aspg-34-4481-1791482621 20261008180341 American Scientific Publishing Group admin@americaspg.com American Scientific Publishing Group Prospects for Applied Mathematics and Data Analysis PAMDA 2836-4449 2025 5 2 Aiyared Iampan School of Science, University of Phayao, 19, Moo 2, Tambon Mae Ka, Amphur Mueang, Phayao 56000, Thailand Said Broumi Laboratory of Information Processing, Faculty of Science Ben MSik, University of Hassan II, Casablanca, Morocco Prediction intervals for temporally dependent data require both conditional quantile estimation and a calibration mechanism capable of responding to distribution shift. An adaptive rolling conformal quantile boosting (ARCQB) formulation is developed in which boosted quantile functions provide a nonlinear base interval and a sequential state variable controls the empirical conformal quantile. For target miscoverage 𝛼, the calibration state follows a projected stochastic recurrence, 𝛼𝑡+1 = ΠA{𝛼𝑡 + 𝛾(𝛼 − 𝑒𝑡 )}, where 𝑒𝑡 is the realized miss indicator. A telescoping identity links the time-averaged miss frequency to the state displacement and projection residuals; in the unprojected bounded case, the calibration error is 𝑂(𝑇−1). The interval width admits the exact decomposition 𝑤𝑡 = 𝑤(0) 𝑡 + 2𝑞𝑡 , separating predictive sharpness from conformal inflation. Numerical evaluation uses a strictly chronological one-hour-ahead design on hourly Beijing air-quality measurements. For nominal 90% coverage, raw boosted quantiles attain 83.18%, static conformal calibration 87.40%, and rolling conformal calibration 89.87%. ARCQB attains 90.05% with mean width 47.14 𝜇gm−3 and the lowest interval score, 72.86. Its maximum seasonal coverage deviation is 0.38 percentage points, compared with 7.99 points for the uncalibrated interval. The numerical behavior is therefore consistent with the feedback relation predicted by the calibration dynamics, while high-pollution regimes remain the principal source of conditional under-coverage. 2025 15 21 4481 https://creativecommons.org/licenses/by/4.0/ 10.54216/PAMDA.050203 https://www.americaspg.com/journal/34/article/4481 https://www.americaspg.com/storage/articles/manuscripts/01KZWDJM8SER9HCTXAA19YB1P7.pdf