Recursive Structures and Reconstruction of Concentration-Sparsity Spaces / 浓稀空间的递归结构与反向构造

We fix the usual complex plane, its original field operations, and the distance family d_c(x,y)=|x-y|^c for 0<c<=1. We organize the scale relations, internal representations, operations across layers, and recovery conditions of concentration-sparsity spaces, including reconstruction of the branch in complex dimension three. We first establish the topology of positive-parameter layers, the discrete sparsity endpoint, the obstruction to collapsing fixed original point pairs, and classifications of affine recursion, infinite readout composition, and hierarchical ultrametric limits. The explicit response f(z)=1/(3-z) then yields a compatible tail space that is biholomorphic to the original complex field and preserves its encoded addition, multiplication, unit, and complete family of distance readouts. Under an explicitly specified unit-relation stimulus interface, each internal label can enter at a finite stage and remain compatible; uniform external concentration of completed outputs is characterized by an exact cumulative condition. For the branch on the original C^3, with its retained Hermitian structure and the same distance-power rule, we prove bidirectional recovery from a six-real-dimensional distance core, reconstruction of the complete difference-quotient zero fiber, and reconstruction through a compatible tail space for a fixed three-complex-dimensional response. A difference-vector estimate for its exact charts gives uniform relative recovery over every bounded internal ball and all positive distance exponents. Separately, for scalar response-error histories, two exact thresholds distinguish ordinary internal convergence from compatibility of joint sparsity readouts on histories with a common limit. Fixed original points, layer representations, compatible tail classes, and unrecorded external endpoint positions are kept distinct throughout. The proved results for specified constructions are separated from the open problem of selecting the response and recursion intrinsically from the original relations. This bilingual v1.1 deposit includes the complete English and Chinese manuscripts, editable LaTeX sources, HTML reading copies, and formula data. The static model, stated hypotheses, conditional interfaces, and open intrinsic-selection problems remain explicitly distinguished.