Introduction of the Energy Discrepancy Index χ in a Hierarchical Universe and Multi-Scale Verification / 階層宇宙におけるエネルギー不整合指標 χ の導入と多スケール検証

Short Abstract This preprint introduces the Energy Discrepancy Index χ = Ėreq / Ėint,max as a diagnostic for testing whether the observed energy requirements of an astrophysical system can be accounted for within the maximum capacity of known internal mechanisms. The framework retains energy conservation while making the assumed system boundary explicit: χ ≤ 1 is consistent with an internally closed approximation, whereas χ > 1 indicates an unresolved energy-budget discrepancy that may require reassessing internal limits, observational estimates, or possible external contributions. The study applies order-of-magnitude evaluations across the solar corona, active stars, black hole X-ray binaries, and active galactic nuclei. It reports indicative ranges from approximately unity in the solar case to larger values in more active or higher-scale systems, and examines possible scale and activity dependence. Established mechanisms, including magnetic reconnection, wave heating, and the Blandford-Znajek process, are incorporated as internal energy-conversion terms rather than rejected. The paper presents χ as a falsifiable observational framework, not as proof of an external energy source. The proposed open-system extension would be unnecessary if improved observations and internal models consistently establish χ ≤ 1 within quantified uncertainties. Future work therefore focuses on multi-wavelength estimates, systematic-error control, scaling tests, and comparisons among theoretical models. Japanese Abstract 本プレプリントは、観測から要求されるエネルギー供給率と、既知の内部機構が供給し得る最大値との比 χ = Ėreq / Ėint,max を「エネルギー不整合指標」として導入し、天体系を内部完結系として扱う近似の成立範囲を検討する。エネルギー保存則そのものは維持しつつ系の境界条件を明示し、χ ≤ 1 を内部完結近似と整合する領域、χ > 1 を内部供給上限・観測推定・外部寄与の可能性を再検討すべきエネルギー収支上の不整合として位置づける。 太陽コロナ、活動星、ブラックホールX線連星、活動銀河核(AGN)