Cross-depth locking

Topic navigation; Research index lists all notes.

• Conditional mixed-bilinear singlet selector — Research record, section 93: Conditional mixed-bilinear singlet selector.

• Correction to Section 93: the Levi--Civita mixed connection contracts to mean curvature and does not by itself supply the nine-component lock operator — Research record, section 94: Correction to Section 93: the Levi--Civita mixed connection contracts to mean curvature and does not by itself supply the nine-component lock operator.

• Quadratic cross-depth route: a first-order $3\leftrightarrow6$ transition can generate the full $(\mathbf3,\mathbf3)$ lock after Schur elimination — Research record, section 95: Quadratic cross-depth route: a first-order $3\leftrightarrow6$ transition can generate the full $(\mathbf3,\mathbf3)$ lock after Schur elimination.

• Cross-depth channel audit: the canonical factorized normal-spinor contraction does not lock the frames — Research record, section 97: Cross-depth channel audit: the canonical factorized normal-spinor contraction does not lock the frames.

• Leading-source orientation no-go: the written scalar and factorized $3\leftrightarrow6$ structures can select $s$ but cannot select the locked frame orientation — Research record, section 98: Leading-source orientation no-go: the written scalar and factorized $3\leftrightarrow6$ structures can select $s$ but cannot select the locked frame orientation.

• Correction to the rank-three requirement: a generic rank-two orbital bivector uniquely selects an $SO(3)$ frame by orientation-preserving Procrustes completion — Research record, section 99: Correction to the rank-three requirement: a generic rank-two orbital bivector uniquely selects an $SO(3)$ frame by orientation-preserving Procrustes completion.

• Circulating isotropic orbitals — Auxiliary model only: Pure excited orbitals can have isotropic position moments while carrying a full-rank circulation that supplies a conditional singlet spin source.

• Controlled orbital gap — Auxiliary model only: An exactly solved orbital-gap model quantifies when a projected spin source survives orbital backreaction.

• Coupled circulating branch — Auxiliary model only: The coupled scalar 3-in-6 model admits excited circulating stationary states with an exact oscillatory linear moment spectrum along the equal-coupling family.

• Fixed circulation energy bound — Auxiliary model only: An exact positive-square identity makes the circulating Hartree states global energy minima at fixed mean circulation.

• Hartree moment spectral stability — Auxiliary model only: Exact moment closure gives real linear frequencies for the isotropic circulating Hartree branch without establishing nonlinear stability or selecting particle indices.

• Orbital projection source test — Auxiliary model only: A pure quadratic six-dimensional orbital can supply a two-plane spin source, but the complete orbital-spin operator does not freeze that orbital automatically.

• Self-consistent moment lock test — Auxiliary model only: The positive rank-one scalar moment model has a unique symmetric ground-state covariance and a calculable mixed-source susceptibility.

• Spin-orbit source audit — Auxiliary model only: A local Dirac spin-orbit term supplies mixed generators, but a single simple bivector retains a twofold spinor degeneracy.