Arithmetic Selector Geometry and Residual Spectral Fusion in Weighted-Exponential Source Systems

This paper develops a quantitative selector-and-fusion theory for finite weighted-exponential source systems carrying discrete translation labels. The first result identifies a dual-provenance carrier for a finite labelled atom set. If \(E_S\) denotes the synthesis operator, then AS=2ESES^* has nonzero spectral projection exactly equal to the atom-span selector, while its commutators with label modulations encode the corresponding signed symmetric-difference boundaries. Thus spectral-support information and labelled-translation geometry are retained by a single operator. A finite-dimensional support-rotation obstruction shows that negative-space compatibility and unitary-invariant positive spectral data alone do not determine the full source selector. An orientation-sensitive full-source defect is therefore required for quantitative selector recovery. For families of nearby source subspaces, canonical polar orthogonalization W=V(V^*V)-1/2 produces mutually orthogonal nearby blocks. The resulting aggregate source-compression error is controlled in Hilbert-Schmidt norm without an explicit factor depending on the number of components. In the canonical common-depth two-ended weighted-exponential model, clustered Cauchy-Hilbert cancellation yields component-count-independent fusion estimates for bounded-order components. A fixed local unitary transfers this cancellation exactly to the signed two-channel source. The residual formulation gives an exact factorization Q=T Q T^* and, with \(G=T^*T\) and \(U=TG-1/2\), U^*QU=G1/2 QG1/2. Consequently, inertia and nonzero positive and negative eigenvalues transfer multiplicatively whenever the fusion Gram is sufficiently close to the identity. This residual route does not require a componentwise lower bound on the local residual-to-endpoint cancellation ratio. For the bounded-order common-depth fixed-scale model, the fusion estimate is formulated uniformly over admissible configuration classes. If \(Σ_R(K,H)\) denotes the supremal fusion defect over configurations whose \(KH\)-connected components have order at most \(R\), then limK→∞H→∞ΣR(K,H)=0. For a concrete source family, asymptotic losslessness requires an admissible diagonal \(K(H)→∞\) along which the component order remains bounded. The theory is formulated first for abstract labelled weighted-exponential systems. A final section records its specialization to bandwidth-one Weil-source models with prime-log labels. The paper does not prove the upstream arithmetic realization or small-boundary hypotheses required for such an application, and no conclusion concerning the Riemann hypothesis is claimed. Version v1.0.