Yuting Wang, Gong-Bo Zhao, Kazuya Koyama, Ruiyang Zhao, Takahiro Nishimichi, Zhongxu Zhai, Héctor Gil-Marín, Hanyu Zhang, Jessica Nicole Aguilar, Steven Ahlen, Florian Beutler, Davide Bianchi, David Brooks, Francisco Javier Castander, Todd Claybaugh, Andrei Cuceu, Axel de la Macorra, Arnaud de Mattia, Biprateep Dey, Peter Doel, Daniel J. Eisenstein, Simone Ferraro, Jaime E. Forero-Romero, Enrique Gaztañaga, Satya Gontcho A Gontcho, Gan Gu, Gaston Gutierrez, ChangHoon Hahn, Klaus Honscheid, Cullan Howlett, Dick Joyce, Stephanie Juneau, Robert Kehoe, David Kirkby, Theodore Kisner, Jean-Paul Kneib, Anthony Kremin, Claire Lamman, Martin Landriau, Laurent Le Guillou, Marc Manera, Aaron Meisner, Roman Miquel, Seshadri Nadathur, Jeffrey A. Newman, Enrique Paillas, Will J. Percival, Francisco Prada, Ignasi Pérez-Ràfols, Alberto J. Rosado-Marín, Ashley J. Ross, Graziano Rossi, Lado Samushia, Eusebio Sanchez, Edward F. Schlafly, David Schlegel, Michael Schubnell, Hee-Jong Seo, Joseph Harry Silber, David Sprayberry, Gregory Tarlé, Xiaoma Wang, Benjamin Alan Weaver, Shuo Yuan

We present joint measurements of the pre- and post-reconstruction power spectra, $P_{\rm pre}$ and $P_{\rm post}$, together with their cross-power spectrum, $P_{\rm cross}$, for the Luminous Red Galaxies (LRGs) in the DESI Data Release 1 (DR1). We jointly analyse these observables with an emulator-based full-shape modeling framework, thereby, for the first time, we extract complementary nonlinear information from the galaxy density field before and after reconstruction in real survey data. Specifically, including $P_{\rm post}$ and $P_{\rm cross}$ in addition to $P_{\rm pre}$ (hereafter $P_{\rm all}$) yields an improvement of approximately $18$-$27\%$ in the $σ_8$ constraint in both $Λ$CDM and $w$CDM, depending on the redshift bin, relative to the $P_{\rm pre}$-only analysis with the cosmic microwave background distance priors (hereafter CMB). In $w$CDM, the joint CMB+$P_{\rm all}$ analysis can tighten the constraints on $w$ by approximately $5$-$15\%$ across the two LRG redshift bins, compared to the CMB+$P_{\rm pre}$ combination. Further incorporating the Type Ia supernova dataset and comparing the cosmological constraints in $w$CDM from each individual power-spectrum component with those from the full combination, we find that $P_{\rm all}$ consistently provides the tightest constraints. From the joint CMB+$P_{\rm all}$+DES-Dovekie dataset, we obtain $Ω_m = 0.314 \pm 0.0048$ and $w = -0.988 \pm 0.023$ for the \texttt{LRG1} sample, and $Ω_m = 0.318 \pm 0.0046$ and $w = -0.988 \pm 0.025$ for \texttt{LRG2}. These results demonstrate that combining pre- and post-reconstruction power spectra with their cross-correlation enables DESI to harvest additional nonlinear information, leading to tighter constraints on cosmological parameters.