VPSCF14: Stable Law-to-Operator Carrier Perturbation Theory Multiplier-Norm Stability, Hard–Soft Carrier Perturbations, Shifted Extended Mosco Limits, and Measurable Elliptic Projection

This paper develops the stability-certificate layer of the Variable-Probability Signature
Clifford Field framework. VPSCF13 established compact-resolvent certificates for stabilized
active operators. VPSCF14 studies how law-dependent VPSCF forms, carrier masks, soft
activations, hard constraints, elliptic law projections, and associated resolvents vary under
perturbation.
All carrier models are represented in a fixed ambient complex Hilbert space. For common-
domain models, multiplier classes M(V, V∗) encode law-to-form continuity, and convergence in
B(V, V∗), under uniform shifted graph-norm equivalence, yields norm-resolvent convergence.
A shifted KLMN branch treats Hermitian relatively form-bounded carrier potentials. Hard
masks are certified through threshold, Sobolev-multiplier, or direct multiplier conditions; soft
carriers are stabilized through Lipschitz activation.
For hard-domain constraints, the paper uses extended lower-semicontinuous quadratic
functionals and shifted convex representatives on the underlying real Hilbert space. Mosco
convergence yields strong convergence of ambient lifted resolvents. The paper also proves
measurable projection onto elliptic law spectrahedra, stability of fixed-in-n spectrahedron
projections, conditional variable-fiber stability, and a projection-to-operator handoff. Compact
spectral conclusions are stated only under certified compactness inherited from VPSCF13