I tripped over this link as I was surfing the net lately and I was floored by the eery similarity with my own independent contemplations. It's Oswald Szemerenyi's book entitled Introduction to Indo-European Linguistics that I never got my hands on to read in a more physical format until I found it on the net. On page 112 (see here), he cites words from C. H. Borgstrøm who published his work on Pre-Indo-European (Pre-IE) in 1949. According to his theory, using the Pre-IE verb "to be" which he reconstructs as *häsä-, Borgstrøm proposes that the third person singular (3ps) was once *häsä-tä (> *est) "he/she/it is" and that the third person plural (3pp) was *häsä-nätä. It's not a mirror image of my theory since he proposed only one vowel in Pre-IE, namely *ä as you probably guessed, but what it shares with mine which I arrived at independently and what was apparently already understood sixty years ago is that Proto-Indo-European (PIE) at some point in its past had dropped unstressed syllables.
My own search for a clear theory of Pre-IE began years ago with the observation that if PIE's accent alternates between one syllable and the next without an intuitive and common motivation, it must have been more regular in the past. I found out that most accent alternation patterns reconstructed for PIE (such as, *wódr̥ "water" vs. *udn-ós "of the water") could be easily regularized to the penultimate accent by presupposing that there was once a Pre-IE vowel positioned after case endings such as genitive *-ós. Along with the more obvious examples of an earlier drop of unstressed vowels (a.k.a syncope) in forms like *bʰr̥tós "carried", I became convinced that PIE must have once had a strong stress accent as in English or Italian and that this early syncope affected all unstressed syllables equally. (PIE itself is reconstructed with tonal accent, as in French. Tonal accents don't erode vowels the way stress accents do.) I normally capitalize the word as Syncope when I'm talking about this specific sound rule in Pre-IE, by the way. I was quickly struck by the idea that PIE's so-called "mobile" accent system was mobile (i.e. untuitive as to where accent is placed in a word) precisely because of Syncope. Where word-final vowels were lost, penultimate accent automatically became ultimate while those forms that did not lose a final-vowel would retain the old penultimate accent. In this way, Syncope in one fell swoop obscured the original, clockwork regularity of the accent. There's a lot more I need to say on Mid IE (MIE) stress accent, a lot more, but let's just say that somewhere down the line I realized that the accent in MIE could also occasionally occur on the antepenultimate syllable if an ending was formed in Old IE by the addition of an agglutinized element, as in the MIE 3ps and 3pp verbal endings with the attachment of deictic *ta (> PIE *to- "that") to the pre-existing Old IE 3ps ending *-a and 3pp ending *-éna. I now refer to this Mid IE rule of accentuation as QAR (Quasi-Penultimate Accent Rule).
I also reasoned that qualitative ablaut (i.e. the alternation between *e and *o in many verb forms), which is afterall the rather wild and unexplained alternation between phonetically polar opposites (between an unrounded, front vowel *e and a rounded, back vowel *o) , was probably once a simpler ablaut involving differences in height contrasts only. So at some point in Pre-IE we can reconstruct mid-central *ə alternating with low-central *a. This is what Allan Bomhard has proposed, although he also believes that Indo-European's ablaut originates at the Nostratic stage preceding PIE by more than 10,000 years. To me, this added assumption is too fantastical to believe. At any rate, this signals that PIE underwent a recent vowel shift of *a to *o, particularly considering the otherwise scarcity of *a in PIE proper. From such a "height-contrasting ablaut" stage, the ablaut system could be further pushed back to a stage where there was no ablaut at all once the initial conditioning factor of ablaut (possibly vowel harmony) can be conclusively found, several millenia before PIE. So in a nutshell, this is why I reconstruct two vowels in Pre-IE, rather than one.
So back to Borgstrøm's 3ps *häsä-tä and 3pp *häsä-nätä, I had come to similar but more detailed conclusions, as you can see. I've reconstructed late Mid IE 3ps *ʔésatai and 3pp *ʔasénatai (n.b. the regular accent by way of the above-mentioned QAR). Unaccented *a was dropped in most circumstances in early Late IE via the Syncope rule, producing *ʔésti and *ʔsénti (traditionally written as *h₁esti and *h₁sénti) . The problem of the 1pp and 2pp forms that Szemerenyi uses to disprove Borgstrøm's proposal have no affect on my somewhat different explanation: MIE 1pp *ʔasména "we are" and 2pp *ʔasténa "you are". However, Szemerenyi admits that Borgstrøm's fatal flaw here comes from "the assumption of a stage with open syllables only".
 The genitive form *udnós is reflected in Sanskrit udnás. Hittite witenas however reflects an alternative genitive case form *wedéns. Personally, I feel that forms with *wed- are in all likelihood the more conservative because of the same *o/*e alternation in other ancient paradigms like that of *pod- "foot" (nominative *pōds "foot" versus genitive *pedós "of the foot"). I would surmise that *udnós is the result of speakers generalizing the more prevailing pattern in PIE of stressed vowels reducing to zerograde when unaccented.
(Jan 24/08) I've decided to alter what I originally stated in footnote #1 above. After a good debate with Phoenix in the comments box on curiosities of Hittite spelling, I'll cave in and admit that Hittite genitive witenas "of water" is likelier to reflect a pronunciation such as /wɪténs/ and thus would be a reflex of *wedéns rather than *wednós as I originally stated (which I based in part on Sanskrit's reflex and partly on my theories concerning Syncope and QAR in Pre-IE). I'll assume for the moment at least that the Anatolian genitive form hasn't been affected by other case forms such as the locative form, *wedéni. I still have a funny suspicion that the underlying paradigm of "water" in PIE contains nomino-accusative *wódr̥, genitive *wedn-ós and locative *wedén-i.