Lu, Chuang & Baayen (2026): Realization of tones in spontaneous Taiwan Mandarin

@cite{lu-chuang-baayen-2026} @cite{chuang-bell-tseng-baayen-2026} @cite{baayen-2019} @cite{heitmeier-chuang-baayen-2026}

Lu, Y., Chuang, Y.-Y., & Baayen, R. H. (2026). The realization of tones in spontaneous spoken Taiwan Mandarin: a corpus-based survey and theory-driven computational modeling. Corpus Linguistics and Linguistic Theory. DOI 10.1515/cllt-2025-0028.

Empirical claims

This study generalises @cite{chuang-bell-tseng-baayen-2026} from a single tone pattern (T2-T4 rise-fall) to all 20 disyllabic tone patterns of Taiwan Mandarin, in a 4,283-token / 313-word-type subset of the Corpus of Spontaneous Taiwan Mandarin (Fon 2004), restricted to the four most frequent tonal contexts (4.4, 3.4, 4.1, 4.0).

1. Word effect generalises across all 20 tone patterns. Withholding word from the GAMM raises AIC by 7,269.27 to 12,345.54 across the four tonal contexts (paper §3.4.1, Fig. 2), far exceeding any other predictor. When word is included, withholding tone_pattern raises AIC by only single- or low-double-digit units, reflecting near-total subsumption of tone_pattern by word.
2. Sense > word. Replacing the word factor smooth with a sense_type factor smooth further reduces AIC (paper §3.4.2, Table 4), replicating @cite{chuang-bell-tseng-baayen-2026} on the broader 20-pattern dataset.
3. Form–meaning isomorphism for all 20 tone patterns. A linear meaning→form mapping from CE centroids to fixed-length pitch vectors recovers the GAMM-derived gold-standard contours of all 20 tone patterns with high cosine similarity (paper §4.4, Fig. 9).
4. T3 tone sandhi as DLM kernel-neutralization. The classically stipulated "T3 → T2 / __ T3" sandhi rule does not appear in the model: T3-T3 and T2-T3 word-token CE centroids in Taiwan Mandarin differ only by elements that lie in the kernel of the trained production map, so their surface pitch contours are identical ("complete neutralization for Taiwan Mandarin", paper §5; consistent with previous corpus measurement showing T3-T3 ≡ T2-T3 surface in spontaneous Taiwan Mandarin).

The substantive theoretical claim is that a word-and-paradigm DLM suffices to predict every disyllabic tone pattern's surface realisation without any explicit phonological tone-sandhi rule. The architecture predicts neutralization where the meaning-space evidence supports it (Taiwan Mandarin, complete) and underdetermines it where evidence does not (Standard Chinese, where T3-T2 / T3-T3 distinction "is hardly visible to the eye" but reportedly incomplete; @cite{lu-chuang-baayen-2026} §5).

Substrate

The LinearDiscriminativeLexicon substrate (Theories/Processing/Lexical/Discriminative/Defs.lean) is reused verbatim — this is the second paper-anchored study consuming it (@cite{chuang-bell-tseng-baayen-2026} is the first). This file specialises the substrate to:

• 100-dimensional pitch vectors (vs. Chuang's 50-dim) — paper §4.1;
• the same 768-dimensional CKIP GPT-2 embedding space (CKIPGPT2HiddenDim, reused from ChuangEtAl2026);
• a 20-element TonePattern20 enum covering all disyllabic combinations of T1/T2/T3/T4/T0.

The fact that the same substrate parameterises cleanly to both 50-dim and 100-dim form vectors validates the dimension-polymorphic carrier typing of LinearDiscriminativeLexicon.

Cross-framework note: vs. autosegmental T3 sandhi

The classical autosegmental account of Mandarin T3 sandhi (@cite{chen-2000}, @cite{duanmu-2007}, and the surrounding literature) posits a structural rule: an underlying /T3 T3/ sequence undergoes spreading or delinking, surfacing as [T2 T3]. Linglib's existing discrete-tone substrate (Theories/Phonology/Tone/Constraints.lean, Theories/Phonology/Autosegmental/Floating.lean) is set up to express such rule-based accounts, although no Mandarin-specific T3-sandhi fragment has been formalised in linglib to date.

The DLM-side account differs both architecturally and predictively:

• Architecturally, no rule is invoked. The neutralisation falls out of the kernel of the trained meaning→form map. t3_sandhi_via_kernel in §4 below states this as a direct application of the structural lemma Phenomena.Tone.Studies.ChuangEtAl2026.dlm_neutralizes_meanings_in_kernel.
• Predictively, the DLM accommodates dialect variation in sandhi completeness without re-stipulating the rule: Taiwan Mandarin's complete neutralisation = the kernel-difference holds; Standard Chinese's incomplete neutralisation = the difference is small but non-zero. The autosegmental account requires either a derivational optionality or a different rule per dialect.

The two formalisations are not in head-on conflict; they generate different predictions on the same data. A formal cross-framework discrimination experiment would require the discrete-tone substrate to commit to a Mandarin T3-sandhi fragment, which it currently does not. Until then, this comparison lives in prose.

Sections

• §1 Substrate import and paper-specific dimensional instantiation
• §2 The 20 disyllabic tone patterns of Mandarin
• §3 Tone sandhi neutralization as DLM kernel application
• §4 DLM dialect flexibility: same architecture, different kernels
• §5 Empirical content (AIC, accuracies, isomorphism metrics)

@[reducible, inline]

abbrev Phenomena.Tone.Studies.LuChuangBaayen2026.LuPitchSampleCount : ℕ

The paper uses 100 evenly-spaced f0 samples per token (paper §4.1). Distinct from @cite{chuang-bell-tseng-baayen-2026}'s 50; the FormVec substrate is dimension-polymorphic so both fit.

Equations

• Phenomena.Tone.Studies.LuChuangBaayen2026.LuPitchSampleCount = 100

@[reducible, inline]

abbrev Phenomena.Tone.Studies.LuChuangBaayen2026.LuPitchContour : Type

Pitch contour at the paper's chosen sampling resolution.

Equations

• Phenomena.Tone.Studies.LuChuangBaayen2026.LuPitchContour = Theories.Processing.Lexical.Discriminative.FormVec Phenomena.Tone.Studies.LuChuangBaayen2026.LuPitchSampleCount

@[reducible, inline]

abbrev Phenomena.Tone.Studies.LuChuangBaayen2026.LuTaiwanMandarinDLM : Type

The paper's specific DLM instantiation: 100-dim pitch vectors, 768-dim contextualised embeddings (CKIPGPT2HiddenDim reused from ChuangEtAl2026). The substrate type LinearDiscriminativeLexicon is in Theories/Processing/Lexical/Discriminative/Defs.lean.

Equations

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

inductive Phenomena.Tone.Studies.LuChuangBaayen2026.TonePattern20 : Type

The 20 disyllabic tone patterns of Mandarin (paper Table 1). Enumerated as the row-major Cartesian-product list of {T1, T2, T3, T4} (full lexical tones) on the first syllable with {T1, T2, T3, T4, T0} (full + neutral) on the second.

The neutral tone (T0) appears only on the second syllable in this paradigm; mono-T0 disyllables are not analysed. The structural fact 20 = 4 × 5 is by enumeration here, not by a typeclass.

• T1_T1 : TonePattern20
• T1_T2 : TonePattern20
• T1_T3 : TonePattern20
• T1_T4 : TonePattern20
• T1_T0 : TonePattern20
• T2_T1 : TonePattern20
• T2_T2 : TonePattern20
• T2_T3 : TonePattern20
• T2_T4 : TonePattern20
• T2_T0 : TonePattern20
• T3_T1 : TonePattern20
• T3_T2 : TonePattern20
• T3_T3 : TonePattern20
• T3_T4 : TonePattern20
• T3_T0 : TonePattern20
• T4_T1 : TonePattern20
• T4_T2 : TonePattern20
• T4_T3 : TonePattern20
• T4_T4 : TonePattern20
• T4_T0 : TonePattern20

@[implicit_reducible]

instance Phenomena.Tone.Studies.LuChuangBaayen2026.instDecidableEqTonePattern20 : DecidableEq TonePattern20

Equations

• Phenomena.Tone.Studies.LuChuangBaayen2026.instDecidableEqTonePattern20 x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

def Phenomena.Tone.Studies.LuChuangBaayen2026.instReprTonePattern20.repr : TonePattern20 → ℕ → Std.Format

Equations

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

@[implicit_reducible]

instance Phenomena.Tone.Studies.LuChuangBaayen2026.instReprTonePattern20 : Repr TonePattern20

Equations

• Phenomena.Tone.Studies.LuChuangBaayen2026.instReprTonePattern20 = { reprPrec := Phenomena.Tone.Studies.LuChuangBaayen2026.instReprTonePattern20.repr }

theorem Phenomena.Tone.Studies.LuChuangBaayen2026.t3_sandhi_via_kernel (D : LuTaiwanMandarinDLM) (centroidOf : TonePattern20 → ChuangEtAl2026.ContextualEmbedding) (h_kernel : D.production (centroidOf TonePattern20.T3_T3 - centroidOf TonePattern20.T2_T3) = 0) : D.production (centroidOf TonePattern20.T3_T3) = D.production (centroidOf TonePattern20.T2_T3)

T3 tone sandhi neutralization via DLM kernel.

The classical phonological rule "T3 → T2 / __ T3" stipulates that underlying /T3 T3/ surfaces as [T2 T3]. Within the DLM, this surface coincidence is derived, not stipulated: if the CE centroids of T3-T3 and T2-T3 word tokens differ by an element of the production map's kernel, then their predicted pitch contours coincide.

The paper (§5) reports that for Taiwan Mandarin this kernel condition holds empirically — complete neutralization. For Standard Chinese the kernel condition holds only approximately, consistent with the literature's report that T3 sandhi neutralization is incomplete in Standard Chinese.

The specific patterns T3_T3 and T2_T3 are load-bearing in the type signature: the theorem witnesses the paper's specific empirical claim, not a generic two-pattern claim. The body is a direct application of the structural sister lemma Phenomena.Tone.Studies.ChuangEtAl2026.dlm_neutralizes_meanings_in_kernel.

theorem Phenomena.Tone.Studies.LuChuangBaayen2026.t3_sandhi_quantitative (D : LuTaiwanMandarinDLM) (centroidOf : TonePattern20 → ChuangEtAl2026.ContextualEmbedding) {ε : ℝ} (h : ‖centroidOf TonePattern20.T3_T3 - centroidOf TonePattern20.T2_T3‖ ≤ ε) : ‖D.production (centroidOf TonePattern20.T3_T3) - D.production (centroidOf TonePattern20.T2_T3)‖ ≤ ‖LinearMap.toContinuousLinearMap D.production‖ * ε

Quantitative refinement of t3_sandhi_via_kernel. The exact- kernel hypothesis is the limiting case; in real data the centroid difference is small but non-zero. Lipschitz continuity bounds the resulting contour difference by ‖production‖ * ε whenever the centroid difference is within ε.

The empirical content of @cite{lu-chuang-baayen-2026} §5 is that for Taiwan Mandarin the trained ‖production‖ and the centroid distance are both small enough that the contour bound is consistent with the reported "essentially identical" surface realizations. For Standard Chinese the bound permits visible contour difference, matching the reported incomplete neutralization.

Lipschitz application of dlm_neighbor_centroids_imply_neighbor_contours.

theorem Phenomena.Tone.Studies.LuChuangBaayen2026.dlm_dialect_flexibility : ∃ (D₁ : LuTaiwanMandarinDLM) (D₂ : LuTaiwanMandarinDLM) (e₁ : ChuangEtAl2026.ContextualEmbedding) (e₂ : ChuangEtAl2026.ContextualEmbedding), D₁.production e₁ = D₁.production e₂ ∧ D₂.production e₁ ≠ D₂.production e₂

DLMs accommodate dialect variation in neutralization without rule modification. The same LinearDiscriminativeLexicon architecture supports both complete neutralization (Taiwan Mandarin: T3-T3 surface = T2-T3 surface) and incomplete neutralization (Standard Chinese: T3-T3 differs subtly from T2-T3) by varying the production map's kernel, not by introducing a separate sandhi rule.

Witness: the all-zero DLM has every MeaningVec mapping to the zero contour — every meaning pair is "neutralized" (degenerate Taiwan-Mandarin extreme); a DLM with a non-degenerate production map distinguishes the same pair (Standard-Chinese-like extreme). The architectural point survives the witness's simplicity: a SINGLE LinearDiscriminativeLexicon type accommodates both regimes; only the trained weights differ.

Cross-framework significance: a rule-based account requires dialect-specific rule modification (or stochastic optionality) to handle this variation. The DLM accommodates it without any framework-level move.

Methods + accuracies as paper-supplied empirical facts

Per CLAUDE.md (Processing-scope guidance), specific numerical fits (R² values, AIC reductions, cross-validation accuracies) are out of scope as Lean theorems. They are recorded here as documented empirical findings with paper-section pointers.

Word emerges as crucial across all 20 tone patterns (paper §3.4.1, Fig. 2, Table 4). Withholding the word factor smooth from the GAMM raises AIC by 7,269.27 to 12,345.54 across the four tonal contexts, substantially exceeding the AIC change for any other predictor including tone_pattern. When word is included, withholding tone_pattern raises AIC by only single- or low-double-digit units per context (paper §3.4.1) — word essentially subsumes tone_pattern.

Sense > word (paper §3.4.2, Table 4). Adding sense_type on top of tone_pattern produces a substantial AIC reduction (−16,077.20 at 4.4, −10,541.88 at 3.4, −9,171.19 at 4.1, −7,895.57 at 4.0; values from Table 4), exceeding the contribution of word alone. Replicates the headline sense-effect of @cite{chuang-bell-tseng-baayen-2026} on the broader 20-pattern dataset.

DLM accuracy across three preprocessing methods (paper §4.4):

• Method I (per-token GAMMs): training 2.8% / test 1.4%
• Method II (per-word-type GAMMs): training 23.5% / test 15.1% (best)
• Method III (per-tonal-context GAMMs): training 12.0% / test 6.9%
• Permutation baseline ≈ 0.4%; majority baseline ≈ 1.3%.

Form–meaning isomorphism across all 20 tone patterns (paper §4.4, Fig. 9, Fig. 10). Cosine similarity / Pearson correlation / Euclidean distance between GAMM-derived gold-standard contours and DLM-predicted contours, respectively for methods I, II, III:

• Method I: cosine 0.73 / corr 0.85 / dist 1.24
• Method II: cosine 0.93 / corr 0.98 / dist 1.50
• Method III: cosine 0.82 / corr 0.96 / dist 1.09

Method II wins on cosine and correlation; loses (slightly) on Euclidean distance. Boxplot of the three measures across methods: paper Fig. 10.

Comparison to lab speech (paper §5, Fig. 11). The DLM-derived contours from spontaneous Taiwan Mandarin closely match @cite{xu-1997}'s lab-speech contours for most tone patterns (T4-T4, T2-T3, T2-T4 nearly identical); some patterns (T1-T1, T1-T2) differ visibly, attributed to dialect (Beijing vs. Taiwan Mandarin) and register (lab vs. spontaneous) effects.

Implications recorded in the paper's discussion

• Word-and-paradigm sufficiency (paper §5): the DLM does not need abstract phonological units (syllable, segment, tone) to predict pitch realisation — meaning vectors → pitch vectors directly. This aligns with @cite{heitmeier-chuang-baayen-2026}'s W&P framing of the DLM and challenges item-and-arrangement architectures that presuppose stored sub-word inventories.
• Anti-arbitrariness of the sign (paper §5): the linear meaning→form mapping generalises to held-out data, falsifying the axiom that the form–meaning relation is fundamentally arbitrary.
• Anti-dual-articulation (paper §5, also @cite{chuang-bell-tseng-baayen-2026}): the same DLM architecture exhibits isomorphism between phonetic form-space and contextualised meaning-space, undermining the assumption that form and meaning are orthogonal grammar modules.
• Tone sandhi as kernel-emergence: see §3 (t3_sandhi_via_kernel).
• Dialect variation as kernel variation: see §4 (dlm_dialect_flexibility).
• Beyond Mandarin: paper §5 cites preliminary results (@cite{chuang-bell-tseng-baayen-2026} discussion) that English two-syllable words also exhibit word-specific pitch components, suggesting form-meaning isomorphism is not a tone-language quirk.