We develop the mathematical and physical formulation of cognitive cost optimization that underlies the path-integral model of consciousness. The goal-directed cognitive process is modeled as imaginary-time evolution (ITE) under a projector Hamiltonian that rewards configurations consistent with a target concept. We establish three results. First, this ITE coincides with a double-bracket flow and is therefore the Riemannian gradient flow of a Hilbert--Schmidt cost whose unique minimum is the solution. Second, a Wick rotation re-expresses this non-unitary descent as an equivalent unitary evolution on the same Hilbert space, which admits an exact discrete path-integral representation in which the oracle and the initial-state diffusion projector play the roles of potential and kinetic energy. Third, we identify the continuum from unconscious to conscious processing with the strength of the unitary interaction between the cognitive system and a neural-environment probe, recovering the Gorini--Kossakowski--Sudarshan--Lindblad (GKSL) decoherence model of Asano et al. in the Markovian weak-coupling limit, and the projective, reportable fixation of an optimized state in the strong-coupling limit, an insight-like "Aha" endpoint. Both regimes share the same ITE and path-integral structure, and only the measurement-interaction strength varies. The Wick rotation is therefore a technique of re-description, not a physical regime change.
Blogger's Review: This paper provides profound insights into the mathematical modeling of cognitive processes, especially within the framework of consciousness's path integral, highlighting the close relationship between imaginary time evolution and physical phenomena. This novel perspective offers new tools for understanding the complexity of consciousness and is worth attention.