, both views of the boundary, the peak records and the source data. From equilibrium to a combined diagnostic Later work uses FreeGS equilibria and studies external-kink stability through an energy-principle calculation. The v5 manuscript reports a wall-free normalized ratio across a 20-profile family. That model result is not an exact identification with a topological constant or a validated confinement law. The Rust code includes an equilibrium reader and distinct ballooning, external-kink, edge, density-limit, vertical-stability, tearing-mode and resistive-wall channels. A favorable result in one channel does not certify the others. The expanded study also flags a mismatch between the saved equilibrium file’s summary and its raw counts. THE NEXT RESEARCH STEP Compare declared observables against independent stability codes and discharge records, while resolving source-version discrepancies and testing numerical refinement. The earlier graph-spectrum experiment remains linked as a separate mathematical object. TEACHING CONSTRUCTION (not project measurements) Specify an equilibrium family and construct the operator associated with the stated perturbation model. A spectral diagnostic is meaningful only with its sign convention, boundary conditions, state variables, and discretization. Keep the saved shaped-model sweep separate from the distinct graph-spectrum experiment. L(λ)u = μu λ: A declared equilibrium or shape parameter. L: The particular discretized operator being investigated. μ: An eigenvalue; its stability interpretation depends on the model and sign convention. For the illustrative diagonal operator diag(1, 1−λ), the smaller eigenvalue reaches zero at λ=1. This is an exact algebraic crossing in a toy model. Without a derivation connecting that operator to plasma dynamics, it says nothing about a confinement threshold. SOURCE RECORD CHIHIRO app: https://chihiro.sh Geometry and diagnostic study: https://www.davisgeometric.com/research/figures/chihiro.html Version 5 preprint: https://doi.org/10.5281/zenodo.21110010 Earlier graph-spectrum records: https://www.davisgeometric.com/research/figures/plasma.html Earlier plasma project material: https://www.davisgeometric.com/projects/plasma.html The Spectral Geometry of Plasma Confinement: A Davis Field Equations Framework for Fusion Stability, Transport Bottlenecks, and Cross-Domain Universality: https://doi.org/10.5281/zenodo.18969038 Source versions: https://www.davisgeometric.com/research/account-source-records.json No new experiments are reported in this research note. Illustrated study: https://www.davisgeometric.com/research/figures/draft-plasma.html