VSAG-CMT IV– Constrained Signature Dynamics and Variational Solver Certification Box-Constrained Signature Flows, Claim Packets, and Anti-Overclaim Logic

This paper is the fourth paper of the VSAG-CMT sequence. It develops constrained
signature dynamics and variational solver certification for the normalized mixed-type signature
field
σ(t, x) ∈ [−1,1].
The paper does not prove global solvability, uniqueness, Fredholmness, transmission, or
propagation for the mixed-type PDE Tσu = f. Its theorem-level output is a certified
variational and solver-output claim architecture for the constrained signature field σ. The
central point is not the general theory of normal-cone flows, nor a global mixed-type PDE
solvability theorem. The central point is a guarded certification architecture: a signature
evolution or numerical output is certified only through typed admissibility records, variational
descent records, diagnostic records, failure records, and downgrade records. The paper
proceeds from analytic constraint geometry to solver-level certification: it fixes the reference
mixed-type spine, records the dependency and nonpromotion ledgers, defines the Hilbert
space box constraint and typed normal-cone conventions, declares the energy/source/calculus
packages, formulates sigma-only constrained dynamics and its proximal scheme, separates
closed-constraint passage from EDI passage and inclusion recovery, treats coupled u,σ
dynamics only as an optional coercive extension record, constructs typed discrete admissibility
modes and their implication lattice, proves finite-dimensional solver existence and typed
range/descent results, proves the anti-clipping obstruction with a post-clipping re-certification
rule, and finishes with claim-packet soundness, neutral/interface downgrade logic, and the
export ledger for the next VSAG-CMT papers.