BC XX: Certified Atlas Transitions for Parameter-Dependent Dictionary Tubes 2026 06 21

This record contains the twentieth technical paper in the Boundary Compensation research programme: Boundary Compensation XX: Certified Atlas Transitions for Parameter-Dependent Dictionary Tubes This paper extends the finite-resolution certification framework developed in earlier Boundary Compensation modules from single positive response operators to parameter-dependent response families. Previous modules established that finite-resolution response data do not determine a unique hidden Schur/Feshbach lift. They determine response-equivalence classes, admissible subclasses, quotient representatives, wall-stratified chambers, perturbation tubes of compatible hidden-lift dictionaries, and finite-dimensional certification procedures for thresholded dictionary charts. BC-XIX studied the certification problem for a single positive response operator under a deterministic operator-norm tolerance, an accessibility threshold, a branch-resolution scale, and a declared admissibility class. BC-XX studies the next problem: how such certificates behave along a parameter-dependent family of positive finite-dimensional response operators. The central thesis is that a parameter-dependent response family does not define one global hidden-lift dictionary chart. It defines a wall-labeled atlas of local certified charts. On maximal certified chambers, thresholded perturbation dictionary tubes admit local support-valued and, after choosing frames, framed chart descriptions. At rank, threshold, cluster-merger, isotypic-rank, or branch-resolution walls, the certification conditions may fail. The paper therefore formulates the certificate reset protocol CERTIFICATE_RESET, derived from the BC-XIX certification inequalities. The main diagnostic principle is: No silent continuation across wall-labeled certificate failure. This means that when the declared soundness conditions fail, the previous certificate cannot be silently continued. Instead, the old certificate must stop, the relevant wall or failure-zone label must be recorded, and a new local certificate may be attempted only in a subsequent certified chamber. The paper distinguishes exact spectral walls from finite-resolution certification-failure zones, defines certified chambers as maximal connected components of threshold-gap regions, formulates local support-valued and framed chart assignments, restricts transition maps to certified overlaps, and develops pathwise safe continuation records. It also provides finite-dimensional diagnostic examples illustrating threshold walls, positive cluster-merger walls, exact-rank walls, isotypic effective-rank walls, and branch-resolution walls. The construction is strictly finite-dimensional and operator-theoretic. It introduces no physical time, no renormalization-group flow, no dynamics, no stochastic noise model, no detector model, no hidden-sector reconstruction, no global framed atlas, no canonical global section, no global trivialization, no Standard Model claim, no cosmological prediction, and no quantum-gravity claim. BC-XX closes the local safe-continuation layer of the Boundary Compensation programme and prepares later atlas-level questions concerning gluing, cocycle data, holonomy-like effects, and metrics on thresholded dictionary tubes.