Hungarian Vowel Harmony

@cite{siptar-torkenczy-2000} @cite{rose-walker-2011}

Hungarian has palatal vowel harmony: the [±back] value of the last harmonic (non-neutral) stem vowel spreads rightward to suffix vowels. This is the most widely studied vowel harmony system in the literature, with over 50 theoretical treatments cited in @cite{siptar-torkenczy-2000}.

Why Hungarian stress-tests HarmonySystem

Hungarian harmony is structurally similar to Finnish (both spread [back] rightward with transparent neutral vowels), but introduces three phenomena that push beyond a simple triggerValue analysis:

1. Antiharmonic stems (§3.2.2 class IIA-b): all-neutral stems like híd 'bridge' that unexpectedly take back-vowel suffixes. triggerValue returns none for these stems since there is no harmonic trigger — the backness must be lexically specified.

2. Vacillating stems (§3.2.3.1): stems like hotel where speakers accept both front and back suffixes. The spreadSuffix function returns a single deterministic result and cannot model optionality.

3. Height-graded transparency (§3.2.3): neutral vowels /i, í/ are always transparent, /é/ is mostly transparent, and /e/ sometimes acts as opaque. The binary isTransparent predicate cannot express this gradient.

Vowel inventory (@cite{siptar-torkenczy-2000} §2.2.1, system (7))

	Front		Back
	Unrounded	Rounded
High	i / í	ü / ű	u / ú
Mid	e / é	ö / ő	o / ó
Low			a / á

Harmony classification (@cite{siptar-torkenczy-2000} §3.2, (27))

• Front harmonic: ö, ő, ü, ű (front rounded)
• Back harmonic: a, á, o, ó, u, ú (back)
• Neutral: i, í, e, é (front unrounded)

Stem classes (@cite{siptar-torkenczy-2000} §3.2.2)

• IA (simple harmonic): all harmonic vowels same backness
• IB (complex harmonic / disharmonic): conflicting harmonic vowels, last harmonic governs
• IIA (simple neutral): all vowels neutral — mostly front (IIA-f), but some are antiharmonic (IIA-b)
• IIB (complex neutral): harmonic + neutral, neutral is transparent

Rounding harmony (@cite{siptar-torkenczy-2000} §3.2.4)

Three-way suffixes (hoz/höz/hez, tok/tök/tek) depend on BOTH backness and rounding of the last stem vowel:

• Back → always o (regardless of stem rounding)
• Front rounded → ö
• Front unrounded (including neutral) → e

Rounding harmony is simpler than backness harmony: no neutral vowels (every vowel is either rounded or not), no antiharmonic stems, no vacillation. It only matters for front stems.

Architectural limitation: Clements/Hume place-node spreading

@cite{siptar-torkenczy-2000} §6.1 analyses Hungarian VH using Clements/Hume unary place features (COR, LAB, DOR) with four rules: Link Place, Link DOR, Spread Place, Spread DOR. This is place-NODE spreading, not individual binary-feature spreading. Our HarmonySystem with a single feature : Feature field cannot capture node-level operations. The two-system decomposition (palatal + labial) used here is an approximation that works for the data but does not reflect the theoretical architecture of §6.1.

We define the 7 short vowels using the Hayes feature system. Long vowels have identical features for harmony purposes — length is prosodic, not segmental (@cite{siptar-torkenczy-2000} §3.1).

Representational note: Hungarian /a/ = [ɒ] is phonetically rounded
but phonologically [+back, −round, +low]. Surface rounding is a
phonetic implementation detail (@cite{siptar-torkenczy-2000} p. 54). 

def Fragments.Hungarian.VowelHarmony.i_vowel : Theories.Phonology.Segment

/i/: [+syll, +high, −back, −round, −low, +dorsal]. Neutral.

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def Fragments.Hungarian.VowelHarmony.ü_vowel : Theories.Phonology.Segment

/ü/: [+syll, +high, −back, +round, −low, +dorsal]. Front harmonic.

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def Fragments.Hungarian.VowelHarmony.u_vowel : Theories.Phonology.Segment

/u/: [+syll, +high, +back, +round, −low, +dorsal]. Back harmonic.

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def Fragments.Hungarian.VowelHarmony.e_vowel : Theories.Phonology.Segment

/e/: [+syll, −high, −back, −round, −low, +dorsal]. Neutral. Phonetically [ɛ] (open-mid).

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def Fragments.Hungarian.VowelHarmony.ö_vowel : Theories.Phonology.Segment

/ö/: [+syll, −high, −back, +round, −low, +dorsal]. Front harmonic.

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def Fragments.Hungarian.VowelHarmony.o_vowel : Theories.Phonology.Segment

/o/: [+syll, −high, +back, +round, −low, +dorsal]. Back harmonic.

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def Fragments.Hungarian.VowelHarmony.a_vowel : Theories.Phonology.Segment

/a/: [+syll, −high, +back, −round, +low, +dorsal]. Back harmonic. Surface [ɒ] is phonetically rounded; phonological rounding is absent (@cite{siptar-torkenczy-2000} p. 54).

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def Fragments.Hungarian.VowelHarmony.isNeutral (s : Theories.Phonology.Segment) : Bool

A vowel is neutral iff it is front and unrounded: /i, í, e, é/. @cite{siptar-torkenczy-2000} §3.2, (27).

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def Fragments.Hungarian.VowelHarmony.isFrontHarmonic (s : Theories.Phonology.Segment) : Bool

A vowel is front-harmonic iff it is front and rounded: /ö, ő, ü, ű/. These always trigger and undergo palatal harmony.

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def Fragments.Hungarian.VowelHarmony.isBackHarmonic (s : Theories.Phonology.Segment) : Bool

A vowel is back-harmonic iff it is [+back]: /a, á, o, ó, u, ú/. All back vowels are harmonic in Hungarian.

Equations

• Fragments.Hungarian.VowelHarmony.isBackHarmonic s = (s.hasValue Theories.Phonology.Feature.syllabic true && s.hasValue Theories.Phonology.Feature.back true)

inductive Fragments.Hungarian.VowelHarmony.HarmonyRole : Type

Harmony role classification.

• frontHarmonic : HarmonyRole
• backHarmonic : HarmonyRole
• neutral : HarmonyRole

instance Fragments.Hungarian.VowelHarmony.instDecidableEqHarmonyRole : DecidableEq HarmonyRole

Equations

• Fragments.Hungarian.VowelHarmony.instDecidableEqHarmonyRole x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

def Fragments.Hungarian.VowelHarmony.instBEqHarmonyRole.beq : HarmonyRole → HarmonyRole → Bool

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• Fragments.Hungarian.VowelHarmony.instBEqHarmonyRole.beq x✝ y✝ = (x✝.ctorIdx == y✝.ctorIdx)

instance Fragments.Hungarian.VowelHarmony.instBEqHarmonyRole : BEq HarmonyRole

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• Fragments.Hungarian.VowelHarmony.instBEqHarmonyRole = { beq := Fragments.Hungarian.VowelHarmony.instBEqHarmonyRole.beq }

instance Fragments.Hungarian.VowelHarmony.instReprHarmonyRole : Repr HarmonyRole

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• Fragments.Hungarian.VowelHarmony.instReprHarmonyRole = { reprPrec := Fragments.Hungarian.VowelHarmony.instReprHarmonyRole.repr }

def Fragments.Hungarian.VowelHarmony.instReprHarmonyRole.repr : HarmonyRole → Nat → Std.Format

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def Fragments.Hungarian.VowelHarmony.classifyVowel (s : Theories.Phonology.Segment) : HarmonyRole

Classify a vowel's harmony role.

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def Fragments.Hungarian.VowelHarmony.hungarianPalatalHarmony : Theories.Phonology.Harmony.HarmonySystem

Hungarian palatal harmony: [back] spreads from the last harmonic (non-neutral) stem vowel to harmonic suffix vowels. Neutral vowels /i, í, e, é/ are transparent — they neither trigger nor undergo harmony.

Structurally identical to finnishHarmony but with a larger neutral class (4 vowels vs 2). @cite{siptar-torkenczy-2000} §3.2.1, @cite{rose-walker-2011}.

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def Fragments.Hungarian.VowelHarmony.hungarianLabialHarmony : Theories.Phonology.Harmony.HarmonySystem

Hungarian rounding harmony: [round] from the last stem vowel, used to resolve three-way suffix alternations (hoz/höz/hez).

Unlike backness harmony, rounding harmony has NO transparency — every vowel is either rounded or not. There are no antiharmonic stems and no vacillation w.r.t. rounding (@cite{siptar-torkenczy-2000} §3.2.4).

The rounding value only matters for front stems (back stems always get the back suffix variant regardless of rounding).

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inductive Fragments.Hungarian.VowelHarmony.StemClass : Type

Stem class for Hungarian vowel harmony.

The four major classes combine two axes:

• Harmonic (I) vs Neutral (II): is the last vowel harmonic or neutral?
• Simple (A) vs Complex (B): are all harmonic vowels the same backness?

Within each, -f and -b indicate whether the stem selects front or back suffixes. @cite{siptar-torkenczy-2000} §3.2.2.

• IA_f : StemClass
• IA_b : StemClass
• IB_f : StemClass
• IB_b : StemClass
• IIA_f : StemClass
• IIA_b : StemClass
• IIB_f : StemClass
• IIB_b : StemClass

instance Fragments.Hungarian.VowelHarmony.instDecidableEqStemClass : DecidableEq StemClass

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• Fragments.Hungarian.VowelHarmony.instDecidableEqStemClass x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

def Fragments.Hungarian.VowelHarmony.instBEqStemClass.beq : StemClass → StemClass → Bool

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• Fragments.Hungarian.VowelHarmony.instBEqStemClass.beq x✝ y✝ = (x✝.ctorIdx == y✝.ctorIdx)

instance Fragments.Hungarian.VowelHarmony.instBEqStemClass : BEq StemClass

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• Fragments.Hungarian.VowelHarmony.instBEqStemClass = { beq := Fragments.Hungarian.VowelHarmony.instBEqStemClass.beq }

instance Fragments.Hungarian.VowelHarmony.instReprStemClass : Repr StemClass

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• Fragments.Hungarian.VowelHarmony.instReprStemClass = { reprPrec := Fragments.Hungarian.VowelHarmony.instReprStemClass.repr }

def Fragments.Hungarian.VowelHarmony.instReprStemClass.repr : StemClass → Nat → Std.Format

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def Fragments.Hungarian.VowelHarmony.StemClass.isBack : StemClass → Bool

Does this stem class select back suffixes?

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• Fragments.Hungarian.VowelHarmony.StemClass.IA_b.isBack = true
• Fragments.Hungarian.VowelHarmony.StemClass.IB_b.isBack = true
• Fragments.Hungarian.VowelHarmony.StemClass.IIA_b.isBack = true
• Fragments.Hungarian.VowelHarmony.StemClass.IIB_b.isBack = true
• x✝.isBack = false

inductive Fragments.Hungarian.VowelHarmony.SuffixType : Type

Suffix alternation types. @cite{siptar-torkenczy-2000} §3.2.1, (28).

Hungarian suffixes alternate in pairs (two-way) or triplets (three-way). Two-way alternation depends on [±back] alone. Three-way alternation depends on [±back] AND [±round] of the last stem vowel.

• twoWay_a_e : SuffixType
• twoWay_á_é : SuffixType
• twoWay_u_ü : SuffixType
• twoWay_ú_ű : SuffixType
• twoWay_ó_ő : SuffixType
• threeWay_o_ö_e : SuffixType

instance Fragments.Hungarian.VowelHarmony.instDecidableEqSuffixType : DecidableEq SuffixType

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• Fragments.Hungarian.VowelHarmony.instDecidableEqSuffixType x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

instance Fragments.Hungarian.VowelHarmony.instBEqSuffixType : BEq SuffixType

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• Fragments.Hungarian.VowelHarmony.instBEqSuffixType = { beq := Fragments.Hungarian.VowelHarmony.instBEqSuffixType.beq }

def Fragments.Hungarian.VowelHarmony.instBEqSuffixType.beq : SuffixType → SuffixType → Bool

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• Fragments.Hungarian.VowelHarmony.instBEqSuffixType.beq x✝ y✝ = (x✝.ctorIdx == y✝.ctorIdx)

instance Fragments.Hungarian.VowelHarmony.instReprSuffixType : Repr SuffixType

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• Fragments.Hungarian.VowelHarmony.instReprSuffixType = { reprPrec := Fragments.Hungarian.VowelHarmony.instReprSuffixType.repr }

def Fragments.Hungarian.VowelHarmony.instReprSuffixType.repr : SuffixType → Nat → Std.Format

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def Fragments.Hungarian.VowelHarmony.resolveA (back : Bool) : Theories.Phonology.Segment

Resolve a two-way a/e suffix given backness.

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• Fragments.Hungarian.VowelHarmony.resolveA back = if back = true then Fragments.Hungarian.VowelHarmony.a_vowel else Fragments.Hungarian.VowelHarmony.e_vowel

def Fragments.Hungarian.VowelHarmony.resolveThreeWay (back round : Bool) : Theories.Phonology.Segment

Resolve a three-way o/ö/e suffix given backness and rounding. @cite{siptar-torkenczy-2000} §3.2.4, (39):

• Back → /o/ (always, regardless of stem rounding)
• Front rounded → /ö/
• Front unrounded (including neutral) → /e/

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theorem Fragments.Hungarian.VowelHarmony.a_is_back : a_vowel.hasValue Theories.Phonology.Feature.back true = true

theorem Fragments.Hungarian.VowelHarmony.o_is_back : o_vowel.hasValue Theories.Phonology.Feature.back true = true

theorem Fragments.Hungarian.VowelHarmony.u_is_back : u_vowel.hasValue Theories.Phonology.Feature.back true = true

theorem Fragments.Hungarian.VowelHarmony.i_is_front : i_vowel.hasValue Theories.Phonology.Feature.back false = true

theorem Fragments.Hungarian.VowelHarmony.e_is_front : e_vowel.hasValue Theories.Phonology.Feature.back false = true

theorem Fragments.Hungarian.VowelHarmony.ö_is_front : ö_vowel.hasValue Theories.Phonology.Feature.back false = true

theorem Fragments.Hungarian.VowelHarmony.ü_is_front : ü_vowel.hasValue Theories.Phonology.Feature.back false = true

theorem Fragments.Hungarian.VowelHarmony.ö_is_round : ö_vowel.hasValue Theories.Phonology.Feature.round true = true

theorem Fragments.Hungarian.VowelHarmony.ü_is_round : ü_vowel.hasValue Theories.Phonology.Feature.round true = true

theorem Fragments.Hungarian.VowelHarmony.i_is_unround : i_vowel.hasValue Theories.Phonology.Feature.round false = true

theorem Fragments.Hungarian.VowelHarmony.e_is_unround : e_vowel.hasValue Theories.Phonology.Feature.round false = true

theorem Fragments.Hungarian.VowelHarmony.a_is_unround : a_vowel.hasValue Theories.Phonology.Feature.round false = true

theorem Fragments.Hungarian.VowelHarmony.i_is_neutral : isNeutral i_vowel = true

theorem Fragments.Hungarian.VowelHarmony.e_is_neutral : isNeutral e_vowel = true

theorem Fragments.Hungarian.VowelHarmony.ö_not_neutral : isNeutral ö_vowel = false

theorem Fragments.Hungarian.VowelHarmony.ü_not_neutral : isNeutral ü_vowel = false

theorem Fragments.Hungarian.VowelHarmony.a_not_neutral : isNeutral a_vowel = false

theorem Fragments.Hungarian.VowelHarmony.o_not_neutral : isNeutral o_vowel = false

theorem Fragments.Hungarian.VowelHarmony.u_not_neutral : isNeutral u_vowel = false

theorem Fragments.Hungarian.VowelHarmony.ö_is_frontHarmonic : isFrontHarmonic ö_vowel = true

theorem Fragments.Hungarian.VowelHarmony.ü_is_frontHarmonic : isFrontHarmonic ü_vowel = true

theorem Fragments.Hungarian.VowelHarmony.a_is_backHarmonic : isBackHarmonic a_vowel = true

theorem Fragments.Hungarian.VowelHarmony.o_is_backHarmonic : isBackHarmonic o_vowel = true

Class IA: Simple harmonic stems

These are the easy cases: all harmonic vowels agree in backness.
`triggerValue` finds the last harmonic vowel and returns its [back]
value. This always gives the correct suffix harmony. 

theorem Fragments.Hungarian.VowelHarmony.tűz_front : Theories.Phonology.Harmony.triggerValue hungarianPalatalHarmony [ü_vowel] = some false

tűz 'fire' (IA-f): last vowel /ü/ → front harmony. Suffixes: tűz-nek (dat), tűz-től (abl).

theorem Fragments.Hungarian.VowelHarmony.ház_back : Theories.Phonology.Harmony.triggerValue hungarianPalatalHarmony [a_vowel] = some true

ház 'house' (IA-b): last vowel /a/ → back harmony. Suffixes: ház-nak (dat), ház-tól (abl).

theorem Fragments.Hungarian.VowelHarmony.koszorú_back : Theories.Phonology.Harmony.triggerValue hungarianPalatalHarmony [o_vowel, o_vowel, u_vowel] = some true

koszorú 'wreath' (IA-b): vowels /o, o, u/ all back → back harmony. Suffixes: koszorú-nak, koszorú-tól.

theorem Fragments.Hungarian.VowelHarmony.tükör_front : Theories.Phonology.Harmony.triggerValue hungarianPalatalHarmony [ü_vowel, ö_vowel] = some false

tükör 'mirror' (IA-f): vowels /ü, ö/ both front harmonic → front. Suffixes: tükör-nek, tükör-től.

theorem Fragments.Hungarian.VowelHarmony.hernyó_back : Theories.Phonology.Harmony.triggerValue hungarianPalatalHarmony [e_vowel, o_vowel] = some true

hernyó 'caterpillar' (IA-b): vowels /e, o/ — /e/ is neutral, /o/ is the last vowel and is back harmonic → back harmony. Suffixes: hernyó-nak, hernyó-tól. @cite{siptar-torkenczy-2000} p. 67.

Class IB: Complex harmonic (disharmonic) stems

Conflicting harmonic vowels within the stem. The LAST harmonic
vowel governs suffix selection. `triggerValue` gets this right
because it extracts the last trigger. 

theorem Fragments.Hungarian.VowelHarmony.sofőr_front : Theories.Phonology.Harmony.triggerValue hungarianPalatalHarmony [o_vowel, ö_vowel] = some false

sofőr 'driver' (IB-f): vowels /o, ő/ — last harmonic is /ő/ [−back] → front suffixes. Suffixes: sofőr-nek, sofőr-től.

theorem Fragments.Hungarian.VowelHarmony.nüansz_back : Theories.Phonology.Harmony.triggerValue hungarianPalatalHarmony [ü_vowel, a_vowel] = some true

nüansz 'nuance' (IB-b): vowels /ü, a/ — last harmonic is /a/ [+back] → back suffixes. Suffixes: nüansz-nak, nüansz-tól. (Note: /ü/ is front harmonic but /a/ is later → back wins.)

Class IIB-b: Complex neutral, back (transparent)

These are the textbook transparency cases. A back harmonic vowel
precedes one or more neutral vowels, and harmony passes THROUGH
the neutral vowels to reach the suffix. `triggerValue` handles this
correctly because neutral vowels are not triggers, so the search
skips them and finds the earlier back vowel. 

theorem Fragments.Hungarian.VowelHarmony.papír_back : Theories.Phonology.Harmony.triggerValue hungarianPalatalHarmony [a_vowel, i_vowel] = some true

papír 'paper' (IIB-b): vowels /a, i/ — /i/ is neutral (not a trigger), so triggerValue finds /a/ → back harmony. Suffixes: papír-nak, papír-tól.

theorem Fragments.Hungarian.VowelHarmony.kávé_back : Theories.Phonology.Harmony.triggerValue hungarianPalatalHarmony [a_vowel, e_vowel] = some true

kávé 'coffee': vowels /a, e/ — /e/ is neutral, triggerValue finds /a/ → back harmony. Suffixes: kávé-nak, kávé-tól. (The book lists kávé under IA-b (p. 67) since the neutrality of final /é/ is disputed — some speakers treat it as harmonic. Our system gives the correct result either way.)

Class IIB-f: Complex neutral, front (transparent from front)

Front harmonic vowel + neutral → front harmony via last trigger. 

theorem Fragments.Hungarian.VowelHarmony.üveg_front : Theories.Phonology.Harmony.triggerValue hungarianPalatalHarmony [ü_vowel, e_vowel] = some false

üveg 'glass' (IIB-f): vowels /ü, e/ — /e/ is neutral, triggerValue finds /ü/ → front harmony. Suffixes: üveg-nek, üveg-től.

Class IIA: Simple neutral stems — THE CRITICAL FAILURE

All-neutral stems contain only /i, í, e, é/. Since none of these are
triggers in `hungarianPalatalHarmony`, `triggerValue` returns `none`.
But Hungarian speakers DO assign a definite backness to these stems:

- IIA-f (*víz* 'water', *szegény* 'poor'): front suffixes ✓
- IIA-b (*híd* 'bridge', *cél* 'aim', *derék* 'waist'): back suffixes ✗

The antiharmonic stems (IIA-b) are **lexically specified** — their
backness cannot be predicted from the phonological form. This is the
fundamental limitation of any purely phonological harmony system. 

theorem Fragments.Hungarian.VowelHarmony.víz_no_trigger : Theories.Phonology.Harmony.triggerValue hungarianPalatalHarmony [i_vowel] = none

víz 'water' (IIA-f): only /i/ → triggerValue returns none. The correct answer is front (víz-nek, víz-től), but the system cannot determine this.

theorem Fragments.Hungarian.VowelHarmony.híd_no_trigger : Theories.Phonology.Harmony.triggerValue hungarianPalatalHarmony [i_vowel] = none

híd 'bridge' (IIA-b, antiharmonic): only /i/ → triggerValue returns none. The correct answer is BACK (híd-nak, híd-tól), but the system returns the same none as for víz.

This is the canonical limitation: antiharmonic stems require lexical specification beyond phonological features.

theorem Fragments.Hungarian.VowelHarmony.víz_híd_indistinguishable : Theories.Phonology.Harmony.triggerValue hungarianPalatalHarmony [i_vowel] = Theories.Phonology.Harmony.triggerValue hungarianPalatalHarmony [i_vowel]

The critical failure: víz and híd are phonologically identical (both have only /i/) but require different suffix harmony. triggerValue cannot distinguish them.

theorem Fragments.Hungarian.VowelHarmony.cél_no_trigger : Theories.Phonology.Harmony.triggerValue hungarianPalatalHarmony [e_vowel] = none

cél 'aim' (IIA-b): only /e/ → triggerValue returns none. Correct answer: back (cél-nak). Another antiharmonic stem.

Vacillation — OPTIONALITY FAILURE

Some stems allow BOTH front and back suffixes. This is common with
mixed stems where back-harmonic vowels are followed by neutral vowels,
especially /e/ (@cite{siptar-torkenczy-2000} §3.2.3.1, Table 11).

`spreadSuffix` is deterministic — it returns a single list. A
vacillating stem like *hotel* would need to return a SET of possible
outputs.

The height effect: transparency correlates with vowel height.
/i, í/ — always transparent → neutral.
/é/ — mostly transparent → mostly neutral, sometimes vacillating.
/e/ — variably transparent → neutral, vacillating, or disharmonic.

The binary `isTransparent` predicate cannot capture this gradient. 

theorem Fragments.Hungarian.VowelHarmony.hotel_predicted_back : Theories.Phonology.Harmony.triggerValue hungarianPalatalHarmony [o_vowel, e_vowel] = some true

hotel has vowels /o, e/. Our system finds /o/ as last trigger → back. In reality speakers vacillate: hotel-nak ~ hotel-nek. The system predicts only back, which is one of the two options.

Demonstrate suffix harmonization using harmonizeOne with an underspecified archiphonemic suffix vowel.

def Fragments.Hungarian.VowelHarmony.archiphoneU : Theories.Phonology.Segment

A generic high vowel archiphoneme (placeholder for suffix /U/).

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theorem Fragments.Hungarian.VowelHarmony.ház_suffix_becomes_back : (Theories.Phonology.Harmony.harmonizeOne hungarianPalatalHarmony true archiphoneU).hasValue Theories.Phonology.Feature.back true = true

ház: harmonize an underspecified suffix vowel to [+back].

theorem Fragments.Hungarian.VowelHarmony.tűz_suffix_becomes_front : (Theories.Phonology.Harmony.harmonizeOne hungarianPalatalHarmony false archiphoneU).hasValue Theories.Phonology.Feature.back false = true

tűz → suffix vowel becomes [−back].

Three-way suffix resolution verified against (39): tűz-höz, szemölcs-höz, sofőr-höz (front rounded → höz) víz-hez, kötény-hez, kódex-hez (front unrounded → hez) ház-hoz, hernyó-hoz, nüansz-hoz (back → hoz)

theorem Fragments.Hungarian.VowelHarmony.tűz_round : Theories.Phonology.Harmony.triggerValue hungarianLabialHarmony [ü_vowel] = some true

theorem Fragments.Hungarian.VowelHarmony.víz_unround : Theories.Phonology.Harmony.triggerValue hungarianLabialHarmony [i_vowel] = some false

theorem Fragments.Hungarian.VowelHarmony.ház_unround : Theories.Phonology.Harmony.triggerValue hungarianLabialHarmony [a_vowel] = some false

theorem Fragments.Hungarian.VowelHarmony.tűz_hoz_suffix : resolveThreeWay false true = ö_vowel

tűz-höz: front (back=false) + rounded (round=true) → /ö/.

theorem Fragments.Hungarian.VowelHarmony.víz_hez_suffix : resolveThreeWay false false = e_vowel

víz-hez: front (back=false) + unrounded (round=false) → /e/.

theorem Fragments.Hungarian.VowelHarmony.ház_hoz_suffix : resolveThreeWay true false = o_vowel

ház-hoz: back (back=true) → /o/ regardless of rounding.

theorem Fragments.Hungarian.VowelHarmony.back_rounding_irrelevant : resolveThreeWay true true = resolveThreeWay true false

Back stems always get /o/ in three-way suffixes, regardless of whether the stem vowel is rounded or not. @cite{siptar-torkenczy-2000} §3.2.4.

theorem Fragments.Hungarian.VowelHarmony.kötény_front : Theories.Phonology.Harmony.triggerValue hungarianPalatalHarmony [ö_vowel, e_vowel] = some false

kötény-hez: vowels /ö, e/. Palatal: last trigger /ö/ → front. Labial: last vowel /e/ → unrounded. Three-way → /e/ (hez).

theorem Fragments.Hungarian.VowelHarmony.kötény_unround : Theories.Phonology.Harmony.triggerValue hungarianLabialHarmony [ö_vowel, e_vowel] = some false

theorem Fragments.Hungarian.VowelHarmony.szemölcs_round : Theories.Phonology.Harmony.triggerValue hungarianLabialHarmony [e_vowel, ö_vowel] = some true

szemölcs-höz: vowels /e, ö/. Palatal: last trigger /ö/ → front. Labial: last vowel /ö/ → rounded. Three-way → /ö/ (höz).

The two harmony dimensions compose to derive three-way suffixes: 1. triggerValue hungarianPalatalHarmony stem → back : Option Bool 2. triggerValue hungarianLabialHarmony stem → round : Option Bool 3. resolveThreeWay back round → suffix vowel

This decomposition mirrors the book's observation that rounding harmony
is a "minor subsidiary pattern" layered on top of backness harmony
(@cite{siptar-torkenczy-2000} §3.2.4). 

theorem Fragments.Hungarian.VowelHarmony.ház_twoDim : have back := Theories.Phonology.Harmony.triggerValue hungarianPalatalHarmony [a_vowel]; have round := Theories.Phonology.Harmony.triggerValue hungarianLabialHarmony [a_vowel]; back = some true ∧ round = some false

Compose both harmony dimensions for a back stem.

theorem Fragments.Hungarian.VowelHarmony.tűz_twoDim : have back := Theories.Phonology.Harmony.triggerValue hungarianPalatalHarmony [ü_vowel]; have round := Theories.Phonology.Harmony.triggerValue hungarianLabialHarmony [ü_vowel]; back = some false ∧ round = some true

Compose both harmony dimensions for a front rounded stem.

theorem Fragments.Hungarian.VowelHarmony.víz_twoDim : have back := Theories.Phonology.Harmony.triggerValue hungarianPalatalHarmony [i_vowel]; have round := Theories.Phonology.Harmony.triggerValue hungarianLabialHarmony [i_vowel]; back = none ∧ round = some false

Compose both harmony dimensions for a front unrounded (neutral) stem. Palatal harmony returns none (no trigger), but labial harmony returns some false — rounding works even when backness doesn't.

Hungarian and Finnish palatal harmony share the same HarmonySystem structure: both spread [back] rightward with transparent neutral vowels.

Key structural differences:
- Neutral class: Finnish /i, e/ (2) vs Hungarian /i, í, e, é/ (4)
- Hungarian has antiharmonic stems (IIA-b); Finnish does not
- Hungarian has a second harmony dimension (rounding, §3.2.4)

Hungarian and Turkish share the two-dimensional structure (palatal +
labial), but differ in targets: Turkish labial harmony targets only
[+high] suffix vowels, while Hungarian labial harmony (rounding
for three-way suffixes) targets all relevant suffix positions. 

theorem Fragments.Hungarian.VowelHarmony.hungarian_finnish_same_feature : hungarianPalatalHarmony.feature = Theories.Phonology.Feature.back

theorem Fragments.Hungarian.VowelHarmony.hungarian_finnish_same_direction : hungarianPalatalHarmony.direction = Theories.Phonology.Harmony.HarmonyDir.rightward

theorem Fragments.Hungarian.VowelHarmony.hungarian_turkish_both_twoDim : hungarianPalatalHarmony.feature = Theories.Phonology.Feature.back ∧ hungarianLabialHarmony.feature = Theories.Phonology.Feature.round

Both Hungarian and Turkish have two-dimensional harmony: palatal ([back]) + labial ([round]).

The height-graded transparency problem (§3.2.3) can be partially addressed using the isBlocker field. By treating /e/ as a blocker rather than transparent, we capture the intuition that /e/ sometimes blocks back harmony from reaching the suffix.

This alternative system makes different predictions for mixed stems:
- *hotel* /o, e/: `/e/` blocks → `triggerValue` returns `none` (no trigger
  in the suffix-adjacent domain). This predicts front harmony by default,
  matching the front option of the vacillation.
- *papír* /a, i/: `/i/` is still transparent → `triggerValue` returns
  `some true` (back harmony), unchanged.

This does NOT model vacillation itself (which requires optionality), but
it shows that the `isBlocker` field can shift predictions in the right
direction for stems where /e/ is opaque. 

def Fragments.Hungarian.VowelHarmony.palatalHarmony_eBlocks : Theories.Phonology.Harmony.HarmonySystem

Alternative palatal harmony with /e/ as a blocker (opaque). Only /i, í/ are transparent; /e, é/ block spreading. @cite{siptar-torkenczy-2000} §3.2.3: /e/ is the least transparent neutral vowel (most likely to block).

Equations

• One or more equations did not get rendered due to their size.

theorem Fragments.Hungarian.VowelHarmony.hotel_eBlocks_no_trigger : Theories.Phonology.Harmony.triggerValue palatalHarmony_eBlocks [o_vowel, e_vowel] = none

With /e/ as blocker: hotel /o, e/ → /e/ blocks, domain = [], no trigger. Contrast: hotel_predicted_back in the transparent model gives some true.

theorem Fragments.Hungarian.VowelHarmony.papír_eBlocks_still_back : Theories.Phonology.Harmony.triggerValue palatalHarmony_eBlocks [a_vowel, i_vowel] = some true

With /e/ as blocker: papír /a, i/ → /i/ is still transparent (high), so triggerValue finds /a/ → back harmony. Unchanged from transparent model.

theorem Fragments.Hungarian.VowelHarmony.hotel_domain_empty : (Theories.Phonology.Harmony.harmonyDomain palatalHarmony_eBlocks [o_vowel, e_vowel]).length = 0

The blocker model shrinks the harmony domain for hotel: domain after the last blocker (/e/) is empty.

theorem Fragments.Hungarian.VowelHarmony.hotel_domain_full : (Theories.Phonology.Harmony.harmonyDomain hungarianPalatalHarmony [o_vowel, e_vowel]).length = 2

The transparent model gives the full domain (no blockers).

theorem Fragments.Hungarian.VowelHarmony.spread_preserves_length : (Theories.Phonology.Harmony.spreadSuffix palatalHarmony_eBlocks true [e_vowel, archiphoneU]).length = 2

spreadSuffix preserves suffix length even with blockers.