Spectral-Gap-Modulated Diffusion for Non-Markovian Risk Calibration: A Scalable Framework Tested on Low-Dimensional Real Markets

This code was developed iteratively, with several real methodological corrections made along the way after external review. Rather than presenting only a final "clean" version, we have kept the scripts that reflect this real process, including: - `backtest_spectral_gap.py` / `backtest_spectral_gap_with_volatility.py`: the original in-sample composite-trigger backtest. This calibration is **in-sample** (thresholds computed from the full sample, then evaluated on that same sample) and its results should **not** be treated as validated out-of-sample performance. It is retained here for transparency and because it motivated the joint-condition trigger design. - `backtest_walkforward.py`: the corrected, honest out-of-sample evaluation. This is the paper's actual evidentiary result for the historical-event backtest (3 of 5 real crisis episodes caught, ~77% false-positive rate). This script itself required one internal bug fix (a burn-in threshold that was accidentally compared against already-subsampled observations) before it produced a trustworthy result — this history is documented in the script's own comments. - `var_es_backtest.py`: the six-model VaR/ES backtest. The diffusion-modulation coupling parameter's sign was found, through direct numerical testing, to have been initially specified backwards relative to its intended economic interpretation; the corrected sign is what is implemented here. - `calibrate_kbg_standalone.py`: the McKean-Vlasov particle simulation combined with its KL-divergence parameter calibration, self-contained in one file (no cross-file imports) specifically to avoid environment/upload issues. Includes the corrected kernel-deformation sign, the corrected diffusion-modulation sign, exclusion of the trivial graph-Laplacian eigenvalue from the empirical Stieltjes transform, and bounded (physically constrained) Prony-series fitting.