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We present a new analysis of the two-decade-old controversy over interpretation of satellite observations of total solar irradiance (TSI) since 1978 and the implications of our findings for TSI as a driver of climate change. Our approach compares the methods of constructing the two most commonly referenced TSI composites (ACRIM and PMOD) that relate successive observational databases and two others recently constructed using a novel statistical approach. Our primary focus is on the disparate decadal trending results of the ACRIM and PMOD TSI composite time series, namely, whether they indicate an increasing trend from 1980 to 2000 and a decreasing trend thereafter (ACRIM) or a continuously decreasing trend since 1980 (PMOD). Construction of the four-decade observational TSI composites from 1978 to the present requires the use of results from two less precise Earth Radiation Budget experiments (Nimbus7/ERB and ERBS/ERBE) during the so-called ACRIM-Gap (1989.5-1991.8), between the end of the ACRIM1 and the beginning of the ACRIM2 experiments. The ACRIM and PMOD composites used the ERB and ERBE results, respectively, to bridge the gap. The well-established paradigm of positive correlation between Solar Magnetic Field Strength (SMFS) and TSI supports the validity of the upward trend in the ERB results and the corresponding decadal upward trend of the ACRIM composite during solar cycles 21 and 22. The ERBE results have a sensor degradation caused downward gap trend, contrary to the SMFS/TSI paradigm, that biased the PMOD composite decadal trend downward during solar cycles 21 and 22. The different choice of gap bridging data is clearly the cause of the ACRIM and PMOD TSI trending difference, agreeing closely in both magnitude and direction. We also analyze two recently proposed statistical TSI composites. Unfortunately their methodology cannot account for the gap degradation of the ERBE experiment and their resulting uncertainties are too large to uniquely distinguish between the trending of the ACRIM and PMOD composites. Our analysis supports the ACRIM TSI increasing trend during the 1980 to 2000 period, followed by a long-term decreasing trend since.