arXiv:2505.22680v3 Announce Type: replace Abstract: When substrate-constrained covariance flow on the Bures--Wasserstein manifold reaches the Williamson boundary, single-mode compression saturates and further admissible covariance evolution is forced into the cross-mode complement. This paper derives how that substrate boundary transition becomes experimentally visible in an embedded spin probe in the living human brain. We formulate a boundary-conditioned transfer theorem: when the substrate enters the deep boundary regime in a coupled mode, the boundary-selected cross-mode continuation of substrate covariance flow enters the reduced spin dynamics as a nonzero inter-spin correlation block. The spin probe does not inherit the substrate boundary as a state; it detects the boundary indirectly through the transferred cross-mode sector of the reduced dynamics. To leading order, this transfer is selective: it acts through an additive cross-diffusion channel while leaving conventional single-mode NMR observables such as \(T_1\), \(T_2\), linewidths, and the ordinary single-quantum response dominated by the thermal background. Projecting the induced spin cross-mode structure into the two-spin algebra, we argue that the experimentally relevant dominant recipient is the double-quantum SU(1,1) pair sector rather than the compact zero-quantum SU(2) exchange sector. We then derive the coherence-transfer pathway through which this double-quantum pair coherence is converted into a detectable signal by the \(45^\circ\)--gradient--\(45^\circ\) readout block.

arXiv:2505.22680v2 Announce Type: replace Abstract: A recently proposed geometric framework predicts that the transition from distributed belief to committed action involves a metric regime change, culminating in a boundary regime where cross-mode structure becomes algebraically necessary for continued state-space compression. This paper examines whether reported magnetic resonance measurements of proton spins in neural tissue provide an empirical probe of this regime. A companion analysis identifies the detected signal as the readout-converted signature of double-quantum SU(1,1) pair coherence, which correlates with short-term memory performance and cardiac-phase dynamics during wakefulness. We show that the mathematical bridge between the abstract transport framework and the physical spin system is the Bures metric, which natively governs both Gaussian Wasserstein geometry and quantum density matrices. We argue that the observed signal is best understood as a probe of entry into a deep boundary regime where single-mode compression is exhausted and collective cross-mode squeezing emerges. Because high-temperature bulk NMR obstructs strictly bipartite entanglement witnesses, we contextualize the signal within a macroscopic multiple-quantum-coherence (MQC) framework. Consequently, the current data provide evidence for the metric-driven onset of collective non-compact SU(1,1) structure, establishing the necessary physical foundation for future macroscopic many-body entanglement certification.