Neural-Network Inversion for the Temporal CT Multi-Source Bundle Problem: Per-Bundle Statistical Limits and Near-Optimal Performance

We study the nonlinear inverse problem arising in Temporal CT, a multi-source computed-tomography architecture in which NS = 3 simultaneously active X-ray sources produce M = 5 mixed Poisson intensity measurements of K = 3 unknown line-integral attenuations per projection bundle. The forward model is a sum of exponentials and creates two distinct sources of performance loss: an irreducible aggregation loss fixed by the measurement geometry, and a reducible algorithmic inefficiency that improved estimators can close. We derive closed-form Cramer-Rao bounds and inflation factors for this problem; At unequal attenuation the inflation ratios vary -- and can be considerably worse. We introduce SNN1, a near-optimal classical per-bundle algorithm that brings endpoint paths to within 1-2% of their CRBs and evaluate a physics-motivated residual neural network across three datasets ordered by increasing sinogram structure: RND (synthetic), SGS (analytical chest phantom), and PIS (patient-image-derived). On SGS the NN beats SNN1 at high attenuation by 33-67% but cannot cross the equal-dose single-source floor; on PIS the evaluation ratio drops below 1.0 at bin 6 and reaches 0.096 at bin 9, confirming that the anatomical prior learned from this patient is concentrated enough to dominate collapsed Fisher information at high attenuation -- a characterization of prior informativeness, not a claim of clinical generalizability beyond the single patient studied. A cross evaluation (SGS-trained on PIS test) shows that a concentrated wrong prior is catastrophically worse than a broad wrong prior, underscoring prior diversity as a critical requirement for any future multi-patient deployment. Quantitative sinogram correlation analysis motivates a companion strip-processing architecture that exploits inter-bundle structure inaccessible to the per-bundle algorithms of this paper (Thread 1).