VPSCF13: Compact Resolvent via Embedding Certificates, Essential Confinement, and Projected Channel Compactness

This paper develops the compact-resolvent certificate layer of the Variable-Probability
Signature Clifford Field framework. Starting from the closed semibounded self-adjoint
operator exported by VPSCF12, it reduces certification of discrete spectral behavior to
compactness of the shifted form-domain embedding. The central principle is that ellipticity,
closedness, and semiboundedness do not imply compact resolvent. VPSCF13 characterizes
compact resolvent at the abstract form-domain embedding level and then develops sufficient
certificate routes for proving that compactness; it does not claim an exhaustive coefficient-level
classification of all VPSCF operators with compact resolvent.
The paper works in a fixed finite-channel complex Hilbert realization
H = L2(Ω; ES ), |S| < ∞, ES ∼= CN .
Clifford, quaternionic, and split-quaternionic structures are admitted into the analytic layer
only through a representation compatibility certificate. All adjoints, eigenvalue bounds,
positivity statements, matrix inequalities, and spectral expansions are interpreted in the
represented complex Hilbert setting fixed by that certificate. The paper uses the active-
operator convention
Hact = (H, D(qact), qact, Lact, ℓcert
act
),
so that certification mode preserves the VPSCF12 operator, whereas design mode uses a
separately certified compact-resolvent variant.
For a densely defined closed Hermitian semibounded form q and its associated self-adjoint
operator L, the identity
D(q) = D((L + γI)1/2), ∥u∥2
q,γ = ∥(L + γI)1/2u∥2
H
yields
(D(q), ∥ · ∥q,γ ) ↪→ H compact ⇐⇒ (L + γI)−1 ∈ K(H).
The subsequent sections develop direct embedding certificates, intermediate compact fac-
torizations, bounded-domain compact-factorization branches, local compactness plus tail
tightness, diagonal and matrix confinement certificates, projected channel compactness, and
weak-escape noncompactness witnesses