These notes support the analysis of candidate V2 operators and numerical results. Each states the mathematical model and derives its consequences. Field amplitudes and profile parameters describe fields on the one nested physical manifold; all physical dimensions remain ordinary macroscopic spatial directions.

Tool Use
Spectral survival recurrence Computes mean return weight and distinguishes exact recurrence, approximate recurrence, and continuous-spectrum dephasing.
Continuum mode memory Eliminates coupled modes exactly before making a decay approximation.
Bounded spectral perturbations Bounds frequency shifts and preserves separated spectral-cluster counts under a stated norm bound.
Gaussian mode geometry Computes the overlap metric of normalized profiles and its Berry curvature.
Simplex spectral zeta Continues a specified positive tower to zero and evaluates its regularized product with scale dependence intact.
Isotropic projection fractions Calibrates squared spatial projections for an explicitly isotropic preparation.
Fixed-null calibration Defines match probabilities, their valid combinations, and finite-simulation uncertainty.

spectral_analysis_checks.py provides finite checks through independent quadrature, matrix evolution, analytic continuation, and exact probability identities. Universal claims rest on the proofs in the notes.