Spectral Imprints and the Structure of Spectral Manifolds—Geometric-Dynamical Foundations of the Kraus Expansion

This paper develops the geometric and dynamical foundations of spectral imprints in the Mind Manifestation Mechanism (MFM). Taking the spectral-imprint postulate of L2 (Proposition 6; DOI: 10.5281/zenodo.23228661) and the spectral-imprint mechanism of L3 (§6; DOI: 10.5281/zenodo.23387375) as input constraints, it constructs a candidate mathematical framework in which Kraus expansions produce instantaneous imprints, stable imprints accumulate into separation interfaces, and the resulting spectral geometry constrains subsequent manifestation. The construction comprises six linked components: a base-layer spectral manifold and a manifestation-layer Kraus spectral manifold, together with the mapping between them; a partitioned spectral manifold equipped with a single topology induced by the Bures metric and partitioned by thresholds into a core region, a boundary region, and a rigid exterior; the chain running from Kraus expansion through instantaneous imprint and threshold stabilization to separation interface and connected components; a three-subspace open-set family and its support regions; an observation-operator field and observation clusters; and a two-level temporal structure formed by continuous base time and discrete mental time steps. Two features define the framework's position. First, geometry is generated rather than presupposed: interfaces, regional partitions, and connected components are products of imprint dynamics and then act back on later dynamics. Second, the same bifurcation admits two end-states, a rigid end and a dissolution end, distinguished not by imprint amplitude but by the direction of intrinsic degrees of freedom. The criterion for overall morphological evolution therefore combines the direction of Choi rank with geometric and topological change; a change confined to Choi rank, with geometry and topology unchanged, counts only as channel-level rearrangement, while channel-level bifurcations are treated as local phase transitions rather than as evidence for a change of the overall end-state. The framework is a candidate construction, not a unique or empirically established model. It does not claim to be direct physical evidence or a one-to-one correlate of EEG or imaging signals. Its function is to supply the deformation legitimacy and generative rules for the companion topics on self, qualia, and memory, and its empirical force is indirect, obtained through the macroscopic predictions of those topics and their L4 tests. Parameters such as the stabilization threshold and interface sharpness are not yet numerically fixed; their calibration is deferred to L4 or to a specific experimental design.