Net Income (Loss) per Share - Schedule of securities were not included in the computation of diluted shares outstanding (Detail) - shares	6 Months Ended	12 Months Ended
	Jun. 30, 2025	Dec. 31, 2024
PubCo Warrants - Public and Private
Antidilutive Securities Excluded from Computation of Earnings Per Share [Line Items]
Number of securities	43,848,750	43,848,750
PubCo Class E Common Stock - Earnouts Units
Antidilutive Securities Excluded from Computation of Earnings Per Share [Line Items]
Number of securities	0	1,075,761
BT OpCo Exchangeable Non-Controlling Interest
Antidilutive Securities Excluded from Computation of Earnings Per Share [Line Items]
Number of securities	0	44,093,024
BT OpCo Earnouts Units
Antidilutive Securities Excluded from Computation of Earnings Per Share [Line Items]
Number of securities	15,000,000	15,000,000
2023 Incentive Plan RSU awards
Antidilutive Securities Excluded from Computation of Earnings Per Share [Line Items]
Number of securities	2,652,452	1,624,687