Residue Spectral Dynamics and Markov Closure in Finite Distinction Systems: Coupled Residue Channels, Transfer Spectra, and Operational Recoverability

Residue Spectral Dynamics and Markov Closure in Finite Distinction Systems: Coupled Residue Channels, Transfer Spectra, and Operational Recoverability Finite effective descriptions retain only a bounded set of distinctions, while unresolved degrees of freedom may continue to influence accessible dynamics through delayed response, hidden modes, environmental records, recurrence, protected invariants, or instability. This work develops FDS-P8 as a spectral-dynamical framework for determining which unresolved distinctions can be excited, persist outside the active record, and later return to affect retained dynamics. The central object is the coupled residue channel R = (G_R, B_R, C_R), where G_R governs internal residue evolution, B_R excites residue from the retained sector, and C_R returns residue to retained observables. The corresponding memory kernel is K(t) = C_R S_R(t) B_R. The paper: separates formal projection from physical pruning, externalization, and thermodynamic erasure; defines excitable, return-active, coupled, silent, accessible, and protected residue; identifies the active residue quotient that determines retained input-output memory; distinguishes internal spectral sets, transfer singularities, initial-residue recoverability, and reachable round-trip recoverability; introduces finite-window operational residue ranks based on Hankel singular values; derives conditional closure bounds from semigroup stability and closure obstructions from marginal modes, spectral edges, and non-normal transient amplification; separates memory decay, finite-state Markov augmentation, and finite-window memory approximation; gives a transfer-recoverability bridge under a registered relation between retained-return and physical-observation channels. The release also includes a deterministic Python script that reproduces the figures, CSV data, and numerical diagnostics reported in the paper. The script requires NumPy, SciPy, pandas, and Matplotlib, has no network dependency, and uses no fitted random parameters. Minimal reproducibility package: FDS_P8_residue_spectral_models.py requirements.txt README.md manuscript PDF Execution: python FDS_P8_residue_spectral_models.py --outdir . The numerical models are normal-form diagnostics of the stated mathematical conditions. They are not presented as universal microscopic models or empirical parameter estimates. P8 does not claim to invent Mori-Zwanzig projection theory, realization theory, balanced truncation, or semigroup methods. Its contribution is the integration of these tools with finite task-relative projection, physical residue qualification, accounting boundaries, operational recoverability, and explicit failure conditions. Official website: distinctiontheory.orgPublic portal for the start guide, papers, claim status, failure registry, prior-art boundary, and citation resources. Canonical GitHub repository:https://github.com/yiningwu-research/Distinction-Theory