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Measurements of current-voltage characteristics from ideal carbon nanotube (CNT) field electron emitters of a small apex radius have shown that these emitters can exhibit a linear Fowler-Nordheim plot [e.g., Dean and Chalamala, Appl. Phys. Lett. 76, 375 (2000)]. From such a plot, a constant (voltage-independent) characteristic field enhancement factor (FEF) can be deduced. Over 15 years later, this experimental result has not yet been convincingly retrieved from first-principles electronic-structure calculations or, more generally, from quantum mechanics (QMs). On the contrary, several QM calculations have deduced that the characteristic FEF should be a function of the macroscopic field applied to the CNT. This apparent contradiction between the experiment and the QM theory has been an unexplained feature of CNT emission science and has raised doubts about the ability of existing QM models to satisfactorily describe experimental CNT emission behavior. In this work, we demonstrate, by means of a density functional theory analysis of single-walled CNTs "floating" in an applied macroscopic field, the following significant result. This is that the agreement between the experiment, classical-conductor CNT models, and QM calculations can be achieved if the latter are used to calculate (from the "real" total-charge-density distributions initially obtained) the distributions of "induced" charge-density, induced local fields, and induced local FEFs. A similar result was previously obtained for so-called vertically aligned CNT/graphene structures. The present work confirms, more reliably and in significantly greater detail than in earlier work on a different system, that this finding applies to the common "post-on-a-conducting plane" situation of CNT field electron emission. This finding also brings out various further theoretical questions that need to be explored. [ABSTRACT FROM AUTHOR]