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Evidence for Bures--Wasserstein Boundary Dynamics in the Living Human Brain

Christian Kerskens

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.

Evidence for Bures--Wasserstein Boundary Dynamics in the Living Human Brain

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 , , 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 --gradient-- readout block.

Paper Structure

This paper contains 56 sections, 2 theorems, 78 equations, 1 table.

Table of Contents

  1. Introduction
  2. Substrate Boundary Dynamics: Summary of the Companion Result
  3. Substrate covariance on the Bures--Wasserstein manifold
  4. Substrate-constrained gradient flow
  5. What changes at the boundary
  6. What the present paper takes from the companion result
  7. Ontological status of the boundary regime.
  8. Transition to the spin-probe problem
  9. Boundary-Conditioned Transfer: From Substrate Geometry to Spin Correlations
  10. Reduced open-system setting
  11. Shared-environment coupling and the origin of spin cross-diffusion
  12. Approximation class.
  13. Spectral bridging: slow boundary control and fast spin response
  14. The kinematic transfer mechanism
  15. Asymptotic onset and pre-boundary leakage
  16. ...and 41 more sections

Key Result

Proposition 3.2

Let two spin subsystems $A$ and $B$ be linearly coupled to a common Gaussian substrate through nondegenerate coupling blocks $C_A$ and $C_B$. Then the boundary-conditioned cross-diffusion block of the reduced spin dynamics is determined by the substrate cross-correlation kernel. In the local linear- Hence, a nonzero boundary-induced change in the substrate cross-mode covariance generically induces

Theorems & Definitions (8)

  • Remark 3.1: Meaning of "boundary-conditioned"
  • Proposition 3.2: Shared-environment cross-diffusion transfer
  • proof
  • Remark 3.3: Non-adiabaticity as a selector, not a carrier
  • Proposition 3.4: Boundary-conditioned spill-over and transfer
  • Remark 3.5: What boundary-conditioned transfer does and does not claim
  • Remark 4.1: Conditional status of the symmetry selection
  • Remark 8.1: Classical versus quantum cross-correlation