Internal Reconstruction in Swahili verb paradigms
Most regular verb stems in Swahili are bi- or poly-syllabic. In finite verb paradigms, the verb stem must be preceded by a tense-aspect marker, which is in turn preceded by a subject-agreement marker:

This system looks rather neat, until one begins to notice some ‘distortions’. First, a small group of verbs are monosyllabic, and the paradigm for these verbs contains an added element, the stressed morpheme-kú-, tucked between the tense-aspect marker and the verb stem:

And second,-ku- does not appear in the habitual (zero) tense (7e).
The first distortion factors out into two sub-questions:

The answer to (8a) turns to be relatively transparent, once one notes that the Swahili word-stress is overwhelmingly penultimate; that one lexical stress must fall on the verb stem; and that mono-syllabic verb stems cannot carry this stress without violating the penultimate stress rule. One thus makes the interim explanatory hypothesis (9), through which the hitherto disparate facts of-ku- and the Swahili stress are linked in a single coherent pattern:

But why-ku-? In historical linguistics, given what we know about the process of grammaticalization, via which lexical stems give rise to grammatical morphology, this question is better recast as:

One now casts around for other contexts where-ku- is found; not any old context, but one whose syntactic and functional distribution may tag them as plausible source of the-ku- in short-verb paradigms. And one might as well note that judgements of plausibility here are heavily theory-driven, depending on what one knows about universals of grammaticalization.
The first relevant observation is that ku- as a verb prefix also marks the infinitive of verbs, and further has the same stress distribution with short verbs:

The next relevant observation is that-ku-marked infinitives appear in complements of modality verbs, again with the same stress distribution in long vs. short verbs:

At this point, the abduction-bent linguist has just about enough facts, both puzzling and regular, to attempt an abductive leap to an explanatory hypothesis. Before outlining the hypothesis, however, let us consider some additional puzzling facts, this time involving verbs in relative clauses.
In their rel-clause form, verbs in three of the tenses — excepting the perfect and habitual (‘zero’) — may take an infix rel-pronoun between the tense marker and the verb stem. For long-stem verbs, the pattern is as follows:


In (13c) we now spot one additional puzzle: The future marker-ta- has somehow acquires an additional syllable in the rel-clause pattern, appearing now as-taka-. As we shall see below, this puzzling fact eventually falls into place, and in fact contributes an important clue for resolving the initial puzzle.
The salutary role of an erstwhile puzzle may be re-cast as a well-known general principle of empirical science:

More puzzling yet, the perfect marker-me-, which earlier (7) followed the pattern of the other three tense-aspects in (13), now joins the habitual (zero) in displaying an irregular rel-clause pattern:

As different as the relativization patterns (15a,b) are, they nonetheless supply us with an added clue, one that turns out to be crucial for abducting a comprehensive hypothesis: In both aspects, the rel-pronoun is a verb suffix. In (15a) it is suffixed to the verb-stem-amba ‘say’ that is now used as the rel-subordinator. In (15b) the rel-pronoun is attached to the relativized verb itself.
Another piece of evidence is the fact that the four tense-aspects in (13a–c) and (15b) may also adopt, optionally, the rel-subordinator pattern with ‘say’ that is obligatory for the perfect (15a).
At this point the hypothesis fairly leaps out of the page at you,2 especially if you know something about the most common diachronic source of tense-aspect markers: They tend to arise, overwhelmingly, from a small group of complement taking modality verbs (as in (12) above), via the process of grammaticalization. The verbs that most commonly contribute to this process are ‘have’, ‘be’ (or ‘be’-like verbs), ‘want’, ‘go’, ‘come’, and ‘finish’.
Further, each of these verbs gives rise, most commonly, to a highly specific tense-aspect:

The facts surveyed thus far now fall into an unmistakably coherent pattern. This pattern leaves a number of loose facts still puzzling and not yet integrated into the overall coherence. Nonetheless, the pattern is there, as hypothesis:

Hypothesis (17) immediately explains two major pieces of our puzzle; first the shape and verb-prefix position of-ku- when it appears in front of short-stem verbs; and second, the position of the relative pronoun — following the tense aspect marker — in the ‘infix’ constructions (13). That is:

As satisfying as hypothesis (17a–e) is, it still leaves the exceptional behavior of both the habitual and perfect outside the newly cohered system. The habitual poses no problem, since it displays neither verb-related characteristics (infinitive on the following verb, infix rel-pronoun) nor any phonological material. It may be thus dispatched summarily by elaborating hypothesis (17) as follows:

The perfect marker-me- is a bit more problematic, since it shows one verb related feature (-ku- before short stems) but not the other (‘infix’ rel-pronoun). As it turned out, however, the answer to the next important question indirectly also resolves the problem of the perfect.
As hypothesis (17a–f) stands now, it predicts one absolutely vital set of hypothesized facts whose falsification would falsify the entire hypothesis:

From Popper’s (1934/1959) perspective, thus, if (18) were not the case, hypothe sis (17) must be rejected, by modus tolens.
Of the four aspectual markers of Swahili, the first three — with least irregular pattern — readily yield verbal etymologies, traceable to ‘be’, ‘have’ and ‘want’, respectively:

The perfect marker-me- turns out to be related to Swahili-mal-iz- ‘finish’ (Bemba-mal- ‘finish’). Its current shape in Swahili is derived historically from the so-called “modified base” form (Bemba-meele) by predictable phonological reduction.
We have then plausible etymological sources; but these sources are not free of problems. To wit:


The answer to these two puzzles, it turns out, has to do with more idiosyncratic historical developments in Swahili, where-na- indeed became first the perfect, then later changed into the present, which was in turn re-interpreted as ‘present progressive’. And-li- was indeed first a past-progressive tense-aspect, and only later was re-interpreted as simple (preterit) past.4
Lastly, we still have the functional-explanatory hypothesis (9) concerning the appearance of-ku- only before short verb-stems. Given the added facts we have unearthed since first abducting (9), and the current shape of hypothesis (17a–f), we must now modify (9) slightly, in order that it fit into the coherence pattern of (17). That is, it must be now expressed in coherent historical terms:

Our sub-hypothesis (17g) of course takes advantage of other pieces of theoretical knowledge, concerning the phonological reduction (‘erosion’) that occurs during grammaticalization and how it impacts de-stressed syllables.
At the end of a protracted process that involved many diverse steps, we wind up with a more general theory of the phenomenon — Swahili tense-aspect morphology — than we had at the beginning. Within this expanded theory, the distortions that gave impetus to our cycle of investigation find their natural resolution, coherence, or theoretical context.
Our investigation cycle in Swahili involved the gathering of additional relevant facts, the successive fitting of those facts into a single overall pattern, and the gradual refinement — via additional abductions — of a complex explanatory hypothesis. The cycle also involved at least one crucial falsificatory test.
The theory we hypothesized and gradually constructed explains the original puzzle, but also others that came along the way. It explains them all within a single — albeit complex — theoretical framework. And it explains them in conformity with a number of well-known general principles observed in gram maticalization elsewhere. Whether our hypothesis will survive further falsificatory testing only time will tell.