Adaptive Fractional-Power Transient Signal Detection via Reconstruction Error in a Space with a Generating Element

We formulate a two-model detector for transient signals using decomposition in a space with a generating element (DSGE). Each observation window is centered under the noise-only and signal-present hypotheses, reconstructed with generating functions, and scored by a calibrated log-ratio of reconstruction errors. A parametrically adaptive transition polynomial (PATP) basis and exponential window weights are selected only on validation data, with the singular midpoint region excluded. In aligned-window Monte-Carlo tests with finite-variance Gaussian, Student, and contaminated noise, the selected PATP detector increased detection probability by 0.032 on average over uniform PATP reconstruction in all 75 test cells and by 0.121 over the best non-reconstruction baseline. A basis-size ablation linked decreasing performance to an eleven-order rise in the condition number. Three public real-data gates tested calibration across acquisition conditions. On partial-discharge antenna data, the detector reached a detection probability of 0.822 versus 0.622 for the best classical comparator at a realized false-alarm rate of 0.053. Electrocardiography and random-arrival scanning remained negative boundaries against morphology-aware matched filtering and exact-template GLRT-like detection. The method targets non-Gaussian or mismatch-prone regimes rather than universally replacing classical detectors. This is the author's original manuscript (preprint), deposited to establish a citable record and a public priority date. It has not been peer reviewed.