Finite-Query Collapse and Modal Exact Bases in the SCI Hierarchy

Abstract

We study the exact-basis problem for Solvability Complexity Index (SCI) computational problem families through finite-query transports. A raw finite-query reduction permits arbitrary encodings and finite transcript reconstructions, with only a continuous output decoder. For the Colbrook-Hansen (CH23) singleton-window spectral/pseudospectral block, this raw preorder collapses the expected two-source structure: the diagonal exact spectral and fixed-$\varepsilon$ pseudospectral sources are raw- and continuous-finite-query equivalent, and, for computable $\varepsilon$ under the evaluation-name representations, TTE-finite-query equivalent, so the six-problem ambient is raw-principal. We then introduce modal finite-query preorders, whose admissibility conditions may restrict encodings, decoders, reconstructions, uniformity, and geometric naturality. We also characterize TTE finite-query transport as computable point transport with a uniform finite interface trace; after forgetting the trace this gives strong Weihrauch reducibility, and the implication is strict. Under a CH23 geometric modality generated by representation inclusions, unitary and graph relabelings, and neutral stabilizations, the same ambient has exactly two minimal exact sources. This gives a calibrated reformulation of the exact-basis problem: natural SCI families should be classified by modality-indexed exact bases and refinement maps, not by one raw preorder alone.