Transpose-odd operator chirality in co-moving multiplicative processes: exact tail-level structure and a detectability obstruction

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While highly theoretical, the paper addresses the structure of heavy tails in multiplicative processes, which are fundamental to modeling asset returns and risk. The distinction between tail exponent (invariant) and tail level (potentially asymmetric) is crucial for accurate VaR/CVaR estimation in non-normal regimes. However, it does not provide a direct trading strategy or empirical financial data analysis.

The paper involves advanced concepts in probability theory, invariant theory, and control theory. Implementing the numerical checks requires precise handling of Haar measures, polar decompositions, and high-dimensional integration/simulation to verify the subtle level asymmetries against the invariant exponents.

The paper provides detailed mathematical derivations and specifies numerical checkpoints (C1-C31) with specific parameter values (e.g., singular values, rotation angles). However, the code is stated to be 'available from the author upon request' rather than publicly hosted, and the work relies on specific assumptions (Assumption R, UI'') that require careful implementation to reproduce the numerical consistency.