The competitive tier model – Element subtraction in German and Pomeranian

Competition of segmental material is inherent to all proposals of phonological template satisfaction. Segments are aligned with a prosodic template and compete for prosodic space. This vowel competition for space is well-known from the Semitic languages. In this paper, we argue that competition is also present in West-Germanic languages, albeit in disguised form. Basing ourselves on new Pomeranian data, we propose a competitive vocalic tier on which elements compete for prosodic slots besides the well-known vocalic tier that allows for element coalescence. An axiomatic model is developed that predicts alternations such as the [ɑi]-[ɪ] and [e]-[ɪ] root alternation (German treten – tritt ‘(he) step(s), Pomeranian gaita-git ‘(he) pour(s)’). The model allows us to solve three riddles in German morphology: 1. the “epenthesis conundrum”, i.e. the anti-correlation between OCP-driven schwa epenthesis between root and suffix in German (rett[ə]t/*rett versus *rät[ə]t/rät) and root alternations in present tense verbs (alternating verbs): retten – er rettet, raten – (er) rät (Neef 1997; Trommer 2010; Scheer 2016); 2. the “imperative conundrum” (Raffelsiefen 2016), which describes a correlation within the class of alternating verbs: the correlation between |A|-subtraction in a subset of alternating verbs (geben – gibt ‘give(s)’) and ending-less imperatives in German (gib! ‘give!’); 3. the vowel shortening conundrum: change in quantity of the root vowel in function of the type of vowel alternation: [e:]-[i] versus [a:]-[ɛ:]. A formal model is presented that provides us with a calculus of coalescence and competition of phonological features.


Introduction
Competition of segmental material is inherent in phonological proposals of template satisfaction (Goldsmith 1976;McCarthy 1978;Marantz 1982;Steriade 1982, and subsequent work): segments are aligned with a prosodic template and compete for prosodic space. For instance, a Semitic vocalic pattern P={ia} is aligned to a CVCVCV grid from left to right in according to the scheme in (1), producing kitaba. A pattern that is not produced is the pattern /kitiba/ where a vowel /i/ pushes /a/ to the third grid point, while spreading itself over two grid point. So there is first alignment (here: from-left-to-right) and then, as a last resort, spreading (again: from-left-to-right). Moreover, forms like kiteba are not produced either, where a and i share a prosodic slot.
(1) pattern: kitaba (2) pattern: *kitiba CVCVCV → CVCVCV CVCVCV → *CVCVCV | | | | melody: i a melody: i a {ia} {ia} In this study we argue that competition in the search for prosodic space is also present in the vowel system of West Germanic languages. The effect is, compared to Arabic, less easy to detect because West Germanic has complex vowels, i.e. it has coalescence of vocalic material, which obscures the competition effect. Competition effects in umlauted forms can be clearly observed in a West Germanic language, Pomeranian, spoken in the state Espirito Santo in Brazil. In comparing High German & Low Saxon plural morphology with Pomeranian, a curious pattern shows up. Consider the following singular-plural pairs, i.e. lemmas such as 'foot-feet', 'hat(s)', and 'book(s)' which is a systematic pattern in Continental Germanic dialects. (3) foot-feet hat(s) book(s) a. fu:s -fy:sə hu:t -hy:tə bu:x -by:çər [u-y] High German b. fo:t -foe:tə ho:d -hoe:(r) bo:k -boe:kə [o-oe] Low Saxon 1 c. faut -fuit haud -huir bauk -buikər [au-ui] Pomeranian In these lexemes, the plural is formed by adding a floating |I|-element as a suffix to the stem (Hamans 1985;Lodge 1986;Lieber 1987;Wiese 1987;Yu 1992;Hermans & Van Oostendorp 2008;Trommer 2010), apart from segmental material. 2 In Low Saxon [o] transforms into [oe], in High German [u] transforms into [y]. Apparently, both variants apply i-umlaut. Now, when turning to Pomeranian in (3)c, the first thing that strikes us, is that no complex vowels are present. |A| and |U| in the singular faut are realized as distinct segments [fa u t], the broken counterpart of the Low German /foot/. Similarly, in the plural fuit, |U| and |I| are also realized as distinct segments [fu i t], the broken counterpart of High German /füüs/. 3 The broken forms suggest that vocalic elements in Pomeranian do not coalesce. Significantly, upon addition of the extra (floating) |I| of the plural ("umlaut"), the |A|-element of the root is not realized: it is "pushed out" upon right-to-left alignment 4 of the melody over the available grid points. |A| is "not parsed", because it is without grid point.
The competition effect in (4) is wide-spread in Pomeranian morphology, both in inflection and derivation, for instance in denominal verbs: blaud 'blood' -bluira 'to bleed' (Postma 2018), where an umlaut feature is added. Competition gives rise to subtractive effects in phonology: the addition of |I| to faut causes a subtraction of |A| from the root.
Modern German exhibits competition effects in plural forms as well, albeit in less clear form, e.g. b [au]m-b[oi]me, as illustrated in (5). The only difference with Pomeranian is that an extra |A| feature is superposed on it. As this element is not affected by the addition of the umlaut, it must been realized in another autosegmental dimension, i.e. it sits on another tier. 5 Under (6) Similar |U| expulsion effects are found in Luxemburgian (Gilles 2015), Saarbrücken dialect (Steitz 1981: 69, apud Klein 2000 and Yiddish (Weinreich [1973(Weinreich [ ]2008Noam Faust, pers. comm.). Klein explicitly observes the subtraction of |U|.
"(T)he Saarbrücken data […] shows that the back rounded vowels in the dialect of Saarbrücken appear as the front unrounded vowels under umlaut.
[…] Any formal cross-linguistic analysis of fronting umlaut must capture the facts of unrounding versus rounding of non-low vowels adduced above as a language-specific property." (Klein 2000: 19).
In sum, Pomeranian shows competitive effect in pure form, as we saw under (4  ', Du. huid(ig) 'at present'), which has a long complex vowel [y:]. In sum, Pomeranian exhibits both competition and coalescence of vocalic material. The question is then, how we disentangle coalescent properties and competing properties, empirically and theoretically. We argue that natural language allows for two vocalic tiers, one where elements are in competition and one where elements are coalescent and where complex vowels are created. We show that the assumption of competition side-byside to coalescence provides us with a clue to shed light on three well-known riddles in 5 Independent evidence that there is an extra |A| on an independent tier, comes from Old Frisian bām. Proto Germanic baum must have had |U| on some tier which has undergone expulsion in Old Frisian. So where was its |A|? Note that Modern Frisian has bjem-in the plural/diminutive and composita. We then have the sequence for Frisian: 1.baum > 2.baam > 3.baim > 4.beem > 5.bjem-. So first an |U| element and then an |I| element are expulsed while |A| remains constant over the various processes. The |I| in stage 3 is a generalized umlaut element which has been reanalyzed as part of the root. |A|, however, was not affected. Hence, an |A| must have been present in a way that was autosegmentally disconnected. 6 GTRP, location L416p. The singular has accent 2 (level tone), the plural accent 1 (falling tone).
German philology. We call the competitive tier the E2 tier. The well-known (coalescent) tier is called the E1 tier.
In general, we assume that two types of tiers are provided by Universal Grammar and the choice between the type of tiers and the segmental filling of those tiers is specified by the lexicon. We summarize the proposal under (8). (8) • Natural language has two autosegmental vocalic tiers: E1 tier is coalescent E2 tier is competitive

• Competitive Principle
Every grid point is linked to maximally one element on the E2 tier. • The umlaut factor (floating |I|-element) in German realizes on the E2 tier. • Most lexical roots realize their vowels on the E1 tier, but some roots realize vowels (also) on the E2 tier.
These assumptions create the possibility of apparent subtraction of phonological elements, under influence of standard concatenative morphological processes, as illustrated under (4)-(5) above. 7 In the rest of this paper, we elaborate the two tier model in more detail. It allows us to tackle three riddles of German philology. At first sight, the existence of two different tiers for the sub-components of vowel comes as a surprise: why would natural language be designed that way? Why would elements be sometimes coalescent and sometimes competitive? The short answer is that we do not know. But we can give the two-tier hypothesis plausibility by a comparison with the consonantal system. Consonants typically have two types of elements: manner and place elements. Element Theory distinguishes (at least) three place elements (IAU) and three manner elements (|ʔ|, |L|, |H|), where |ʔ| stands for obstruent, |L|(ow) for nasality/voice (also called |N|), and |H|(igh) for aspiration/frication. Consonants are combinations of manner features but, according to standard ET, consonants can also have place features, for instance, [b] is {|ʔL|-|U|}. Vowels, on the other hand, are seemingly mere combinations of place features; no manner feature is assigned. This creates the asymmetric situation given under (9).
(9) scheme in e.g. Backley (2010) [e] We hypothesize that this asymmetry in the phonological system is only apparent. Also vowels have manner. This fact is only obscured by the fact that vowels use place features for manner purposes, i.e. the empty manner slot in (9) is taken by a place feature. As we will see in section 7, this manner dimension in vowels shows up as a lax-tense opposition. Laxness constitutes a manner of realization. In the mean time, the symmetrization of (9) should not be interpreted as if manner as such would be competitive, or, as if manner must be identified with the E2 tier. On the contrary: manner elements do combine (cf. Backley 2011). The consonantal tier is coalescent, just as the vocalic tier. There are indications, however, that one manner or place element in consonants may participate in the competitive E2 tier. It is the cause of consonant gradation in some West Germanic inflectional paradigms (see section 8.2). The article is structured as follows. First we will sketch three riddles in German grammar, the imperative conundrum, the epenthesis conundrum, and the shortening conundrum in section 2. These riddles all refer to vowel alternations in verbal roots, a→e and e→i. Section 3 is a brief historical background. Section 4 is theoretical and sketches the competitive tier model (CTM) in axiomatic form. It designs the mathematical structure of vowel coalescence: a bounded (semi)lattice. It introduces the theoretical background of schwa, i.e. the element |@|. The application to German vowel alternation is given in section 5.1. Section 5.2 applies the model to the imperative conundrum, while it is applied to the epenthesis conundrum in section 5.3. Section 5.4 generalizes the model to weak verbs (the regular verb class), which is a trivial special case of the alternating verbs. Section 5.5 deals with some strong verbs that have epenthesis. Section 5.6 discusses an exceptional verb class with so-called rückumlaut. Section 5.7 discusses seven counterexamples of verbs where umlaut is blocked. After the discussion of another apparent counterexample of the competitive nature in section 6, we discuss the more difficult third riddle: the shortening conundrum (section 7), which we will not completely solve. In section 8, we return to Pomeranian and apply the theory to the three conundrums in this language. The dialectology of the umlaut factor is discussed in section 9. In section 10, we confront the model with two previous accounts of the epenthesis conundrum: the declarative approach in Neef (1997), and the phonological approach in Trommer (2010). The paper closes off by summarizing the results, giving conclusions, some remaining problems, and a speculation on the origin and nature of the competitive tier.

Three conundrums
In this section we describe three (related) riddles of German philology that link a seemingly mysterious distribution of schwa and stem alternating behavior in verbal classes. One is the imperative conundrum, the other the epenthesis conundrum, the third is the shortening conundrum. The data set covered by the model is the morphological data of written Standard German, as we find it in prescriptive 20th century grammars, for instance Duden (1984), Griesbach (1991), Helbig & Buscha (2001). 8 Before turning to the riddles, we must discuss a dimension of German morphology that concerns these. This is the phenomenon that the present tense shows root vowel alternation for some verbs but not in others. In 23sg present tense, the root vowel changes in some verbs compared to the other forms of the paradigm. We call these "alternating verbs". We list two typical examples in (10) next to three non-alternating verbs, sparen 'to save', singen 'to sing', and denken 'to think', which lack root alternation in the present tense. All alternating verbs are strong verbs, i.e. they also show vowel alternation over the tenses, schlafen -schlief -geschlafen, and nehmen -nahm -genommen, but non-alternating verbs are either weak (sparen), strong (singen), or mixed (denken). 10

The imperative conundrum
The imperative conundrum (Raffelsiefen 2016) concerns the shape of the imperative in alternating verbs such as raten and treten. While in non-alternating verbs, the sg imperative is formed by the verbal stem + schwa, a decision has been made which root variant is chosen in the class of alternating verbs. The stem vowel of imperative in the a-e class (the i-umlaut class) is derived from the stem vowel of the non-23sg form: rate-rätst-rät has rat(e)! as its imperative. In the e-i class (the a-umlaut class), on the other hand, the form of the imperative takes the stem vowel of the 23sg form as its basis: trete-trittst-tritt has tritt! as its imperative. 11 Moreover, there is an effect on the shape of the suffix: obligatorily null in the case of the e-i class, optional (ə) in the case of the a-e class, 12 and /ə/ in the non-alternating class. 13 . 10 Though absent in modern standard High German, there is a weak class with alternation in the present tense (23sg). It is accompanied with consonant gradation, e.g. Pomeranian köipa-hai köft 'he buy(s)'. Dutch hebben-heeft 'to have/has'. They are a special case of the class with rückumlaut. We sketch an analysis in section 8.2. 11 There is one exception: werden 'to become', which has werde! as its imperative, i.e. takes the non-23 stem.
At the same time, it does not drop the final -e. In fact, this verbs confirms the relation between stem-alternation and zero-ending in the e-i class. It is probable that werde is a subjunctive form used as a suppletive form for an ill-formed imperative. 12 Historically, the imperative suffix was a ø-suffix in strong verbs but it gradually adopted the e-ending of the weak verbs. The present day grammars of Standard German capture this as an optionality in the written language. 13 The underlying ending is -ə, both historically and synchronically, but there is a synchronic rule of schwa apocope, blocked in verbs with roots ending in a dental stop warte! 'wait', or in bisyllabic roots feiere! 'celebrate', or syllabic nasals atme! 'breathe', which obligatorily show the underlying form.

The Epenthesis conundrum
Just like English, German shows epenthesis when an ending clashes with the last consonant of a verbal root. In such cases a dummy vowel is added between final consonant and ending, e.g. spalt+t ('he splits') to be compared with English pass+s→pass[ə]s and hand+d→hand [ɪ]d. This happens in various locations in the grammar, not only in the verbal system, for instance in adjectives, e.g. German heiß-heiß [ə]st 'hot-hottest'. At first glance, this is a rather late post-lexical phenomenon. However, this process depends on deep grammatical properties as can be inspected from the closely related Dutch/Frisian/Flemish, raad+t→raadt 'guess(es)', vers+st→ verst 'fresh(est)', etc., which lack such epenthetic vowels. We come back to this. The curious thing is that the process is not even unconditioned within German. German shows a dependency between "late" schwa epenthesis and rather "early" morphophonological properties of the root. Without clash between dental plosive in ending and root and, there is no epenthesis, e.g. spiel+t→spielt. If there is a clash in roots that end in a dental plosive, as in spalten 'split', there must be epenthesis: spalt+t→spaltət. If a verb accidentally fluctuates between the alternating class and the non-alternating class, as fechten 'to fight/swordplay' does, the schwa epenthesis co-varies, as illustrated under (12)e. Mixed cases such as *fecht or *fichtet do not occur. This indicates that the relation between root alternation and absence of epenthesis is produced by synchronic grammar. So let us summarize the epenthesis conundrum under (13). In the next section we present the riddle concerning the length of the verbal stem.

Shortening conundrum
While the stem vowels undergo a change in quality in 23sg present tense, they undergo a change in quantity in a subset of them. Verbs without vowel alternation do not shorten, only alternating verbs do. The pattern is, however, a mystery: verbs with e→i alternation have it: the vowel is a long [e:] in geben, nehmen, treten, and is a short [i]  This closes off our presentation of the three riddles that we aim to solve.

Background
In their discussion of apophony, Ségéral & Scheer (1998) discuss German ablauting verbs such as binden-band-gebunden 'bind-bound-bound' and identify a universal apophonic sequence I-A-U. If we assume autosegmental theory with a consonantal tier (not represented here) and a vocalic tier (Goldsmith 1976;McCarthy 1979), and if we assume the melodic content to be captured in Element Theory (Kaye et al. 1985;Harris 1994;Backley 2011), we can represent this as in (15). 15 We treat these three root forms as lexically related, not morphologically derived, and assume their elements to sit on the E1 tier. There is also a more complex case of I-A-U ablaut, as in helfen-half-geholfen 'to help', where the ablauting I-A-U sequence is superposed on, what Scheer calls, a "parasitic element", represented under (16), where the parasitic element is |A|, realized on another tier (see also Postma 1996Postma , 2014. As it is unaffected by changes on the tier that undergoes ablaut, we assume it to be in distinct planes (multiplanar approach, Archangeli 1985). While the ablauting vowel undergoes a change, the parasitic element remains constant.
This parasitic element in the case of helfen derives historically from an |A|-element, which was part of a suffix and anchored to the root. 16 Just as in the case in the previous section, we assume that these two levels represent two different tiers with distinct properties, the E1 tier and E2 tier, as in (17).
This parasitic element on the E2 tier has a relation with the suffix: the |A| in (17) originates from a suffix, which contained [a], e.g. OHG 13pl present tense hilf-an. This |A| has spread to the stem and was later generalized over all persons and tenses: it became part of the root. This generalized |A| features the change from gehulfen to modern geholfen. 17 It also shows that, synchronically, the relation in the present tense paradigm between /hilf-/ and /helf-/ is not A-umlaut anymore, i.e. addition of |A| to /hilf/, creating /helf/ (which is diachronically the case), but subtraction of |A| from /helf/ creating /hilf/ in synchrony. If hilf were the base form, we would expect the ablaut sequence hilf-half-hulf, counter to fact. Notice that addition of |I| (i-umlaut) and subtraction of |A| in (18) pattern as if they were one and the same morphological process in synchrony. Hence, we must generalize over I-addition and A-subtraction in the 23sg forms of the present tense paradigm.
In the next section we expound our model in an axiomatic way. It might be skipped on first reading and be used under backtracking, although we recommend to study it, perhaps after having seen it operational in section 5 and further.

The Competitive Tier Model: Competition and coalescence in Element Theory
Segmental content of the grid points have been described by various theories of feature composition in the tradition of the Prague school, via Chomsky & Halle (1968) to modern theories of feature geometry. Here we develop a model 18 that adopts a competitive tier hypothesis, next to the well-known coalescent tier, and next to various standardly adopted phonological principles. This paper is not a new phonology. It is a model. It is an operational model of the two-tier hypothesis, a "pseudo code", i.e. we also give a place to extant phonological principles, mostly with a twist, not for the sake of modifying them, but in order for the principle(s) to get implemented in a consistent way. Our intention has been to stay as close as possible to the principles and insights of modern rule-based phonology.
We take Element Theory as a starting point (Kaye, Lowenstamm & Vergnaud 1985;Backley 2011). Element Theory is a theory of phonological features that are strictly monovalued ("privative"): elements can be there or there is nothing, i.e. no negative values. As we mainly discuss vowels, we focus on the space spanned by place features. For further reference, we list some place features under (19). Vowels are built up by the basic features |A|, |I|, and |U|. An extra element |@| is assumed that functions as a neutral element under (19)f (Harris 1994;Van Oostendorp 2005), discussed in section 7. In isolation, |@| has [ə] as its exponence. Finally, we assume floating features (Lodge 1986;Wiese 1996), e.g. the umlaut factor |I|, illustrated under (19)g, notated as a fraktur character .
A floating element lacks an active prosodic anchoring point of its own (because of its lexical specification). Such a melody without prosodic space is in search of space to realize. This is parallel to Rubach's 1986 theory of yers as vowels that are not connected to a unit of prosody. 19 Notice that we don't assume the floating nature to reside in a delinking process, as under (20)a. The deactivation resides in the nature of the grid point, as in (20)b. The modeltheoretical reason is that that we want the nature of the grid point to be shared by both tiers, i.e. it is a source of interaction between the tiers: the elements on both tiers are floating, (20) c. Furthermore, all processes are defined on grid points and executed on the distinct tiers, rather than being independent tier processes. It is the more restrictive option in a model with two tiers. There is also a principled reason: association lines are part of the productive phonological calculus, while the nature of the grid point belongs to its lexical specification. Elements are associated to grid points according to alignment, which operates Edge-In (Yip 1988), i.e. from the affix inward. In the cases we consider here, it will be mostly right-toleft. Finally, we assume that only the element that is underspecified, |@|, may stand on a deactivated grid point, provided that it is licensed (see below). This is represented in (19)h. A deactivated grid point is a weak position that resists hosting elements that are phonological complex (see below for a definition), and hosts only dependent elements (Walker 2011).
Element Theory (ET) is a calculus of elements which is basically standard set theory. Segments are sets of elements. ET assumes that elements do not constitute the sounds themselves but are (place) operators working on grid points, thus producing a grid point with segmental exponence. For instance, the element |U| working on a grid point •, is assumed to produce [u]. We use the standard notation |U|-•. Similarly, |I| and |A| produce [i] and [a], respectively, when working on a grid point. |A| and |I| working together on • produce [e], etc. In ET, the sound [e] is more complex than [i] or [a], while [oe] is the most complex of all vowels. We here elaborate an implementation of the place features in ET as a (semi)-lattice, i.e. we implement it along the lines pursued in Harbour's (2014) theory of number features. Element composition creates a mathematical structure known as a bounded join semi-lattice. On full lattices, two basic operations are defined, JOIN (or "union") and MEET (or "intersection") being the only connectors that are operative on lattices. Empirically, we need JOIN to capture the anchoring of floating elements, e.g. i-umlaut, producing [e] out of [a]. We need MEET to account for reduction phenomena, e.g. schwa formation. In the standard (and most simple) realization, JOIN (+) and MEET (.) are commutative, (21)a and (22)a, as well as associative. MEET has |OE| as its neutral element. JOIN has |@| as its neutral element, as in (21)-(22).
These axioms produce the following laws, which we here list for future reference.
The subsequent application of MEET X and JOIN X (in either order) on Y produces absorption of Y, as the absorption laws in (23)ab show. 20 All operations are idempotent, which means that applying once has the same result as applying more times: P n = P. As said, these elements form a well-known mathematical pattern that is called a lattice (Partee, Van der Meulen & Wall 1990: 277ff). The lattice structure can be (incompletely) represented in a Hasse diagram, given under (24)a and its exponence counterpart in (24)b.

IA+I+UI +UI
III l@I b.

[e] [o]
[i] [u] [a] 20 An anonymous reviewer suggests to exploit this algebraic relation as a model of expulsion. If we follow this intriguing suggestion, absorption might arise upon alignment in an ambiguous stress context, i.e. whenever both (27) and (28)  There is a well-known relation between stress and vowel complexity, in the sense that stress tends to require complex segments, and simple segments require less stress (Trubetzkoy 1969;Dresher & Van der Hulst 1998: 322). Although this may be the result of historical processes, not a requirement of the synchronic grammar, schwa has synchronic requirements on the phonological structure it sits in. It cannot carry the stress on its own, but leans on a stressed syllable, i.e. it is in the weaker part of a foot. We capture this dependency ("stress-bound-without-being-under-the-stress"), by |@| being stress-licensed. A formal definition is given under (26). It captures the fact that schwa is parasitic on another stressed vowel: |@| is either spreading/moving to it when it is in the same phase (in the sense of Lowenstamm 2010; Enguehard 2016), or, it can stay on its own if it is adjacent to the stressed syllable. Adjacency depends on linearity, which is only provided upon closure of the phase. Adjacency is, therefore, defined in relation to a phase domain in the sense of Lowenstamm, i.e. when it is in another morphosyntactic phase than the stress.

(26)
Schwa Licensing |@| is licensed iff |@| … is associated to a grid point with stress or is adjacent to the stress. 22 Spreading/alignment, being ways to establish licensing, typically occurs in strong verbs (whose roots are not phases), the second way (adjacency) is with weak verbs (whose stems are phases, discussed in section 5.4).
We further define when the two operations MEET and JOIN operate. We assume a deterministic model where the structure defines which operation is active. On the E2 tier, elements are in competition i.e. they do not operate on each other: one expulses the other. The competitive nature of the E2 tier implies that JOIN and MEET both operate on the E1 tier. We assume that they operate depending on stress. Taking into account the relation between stress and complexity and the relation between destressing and neutralization (Trubetzkoy 1969), we adopt (27)-(28).
For the sake of completeness, we repeat under (29) how alignment proceeds (here: from right to left), first the anchoring of the rightmost |Y|, while |X| remains without prosodic space. |X| is not deleted (i.e. no stray erasure, Steriade 1982) or "overwritten" but remains without exponence when lacking a grid point.
A stray element only leads to well-formedness when it occurs at the edge (#) of the spellout domain (Goldsmith's 1990  A grid point that is not inclusive is called exclusive. This concept is relevant in most interactions that involve interpretation upon spellout, such as the lax-tense opposition, the short-long opposition, schwa epenthesis, Tier Conflation, etc., as we will see. There is an optional procedure of Tier Conflation (TC) upon spellout (McCarthy 1986) or, in a more modern formulation, upon closure of a phase (Lowenstamm 2010;Creemers, Don & Fenger 2016). Logically, there are two possibilities: conflation towards the E1 tier or conflation towards the E2 tier. The former possibility, conflation to the E1 tier, leads to coalescent forms. The latter leads to competition and element expulsion. We do not discuss the latter in this paper. When we speak of Tier Conflation in this paper, we simply mean conflation to the E1 tier, as represented under (32). Fully in line with (27) This has as a consequence that TC either applies on all or no grid points within a phase. The presence of an expulsed element, therefore, blocks TC and triggers phonetic analysis of the full tier structure rather than the conflated structure. The second restriction on TC is that inclusive grid points, i.e. when the E2 material is included in the E1 material, resist application of TC, because it would imply loss of information ("Full Interpretation" or "Containment"). For instance, the grid point |I|-•-|I| cannot conflate to •-|I|, but |A|-•-|I| may conflate to •-|A+I|.
We symbolize the umlaut factor with a fraktur character , and assume a structure as under (33) for German, which includes a floating umlaut factor |I| on the E2 tier and |@| on the E1 tier. The empirical justification is postponed to section 5.3. We also added the imperative morpheme.
We follow Leben (1973), Goldsmith (1976), McCarthy (1978) and assume the Obligatory Contour Principle (OCP), which penalizes two adjacent identical elements on one tier, and provides a resolution at the same time, given under (34). Notice that we take the OCP to be defined on the E2 tier (it is not active on the E1 tier because of the idempotence of JOIN and MEET) and, hence, is defined in function of at least one grid point. 24 In order to handle the shortening riddle, we define the lax/tense dimension.
We elaborate the lax-tense dimension in section 7 especially its dynamic nature vis-a-vis Tier Conflation.
A brief discussion of the inflectional morphology cannot be omitted, as umlaut is sensitive to different morpheme boundaries, traditionally indicated by + and #. We here assume the phase-theoretical approach to morphological domains, as elaborated for derivational morphology in Creemers, Don & Fenger (2016), henceforth CDF. Building further on Borer (2005), Marantz (1997), Embick (2010) and, most notably, Lowenstamm (2010), CDF assume, apart from root affixes, affixal cyclic heads: the first cyclic head or simply first head (F1) is special, not by definition but by consequence of the theory: it may influence spellout, stress, and meaning of the root domain, while further heads (F2, etc.) cannot, because their complement is impenetrable. This classification of heads is not a stipulation but it follows from the principles under (36). (36) • Lexemes and affixes are roots, 25 be they unbound (spott-) or bound (be-). Roots may combine and form complex roots, such as the root phrase √P (bespott-). • Head affixes ("cyclic functional heads") mark potential phase boundaries.
• When a cyclic head x is merged, cyclic domains in the complement of x are spelled out (Embick 2010: 51). • Spellout = linearization/concatenation + phonological calculus The "phonological calculus" in the fourth dot of (36) refers to the processes we discuss in this study: alignment, spreading, tier conflation, etc. The rules under (36) imply that these phonological processes cannot operate across a phase. The principles under (36) are exemplified under (37) with a maximally articulated verb like vergrößern 'to amplify' or bespötteln 'to mock repeatedly'. The root AFX and/or the cyclic head F1 may be absent or without lexical material. (37) Illustrative structure of bespöttelt '(he) mocks repeatedly' and vergrößert 'increases' Let us follow the construal bottom-up. First, one or more roots are combined into a complex root phrase, indicated by √P. Upon subsequent merger of the cyclic head F1 (empty or filled with e.g. -+l-, -el-, etc), the lower domain is checked for spellout. In this case, no spellout occurs because no cyclic head is probed in the complement of F1. Upon merger of the next cyclic head F2 (filled with agr1), the lower cyclic domain be+spott+ l-is spelled out. Within this phase, an occasional umlaut factor in the theme head F1 can modify the root: bespöttel-, vergrößer-. The umlaut factor present in 23sg agreement affixes ( st), on the other hand, cannot change the spellout of F1, which becomes visible in, say, brumm+el-in brummeln 'to mutter' without umlaut, which does not change to brümmelt in 23sg, because the affix is outside the phasal boundary ("weak verb" cf. Kayne 2016). We discuss in section 4 how floating umlaut material is handled in these cases. In strong verbs, say (be)schlafen, the derivational cyclic head F1 is absent and the agreement affix, being the first cyclic head, can change the spellout of the root, e.g. umlaut applies in 23sg present tense (s)t. The existence of a projection F1 ("theme") thus safeguards the integrity of the verbal root from being affected by the agreement affixes. There is a second effect of the phasehood of the F1 domain: the spellout of F1 makes agreement markers in F2 adjacent to the stem in weak verbs, while this is not the case in strong verbs, where adjacency is not defined (yet). Phasal spellout of a domain captures the existence and distribution of a phonological word boundary #. This is the only morphological ingredient we need in this basically phonological model of root alternations. 26 This finishes our axiomatic overview. In the next sections we apply competition and coalescence of elements to capture 1. alternating verbs, 2. the imperative conundrum, 3. the epenthesis conundrum, and 4. the shortening conundrum.

Alternating verbal paradigms
In this section, we apply the model developed above to the problem of the alternating verbal paradigms, i.e. vowel alternation in the strong verb class present tense. It explains the alternation such as ich schlafe -er schläft 'I/you sleep', and ich helfe -er hilft 'I/he helps'. After our preparatory work, the calculus is rather straightforward. The representations of the two verbs become as under (38)-(39). Upon morphological concatenation of root and the floating 23sg suffix, realignment takes place. The scheme ignores vowel length, which will be discussed in section 7.
In the case of helfen, the root lexicalizes both the E1 tier and the E2 tier because of the historical A-umlaut, cf. (16).
Some comments are in place. Under (38) we have the more simple I-umlaut. This I-umlaut comes about by adding a floating I-element in the 23sg of the present tense by some morphological (inflectional) procedure. We assume |I| is on the E2 tier, which is competitive, but this property is vacuous in the case of schlafen 'to sleep' and raten 'to guess', as the root's E2 tier is empty. The floating |I|-element, which does not have a grid point of its own, anchors to the (stressed) stem but stays on its own tier. The result is an alternating stem vowel [a] to [e]. The |@|-element on the E1 tier coalesces with the material present. Let us now look at the more challenging case of helfen 'to help', a verb which had, historically, A-umlaut (<hilfan). As we have seen from the introduction, these verbs have generalized an A-element that used to be part of the ending. This |A| has become part of the root (producing geholfen) but this (originally suffixal) |A|-element has not changed its E2 tier nature. We then have the structure in (39). Now, consider what happens if we add the same suffixal floating |I|-element of 23sg present tense. As it is an E2 tier element, and since E2 tier elements compete, the anchoring of the suffixal |I| to the stem and upon right-to-left alignment, pushes out the A-element. As a result, we get a seemingly subtractive effect, although it comes about by a normal additive procedure with the standard floating I-suffix. Expulsion (of |A|) is the CTM counterpart of Stray Erasure (Steriade 1982). This generalizes over the two paradigms in (18). Notice that (38) is in line with the rather standard assumption that umlaut is a floating coronal feature that must spread to the stem (Hermans & Van Oostendorp 2008; Trommer 2010).

Application to the imperative conundrum
Let us now apply the model to the imperative conundrum, given in (11) This riddle concerns the choice of the root vowel as well as the choice of the ending. From a historical point of view, the shape of the stem is perfectly understandable: the imperative takes the verbal bare root + imperative suffix, i.e. it lacks both the suffixal |I|-element in the rate -rätst-rät class, resulting in rat!, and it lacks the (originally suffixal) |A|-element in helfen, resulting in hilf! So, from an historical perspective, there is no conundrum. However, as we have seen above, the synchronic situation is that this |A|element is part of the root for the present-day native speaker. And secondly, we have seen in section 5.1 that the alternating verbs do not come about by A-umlaut, but by adding a floating |I|-element, just as in the case of the I-umlauting verbs, which expels |A|. So, we are looking for a synchronic account. Our model developed in the previous pages provides a way to see the imperative forms as being produced by a synchronic, productive, regular process. In order to get the desirable results, we only have to assume an imperative morpheme that sits on the E2 tier. Since the German imperative morpheme is schwa in regular verbs (spielen/spiele! 'play. inf/play.imp'), let us assume it is schwa in strong verbs as well. Let us see what happens when a floating imperative schwa is added. In (41)-(42) we represented the alignment strategy of root and floating suffix.
For /a/-stem verbs such as raten 'to guess' and schlafen 'to sleep', etc, |@| simply realizes on the E2 tier of the verbal root. No element is present on this tier and so, there is no problem in this case. The more interesting case is the A-umlauted verbs. As this |A| is synchronically part of the stem as an E2 tier vowel, it goes in direct competition with |@|. Upon right-to-left alignment, the |A| remains without anchoring, it is driven out by the imperative-schwa, as it were. Hilf results. In (43)-(44), we analyze the (older) variants with spreading of |@|, schlafe! and *gibe!. The spread variant, though slightly stilted in modern German, is without problem in the case of schlafen, because there is no element on the E2 tier. In the A-umlauted class, spreading is simply not invoked upon right-to-left alignment as it is a last resort (McCarthy 1979) because two elements are present (|A| and |@|) and two grid points. Moreover, the result would be ill-formed because one grid point is associated with two E2 tier elements. These spread variants are indeed absent: *hilfe! 27 'help.imp', *gibe! 'give.imp', *tritte! 'step.imp'.
This solves the imperative conundrum. Notice that the system allows for a combination of the strategies in (43) and (44) This option is realized in the verb lassen 'to let' (and fallen 'to fall' for modern speakers), which is an alternating verb of the i-umlaut type (lassen-lässt) but also lacks the option of schwa in the imperative (lass!/*lasse!). It is evidence that a double |A| element is present on both E1 tier and E2 tier in lassen 'to let'.

Application to the epenthesis conundrum
Before we apply the theory to the epenthesis conundrum, let us first give an an empirical justification to our representation of the umlaut factor in (33). We take our evidence from West-Germanic dialectology. Notice first that Dutch, Frisian, Pomeranian, the Low-Saxon and Limburgian dialects in the Netherlands do not have epenthetic vowels in the contexts under scrutiny at all, and resolve the -t+t-clash by degemination (Booij 1995: 68) or -as happens in the Low Saxon dialects of Twente -by simply having a "thick T" in this context (Van der Velde 1994: 70). Dutch does not have umlauted 23sg forms, but Pomeranian, Low Saxon, and Limburgian do exhibit vowel alternation but do not show epenthesis. Frisian has sometimes alternating forms in 23sg (ik meitsje-du makkest-hy makket 'I/you/he make(s)'), but no morphological umlaut. Moreover, Frisian schwa between root and ending is a property of a separate verbal class (the so-called je-verbs), not a phonological effect. Hence we listed Frisian in the [-epenthesis, -umlaut] class together with Dutch. Finally, I added Yiddish which lost both umlaut in 23sg present tense and epenthesis (Albright 2006 The chart under (46) shows that epenthetic schwa comes about in a subset of the languages with i-umlauting forms in 23sg present tense. In languages without I-umlaut in 23sg, epenthetic schwa is absent. This shows that schwa-epenthesis is dependent on 23-umlaut in the language despite the fact that epenthesis does not show up precisely when 23-umlaut applies in a specific verb. This can be explained by assuming that, synchronically, schwa epenthesis lexicalizes the vocalic slot in a 23sg morpheme, i.e. , unless the melody has incorporated into the verbal stem upon alignment. It points to the structural identity of the possibility of umlaut and the possibility of schwa epenthesis. This suggests the following Element-Theoretical implement-ation of umlaut in the Competitive Tier Model: a floating |@| on the E1 tier with an |I|-coloring on the E2 tier, cf.  (48), which activates the grid point.
However, this cannot be correct because it would provide |I| with prosodic space and removes its floating nature. Moreover, changing an underlying gridpoint is not part of our synchronic productive phonological toolbox. Instead, we retain the deactivated grid point • and make the assumption that epenthesis in German is not truly an independent process, but is a consequence of tier conflation, defined under (32) and applied to umlaut under (49). 28 Because the floating position is without stress, MEET applies. As we have shown in section 1, verbs with I-umlaut (raten 'to guess', lassen 'to let'), and verbs with A-umlaut (geben 'to give', treten 'to step', etc.) have the |I|-element in the ending, i.e. it is +t and +st. The representations are given under (50)-(52). The alignment structure is in (50).
The spreading configuration is under (51). This produces an ill-formed structure as |I| may not sit on a deactivated grid point, cf. section 4.

|A| |A|
The structure would become (52) upon tier conflation. Let us finally discuss tier conflation. This process gives rise to the ungrammatical (55). As treten is a strong verb, its root is not a phase. Hence licensing of schwa by adjacency is not possible because adjacency is not defined.
Notice that there is no conundrum in second person plural present tense. The forms are ratet and tretet. We only have to assume that the 2pl suffix has no umlaut factor, only |@| + t on a deactivated grid point on the E1 tier. The analysis for tretet is given under (56). The analyses for ratet 'you.pl guess', as well for non-epenthetic forms such as spielt 'you. pl play' are mutatis mutandis.
In the next section, we apply the analysis to weak verbs, and show why these do show epenthesis.
Let us start analyzing the imperative. Spreading of |@| to the stem is impossible as the stem is a phase. Staying in situ is licit under adjacency, as the stem has been spelled out. Tier conflation is licit as well. We actually think that the latter analysis is the only correct one, which is given under (57).
At first glance, spreading of |@| to the stem seems to be licit because there is no material present on the E2 tier. However, spreading to the stem, being a phase, is forbidden. On the other hand, spreading to the stress is not needed because the schwa is already licensed in situ under adjacency, as /zahl-/ is a phase. This makes (57) with tier conflation the only possible analysis. The 23sg suffixes are parallel. Alignment is not possible because the stem of a weak verb is a phase. Spreading is not possible, because |I| may not sit on a deactivated grid point. Staying in situ is illicit because |I| cannot sit on a deactivated grid point (*spaltit). Tier conflation is the only option.
TC gives us |@| on the E1-tier (under MEET because of -stress), which is licit on a deactivated grid point provided it is licensed. This is the case, as |@| is adjacent to the stem because a weak verbal stem is a phase. An attractive feature of our model is that it generalizes over strong and weak verbs. They have the same underlying endings in the present tense, though their realization is different. This solves the epenthesis conundrum in CTM.

Strong verbs with epenthesis -OCP effects
Let us now observe a limited class of strong verbs that surprisingly have epenthesis in 23sg.
These are typically verbs with [i] in the root, be they lax or tense monophthongs (bitten 'to request', bieten 'to offer'), diphthongs (leiden [ai] 'to suffer'), or vowel nasal combinations (binden 'to bind', singen 'to sing'): they realize epenthetic (er) bittet/bietet/leidet/bindet '(he) requests offers/leads/binds'. In these circumstances, there is an adjacent realization of two |I|-elements, one of the root and one of the suffix. So, we hypothesize that the Obligatory Contour Principle (OCP) is at work. Evidence for this is that heterorganic diphthongs [ai], [oi] and [ui] in other domains in the grammar, e.g. noun pluralization and diminution, are unaffected by mutation in German. Let us see how it works. Upon alignment and expulsion, as given under (59), a OCP violation is still there, as expulsion does not delete an element (no "stray erasure"). So the OCP violation is still present. On the other hand, the context of OCP resolution (OCPr) is lost, because OCP resolution makes reference to grid points, and hence does not operate on this context with extrametrical material, cf. (34).
While alignment gives rise to ill-formedness *biett, spreading gives a well-formed structure, as represented under (60). As the context of OCP resolution is present (which refers to grid points), one |I| is deleted, and we are only left with spreading |@|, which is licit: bietet. When the OCP works in situ, the structure becomes well-formed upon spreading. (60) This shows that this strong class with epenthesis is allowed by the model by virtue of OCP resolution.

The -jan class ("Rückumlaut")
There is a class of verbs in German with present-past tense vowel change (brenn-/brann-'burn', nenn-/nann-'to mention', renn-/rann-'to run', denk-dacht-'to think', …), that belongs to the epenthetic category. These stems typically have /e/+/n/ in the root. 33 This class includes two stems that end in a dental plosive: send-/sand-'to send' and wend-/wand-'to turn', which can serve as a testing ground for epenthetic behaviour: they have epenthetic schwa: sende/sendet and wende/wendet. The question is how these verbs work. Now, these verbs belong to a class that underwent the so called rückumlaut. Rückumlaut happened in verbs that made the present tense with -jan=|I|+an, with |I| a present tense marker, not part of the stem, and hence not showing up in the past. Traditional historical grammar has taught us that his /j/ glide in the suffix caused two changes: 1. the glide's (floating) |I| caused umlaut of the present tense stem and 2. its consonantal nature caused gemination of the final root consonant (Vennemann 1986). Because of historical Rückumlaut, an |I|suffix that is now reanalyzed as part of the root, and to make a contrast to the verbs like treten, geben, nehmen (which had A-umlaut), let us explore the null hypothesis that these root verbs have their |I| element on the E2 tier and |A| on the E1 tier in their present-day lexical specification. Let us first consider what happens synchronically under concatenation of a 23sg sufiix. Under alignment, |I| expulses the |I| element but, after expulsion, the OCP blocks this structure, while the context of OCP resolution, which refers to grid points, is lost, cf.
(34) in section 4. So the OCP clash cannot be resolved under alignment. The configuration is given under (63) The theory predicts (65) to be well-formed if and only if the stem licenses the epenthetic schwa in situ. Interestingly, these verbs do not only display vowel change, they are suffixal as well, i.e. they are both strong and weak: senden-sandte-gesandt. Being weak, the root is a phase because of (26), and can license the schwa in the suffix under adjacency. 35 So, |@| is licensed without spreading to the root. This shows that the model predicts sendet to be well-formed, more specifically, two underlying representations of it: one that is well-formed when /send-/ is strong, and one that is well-formed when /send-/ is weak. We would like to decide which structure sendet has: the spreading strategy of bindet, or the analysis with tier conflation of the weak verbs. Now, all the verbs in this -jan class were lost in Modern German, unless they had /n/ in the coda. 36 If this /n/ is alone in the cluster, it is dropped in the past tense: e.g. denkendacht, but when it is part of an historical geminate, it is retained. Now, let us suppose that (moraic) nasals do not only have articulation on their consonantal tier (not drawn) but may also have specification on the competetive E2 tier. So let us assume that the /n/ in these verbs have |N| on the E2 tier. 37 This intervention has as a consequence that the OCP between the two |I|s does not apply. Moreover, the umlaut factor would cause its expulsion upon alignment, but the fact that |N| is part of a consonantal cluster or geminate resists this (Honeybone 2005). To solve this clash, TC and in situ spellout of the umlaut factor is the only option. This is only possible upon tier conflation: senden-sandte-gesandt must be a weak verb. (The |N| goes to the consonantal tier upon TC, not drawn.)

Strong verbs without alternation (heteroclisis)
Seven ablauting verbs in German lack umlaut in the present tense: schaffen-schuf-geschaffen 'to create ', rufen-rief-gerufen 'to call' and hau(w)en-hieb-gehauen 'to hew', saugen-soggesaugen 'to suck', schnauben-schnob 'to sniff', kommen-kam-gekommen 'to come' and t[u:] n-tat-getan 'to do'. These do not have 23sg with umlaut, hence schafft not schäfft, tut not 35 A domain cannot be a phase and not-a-phase, obviously. This implies that an extra head must be present, which is either zero, or coincides with part of the stem, say, that sen-is the root and send-the weak stem. 36 The class with rückumlaut was expanded in English (e.g. tell-told, sell-sold) and became prolific in Low Saxon, e.g. in the dialect of Ravensberg (Jellinghaus 1885). 37 Standardly used manner elements are |ʔ| -stopness, |h| -noise, |N| -nasality, |H| -stiff vocal cords, |L| -slack vocal cords. An comprehensive model of manner in ET is as yet lacking. An elegant proposal with 3 manner elements (headed and unheaded, 6 in total) is Backley (2010: 161) and seems to be compatible with CTM. An even more reduced system of 2 place and 2 manner elements is found in Pöchtrager (2018).
tüt, kommt not kömmt, etc. These seven verbs have typically two |U| elements on adjacent grid points in the nucleus, be they vocalic, consonantal, or both. We here assume the ET assumption that velars and labials carry |U| (Backley 2011: 77ff.). Apparently, such double grid points with |U| can be umlaut-blocking, "umlauthinderend" (Paul 1918: 25, Anm. 6.), "inalterable" (Hayes 1986), or with "inhibition effects" (Honeybone 2005), just like the n-geminates of the previous section. On the other hand, some other roots with double |U|, such as laufen-läuft, do have umlaut. This suggests that a different distribution of |U| elements over the two tiers is involved. 38 The OCP allows double |U| under the assumption that these two |U|s are not on the same tier, which is probably the reason why laufen has regular umlaut. If they are on the same tier, the OCP must have reduced it by a leftto-right spread configuration with inhibition effects. So, let us assume that in the seven deviant roots have a spread |U|s on the E2 tier by their lexical specification of their root (left to right alignment). We give a representation under (67). Apparently, realignment of spread elements is not allowed. Tier Conflation is needed to prevent realignment of the spread |U|. Notice that TC of the consonantal |U| goes to the C-tier (not drawn).
(67) saug--t +stress -stress saugt E2 tier |A||U| |I| | | | TC In view of the correlation between the short-long dimension and the lax-tense dimension, 44 we have to decide in what terms we capture shortening. At first glance, the optionality in lax/tense in schlafen-schläft without shortening ([a:]-[e:/ɛ:]), suggests shortening in terms of length irrespective of quality. Nevertheless, we cast the shortening conundrum in terms of laxing as it is more in line with existing theorizing on the lax-tense dimension in ET. So we will explore (75). (75) • Element expulsion correlates with laxing Consider the traditional vowel charts under (76), which is organized along the tense/lax dimension, taken from Swets (2004: 99 In the absence of this E2 tier element, the vowel is tense. Evidence for an element on an independent tier that causes laxness is the existence of (retracted/advance) tongue root harmony in some languages (Clements 1981, Lowenstamm & Prunet 1988, Backley & Takahashi 1996 ("headedness agreement")). Cast in CTM, tongue root harmony is the spreading of |@| on the E2 tier. Instead of the restriction to |@| on the E2 tier, we explore a generalized version of lax and tense under (79). Let us, therefore, review the circumstances that restrict Tier Conflation. Two conditions on TC hold: 1. TC may not disrupt alignments, and 2. phonological information does not get lost (Containment). The first condition provides the reason why the presence of an expulsed element triggers a lax vowel: the expulsed element lacks a grid point and cannot undergo TC. Hence, upon TC, the alignment would be disrupted (and hence expulsion would be lifted). This makes that tier conflation is blocked in expulsion contexts. Put differently, in the presence of an expulsed element, there must be an adjacent element on the E2 tier, and hence the corresponding vowel is lax. This produces the shortening conundrum in terms of Harris-type laxing. This approach of the shortening conundrum in terms of the lax-tense dimension cannot be the whole story, however, because lax-tense and short-long are not in a bijective relation. This shows up exclusively in the case of schlafen-schl[e:]ft/sl[ɛ:]ft, a context without element expulsion, with no shortening, even when the vowel is lax. It shows that length must have a definition that is different but not disconnected from the tense-lax dimension. Let us first study the inclusive-exclusive dimension and its relation to length. Consider the definition of (31), repeated under (81) (78), cannot undergo TC and are, therefore, inherently lax, i.e. they instantiate (80)a. In general, inclusive grid points, which are short because of (81), are lax because of (79). The reverse is not true, exclusive grid points are not tense, because TC is, in principle optional. This brings us to the corrollary in (83) (83) is slightly more tolerant, as it also identifies |X|-•-|X| configurations as lax, for instance the |I|-•-|I| configuration in gibt and the |A|-•-|A| configuration in lassen. Notice that also |I|-•-|I+A| configurations are identified as lax, with X=|I| and Y=|I+A|. On the other hand, all structures without any material on the E2 tier, for instance •-|A| in schlafen are identified as tense.

Vocalic alternations
Pomeranian (Tressmann 2006;Postma 2018) has virtually lost the a → e class of alternating verbs, just as the Low Saxon dialect (Gronings) in the Netherlands did (Reker 1989): there are no raten-rät verbs. 45 The expulsion type e→i has survived in 6 verbs: geewa -hai giwt 'to give -he gives', gaita -git '(he) pour(s)', which has the same analysis as in German, where |A| has been pushed out (and the vowel shortened). In addition to this, a new class emerged. Under (84) we give some typical cases of the new alternating verbs. The drawn structures concern 3sg present tense contexts. A parallel rule (mutatis mutandis) holds for 2sg contexts. Dutch and Frisian (and perhaps Yiddish) are underlyingly different from these umlaut varieties because they do not have an umlaut factor as part of the 23sg morpheme, i.e. they do not have an deactivated grid point and do not need these spellout rules.

Two previous accounts of the epenthesis conundrum
The description of German alternating verbs, epenthesis upon OCP clashes, the varying shape of the imperative morpheme, and their correlating character, are part of most grammars of Modern German, as well as every historical grammar of Middle High German. Nevertheless, there are only few attempts to account for the correlation theoretically.
Most scholars are satisfied with a diachronic description of how these effects emerged, and ignore the question how they survive the wild tides of language change. As we have seen, even under language variation, e.g. ficht versus fechtet, the correlation retains (*fichtet/*fecht), which indicates that a synchronic force must be at stake. Ignoring the lexicalist storage approach, there are only two serious synchronic attempts. The first is Neef's word design model (1997), which hypothesizes well-formedness conditions that apply on forms at the spellout level, briefly discussed in section 10.1. And there is Trommer's (2010) derivational approach, which is evaluated in section 10.2. Neef (1997) is the first who accounts for the anticorrelation between root alternation and epenthetic schwa in a synchronic framework. It is not a derivational approach but an account of the anticorrelation in terms of word design in paradigms. Neef assumes that particular forms in a paradigm have surface correspondences to sister forms, for instance, two forms must be equal or they must be distinct. In the case of 3sg present tense, there is, according to Neef, the paradigmatic requirement that 3sg present tense is distinct from the verbal root. Upon a potential tt→t degemination in verbal stems terminating in a dental plosive, identity of root and 3sg present tense looms and schwa insertion is chosen to circumvent it. This gives correct predictions to 3sg, cf. (97)abc.
As an explanation, Neef proposes a second requirement for 2sg: its word design is based on 3sg: replace final t by st. We ignore here that this procedure is not without exception, (kannst<*kann, bist *< ist), as most theories have a problem with the copula. It does not work on the surface forms in dental final roots either: er hält→ du *hälst/hältst, it only works on the underlying form er /hältt/. Despite these problems, the relational status of 2sg and 3sg is clearly within the possibilities of the word design model. What is more (106) The E1 tier is CV aligned; the E2 tier is VC aligned In this view, the E2 tier is the source of closed syllables and the emergence of lax vowels, which we identified as a vocalic manner realized on the E2 tier. We represent this realignment by a formal feeding mechanism of the E2 tier in diachrony under (107).
(107) Closed syllable formation and laxing E2 tier First we have a (CV)(CV) pattern with X and Y representing their vocalic element content on the E1 tier, say Gothic badi 'bed'. When, as part of a diachronic change, its grid point is deactivated under simultaneous flipping of the vowel to the other tier, it receives pre-consonantal realization under synchronic alignment. This double change might be caused by the fact that there is a universal preference for CV alignment to be taken "Edge-In" in the sense of Yip (1988) which is, in the case of a suffix and rightto-left alignment, a VC order, producing modern English bed. The typical expulsion effect shows up whenever this process happens twice (or -in synchrony -a floating feature). What makes this interpretation especially attractive, is the link it lays between the E2 nature of an element and its suffixal origin (Edge-In from the suffix). In the mean time, these considerations do not shed light on the competitive character of the E2 tier. The competitive nature of the E2 tier seems independent, but it might be related to the affixal origin of E2-material as well, as affixes often compete for prosodic space. Nothing hinges on this preliminary interpretation of CTM as reversely aligned (CV) units, but the phonological calculus presented in this paper and most empirical details are compatible with it.
Let us finally discuss a problem of the model. It does not concern the competitive tier model itself, but its implementation with the Hasse diagram in (24). This diagram predicts a symmetric behaviour of the three basic elements IAU. However, there are various empirical symmetry breaking effects, especially with respect to sonority. In the first place, expulsion of |A| is the most common, expulsion of |U| is somewhat rarer, while the expulsion of |I| is not attested. This comes as a surprise in view of the symmetry in the model and cannot be captured by it. Secondly, the Harris-type configuration |@|-•-|X| is always lax as predicted (because of inclusivity), with the exception of when |X|=|A|, for instance in the aligned imperative schlaf!, which has |@|-•-|A| but realizes as tense ]ter/*g[ʏ:]ter 'goods'. This problem shows that the IAUsymmetric model has defects, or rather, it does not cover the empirical facts with respect to symmetry breaking by sonority. These problems are, however, not inherent to the theoretical assumption of the Competive Tier itself, only of the CTM model presented with a sonority symmetric element lattice. Other lattices are worthwhile being explored, for instance the interchange of |A| and |@| in the lattice in (24)a. Various researchers have recently suggested that |A| is not an element like the others, but an underspecified one (Cavirani & Van Oostendorp 2017; Faust 2017) or even pure prosodic space (Pöchtrager 2018).