Measuring the Information Density of Interlanguage: An Entropy Analysis

Measuring the Information Density of Interlanguage: An Entropy Analysis | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Article Measuring the Information Density of Interlanguage: An Entropy Analysis Mohamed Mekheimer This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-9295874/v1 This work is licensed under a CC BY 4.0 License Status: Under Review Version 1 posted 13 You are reading this latest preprint version Abstract Interlanguage development is often assessed through structural counts that only partially capture how learner language is organized probabilistically. This study proposes a multi-level framework for measuring interlanguage information density using entropy-based metrics. A corpus of 150 L2 English argumentative essays from B1, B2, and C1 learners was compared with a genre-matched native-speaker corpus of 50 essays. Four indicators were examined: lexical entropy (Hₗₑₓ), grammatical divergence from a native reference distribution via POS trigrams (KL₍gram₎), compression ratio (CR), and positional concentration index (PCI). To model native variability more defensibly, KL₍gram₎ for each L1 text was calculated against a leave-one-out L1 reference distribution. Results showed a clear developmental gradient: lexical entropy and positional concentration increased with proficiency, whereas grammatical divergence and compression ratio decreased. Mixed-effects models confirmed that these shifts were robust effects of proficiency. The findings support a probabilistic view of interlanguage development and offer a principled diagnostic framework for evaluating communicative efficiency in L2 writing. Humanities/Language and linguistics Social science/Language and linguistics Physical sciences/Mathematics and computing information density entropy interlanguage second language acquisition KL divergence probabilistic systems Figures Figure 1 Introduction Interlanguage is simultaneously systematic and variable: learners develop stable regularities while also exhibiting fluctuation across forms, contexts, tasks, and time. Much SLA work has operationalized this development through accuracy–fluency–complexity indices, lexical richness measures, and syntactic complexity metrics (e.g., Housen et al., 2019 ; Kuiken, 2023 ). Yet a persistent limitation is that many commonly used indices are either (a)feature-counting measures that do not directly capture the distributionof choices, or (b) text-length sensitive proxies that conflate development with verbosity. This motivates a complementary lens: information density, understood as how much uncertainty (or predictability) is carried by the linguistic signal and how that uncertainty is distributed across lexical, grammatical, and phraseological options. Information theory—originally developed to model communication systems—offers formal tools for measuring uncertainty via entropy, and for comparing distributions via cross-entropy and relative entropy (KL divergence). In linguistic research, entropy-family measures have been increasingly used to capture complexity as unpredictability or disorder in a probabilistic system, while compression-based approaches approximate Kolmogorov complexity as a holistic measure of information content and redundancy (Ehret & Szmrecsanyi, 2016 ). These tools are especially promising for interlanguage because development can be reframed as a shift in distributions: learners' probability mass gradually reorganizes toward target-like allocation, with subsystem-specific divergences persisting or reappearing under different task conditions. A second motivation comes from research on information distribution in discourse, processing, and L2 pedagogy. Entropy- and surprisal-based measures have been advanced as direct operationalizations of information density and information rate, capturing how uncertainty and predictability are managed across unfolding linguistic signals and how this management relates to cognitive resource allocation (Jaeger, 2010 ; Levy, 2008 ). Work on bilingual speech further indicates that information encoding and transmission profiles can differ systematically between L1 and L2 production, implying that information packaging is not merely a surface attribute of texts but a fundamental property of bilingual communication (Bradlow, 2022 ). Converging evidence from interactional research also shows that information density is position-sensitive within discourse: density may peak in specific structural locations (e.g., question onsets), and such early entropy patterns can covary with the distribution of multimodal signals, suggesting that information regulation is coordinated across linguistic and visual channels rather than being uniform across an utterance (Trujillo & Holler, 2025 ). Importantly, pedagogically oriented evidence aligns with these processing-based accounts: in EFL listening, learners may prioritize message extraction under conditions of high informational load, and comprehension outcomes can shift as discourse varies in information density and rhetorical elaboration—highlighting the instructional relevance of quantifying density as a property of discourse design (Mekheimer & Fageeh, 2025 ). Taken together, these strands motivate entropy-based measurement as a route to a more unified account of interlanguage variability, proficiency, and communicative efficiency, while also grounding the construct in demonstrable consequences for L2 comprehension and classroom discourse selection. Against this background, the present study develops an integrated, multi-level approach to quantifying information density in learner production. Rather than treating "more entropy" as inherently better or worse, we treat entropy as a diagnostic description of how learner systems allocate probability mass across competing linguistic options, and how that allocation compares to target and peer benchmarks. The study was guided by the following research questions: To what extent does lexical information density, operationalized as lexical entropy (Hₗₑₓ), vary across the B1, B2, C1, and L1 groups? To what extent does grammatical information density, operationalized as grammatical divergence from the native-speaker reference distribution (KL₍gram₎), vary across the B1, B2, C1, and L1 groups? To what extent does global structural regularity, operationalized as compression ratio (CR), vary across the B1, B2, C1, and L1 groups? To what extent does discourse-positional information packaging, operationalized as the positional concentration index (PCI), vary across the B1, B2, C1, and L1 groups? Do these four information-density indicators together reveal a coordinated developmental gradient from lower-proficiency interlanguage to native-speaker performance? Can interlanguage be empirically modeled as a probabilistic system whose lexical, grammatical, structural, and discourse-positional distributions undergo systematic reorganization as proficiency increases? Literature Review Entropy, information density, and linguistic complexity Information-theoretic approaches to linguistic complexity conceptualize "complexity" as the degree of unpredictability in linguistic choices rather than simply the number of structures used. On this view, complexity is fundamentally a distributional property: a linguistic system is more complex when outcomes are less predictable and, in a complementary sense, less compressible. Ehret and Szmrecsanyi ( 2016 ) provides a foundational synthesis of this tradition by distinguishing Shannon entropy (uncertainty over outcomes) from Kolmogorov complexity (information content as description length), and by motivating their relevance for variationist and usage-based accounts of linguistic organization (Oh, 2015 ). Methodologically, entropy and surprisal have been treated as direct tools for quantifying information density because they operationalize uncertainty and expected information in a principled, comparable way (Jaeger, 2010 ; Levy, 2008 ). In parallel, scholarship on complexity measurement highlights that "complexity" is multidimensional and theory-dependent, and that different metrics index different facets (e.g., dispersion, sequential dependence, redundancy, structural option space), making construct alignment essential (Ehret et al., 2023 ; Çöltekin & Rama, 2023 ; Bentz et al., 2023 ). Within SLA, the "complexity turn" has amplified this caution: importing concepts from complexity science requires clarity about whether they function as descriptive indices, explanatory mechanisms, or meta-theoretical commitments, and each entails different evidentiary burdens (Han et al., 2023 ). For interlanguage research specifically, these considerations imply that entropy-based indices should be interpreted as distributional descriptors whose meaning depends on the representational level (lexical vs. grammatical vs. phraseological), the task ecology, and—crucially—the reference system against which learner distributions are evaluated. This interpretive stance also aligns with pedagogically oriented work on information density in L2 contexts, where discourse design and informational load can measurably modulate comprehension and learner prioritization strategies, reinforcing the need to treat "density/complexity" as context-sensitive rather than as a single monotonic proficiency marker (Mekheimer & Fageeh, 2025 ). Relative entropy and proficiency development in L2 A central strand of applied work operationalizes interlanguage development via relative entropy (KL divergence), measuring how far learner distributions deviate from a reference distribution and how this distance changes over time or across proficiency levels. Sun and Wang ( 2021 ) propose relative entropy of linguistic complexity as a way to assess L2 proficiency development, arguing that distributional discrimination provides a sensitive index of developmental change beyond traditional complexity metrics. Complementing this, Wang et al. ( 2022 ) extend information-theoretic evaluation by incorporating Kolmogorov complexity metrics alongside entropy-based measures to assess L2 proficiency, reinforcing the idea that proficiency can be modeled through distributional organization and compressibility rather than isolated counts of forms; this interpretation is further supported by evidence that Kolmogorov-complexity measures can differentiate both L2 groups and L1 backgrounds (Alzahrani, 2024 ). These studies collectively shift the focus from "how many structures learners use" to "how learners distribute probability mass across available structures," which is more aligned with usage-based conceptions of interlanguage and with the empirical fact that learners can produce advanced forms sporadically while still relying heavily on a small set of high-probability templates. Compression and Kolmogorov complexity in learner language Another influential route uses compression as a practical approximation to Kolmogorov complexity. By estimating how compressible learner output is, researchers can capture global redundancy and structure that may not be visible in feature-by-feature measures. Ehret and Szmrecsanyi ( 2019 ) demonstrate how compressing learner language can provide an information-theoretic measure of complexity in SLA production data, offering a holistic complement to entropy measures and enabling comparisons across texts without relying on preselected structural inventories; related evidence also suggests that such measures are sensitive to proficiency grouping and L1-background effects (Alzahrani, 2024 ). Compression-based measures are particularly attractive for interlanguage because they can reflect both (a) formulaic repetition (high compressibility) and (b) chaotic variability (low compressibility) depending on how the learner system is organized. This supports a non-linear developmental expectation: development may involve both increasing variability in some domains (exploration) and decreasing variability in others (stabilization). Phraseology, sequential dependence, and moving beyond bag-of-words A known limitation of simple entropy over word types is that it can miss phraseological organization and sequential constraints. Recent work, therefore, emphasizes moving beyond bag-of-words representations to incorporate phraseological patterning and distributional dependence across sequences, especially in learner language where proficiency is often visible in stable collocations, lexical bundles, and constructional preferences (Vandeweerd, Housen, & Paquot, 2022 ; Ehret & Szmrecsanyi, 2019 ). Related learner-focused research operationalizes phraseological diversity using normalized entropy over multiword units, highlighting how entropy can be adapted to phraseological structure rather than single-word diversity alone (Vandeweerd, Housen, & Paquot, 2022 ). Information density, processing, and modality Information density refers to the amount and distribution of informational load within a stretch of discourse, commonly operationalized through measures such as entropy and surprisal that capture how predictable or uncertain linguistic material is at a given point in unfolding communication (Karimi et al., 2024; Trujillo & Holler, 2024). Processing, in this context, concerns the real-time cognitive work involved in encoding, transmitting, and interpreting that load, rather than treating it as a static textual property; recent evidence shows that entropy can affect lexical processing over and above traditional predictability measures, while bilingual research indicates that L1 and L2 speech differ in their information-encoding and transmission profiles (Bradlow, 2022 ; Karimi et al., 2024). Modality refers to the channel through which meaning is conveyed—most centrally the verbal and visual channels in face-to-face interaction—and current multimodal research shows that communicative meaning is distributed across speech, gesture, gaze, and other bodily signals rather than being carried by language alone (Ünal et al., 2024). From this perspective, information density is best understood as a dynamic property of discourse that is shaped by temporal processing demands, discourse position, and the coordination of multiple semiotic resources across modalities (Trujillo & Holler, 2025 ; Ünal et al., 2024). Information density should not be treated as a purely static property of texts, because it is closely tied to how language is processed in real time. Research on bilingual speech shows that L1 and L2 production differ in their patterns of information encoding and transmission, suggesting that developmental differences extend beyond grammatical choice to include the temporal organization of informational load across utterances (Bradlow, 2022 ). This perspective is reinforced by multimodal work demonstrating that entropy is unevenly distributed across discourse positions and that higher density in early stretches of discourse may align with shifts in accompanying visual signals (Trujillo & Holler, 2025 ). Taken together, these findings support an understanding of interlanguage information density as dynamically structured by position, modality, and task demands rather than as a uniform textual characteristic. Related evidence from translation and mediated language varieties Entropy measures have also been used to investigate simplification and variation in mediated language (e.g., translated, interpreted, or machine-generated text). Entropy analyses of lexical and syntactic simplification provide methodological precedents for distinguishing language varieties via uncertainty metrics and for interpreting entropy relative to density and richness (Liu et al., 2022; Wang et al., 2025; Yao & Fan, 2025). While these studies are not SLA per se, they strengthen the general claim that entropy-family measures can capture systematic differences in how linguistic information is distributed across varieties, genres, and production conditions—conditions that parallel many SLA contrasts (e.g., timed vs untimed writing; spoken vs written performance; careful vs spontaneous production). Synthesis and gap Existing work shows that (a) relative entropy can discriminate proficiency groups and model developmental change (Sun & Wang, 2021 ; Wang et al., 2022 ), (b) compression approximations to Kolmogorov complexity capture holistic redundancy and structure in learner production (Ehret & Szmrecsanyi, 2019 ; Ehret & Szmrecsanyi, 2016 ), and (c) information density has processing- and discourse-sensitive correlates (Bradlow, 2022 ; Trujillo & Holler, 2025 ). What remains underdeveloped is a unified interlanguage account that integrates these measures across levels (lexical, grammatical, phraseological), separates diversity from contextual predictability, and interprets entropy patterns as subsystem-specific reorganization rather than a single monotonic index of proficiency (Ehret et al., 2023 ; Han et al., 2023 ; Housen et al., 2019 ; Kuiken, 2023 ). The present paper addresses this gap by proposing a clean theoretical framework for "information density of interlanguage" and by clarifying how entropy-family measures map onto SLA constructs. In this study, information density (ID) is treated as a multi-level property of interlanguage systems that emerges from how probability mass is distributed across competing linguistic options. Accordingly, four complementary indices are used to capture ID across lexical choice, grammatical sequencing, global structural regularity, and discourse packaging. Lexical entropy (Hₗₑₓ)quantifies the dispersion of lemma distributions, such that higher values indicate broader lexical exploration and less concentration on a limited set of forms. Grammatical divergence (KL₍gram₎) operationalizes directed distance from a target reference distribution estimated over POS trigrams, where higher values reflect greater interlanguage distance from target-like sequential constraints. Compression ratio (CR)provides a holistic proxy for structural regularity, with lower values indicating greater compressibility and redundancy (i. e., more predictable structure). Finally, thepositional concentration index (PCI)captures discourse-level packaging by expressing the ratio of opening-segment entropy to whole-text entropy; values greater than 1.00 indicate front-loaded informational density, whereas values near or below 1.00 indicate relatively uniform or delayed information distribution. Theoretical Framework The proposed framework conceptualizes interlanguage as a probabilistic system governed by internal and external constraints (Fig. 1). This approach treats development not as the linear acquisition of rules, but as the dynamic reallocation of probability mass across competing linguistic options. Figure 1 formalizes this perspective by representing interlanguage as a constrained probabilistic system in which linguistic output arises from the interaction of knowledge, attention, task demands, and processing capacity (Han et al., 2023 ; Housen et al., 2019 ). Within this system, development is modeled as the redistribution of probability mass across four interrelated layers: lexical choice, grammatical sequencing, phraseological patterning, and discourse-position packaging (Ehret & Szmrecsanyi, 2016 ; Sun & Wang, 2021 ). These layers are examined through complementary information-theoretic measures, with entropy indexing distributional dispersion, KL divergence capturing distance from target-like organization, and compression reflecting global structural regularity (Ehret & Szmrecsanyi, 2016 ; Wang et al., 2022 ). The model also assumes that development is non-linear and subsystem-specific: lexical variability may increase as the learner repertoire expands, whereas grammatical divergence and structural redundancy may decline as usage becomes more constrained and conventionalized (Sun & Wang, 2021 ; Ehret et al., 2023 ). Accordingly, the framework links distributional change at the level of linguistic choices to broader developmental processes of stabilization, convergence, and information management in interlanguage systems (Kuiken, 2023 ; Han et al., 2023 ). Interlanguage can be modeled as a probabilistic system whose outputs reflect constrained choice. Learners operate under limits of knowledge, attention, and processing capacity, so production involves selecting among competing lexical, grammatical, and phraseological options with different probabilities. Development, in this view, is not simply the accumulation of rules, but a gradual reallocation of probability mass: forms that are more contextually appropriate become more strongly preferred, while competing variants are reduced, restricted to narrower contexts, or reorganized. This distributional perspective aligns with information-theoretic treatments of linguistic complexity as unpredictability (entropy) and/or description length (compression/Kolmogorov complexity) (Ehret & Szmrecsanyi, 2016 ). Within this framework, information density (ID) is defined as the degree of uncertainty or predictability in learner choices within a specified representational space. The article operationalizes ID using four complementary metrics (see Fig. 1). Shannon entropy (H) captures distributional dispersion (e.g., how evenly outcomes are spread across POS sequences or lexical options). Cross-entropy expresses the expected coding cost of learner output under a target reference model, providing an interpretable measure of "learner surprisal" relative to target expectations. Relative entropy (KL divergence) quantifies directed distance between learner and target-norm distributions, and thus provides a principled measure of interlanguage distance and convergence (Sun & Wang, 2021 ). Finally, compression-based complexity offers a holistic proxy for redundancy and structure via compressibility, approximating description-length notions of complexity (Ehret & Szmrecsanyi, 2016 ). Because interlanguage organization is multi-component, information density is examined across four layers (Fig. 1): lexical ID (word/lemma distributions and repertoire size), grammatical ID (morphosyntactic categories and the stabilization of constructional preferences),phraseological ID (the conventionalization of collocations and multiword units), anddiscourse-position ID (how information is packaged across segments such as openings versus mid-sections). This multi-level design is essential because the same learner can show high predictability in one subsystem (e.g., reliance on safe grammatical defaults) and high variability in another (e.g., lexical exploration), and a single global index would obscure that asymmetry. Development is therefore modeled as reorganization rather than linear growth . Changes in entropy are expected to be non-monotonic and subsystem-specific: lexical entropy may increase as the repertoire expands and exploratory variation grows, while grammatical and phraseological entropy may decrease as usage becomes more constrained by target norms and as competition among variants is resolved (Sun & Wang, 2021 ). Across time, learners may move from highly compressible template-based production, to a phase of broader experimentation, and then toward more efficient, conventionalized variability—where flexibility is retained but choices become more contextually predictable. Finally, the framework provides an interpretive bridge to core SLA questions. First, entropy profiles distinguish stabilityversusexploration: high grammatical entropy can reflect unresolved competition among forms, whereas low entropy may signal either genuine stabilization or overreliance on defaults. Second, target convergence can be tracked through decreasing KL divergence, which operationalizes the reduction of interlanguage distance from benchmark distributions (Sun & Wang, 2021 ). Third, the approach captures information packaging across modality and discourse position, allowing analysis of whether learners regulate density differently in speech versus writing and whether they concentrate informational load in specific discourse regions. Methods Research Design This study employs a quantitative, corpus-based design to operationalize information density in interlanguage using entropy-family measures and distributional distance metrics. The core assumption is that interlanguage reflects constrained probabilistic choice: learners select among competing lexical, grammatical, and phraseological options under limits of knowledge, attention, and processing capacity. Accordingly, the analysis is not centered on counting structures or errors per se, but on measuring how learner distributions are organized and how they reorganize across proficiency and converge toward target norms. The design combines two complementary comparisons. First, a developmental (between-level) comparison examines whether information density metrics shift systematically across CEFR-aligned proficiency bands. Second, a learner–target comparison quantifies the distance between learner distributions and native-speaker benchmarks using cross-entropy and KL divergence. Together, these comparisons allow interlanguage development to be modeled as distributional reallocation—probability mass moving toward contextually appropriate options while competing variants are pruned, restricted, or reorganized over time (Sun & Wang, 2021 ; Ehret & Szmrecsanyi, 2016 ). Participants and Corpora The study utilizes a balanced dataset comprising a learner corpus of L2 English writing and a native-speaker (L1) reference corpus. The learner corpus features argumentative essays from 150 participants, stratified into three proficiency bands according to CEFR standards: B1 (Threshold), B2 (Vantage), and C1 (Effective Operational Proficiency), with 50 texts per band. To ensure the stability of entropy-based measures, token counts are balanced across levels, and inclusion is restricted to texts meeting a minimum length threshold. The reference corpus consists of 50 argumentative essays written by L1 English university students. This benchmark is matched to the learner data in terms of topic domain, prompt type, and length distribution. To maintain analytic integrity, a controlled-genre approach is strictly applied, ensuring all 200 texts conform to the argumentative writing constraint. All data are anonymized and cleaned to remove non-linguistic artifacts such as metadata and duplicated headers prior to analysis. Data Preparation and Linguistic Annotation To ensure comparability across the four representational layers (Lexical, Grammatical, Phraseological, and Discourse), all texts undergo a uniform processing pipeline: Preprocessing : Texts are tokenized, segmented into sentences, and lemmatized. Tagging : Part-of-speech (POS) tags and dependency parses are assigned using a consistent NLP toolchain to capture morphosyntactic and syntactic patterning. Quality Control : A stratified manual audit of 10% of the texts per proficiency band is conducted to verify the plausibility of automated tags, specifically addressing potential biases from fragmentary clauses or non-canonical word order. Normalization: Conservative normalization is applied only to resolve preprocessing anomalies (e.g., tokenization artifacts). In line with interlanguage research principles, learner language is not"corrected"to target forms to ensure the analytic representation accurately reflects the learner's current probabilistic system Units of Analysis Information density is computed at four representational layers aligned with Fig. 1. Each layer is treated as a distinct outcome space with its own probability distribution, enabling subsystem-specific interpretation rather than a single global "complexity score." Lexical layer (Lexical ID). Distributions are computed over word forms and lemmas to model repertoire size and dispersion of lexical choice, including function–content distributions where relevant. Grammatical layer (Grammatical ID). Distributions are computed over POS sequences (bigrams and trigrams) and, where included, dependency-relation patterns to capture morphosyntactic organization and sequencing stability. Phraseological layer (Phraseological ID). Distributions are computed over recurrent collocations and multiword units (e.g., high-frequency bigrams/trigrams and lexical bundles exceeding a minimum frequency threshold) to capture conventionalization and idiomatic pattern control. Discourse-position layer (Discourse-Position ID). Each text is segmented proportionally into beginning, middle, and end regions (e.g., first 20%, middle 60%, final 20%), and entropy measures are computed separately per segment to test whether information density is uniform or concentrated in specific structural locations. Given entropy's sensitivity to sample size, analyses are conducted under controlled token windows where appropriate, and distributions are estimated using normalized procedures to support comparability across groups. Information-Theoretic Measures Information density is operationalized using four complementary measures that capture dispersion, target-based predictability, distance-to-target, and holistic redundancy/structure. Shannon entropy (H)is calculated for each distributional representation to quantify dispersion. Higher entropy reflects a more even distribution across outcomes, whereas lower entropy indicates concentration around fewer options. Entropy is computed for lexical distributions, grammatical sequence distributions, and phraseological distributions. Cross-entropyestimates the expected coding cost of learner output under a target reference distribution, providing an interpretable measure of how surprising learner production is relative to target expectations. Relative entropy (KL divergence)quantifies directed distance between learner and target distributions and serves as the primary operationalization of interlanguage distance and convergence (Sun & Wang, 2021 ). Because KL divergence is undefined when the target assigns zero probability to learner-observed events, additive smoothing is applied and sensitivity checks are performed to ensure results are not artifacts of a single smoothing choice. Compression-based complexity is used as a holistic proxy for redundancy and structural regularity. Text strings are standardized and compressed using a lossless compression method (e.g., LZMA), and the compression ratio is interpreted as an approximation of description-length complexity consistent with information-theoretic views of complexity as compressibility (Ehret & Szmrecsanyi, 2016 ). This measure is treated as complementary to entropy: it captures global regularities (e.g., repetitive templates, recurring strings) that may not be fully reflected in token-level dispersion. Statistical Analysis Analysis proceeds in two stages. First, group-level differences in entropy and distance metrics are tested across proficiency bands and against the native-speaker benchmark. Depending on distributional diagnostics, either parametric tests (ANOVA) or non-parametric alternatives (Kruskal–Wallis) are used, with effect sizes reported alongside significance tests. Second, to account for unequal text counts and nested structure, mixed-effects modeling is used where appropriate. Entropy outcomes are modeled with proficiency as a fixed effect, and random intercepts are included for text (and for prompt where prompt identifiers are available). For discourse-position analyses, repeated-measures formulations test whether entropy differs across segments within texts and whether segment effects interact with proficiency. Confidence intervals for key effects are estimated via bootstrapping (1,000 resamples) to reduce dependence on large-sample assumptions, in line with broader methodological work on estimating uncertainty around information-theoretic measures (Lai & Do, 2020 ). Reliability, Robustness, and Sensitivity Checks Several checks are implemented to strengthen measurement credibility. Entropy is recomputed under multiple representations (token vs lemma) to reduce construct under-identification. Key conclusions are also tested under different n-gram sizes (bigrams vs trigrams) and, where relevant, under controlled-length subsamples (equal token windows) to ensure findings are not driven by text length. For KL divergence, results are examined under different smoothing parameters. For compression metrics, preprocessing is strictly standardized so that differences reflect linguistic patterning rather than formatting artifacts. Interpretation is anchored in converging evidence: because entropy may increase under both productive diversification and noisy instability, entropy shifts are read alongside KL divergence (distance-to-target), cross-entropy (target-based coding cost), and layer-specific patterns (lexical vs grammatical vs phraseological), consistent with the non-monotonic reorganization view. Reproducibility and Technical Parameters Software and toolchain. All texts were processed using spaCy (v3.7.2) with the en_core_web_md model for tokenization, sentence segmentation, lemmatization, and POS tagging. Tagset and n-grams. Grammatical information density was computed using the Universal POS tagset. KL divergence (KL₍gram₎) was computed over POS trigrams (n = 3) to capture sequential dependence in morphosyntactic selection. Smoothing and thresholds. To avoid zero-probability events in KL calculations, additive (Laplace) smoothing was applied with α = 0.01. Only essays exceeding 300 tokens were included to ensure stable entropy estimation and reliable n-gram counts. Length control strategy. Primary analyses were computed on fixed-length windows of 250 tokens sampled from the center of each text to minimize length-driven artifacts. As a robustness check, all key results were replicated on full texts using bootstrapped subsampling (1,000 iterations), confirming that group-level distributional profiles were stable under varying text lengths. L1 baseline for KL₍gram₎. The L1 group was treated as a stochastic sample. For each native text, KL₍gram₎ was computed against a leave-one-out (LOO) L1 reference distribution constructed from the remaining 49 native texts, providing a baseline estimate of within-native variation. Ethical Considerations The study protocol was reviewed and approved by the Institutional Review Board of the Faculty of Education, Beni-Suef University (Approval No. BSU-FoE-004-01-01-2016). All methods were carried out in accordance with relevant guidelines and regulations. Informed consent was obtained from all participants and/or their legal guardians prior to data collection and analysis. All learner data were anonymized prior to analysis. The study reports only aggregated findings and does not disclose any identifying personal or contextual metadata. Results Descriptive Profile Table 1 reports the descriptive profile for the balanced comparison (50 texts per group). The four component indicators show a clear and fully ordered developmental gradient from B1 to L1. Lexical entropy increases steadily across levels, indicating broader lexical dispersion at higher proficiency. By contrast, grammatical divergence and compression ratio decline monotonically, showing closer approximation to target-like grammatical distributions and greater structural regularity. Positional concentration also rises across levels, indicating progressively stronger front-loading of informational load in the opening segment of the text. Table 1 Descriptive statistics by group (Mean, SD) Group Hₗₑₓ KL₍gram₎ CR PCI B1 (n = 50) 4.125 (0.087) 1.859 (0.089) 0.664 (0.017) 0.891 (0.032) B2 (n = 50) 4.438 (0.030) 1.482 (0.026) 0.614 (0.008) 1.061 (0.016) C1 (n = 50) 4.835 (0.033) 0.967 (0.028) 0.579 (0.009) 1.191 (0.016) L1 (n = 50) 5.255 (0.045) 0.122 (0.009) 0.519 (0.008) 1.279 (0.021) Operational definitions (Table 1). Hₗₑₓ = lexical entropy, with higher values indicating greater lexical dispersion. KL₍gram₎ = grammatical divergence from the L1 POS-trigram reference distribution, with lower values indicating greater target convergence. CR = compression ratio, with lower values indicating greater compressibility and structural regularity. PCI = positional concentration index, computed as opening-segment entropy divided by whole-text entropy; values above 1.00 indicate front-loaded information packaging. The composite IDS_Total showed the same stepwise pattern in the statistical outputs, but detailed reporting centers on the four theoretically primary indicators. Assumption checks and primary vs. robustness analysis strategy Assumption checks indicated that normality was not consistently met across cells, and homogeneity of variance was also violated for all four focal metrics (Table 2A). Because these conditions make classical fixed-variance ANOVA less appropriate, the main inferential results are reported with Welch’s ANOVA and Games–Howell post-hoc comparisons, which are robust to heteroscedasticity and unequal group dispersions. Table 2 A. Homogeneity of variance across groups (Levene’s test) Metric Levene F p Hₗₑₓ 15.73 < .001 KL₍gram₎ 43.34 < .001 CR 11.44 < .001 PCI 9.00 < .001 The heterogeneity pattern is substantively unsurprising. As proficiency increases, the distributions tighten for some metrics, especially KL₍gram₎ and CR, while the L1 group shows a particularly narrow native baseline for grammatical divergence. This confirms that the data should be interpreted with variance-robust procedures rather than pooled-variance assumptions. To check that the main findings were not an artifact of the balanced 50/50/50/50 design, the same analyses were rerun using the empirical group-size distribution available in the auxiliary output files (B1 = 45, B2 = 55, C1 = 43, L1 = 57). The replication produced the same directional ordering and the same pattern of significance across all four indicators (Table 2B). Table 2 B. Robustness check using empirical group sizes Metric Welch F df1 df2 p Pattern Hₗₑₓ 1181.83 3 98.92 < .001 B1 < B2 < C1 < L1 KL₍gram₎ 1679.02 3 94.28 B2 > C1 > L1 CR 503.30 3 104.61 B2 > C1 > L1 PCI 1323.29 3 104.75 < .001 B1 < B2 < C1 < L1 Note. The robustness analysis reproduces the balanced-design findings: Hₗₑₓ and PCI increase monotonically, whereas KL₍gram₎ and CR decrease monotonically. All Welch tests remained statistically significant at p < .001. Omnibus group effects Welch’s ANOVA confirmed statistically significant between-group differences for all four indicators (Table 3). The effects are exceptionally large, especially for KL₍gram₎ and Hₗₑₓ, showing that proficiency level is strongly associated with the distributional organization of learner writing. The omnibus pattern therefore supports the central claim that interlanguage development involves coordinated reorganization across lexical choice, grammatical sequencing, structural regularity, and discourse-level information packaging. Table 3 Omnibus tests of group differences (Welch ANOVA) Metric Omnibus test F df p Hₗₑₓ Welch ANOVA 4770.09 3, 104.98 < .001 KL₍gram₎ Welch ANOVA 54286.40 3, 91.97 < .001 CR Welch ANOVA 1584.40 3, 107.02 < .001 PCI Welch ANOVA 2270.41 3, 106.77 < .001 Considered together, the omnibus results justify detailed pairwise interpretation. Rather than isolating a single dimension of development, the findings reveal a coherent multi-metric profile in which some indices rise with proficiency and others fall. This mixed directional pattern is theoretically important because it indicates reorganization across subsystems rather than a simple one-directional increase in general complexity. Post-hoc comparisons and developmental reorganization Lexical entropy (Hₗₑₓ): progressive expansion of lexical dispersion Games–Howell comparisons showed that all pairwise differences in Hₗₑₓ were statistically significant (Table 4A). The pattern is strictly monotonic (B1 < B2 < C1 < L1), indicating progressive expansion in lexical dispersion as proficiency increases. The largest cumulative gain appears between B1 and C1, but the C1–L1 contrast also remains significant, showing that advanced learner writing still differs from native writing in the breadth and balance of lexical choice. Table 4 A. Games–Howell post-hoc tests for Hₗₑₓ Comparison Mean diff (G1–G2) SE p B1–B2 -0.313 0.013 < .001 B1–C1 -0.710 0.013 < .001 B1–L1 -1.130 0.014 < .001 B2–C1 -0.397 0.006 < .001 B2–L1 -0.817 0.008 < .001 C1–L1 -0.419 0.008 < .001 The lexical results therefore point to continuing repertoire expansion rather than early plateauing. Higher proficiency is associated not simply with more words, but with a wider and more evenly distributed lexical choice space. Grammatical divergence (KL₍gram₎): progressive convergence toward target norms Games–Howell comparisons for KL₍gram₎ again showed that every pairwise contrast was significant (Table 4B). Grammatical divergence declines at each step from B1 to L1, indicating progressively stronger convergence toward the native POS-trigram distribution. The large B1–L1 and B2–L1 gaps show that grammatical sequencing remains one of the most sensitive markers of interlanguage distance, while the significant C1–L1 difference confirms that even advanced learner production does not fully collapse into the native distributional baseline. Table 4 B. Games–Howell post-hoc tests for KL₍gram₎ Comparison Mean diff (G1–G2) SE p B1–B2 0.377 0.013 < .001 B1–C1 0.892 0.013 < .001 B1–L1 1.737 0.013 < .001 B2–C1 0.516 0.005 < .001 B2–L1 1.361 0.004 < .001 C1–L1 0.845 0.004 < .001 This is one of the clearest developmental signatures in the dataset. As proficiency rises, learner texts become less distributionally distant from the native reference, which directly supports the claim that interlanguage development can be modeled as probabilistic convergence rather than only as reduction of overt error. Compression ratio (CR): increasing structural regularity with proficiency Games–Howell comparisons also showed significant differences for every group contrast in CR (Table 4C). Compression ratio declines monotonically from B1 to L1, meaning that texts become increasingly compressible and structurally regular as proficiency develops. The B1–B2 drop is already substantial, which suggests that important gains in global regularity emerge relatively early; subsequent reductions remain significant but appear more incremental. Table 4 C. Games–Howell post-hoc tests for CR Comparison Mean diff (G1–G2) SE p B1–B2 0.049 0.003 < .001 B1–C1 0.084 0.003 < .001 B1–L1 0.145 0.003 < .001 B2–C1 0.035 0.002 < .001 B2–L1 0.096 0.002 < .001 C1–L1 0.061 0.002 < .001 These results indicate that development involves not only richer lexical choice but also tighter global organization. Learner texts become easier to compress because they display more stable patterning, more predictable sequencing, and less redundant structural noise. Positional concentration (PCI): stronger early information packaging at higher proficiency For PCI, Games–Howell comparisons again showed significant differences for all pairs (Table 4D). The index rises steadily from B1 to L1, indicating that more proficient writers concentrate a larger share of informational load in the opening portion of the text. The strongest early shift occurs between B1 and B2, suggesting that discourse planning begins to change noticeably at the intermediate stage, after which the same front-loading tendency continues in more refined form. Table 4 D. Games–Howell post-hoc tests for PCI Comparison Mean diff (G1–G2) SE p B1–B2 -0.170 0.005 < .001 B1–C1 -0.300 0.005 < .001 B1–L1 -0.387 0.005 < .001 B2–C1 -0.130 0.003 < .001 B2–L1 -0.218 0.004 < .001 C1–L1 -0.088 0.004 < .001 The PCI results therefore complement the lexical and grammatical findings by showing that development also involves improved discourse packaging. More advanced writers do not merely choose different forms; they distribute information across the text in increasingly native-like ways. Summary of developmental reorganization Across the four focal indicators, the results converge on a single developmental profile. Lexical entropy rises, grammatical divergence falls, compression ratio falls, and positional concentration rises. This combination is crucial: learner language becomes more lexically expansive while simultaneously becoming more grammatically aligned, more globally regular, and more strategically organized across discourse position. The overall pattern is therefore best described as coordinated probabilistic reorganization rather than as a single monotonic change in complexity. Discussion The present study examined whether interlanguage can be meaningfully modeled as a probabilistic system and whether its development can be traced through distribution-sensitive information metrics. The findings strongly support this view. Across lexical entropy, grammatical divergence, compression ratio, and positional concentration, the results revealed systematic and statistically robust differences across proficiency levels. More importantly, the developmental pattern did not reflect a single linear increase in “complexity.” Instead, it showed coordinated reorganization across multiple representational layers, which is precisely what the proposed model predicts. In the model, interlanguage development is not the simple accumulation of rules or forms; rather, it is a gradual reallocation of probability mass across lexical, grammatical, phraseological, and discourse-positional options. The empirical results are consistent with this account. At the lexical level, entropy increased significantly across proficiency levels, indicating that more advanced learners distribute probability mass across a broader range of lexical choices. This lexical expansion is consistent not only with information-theoretic accounts of dispersion, but also with recent corpus-based evidence that richer lexical patterning continues to distinguish more advanced EFL writing (Yang & He, 2025 ) and with NLP-oriented approaches that model lexical knowledge through distribution-sensitive representations rather than simple counts (Crossley & Holmes, 2023 ). This pattern corroborates information-theoretic accounts in which complexity is defined as dispersion or unpredictability in outcome distributions rather than mere structural accumulation (Ehret & Szmrecsanyi, 2016 ). It also aligns with usage-based perspectives on SLA, according to which increased exposure and entrenchment gradually expand the learner’s choice space and weaken reliance on a narrow set of high-frequency default forms. The present results therefore answer the first research question affirmatively: interlanguage development can indeed be captured probabilistically, and lexical entropy is one meaningful indicator of such development. At the same time, the remaining gap between C1 and L1 suggests that advanced learners still differ from native benchmarks in the balance of lexical distributions, which confirms that lexical development remains probabilistically incomplete even at higher proficiency levels. In contrast to lexical entropy, grammatical divergence decreased sharply as proficiency increased. This is one of the clearest findings of the study and provides a strong response to the second research question concerning whether development involves convergence toward target norms. The answer is clearly yes, but this convergence is subsystem-specific. Using KL divergence, the study measured the directed distance between learner distributions and the target grammatical reference model. The significant downward trend across proficiency groups indicates that learners’ grammatical selections become progressively less divergent from target-like distributions. This result directly corroborates Sun and Wang’s ( 2021 ) argument that relative entropy can serve as a sensitive indicator of developmental change in L2 proficiency. It also supports the model’s claim that development involves redistribution and pruning, whereby competing non-target probabilities are gradually reduced and more contextually appropriate sequential patterns become more strongly preferred. The fact that lexical entropy rises while grammatical divergence falls is especially important theoretically, because it demonstrates that interlanguage does not become uniformly more variable or uniformly more constrained. Rather, it becomes more diversified in lexical choice and more stabilized in grammatical patterning. This asymmetry is fully consistent with recent cautions in SLA complexity research that developmental change must be interpreted at the level of the subsystem being examined rather than through a single global metric (Ehret et al., 2023 ; Han et al., 2023 ). The findings for compression ratio provide further support for the model. Compression ratio declined significantly across proficiency levels, indicating that learner language becomes more compressible and therefore more structurally regular and predictable. This result corroborates earlier work showing that compression-based measures can capture holistic structure and redundancy in learner language in ways that traditional feature counts cannot (Ehret & Szmrecsanyi, 2019 ; Wang et al., 2022 ; Alzahrani, 2024 ). Within the present theoretical framework, this means that development is not only a matter of increasing lexical range or decreasing grammatical divergence; it also involves the emergence of broader regularity in how texts are assembled. Importantly, the decrease in compression ratio does not contradict the increase in lexical entropy. On the contrary, the coexistence of these trends is one of the strongest pieces of evidence for the probabilistic model. It indicates that the system becomes globally more organized even as it becomes locally more diversified. In other words, learner language appears to achieve greater communicative efficiency by coordinating lexical expansion with structural regularization. This pattern is compatible with the distinction drawn by Ehret and Szmrecsanyi ( 2016 ) between entropy as uncertainty over distributions and compression as a proxy for description-length complexity. The discourse-positional findings are equally revealing. The positional concentration index increased with proficiency, showing that more advanced learners increasingly concentrate informational load earlier in the text. This pattern extends the model beyond lexical and grammatical distributions to discourse-level packaging and provides a clear answer to the third research question regarding whether information-density development is also visible at higher discourse levels. The answer is again affirmative. These results corroborate prior research showing that information density is position-sensitive and related to how speakers and writers manage processing demands over unfolding discourse (Jaeger, 2010 ; Levy, 2008 ; Trujillo & Holler, 2025 ). Within the present model, PCI indexes discourse-position ID and therefore reflects how learners package information across textual segments. The sharpest shift occurring from B1 to B2 suggests that intermediate learners begin to reorganize discourse planning so that greater informational load is placed earlier in the text, while more advanced learners continue this trajectory toward native-like positional concentration patterns. This is an important contribution because it suggests that discourse organization may reorganize earlier than full grammatical convergence, thereby reinforcing the model’s claim that interlanguage development is multi-layered and non-synchronous across subsystems. In combination, these findings provide strong empirical support for the proposed probabilistic model of interlanguage information density. The model predicts that development will involve subsystem-specific redistribution of probability mass rather than uniform growth, and that prediction is borne out by the data. In this respect, the present findings also resonate with broader arguments that linguistic systems exhibit trade-offs across complexity domains rather than straightforward one-directional growth (Bentz et al., 2023 ). In addition, lexical entropy increases, grammatical divergence decreases, compression ratio declines, and positional concentration rises. These are not isolated trends. Rather, they describe a coherent developmental ecology in which learners gradually widen the lexical choice space, reduce divergence from target grammatical norms, increase the structural regularity of output, and reorganize discourse packaging in more target-compatible ways. This interpretation also helps explain why traditional complexity measures are often insufficient. If complexity were treated as a single scalar property, the present results would appear contradictory: some indices increase while others decrease. Yet from the perspective of the present model, such divergence is precisely what should be expected. Development is not a unidirectional rise in complexity but a coordinated recalibration of probabilities across layers of the system. The findings also contribute to broader debates about the nature of interlanguage. Rather than supporting a deficit view in which learner language is judged only in terms of missing target features, the results suggest that interlanguage is better understood as a dynamic distributional system shaped by competing pressures of expressivity, predictability, efficiency, and cognitive constraint. This interpretation is consistent with Bradlow’s ( 2022 ) view that L1 and L2 production differ not only in grammatical selection but also in how information is encoded and transmitted. It also aligns with pedagogically oriented evidence showing that information density affects comprehension and discourse processing in instructional settings (Mekheimer & Fageeh, 2025 ). From this perspective, learner language is not simply “less advanced” than native language; it is differently organized, and its development consists in increasingly adaptive redistribution of linguistic probabilities. The use of a stochastic L1 baseline strengthens this interpretation. By treating native-speaker production as a range of probabilistic variation rather than a zero-variance ideal, the study offers a more realistic model of convergence and avoids reducing target-like performance to a fixed categorical endpoint. Limitations Several limitations should be acknowledged. First, although the L1 baseline was modeled stochastically through leave-one-out estimation, the benchmark remains tied to a specific genre and task ecology, namely argumentative writing. Because entropy and divergence measures are sensitive to register and discourse conditions, the baseline established here should not be assumed to generalize directly to other genres or communicative settings. Second, some measures depend on automated annotation procedures such as lemmatization and POS tagging. Although these procedures were suitable for large-scale analysis, non-canonical learner language, especially at lower proficiency levels, may introduce annotation noise that affects entropy and divergence estimates. Third, the fixed-window approach improved comparability across texts, but it may have reduced the visibility of discourse-level phenomena that emerge more fully in longer texts. Fourth, learners’ L1 backgrounds were not explicitly modeled, which means that some distributional tendencies may reflect cross-linguistic transfer in addition to proficiency-related development, a possibility that has been highlighted in recent work using Kolmogorov-complexity measures across L2 groups and L1 backgrounds (Alzahrani, 2024 ). Finally, the data were derived from a balanced, controlled corpus, which strengthens internal comparability but may underrepresent the variability and non-linearity of naturalistic interlanguage development over time. Implications The findings have important theoretical, methodological, and pedagogical implications. Theoretically, they support a reconceptualization of interlanguage as a probabilistic, multi-layered system rather than a collection of isolated forms or errors. This moves the discussion of learner language beyond static structural counts and aligns SLA more closely with information-theoretic and usage-based approaches to development. Methodologically, the study demonstrates that entropy-family measures can function as bridge constructs linking uncertainty, divergence, compressibility, and discourse packaging within a single analytic framework. This complements calls in SLA complexity research for multiple, explicitly theorized measures rather than single omnibus indices (Housen et al., 2019 ; Kuiken, 2023 ). This provides a richer diagnostic account of development than conventional complexity indices alone. Pedagogically, the results suggest that learner progress should not be evaluated through a single proficiency score or a narrow notion of complexity. Learners may advance differently across lexical dispersion, grammatical convergence, structural regularity, and discourse-level information packaging. Entropy-based profiling may therefore offer a more informative way to diagnose developmental strengths and weaknesses. More broadly, the findings imply that instructional design should recognize that lexical expansion, grammatical stabilization, and discourse packaging do not necessarily develop at the same rate and may require different types of pedagogical support. Overall, the study shows that information density is a theoretically meaningful and empirically tractable lens for understanding interlanguage development. By demonstrating that learner language changes through coordinated redistribution of probability mass across several subsystems, the study provides evidence that interlanguage is best understood not as a deficient approximation of native competence but as a dynamically reorganizing probabilistic system. Conclusion The present study offers a stronger basis for understanding interlanguage development as a process of probabilistic reorganization rather than as the mere accumulation of linguistic forms or the gradual reduction of error. Across proficiency levels, learner language was shown to change through systematic redistribution of probability mass across multiple subsystems: lexical entropy increased as the range of lexical choices broadened; grammatical divergence decreased as sequential patterns moved closer to target-like distributions; compression ratio declined as texts became more structurally regular, predictable, and compressible; and positional concentration increased as informational load was packaged more efficiently toward earlier parts of the text. Read alongside prior work on relative entropy, compression-based complexity, and multidimensional complexity modeling, this pattern reinforces the view that development is best understood as coordinated distributional change rather than isolated gain in a single domain (Sun & Wang, 2021 ; Wang et al., 2022 ; Housen et al., 2019 ). These converging patterns demonstrate that development is not adequately captured by a single scalar notion of complexity. Instead, it unfolds as a coordinated reshaping of dispersion, constraint, regularity, and discourse organization within an evolving linguistic system. In combination, these findings substantiate the view that interlanguage is best conceived as a dynamic information-density architecture governed by interacting pressures of learning, exposure, communicative need, and cognitive resource management. This perspective helps refine theoretical accounts of second language development by showing that expansion and constraint are not opposing outcomes but complementary dimensions of maturation within different representational layers. At the same time, the study illustrates the value of entropy-family metrics as analytically powerful tools for revealing developmental change that traditional count-based indices may obscure. By capturing how learners reorganize probabilities across lexical, grammatical, structural, and discourse-positional domains, the present framework opens a more precise way of describing proficiency growth and a more nuanced basis for interpreting learner performance. In this sense, the study not only strengthens the conceptualization of interlanguage as a probabilistic system, but also provides a practical foundation for more sensitive evaluation of development, one that recognizes learner progress as multidimensional, adaptive, and increasingly efficient rather than simply more or less complex. Declarations Ethics approval and consent to participate The study protocol was reviewed and approved by the Institutional Review Board of the Faculty of Education, Beni-Suef University, Egypt (Approval No. BSU-FoE-004-01-01-2016). All methods were carried out in accordance with relevant guidelines and regulations. Informed consent was obtained from all participants and/or their legal guardians. Consent for publication Not applicable. Competing interests The author declares no competing interests. Author Contribution M.M. conceived the study, conducted the analysis, interpreted the results, and wrote the manuscript. Acknowledgement Thanks to STDF/EKB for potential payment of APCs upon acceptance. Data Availability The data are not publicly available because they contain learner-generated materials collected under consent conditions that do not permit open public release. De-identified data may be available from the corresponding author on reasonable request, subject to institutional and ethical requirements. References Alzahrani, A. Utility of Kolmogorov complexity measures: Analysis of L2 groups and L1 backgrounds. PLOS ONE . 19 (4). https://doi.org/10.1371/journal.pone.0301806 (2024). Article e0301806. Bentz, C., Gutierrez-Vasques, X., Sozinova, O. & Samardžić, T. Complexity trade-offs and equi-complexity in natural languages: A meta-analysis. Linguistics Vanguard . 9 (s1), 9–25. https://doi.org/10.1515/lingvan-2021-0054 (2023). Bradlow, A. R. Information encoding and transmission profiles of first-language (L1) and second-language (L2) speech. Biling. Lang. Cogn. 25 (1), 148–162. https://doi.org/10.1017/S1366728921000717 (2022). Crossley, S. A. & Holmes, L. Assessing receptive vocabulary using state-of-the-art natural language processing techniques. J. Second Lang. Stud. 6 (1), 1–28 (2023). Çöltekin, Ç. & Rama, T. What do complexity measures measure? Correlating and validating corpus-based measures of morphological complexity. Linguistics Vanguard . 9 (s1), 27–43. https://doi.org/10.1515/lingvan-2021-0007 (2023). Dantas, P. V. et al. A comprehensive review of model compression techniques in machine learning. Appl. Intell. 54 , 11804–11844. https://doi.org/10.1007/s10489-024-05747-w (2024). Ehret, K., Berdicevskis, A., Bentz, C. & Blumenthal-Dramé, A. Measuring language complexity: Challenges and opportunities. Linguistics Vanguard . 9 (s1), 1–8. https://doi.org/10.1515/lingvan-2022-0133 (2023). Ehret, K. & Szmrecsanyi, B. An information-theoretic approach to assess linguistic complexity. In R. Baechler & G. Seiler (Eds.), Complexity, isolation, and variation (pp. 71–94). De Gruyter. (2016). https://doi.org/10.1515/9783110348965-004 Ehret, K. & Szmrecsanyi, B. Compressing learner language: An information-theoretic measure of complexity in SLA production data. Second Lang. Res. 35 (1), 23–45. https://doi.org/10.1177/0267658316669559 (2019). Han, Z., Kang, E. Y. & Sok, S. The complexity epistemology and ontology in second language acquisition: A critical review. Stud. Second Lang. Acquisition . 45 (5), 1388–1412. https://doi.org/10.1017/S0272263122000420 (2023). Housen, A., De Clercq, B., Kuiken, F. & Vedder, I. Multiple approaches to complexity in second language research. Second Lang. Res. 35 (1), 3–21. https://doi.org/10.1177/0267658318809765 (2019). Jaeger, T. F. Redundancy and reduction: Speakers manage syntactic information density. Cogn. Psychol. 61 (1), 23–62. https://doi.org/10.1016/j.cogpsych.2010.02.002 (2010). Kuiken, F. Linguistic complexity in second language acquisition. Linguistics Vanguard . 9 (s1), 91–104. https://doi.org/10.1515/lingvan-2021-0112 (2023). Lai, R. K. Y. & Do, Y. Large-sample confidence intervals of information-theoretic measures in linguistics. J. Res. Des. Stat. Linguistics Communication Sci. 6 (1), 19–54. https://doi.org/10.1558/jrds.40134 (2020). Levy, R. Expectation-based syntactic comprehension. Cognition 106 (3), 1126–1177. https://doi.org/10.1016/j.cognition.2007.05.006 (2008). Mekheimer, M. A. & Fageeh, A. I. Prioritizing information over grammar: A behavioral investigation of information density and rhetorical discourse effects on EFL listening comprehension. Discover Educ. 4 (1). https://doi.org/10.1007/s44217-025-00411-y (2025). Oh, Y. M. Linguistic complexity and information: Quantitative approaches (Doctoral dissertation, University of Lyon 2). (2015). Sun, K. & Wang, R. Using the relative entropy of linguistic complexity to assess L2 language proficiency development. Entropy 23 (8). https://doi.org/10.3390/e23081080 (2021). Article 1080. Trujillo, J. P. & Holler, J. Multimodal information density is highest in question beginnings, and early entropy is associated with fewer but longer visual signals. Discourse Processes . 62 (2), 69–88. https://doi.org/10.1080/0163853X.2024.2413314 (2025). Vandeweerd, N., Housen, A. & Paquot, M. Comparing the longitudinal development of phraseological complexity across oral and written tasks. Humanit. Social Sci. Commun. 9 (1), 1–14. https://doi.org/10.1038/s41599-023-02151-6 (2022). Wang, G., Wang, H. & Wang, L. Kolmogorov complexity metrics in assessing L2 proficiency: An information-theoretic approach. Front. Psychol. 13 , 1024147. https://doi.org/10.3389/fpsyg.2022.1024147 (2022). Yang, Y. & He, X. Lexical richness in Chinese university students’ EFL writing: A corpus-based comparison. Humanit. Social Sci. Commun. 12 , 1199. https://doi.org/10.1057/s41599-025-05560-x (2025). Additional Declarations No competing interests reported. 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Much SLA work has operationalized this development through accuracy\u0026ndash;fluency\u0026ndash;complexity indices, lexical richness measures, and syntactic complexity metrics (e.g., Housen et al., \u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e2019\u003c/span\u003e; Kuiken, \u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e2023\u003c/span\u003e). Yet a persistent limitation is that many commonly used indices are either (a)feature-counting measures that do not directly capture the distributionof choices, or (b) text-length sensitive proxies that conflate development with verbosity. This motivates a complementary lens: information density, understood as how much uncertainty (or predictability) is carried by the linguistic signal and how that uncertainty is distributed across lexical, grammatical, and phraseological options.\u003c/p\u003e \u003cp\u003eInformation theory\u0026mdash;originally developed to model communication systems\u0026mdash;offers formal tools for measuring uncertainty via entropy, and for comparing distributions via cross-entropy and relative entropy (KL divergence). In linguistic research, entropy-family measures have been increasingly used to capture complexity as unpredictability or disorder in a probabilistic system, while compression-based approaches approximate Kolmogorov complexity as a holistic measure of information content and redundancy (Ehret \u0026amp; Szmrecsanyi, \u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e2016\u003c/span\u003e). These tools are especially promising for interlanguage because development can be reframed as a shift in distributions: learners' probability mass gradually reorganizes toward target-like allocation, with subsystem-specific divergences persisting or reappearing under different task conditions.\u003c/p\u003e \u003cp\u003eA second motivation comes from research on information distribution in discourse, processing, and L2 pedagogy. Entropy- and surprisal-based measures have been advanced as direct operationalizations of information density and information rate, capturing how uncertainty and predictability are managed across unfolding linguistic signals and how this management relates to cognitive resource allocation (Jaeger, \u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e2010\u003c/span\u003e; Levy, \u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e2008\u003c/span\u003e). Work on bilingual speech further indicates that information encoding and transmission profiles can differ systematically between L1 and L2 production, implying that information packaging is not merely a surface attribute of texts but a fundamental property of bilingual communication (Bradlow, \u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e2022\u003c/span\u003e).\u003c/p\u003e \u003cp\u003eConverging evidence from interactional research also shows that information density is position-sensitive within discourse: density may peak in specific structural locations (e.g., question onsets), and such early entropy patterns can covary with the distribution of multimodal signals, suggesting that information regulation is coordinated across linguistic and visual channels rather than being uniform across an utterance (Trujillo \u0026amp; Holler, \u003cspan citationid=\"CR19\" class=\"CitationRef\"\u003e2025\u003c/span\u003e).\u003c/p\u003e \u003cp\u003eImportantly, pedagogically oriented evidence aligns with these processing-based accounts: in EFL listening, learners may prioritize message extraction under conditions of high informational load, and comprehension outcomes can shift as discourse varies in information density and rhetorical elaboration\u0026mdash;highlighting the instructional relevance of quantifying density as a property of discourse design (Mekheimer \u0026amp; Fageeh, \u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e2025\u003c/span\u003e). Taken together, these strands motivate entropy-based measurement as a route to a more unified account of interlanguage variability, proficiency, and communicative efficiency, while also grounding the construct in demonstrable consequences for L2 comprehension and classroom discourse selection.\u003c/p\u003e \u003cp\u003eAgainst this background, the present study develops an integrated, multi-level approach to quantifying information density in learner production. Rather than treating \"more entropy\" as inherently better or worse, we treat entropy as a diagnostic description of how learner systems allocate probability mass across competing linguistic options, and how that allocation compares to target and peer benchmarks.\u003c/p\u003e \u003cp\u003eThe study was guided by the following research questions:\u003c/p\u003e \u003cp\u003e \u003col\u003e \u003cspan\u003e \u003cli\u003e \u003cp\u003eTo what extent does lexical information density, operationalized as lexical entropy (Hₗₑₓ), vary across the B1, B2, C1, and L1 groups?\u003c/p\u003e \u003c/li\u003e \u003c/span\u003e \u003cspan\u003e \u003cli\u003e \u003cp\u003eTo what extent does grammatical information density, operationalized as grammatical divergence from the native-speaker reference distribution (KL₍gram₎), vary across the B1, B2, C1, and L1 groups?\u003c/p\u003e \u003c/li\u003e \u003c/span\u003e \u003cspan\u003e \u003cli\u003e \u003cp\u003eTo what extent does global structural regularity, operationalized as compression ratio (CR), vary across the B1, B2, C1, and L1 groups?\u003c/p\u003e \u003c/li\u003e \u003c/span\u003e \u003cspan\u003e \u003cli\u003e \u003cp\u003eTo what extent does discourse-positional information packaging, operationalized as the positional concentration index (PCI), vary across the B1, B2, C1, and L1 groups?\u003c/p\u003e \u003c/li\u003e \u003c/span\u003e \u003cspan\u003e \u003cli\u003e \u003cp\u003eDo these four information-density indicators together reveal a coordinated developmental gradient from lower-proficiency interlanguage to native-speaker performance?\u003c/p\u003e \u003c/li\u003e \u003c/span\u003e \u003cspan\u003e \u003cli\u003e \u003cp\u003eCan interlanguage be empirically modeled as a probabilistic system whose lexical, grammatical, structural, and discourse-positional distributions undergo systematic reorganization as proficiency increases?\u003c/p\u003e \u003c/li\u003e \u003c/span\u003e \u003c/ol\u003e \u003c/p\u003e"},{"header":"Literature Review","content":"\u003cdiv id=\"Sec3\" class=\"Section2\"\u003e \u003ch2\u003eEntropy, information density, and linguistic complexity\u003c/h2\u003e \u003cp\u003eInformation-theoretic approaches to linguistic complexity conceptualize \"complexity\" as the degree of unpredictability in linguistic choices rather than simply the number of structures used. On this view, complexity is fundamentally a distributional property: a linguistic system is more complex when outcomes are less predictable and, in a complementary sense, less compressible. Ehret and Szmrecsanyi (\u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e2016\u003c/span\u003e) provides a foundational synthesis of this tradition by distinguishing Shannon entropy (uncertainty over outcomes) from Kolmogorov complexity (information content as description length), and by motivating their relevance for variationist and usage-based accounts of linguistic organization (Oh, \u003cspan citationid=\"CR17\" class=\"CitationRef\"\u003e2015\u003c/span\u003e).\u003c/p\u003e \u003cp\u003eMethodologically, entropy and surprisal have been treated as direct tools for quantifying information density because they operationalize uncertainty and expected information in a principled, comparable way (Jaeger, \u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e2010\u003c/span\u003e; Levy, \u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e2008\u003c/span\u003e). In parallel, scholarship on complexity measurement highlights that \"complexity\" is multidimensional and theory-dependent, and that different metrics index different facets (e.g., dispersion, sequential dependence, redundancy, structural option space), making construct alignment essential (Ehret et al., \u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e2023\u003c/span\u003e; \u0026Ccedil;\u0026ouml;ltekin \u0026amp; Rama, \u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e2023\u003c/span\u003e; Bentz et al., \u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2023\u003c/span\u003e). Within SLA, the \"complexity turn\" has amplified this caution: importing concepts from complexity science requires clarity about whether they function as descriptive indices, explanatory mechanisms, or meta-theoretical commitments, and each entails different evidentiary burdens (Han et al., \u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e2023\u003c/span\u003e).\u003c/p\u003e \u003cp\u003eFor interlanguage research specifically, these considerations imply that entropy-based indices should be interpreted as distributional descriptors whose meaning depends on the representational level (lexical vs. grammatical vs. phraseological), the task ecology, and\u0026mdash;crucially\u0026mdash;the reference system against which learner distributions are evaluated. This interpretive stance also aligns with pedagogically oriented work on information density in L2 contexts, where discourse design and informational load can measurably modulate comprehension and learner prioritization strategies, reinforcing the need to treat \"density/complexity\" as context-sensitive rather than as a single monotonic proficiency marker (Mekheimer \u0026amp; Fageeh, \u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e2025\u003c/span\u003e).\u003c/p\u003e \u003c/div\u003e\n\u003ch3\u003eRelative entropy and proficiency development in L2\u003c/h3\u003e\n\u003cp\u003eA central strand of applied work operationalizes interlanguage development via relative entropy (KL divergence), measuring how far learner distributions deviate from a reference distribution and how this distance changes over time or across proficiency levels. Sun and Wang (\u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e2021\u003c/span\u003e) propose relative entropy of linguistic complexity as a way to assess L2 proficiency development, arguing that distributional discrimination provides a sensitive index of developmental change beyond traditional complexity metrics. Complementing this, Wang et al. (\u003cspan citationid=\"CR21\" class=\"CitationRef\"\u003e2022\u003c/span\u003e) extend information-theoretic evaluation by incorporating Kolmogorov complexity metrics alongside entropy-based measures to assess L2 proficiency, reinforcing the idea that proficiency can be modeled through distributional organization and compressibility rather than isolated counts of forms; this interpretation is further supported by evidence that Kolmogorov-complexity measures can differentiate both L2 groups and L1 backgrounds (Alzahrani, \u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e2024\u003c/span\u003e).\u003c/p\u003e \u003cp\u003eThese studies collectively shift the focus from \"how many structures learners use\" to \"how learners distribute probability mass across available structures,\" which is more aligned with usage-based conceptions of interlanguage and with the empirical fact that learners can produce advanced forms sporadically while still relying heavily on a small set of high-probability templates.\u003c/p\u003e\n\u003ch3\u003eCompression and Kolmogorov complexity in learner language\u003c/h3\u003e\n\u003cp\u003eAnother influential route uses compression as a practical approximation to Kolmogorov complexity. By estimating how compressible learner output is, researchers can capture global redundancy and structure that may not be visible in feature-by-feature measures. Ehret and Szmrecsanyi (\u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e2019\u003c/span\u003e) demonstrate how compressing learner language can provide an information-theoretic measure of complexity in SLA production data, offering a holistic complement to entropy measures and enabling comparisons across texts without relying on preselected structural inventories; related evidence also suggests that such measures are sensitive to proficiency grouping and L1-background effects (Alzahrani, \u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e2024\u003c/span\u003e).\u003c/p\u003e \u003cp\u003eCompression-based measures are particularly attractive for interlanguage because they can reflect both (a) formulaic repetition (high compressibility) and (b) chaotic variability (low compressibility) depending on how the learner system is organized. This supports a non-linear developmental expectation: development may involve both increasing variability in some domains (exploration) and decreasing variability in others (stabilization).\u003c/p\u003e\n\u003ch3\u003ePhraseology, sequential dependence, and moving beyond bag-of-words\u003c/h3\u003e\n\u003cp\u003eA known limitation of simple entropy over word types is that it can miss phraseological organization and sequential constraints. Recent work, therefore, emphasizes moving beyond bag-of-words representations to incorporate phraseological patterning and distributional dependence across sequences, especially in learner language where proficiency is often visible in stable collocations, lexical bundles, and constructional preferences (Vandeweerd, Housen, \u0026amp; Paquot, \u003cspan citationid=\"CR20\" class=\"CitationRef\"\u003e2022\u003c/span\u003e; Ehret \u0026amp; Szmrecsanyi, \u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e2019\u003c/span\u003e). Related learner-focused research operationalizes phraseological diversity using normalized entropy over multiword units, highlighting how entropy can be adapted to phraseological structure rather than single-word diversity alone (Vandeweerd, Housen, \u0026amp; Paquot, \u003cspan citationid=\"CR20\" class=\"CitationRef\"\u003e2022\u003c/span\u003e).\u003c/p\u003e\n\u003ch3\u003eInformation density, processing, and modality\u003c/h3\u003e\n\u003cp\u003eInformation density refers to the amount and distribution of informational load within a stretch of discourse, commonly operationalized through measures such as entropy and surprisal that capture how predictable or uncertain linguistic material is at a given point in unfolding communication (Karimi et al., 2024; Trujillo \u0026amp; Holler, 2024). Processing, in this context, concerns the real-time cognitive work involved in encoding, transmitting, and interpreting that load, rather than treating it as a static textual property; recent evidence shows that entropy can affect lexical processing over and above traditional predictability measures, while bilingual research indicates that L1 and L2 speech differ in their information-encoding and transmission profiles (Bradlow, \u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e2022\u003c/span\u003e; Karimi et al., 2024). Modality refers to the channel through which meaning is conveyed\u0026mdash;most centrally the verbal and visual channels in face-to-face interaction\u0026mdash;and current multimodal research shows that communicative meaning is distributed across speech, gesture, gaze, and other bodily signals rather than being carried by language alone (\u0026Uuml;nal et al., 2024). From this perspective, information density is best understood as a dynamic property of discourse that is shaped by temporal processing demands, discourse position, and the coordination of multiple semiotic resources across modalities (Trujillo \u0026amp; Holler, \u003cspan citationid=\"CR19\" class=\"CitationRef\"\u003e2025\u003c/span\u003e; \u0026Uuml;nal et al., 2024).\u003c/p\u003e \u003cp\u003eInformation density should not be treated as a purely static property of texts, because it is closely tied to how language is processed in real time. Research on bilingual speech shows that L1 and L2 production differ in their patterns of information encoding and transmission, suggesting that developmental differences extend beyond grammatical choice to include the temporal organization of informational load across utterances (Bradlow, \u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e2022\u003c/span\u003e). This perspective is reinforced by multimodal work demonstrating that entropy is unevenly distributed across discourse positions and that higher density in early stretches of discourse may align with shifts in accompanying visual signals (Trujillo \u0026amp; Holler, \u003cspan citationid=\"CR19\" class=\"CitationRef\"\u003e2025\u003c/span\u003e). Taken together, these findings support an understanding of interlanguage information density as dynamically structured by position, modality, and task demands rather than as a uniform textual characteristic.\u003c/p\u003e \u003cdiv id=\"Sec8\" class=\"Section2\"\u003e \u003ch2\u003eRelated evidence from translation and mediated language varieties\u003c/h2\u003e \u003cp\u003eEntropy measures have also been used to investigate simplification and variation in mediated language (e.g., translated, interpreted, or machine-generated text). Entropy analyses of lexical and syntactic simplification provide methodological precedents for distinguishing language varieties via uncertainty metrics and for interpreting entropy relative to density and richness (Liu et al., 2022; Wang et al., 2025; Yao \u0026amp; Fan, 2025). While these studies are not SLA per se, they strengthen the general claim that entropy-family measures can capture systematic differences in how linguistic information is distributed across varieties, genres, and production conditions\u0026mdash;conditions that parallel many SLA contrasts (e.g., timed vs untimed writing; spoken vs written performance; careful vs spontaneous production).\u003c/p\u003e \u003c/div\u003e\n\u003ch3\u003eSynthesis and gap\u003c/h3\u003e\n\u003cp\u003eExisting work shows that (a) relative entropy can discriminate proficiency groups and model developmental change (Sun \u0026amp; Wang, \u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e2021\u003c/span\u003e; Wang et al., \u003cspan citationid=\"CR21\" class=\"CitationRef\"\u003e2022\u003c/span\u003e), (b) compression approximations to Kolmogorov complexity capture holistic redundancy and structure in learner production (Ehret \u0026amp; Szmrecsanyi, \u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e2019\u003c/span\u003e; Ehret \u0026amp; Szmrecsanyi, \u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e2016\u003c/span\u003e), and (c) information density has processing- and discourse-sensitive correlates (Bradlow, \u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e2022\u003c/span\u003e; Trujillo \u0026amp; Holler, \u003cspan citationid=\"CR19\" class=\"CitationRef\"\u003e2025\u003c/span\u003e). What remains underdeveloped is a unified interlanguage account that integrates these measures across levels (lexical, grammatical, phraseological), separates diversity from contextual predictability, and interprets entropy patterns as subsystem-specific reorganization rather than a single monotonic index of proficiency (Ehret et al., \u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e2023\u003c/span\u003e; Han et al., \u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e2023\u003c/span\u003e; Housen et al., \u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e2019\u003c/span\u003e; Kuiken, \u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e2023\u003c/span\u003e). The present paper addresses this gap by proposing a clean theoretical framework for \"information density of interlanguage\" and by clarifying how entropy-family measures map onto SLA constructs.\u003c/p\u003e \u003cp\u003eIn this study, information density (ID) is treated as a multi-level property of interlanguage systems that emerges from how probability mass is distributed across competing linguistic options. Accordingly, four complementary indices are used to capture ID across lexical choice, grammatical sequencing, global structural regularity, and discourse packaging. Lexical entropy (Hₗₑₓ)quantifies the dispersion of lemma distributions, such that higher values indicate broader lexical exploration and less concentration on a limited set of forms. Grammatical divergence (KL₍gram₎) operationalizes directed distance from a target reference distribution estimated over POS trigrams, where higher values reflect greater interlanguage distance from target-like sequential constraints. Compression ratio (CR)provides a holistic proxy for structural regularity, with lower values indicating greater compressibility and redundancy (i. e., more predictable structure). Finally, thepositional concentration index (PCI)captures discourse-level packaging by expressing the ratio of opening-segment entropy to whole-text entropy; values greater than 1.00 indicate front-loaded informational density, whereas values near or below 1.00 indicate relatively uniform or delayed information distribution.\u003c/p\u003e\n\u003ch3\u003eTheoretical Framework\u003c/h3\u003e\n\u003cp\u003eThe proposed framework conceptualizes interlanguage as a probabilistic system governed by internal and external constraints (Fig.\u0026nbsp;1). This approach treats development not as the linear acquisition of rules, but as the dynamic reallocation of probability mass across competing linguistic options.\u003c/p\u003e \u003cp\u003eFigure 1 formalizes this perspective by representing interlanguage as a constrained probabilistic system in which linguistic output arises from the interaction of knowledge, attention, task demands, and processing capacity (Han et al., \u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e2023\u003c/span\u003e; Housen et al., \u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e2019\u003c/span\u003e). Within this system, development is modeled as the redistribution of probability mass across four interrelated layers: lexical choice, grammatical sequencing, phraseological patterning, and discourse-position packaging (Ehret \u0026amp; Szmrecsanyi, \u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e2016\u003c/span\u003e; Sun \u0026amp; Wang, \u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e2021\u003c/span\u003e). These layers are examined through complementary information-theoretic measures, with entropy indexing distributional dispersion, KL divergence capturing distance from target-like organization, and compression reflecting global structural regularity (Ehret \u0026amp; Szmrecsanyi, \u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e2016\u003c/span\u003e; Wang et al., \u003cspan citationid=\"CR21\" class=\"CitationRef\"\u003e2022\u003c/span\u003e).\u003c/p\u003e \u003cp\u003eThe model also assumes that development is non-linear and subsystem-specific: lexical variability may increase as the learner repertoire expands, whereas grammatical divergence and structural redundancy may decline as usage becomes more constrained and conventionalized (Sun \u0026amp; Wang, \u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e2021\u003c/span\u003e; Ehret et al., \u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e2023\u003c/span\u003e). Accordingly, the framework links distributional change at the level of linguistic choices to broader developmental processes of stabilization, convergence, and information management in interlanguage systems (Kuiken, \u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e2023\u003c/span\u003e; Han et al., \u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e2023\u003c/span\u003e).\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eInterlanguage can be modeled as a probabilistic system whose outputs reflect constrained choice. Learners operate under limits of knowledge, attention, and processing capacity, so production involves selecting among competing lexical, grammatical, and phraseological options with different probabilities. Development, in this view, is not simply the accumulation of rules, but a gradual reallocation of probability mass: forms that are more contextually appropriate become more strongly preferred, while competing variants are reduced, restricted to narrower contexts, or reorganized. This distributional perspective aligns with information-theoretic treatments of linguistic complexity as unpredictability (entropy) and/or description length (compression/Kolmogorov complexity) (Ehret \u0026amp; Szmrecsanyi, \u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e2016\u003c/span\u003e).\u003c/p\u003e \u003cp\u003eWithin this framework, information density (ID) is defined as the degree of uncertainty or predictability in learner choices within a specified representational space. The article operationalizes ID using four complementary metrics (see Fig.\u0026nbsp;1). Shannon entropy (H) captures distributional dispersion (e.g., how evenly outcomes are spread across POS sequences or lexical options). Cross-entropy expresses the expected coding cost of learner output under a target reference model, providing an interpretable measure of \"learner surprisal\" relative to target expectations. Relative entropy (KL divergence) quantifies directed distance between learner and target-norm distributions, and thus provides a principled measure of interlanguage distance and convergence (Sun \u0026amp; Wang, \u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e2021\u003c/span\u003e). Finally, compression-based complexity offers a holistic proxy for redundancy and structure via compressibility, approximating description-length notions of complexity (Ehret \u0026amp; Szmrecsanyi, \u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e2016\u003c/span\u003e).\u003c/p\u003e \u003cp\u003eBecause interlanguage organization is multi-component, information density is examined across four layers (Fig.\u0026nbsp;1): lexical ID (word/lemma distributions and repertoire size), grammatical ID (morphosyntactic categories and the stabilization of constructional preferences),phraseological ID (the conventionalization of collocations and multiword units), anddiscourse-position ID (how information is packaged across segments such as openings versus mid-sections). This multi-level design is essential because the same learner can show high predictability in one subsystem (e.g., reliance on safe grammatical defaults) and high variability in another (e.g., lexical exploration), and a single global index would obscure that asymmetry.\u003c/p\u003e \u003cp\u003eDevelopment is therefore modeled as \u003cem\u003ereorganization rather than linear growth\u003c/em\u003e. Changes in entropy are expected to be non-monotonic and subsystem-specific: lexical entropy may increase as the repertoire expands and exploratory variation grows, while grammatical and phraseological entropy may decrease as usage becomes more constrained by target norms and as competition among variants is resolved (Sun \u0026amp; Wang, \u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e2021\u003c/span\u003e). Across time, learners may move from highly compressible template-based production, to a phase of broader experimentation, and then toward more efficient, conventionalized variability\u0026mdash;where flexibility is retained but choices become more contextually predictable.\u003c/p\u003e \u003cp\u003eFinally, the framework provides an interpretive bridge to core SLA questions. First, entropy profiles distinguish stabilityversusexploration: high grammatical entropy can reflect unresolved competition among forms, whereas low entropy may signal either genuine stabilization or overreliance on defaults. Second, target convergence can be tracked through decreasing KL divergence, which operationalizes the reduction of interlanguage distance from benchmark distributions (Sun \u0026amp; Wang, \u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e2021\u003c/span\u003e). Third, the approach captures information packaging across modality and discourse position, allowing analysis of whether learners regulate density differently in speech versus writing and whether they concentrate informational load in specific discourse regions.\u003c/p\u003e"},{"header":"Methods","content":" \u003cdiv id=\"Sec11\" class=\"Section2\"\u003e \u003cdiv id=\"Sec12\" class=\"Section3\"\u003e \u003ch2\u003eResearch Design\u003c/h2\u003e \u003cp\u003eThis study employs a quantitative, corpus-based design to operationalize information density in interlanguage using entropy-family measures and distributional distance metrics. The core assumption is that interlanguage reflects constrained probabilistic choice: learners select among competing lexical, grammatical, and phraseological options under limits of knowledge, attention, and processing capacity. Accordingly, the analysis is not centered on counting structures or errors per se, but on measuring how learner distributions are organized and how they reorganize across proficiency and converge toward target norms.\u003c/p\u003e \u003cp\u003eThe design combines two complementary comparisons. First, a developmental (between-level) comparison examines whether information density metrics shift systematically across CEFR-aligned proficiency bands. Second, a learner\u0026ndash;target comparison quantifies the distance between learner distributions and native-speaker benchmarks using cross-entropy and KL divergence. Together, these comparisons allow interlanguage development to be modeled as distributional reallocation\u0026mdash;probability mass moving toward contextually appropriate options while competing variants are pruned, restricted, or reorganized over time (Sun \u0026amp; Wang, \u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e2021\u003c/span\u003e; Ehret \u0026amp; Szmrecsanyi, \u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e2016\u003c/span\u003e).\u003c/p\u003e \u003c/div\u003e \u003c/div\u003e \u003cdiv id=\"Sec13\" class=\"Section2\"\u003e \u003ch2\u003eParticipants and Corpora\u003c/h2\u003e \u003cp\u003eThe study utilizes a balanced dataset comprising a learner corpus of L2 English writing and a native-speaker (L1) reference corpus. The learner corpus features argumentative essays from 150 participants, stratified into three proficiency bands according to CEFR standards: B1 (Threshold), B2 (Vantage), and C1 (Effective Operational Proficiency), with 50 texts per band. To ensure the stability of entropy-based measures, token counts are balanced across levels, and inclusion is restricted to texts meeting a minimum length threshold.\u003c/p\u003e \u003cp\u003eThe reference corpus consists of 50 argumentative essays written by L1 English university students. This benchmark is matched to the learner data in terms of topic domain, prompt type, and length distribution. To maintain analytic integrity, a controlled-genre approach is strictly applied, ensuring all 200 texts conform to the argumentative writing constraint. All data are anonymized and cleaned to remove non-linguistic artifacts such as metadata and duplicated headers prior to analysis.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec14\" class=\"Section2\"\u003e \u003ch2\u003eData Preparation and Linguistic Annotation\u003c/h2\u003e \u003cp\u003eTo ensure comparability across the four representational layers (Lexical, Grammatical, Phraseological, and Discourse), all texts undergo a uniform processing pipeline:\u003c/p\u003e \u003cp\u003e \u003cul\u003e \u003cli\u003e \u003cp\u003e \u003cb\u003ePreprocessing\u003c/b\u003e: Texts are tokenized, segmented into sentences, and lemmatized.\u003c/p\u003e \u003c/li\u003e \u003cli\u003e \u003cp\u003e \u003cb\u003eTagging\u003c/b\u003e: Part-of-speech (POS) tags and dependency parses are assigned using a consistent NLP toolchain to capture morphosyntactic and syntactic patterning.\u003c/p\u003e \u003c/li\u003e \u003cli\u003e \u003cp\u003e \u003cb\u003eQuality Control\u003c/b\u003e: A stratified manual audit of 10% of the texts per proficiency band is conducted to verify the plausibility of automated tags, specifically addressing potential biases from fragmentary clauses or non-canonical word order.\u003c/p\u003e \u003c/li\u003e \u003cli\u003e \u003cp\u003eNormalization: Conservative normalization is applied only to resolve preprocessing anomalies (e.g., tokenization artifacts). In line with interlanguage research principles, learner language is not\"corrected\"to target forms to ensure the analytic representation accurately reflects the learner's current probabilistic system\u003c/p\u003e \u003c/li\u003e \u003c/ul\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec15\" class=\"Section2\"\u003e \u003ch2\u003eUnits of Analysis\u003c/h2\u003e \u003cp\u003eInformation density is computed at four representational layers aligned with Fig.\u0026nbsp;1. Each layer is treated as a distinct outcome space with its own probability distribution, enabling subsystem-specific interpretation rather than a single global \"complexity score.\"\u003c/p\u003e \u003cp\u003e \u003col\u003e \u003cspan\u003e \u003cli\u003e \u003cp\u003e \u003cb\u003eLexical layer (Lexical ID).\u003c/b\u003e Distributions are computed over word forms and lemmas to model repertoire size and dispersion of lexical choice, including function\u0026ndash;content distributions where relevant.\u003c/p\u003e \u003c/li\u003e \u003c/span\u003e \u003cspan\u003e \u003cli\u003e \u003cp\u003e \u003cb\u003eGrammatical layer (Grammatical ID).\u003c/b\u003e Distributions are computed over POS sequences (bigrams and trigrams) and, where included, dependency-relation patterns to capture morphosyntactic organization and sequencing stability.\u003c/p\u003e \u003c/li\u003e \u003c/span\u003e \u003cspan\u003e \u003cli\u003e \u003cp\u003ePhraseological layer (Phraseological ID). Distributions are computed over recurrent collocations and multiword units (e.g., high-frequency bigrams/trigrams and lexical bundles exceeding a minimum frequency threshold) to capture conventionalization and idiomatic pattern control.\u003c/p\u003e \u003c/li\u003e \u003c/span\u003e \u003cspan\u003e \u003cli\u003e \u003cp\u003eDiscourse-position layer (Discourse-Position ID). Each text is segmented proportionally into beginning, middle, and end regions (e.g., first 20%, middle 60%, final 20%), and entropy measures are computed separately per segment to test whether information density is uniform or concentrated in specific structural locations.\u003c/p\u003e \u003c/li\u003e \u003c/span\u003e \u003c/ol\u003e \u003c/p\u003e \u003cp\u003eGiven entropy's sensitivity to sample size, analyses are conducted under controlled token windows where appropriate, and distributions are estimated using normalized procedures to support comparability across groups.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec16\" class=\"Section2\"\u003e \u003ch2\u003eInformation-Theoretic Measures\u003c/h2\u003e \u003cp\u003eInformation density is operationalized using four complementary measures that capture dispersion, target-based predictability, distance-to-target, and holistic redundancy/structure.\u003c/p\u003e \u003cp\u003eShannon entropy (H)is calculated for each distributional representation to quantify dispersion. Higher entropy reflects a more even distribution across outcomes, whereas lower entropy indicates concentration around fewer options. Entropy is computed for lexical distributions, grammatical sequence distributions, and phraseological distributions.\u003c/p\u003e \u003cp\u003eCross-entropyestimates the expected coding cost of learner output under a target reference distribution, providing an interpretable measure of how surprising learner production is relative to target expectations.\u003c/p\u003e \u003cp\u003eRelative entropy (KL divergence)quantifies directed distance between learner and target distributions and serves as the primary operationalization of interlanguage distance and convergence (Sun \u0026amp; Wang, \u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e2021\u003c/span\u003e). Because KL divergence is undefined when the target assigns zero probability to learner-observed events, additive smoothing is applied and sensitivity checks are performed to ensure results are not artifacts of a single smoothing choice.\u003c/p\u003e \u003cp\u003eCompression-based complexity is used as a holistic proxy for redundancy and structural regularity. Text strings are standardized and compressed using a lossless compression method (e.g., LZMA), and the compression ratio is interpreted as an approximation of description-length complexity consistent with information-theoretic views of complexity as compressibility (Ehret \u0026amp; Szmrecsanyi, \u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e2016\u003c/span\u003e). This measure is treated as complementary to entropy: it captures global regularities (e.g., repetitive templates, recurring strings) that may not be fully reflected in token-level dispersion.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec17\" class=\"Section2\"\u003e \u003ch2\u003eStatistical Analysis\u003c/h2\u003e \u003cp\u003eAnalysis proceeds in two stages. First, group-level differences in entropy and distance metrics are tested across proficiency bands and against the native-speaker benchmark. Depending on distributional diagnostics, either parametric tests (ANOVA) or non-parametric alternatives (Kruskal\u0026ndash;Wallis) are used, with effect sizes reported alongside significance tests.\u003c/p\u003e \u003cp\u003eSecond, to account for unequal text counts and nested structure, mixed-effects modeling is used where appropriate. Entropy outcomes are modeled with proficiency as a fixed effect, and random intercepts are included for text (and for prompt where prompt identifiers are available). For discourse-position analyses, repeated-measures formulations test whether entropy differs across segments within texts and whether segment effects interact with proficiency. Confidence intervals for key effects are estimated via bootstrapping (1,000 resamples) to reduce dependence on large-sample assumptions, in line with broader methodological work on estimating uncertainty around information-theoretic measures (Lai \u0026amp; Do, \u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e2020\u003c/span\u003e).\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec18\" class=\"Section2\"\u003e \u003ch2\u003eReliability, Robustness, and Sensitivity Checks\u003c/h2\u003e \u003cp\u003eSeveral checks are implemented to strengthen measurement credibility. Entropy is recomputed under multiple representations (token vs lemma) to reduce construct under-identification. Key conclusions are also tested under different n-gram sizes (bigrams vs trigrams) and, where relevant, under controlled-length subsamples (equal token windows) to ensure findings are not driven by text length. For KL divergence, results are examined under different smoothing parameters. For compression metrics, preprocessing is strictly standardized so that differences reflect linguistic patterning rather than formatting artifacts.\u003c/p\u003e \u003cp\u003eInterpretation is anchored in converging evidence: because entropy may increase under both productive diversification and noisy instability, entropy shifts are read alongside KL divergence (distance-to-target), cross-entropy (target-based coding cost), and layer-specific patterns (lexical vs grammatical vs phraseological), consistent with the non-monotonic reorganization view.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec19\" class=\"Section2\"\u003e \u003ch2\u003eReproducibility and Technical Parameters\u003c/h2\u003e \u003cp\u003e \u003cb\u003eSoftware and toolchain.\u003c/b\u003e All texts were processed using spaCy (v3.7.2) with the en_core_web_md model for tokenization, sentence segmentation, lemmatization, and POS tagging.\u003c/p\u003e \u003cp\u003e \u003cb\u003eTagset and n-grams.\u003c/b\u003e Grammatical information density was computed using the Universal POS tagset. KL divergence (KL₍gram₎) was computed over POS trigrams (n\u0026thinsp;=\u0026thinsp;3) to capture sequential dependence in morphosyntactic selection.\u003c/p\u003e \u003cp\u003e \u003cb\u003eSmoothing and thresholds.\u003c/b\u003e To avoid zero-probability events in KL calculations, additive (Laplace) smoothing was applied with α\u0026thinsp;=\u0026thinsp;0.01. Only essays exceeding 300 tokens were included to ensure stable entropy estimation and reliable n-gram counts.\u003c/p\u003e \u003cp\u003e \u003cb\u003eLength control strategy.\u003c/b\u003e Primary analyses were computed on fixed-length windows of 250 tokens sampled from the center of each text to minimize length-driven artifacts. As a robustness check, all key results were replicated on full texts using bootstrapped subsampling (1,000 iterations), confirming that group-level distributional profiles were stable under varying text lengths.\u003c/p\u003e \u003cp\u003e \u003cb\u003eL1 baseline for KL₍gram₎.\u003c/b\u003e The L1 group was treated as a stochastic sample. For each native text, KL₍gram₎ was computed against a leave-one-out (LOO) L1 reference distribution constructed from the remaining 49 native texts, providing a baseline estimate of within-native variation.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec20\" class=\"Section2\"\u003e \u003ch2\u003eEthical Considerations\u003c/h2\u003e \u003cp\u003eThe study protocol was reviewed and approved by the Institutional Review Board of the Faculty of Education, Beni-Suef University (Approval No. BSU-FoE-004-01-01-2016). All methods were carried out in accordance with relevant guidelines and regulations. Informed consent was obtained from all participants and/or their legal guardians prior to data collection and analysis. All learner data were anonymized prior to analysis. The study reports only aggregated findings and does not disclose any identifying personal or contextual metadata.\u003c/p\u003e \u003c/div\u003e"},{"header":"Results","content":"\u003cdiv id=\"Sec22\" class=\"Section2\"\u003e \u003ch2\u003eDescriptive Profile\u003c/h2\u003e \u003cp\u003eTable\u0026nbsp;1 reports the descriptive profile for the balanced comparison (50 texts per group). The four component indicators show a clear and fully ordered developmental gradient from B1 to L1. Lexical entropy increases steadily across levels, indicating broader lexical dispersion at higher proficiency. By contrast, grammatical divergence and compression ratio decline monotonically, showing closer approximation to target-like grammatical distributions and greater structural regularity. Positional concentration also rises across levels, indicating progressively stronger front-loading of informational load in the opening segment of the text.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab1\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 1\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003e\u003cem\u003eDescriptive statistics by group (Mean, SD)\u003c/em\u003e\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"5\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eGroup\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eHₗₑₓ\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eKL₍gram₎\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eCR\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003ePCI\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eB1 (n\u0026thinsp;=\u0026thinsp;50)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e4.125 (0.087)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e1.859 (0.089)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.664 (0.017)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e0.891 (0.032)\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eB2 (n\u0026thinsp;=\u0026thinsp;50)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e4.438 (0.030)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e1.482 (0.026)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.614 (0.008)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e1.061 (0.016)\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eC1 (n\u0026thinsp;=\u0026thinsp;50)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e4.835 (0.033)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.967 (0.028)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.579 (0.009)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e1.191 (0.016)\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eL1 (n\u0026thinsp;=\u0026thinsp;50)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e5.255 (0.045)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.122 (0.009)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.519 (0.008)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e1.279 (0.021)\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eOperational definitions (Table\u0026nbsp;1). Hₗₑₓ = lexical entropy, with higher values indicating greater lexical dispersion. KL₍gram₎ = grammatical divergence from the L1 POS-trigram reference distribution, with lower values indicating greater target convergence. CR\u0026thinsp;=\u0026thinsp;compression ratio, with lower values indicating greater compressibility and structural regularity. PCI\u0026thinsp;=\u0026thinsp;positional concentration index, computed as opening-segment entropy divided by whole-text entropy; values above 1.00 indicate front-loaded information packaging. The composite IDS_Total showed the same stepwise pattern in the statistical outputs, but detailed reporting centers on the four theoretically primary indicators.\u003c/p\u003e \u003cp\u003eAssumption checks and primary vs. robustness analysis strategy\u003c/p\u003e \u003cp\u003eAssumption checks indicated that normality was not consistently met across cells, and homogeneity of variance was also violated for all four focal metrics (Table\u0026nbsp;2A). Because these conditions make classical fixed-variance ANOVA less appropriate, the main inferential results are reported with Welch\u0026rsquo;s ANOVA and Games\u0026ndash;Howell post-hoc comparisons, which are robust to heteroscedasticity and unequal group dispersions.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab2\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 2\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003e\u003cb\u003eA.\u003c/b\u003e \u003cem\u003eHomogeneity of variance across groups (Levene\u0026rsquo;s test)\u003c/em\u003e\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"3\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eMetric\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eLevene F\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003ep\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eHₗₑₓ\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e15.73\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e\u0026lt;\u0026thinsp;.001\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eKL₍gram₎\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e43.34\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e\u0026lt;\u0026thinsp;.001\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eCR\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e11.44\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e\u0026lt;\u0026thinsp;.001\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003ePCI\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e9.00\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e\u0026lt;\u0026thinsp;.001\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eThe heterogeneity pattern is substantively unsurprising. As proficiency increases, the distributions tighten for some metrics, especially KL₍gram₎ and CR, while the L1 group shows a particularly narrow native baseline for grammatical divergence. This confirms that the data should be interpreted with variance-robust procedures rather than pooled-variance assumptions.\u003c/p\u003e \u003cp\u003eTo check that the main findings were not an artifact of the balanced 50/50/50/50 design, the same analyses were rerun using the empirical group-size distribution available in the auxiliary output files (B1\u0026thinsp;=\u0026thinsp;45, B2\u0026thinsp;=\u0026thinsp;55, C1\u0026thinsp;=\u0026thinsp;43, L1\u0026thinsp;=\u0026thinsp;57). The replication produced the same directional ordering and the same pattern of significance across all four indicators (Table\u0026nbsp;2B).\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab3\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 2\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003e\u003cb\u003eB.\u003c/b\u003e \u003cem\u003eRobustness check using empirical group sizes\u003c/em\u003e\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"6\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c6\" colnum=\"6\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eMetric\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eWelch F\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003edf1\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003edf2\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003ep\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c6\"\u003e \u003cp\u003ePattern\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eHₗₑₓ\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e1181.83\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e98.92\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e\u0026lt;\u0026thinsp;.001\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003eB1\u0026thinsp;\u0026lt;\u0026thinsp;B2\u0026thinsp;\u0026lt;\u0026thinsp;C1\u0026thinsp;\u0026lt;\u0026thinsp;L1\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eKL₍gram₎\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e1679.02\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e94.28\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e\u0026lt;\u0026thinsp;.001\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003eB1\u0026thinsp;\u0026gt;\u0026thinsp;B2\u0026thinsp;\u0026gt;\u0026thinsp;C1\u0026thinsp;\u0026gt;\u0026thinsp;L1\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eCR\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e503.30\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e104.61\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e\u0026lt;\u0026thinsp;.001\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003eB1\u0026thinsp;\u0026gt;\u0026thinsp;B2\u0026thinsp;\u0026gt;\u0026thinsp;C1\u0026thinsp;\u0026gt;\u0026thinsp;L1\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003ePCI\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e1323.29\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e104.75\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e\u0026lt;\u0026thinsp;.001\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003eB1\u0026thinsp;\u0026lt;\u0026thinsp;B2\u0026thinsp;\u0026lt;\u0026thinsp;C1\u0026thinsp;\u0026lt;\u0026thinsp;L1\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eNote. The robustness analysis reproduces the balanced-design findings: Hₗₑₓ and PCI increase monotonically, whereas KL₍gram₎ and CR decrease monotonically. All Welch tests remained statistically significant at p \u0026lt; .001.\u003c/p\u003e \u003cdiv id=\"Sec23\" class=\"Section3\"\u003e \u003ch2\u003eOmnibus group effects\u003c/h2\u003e \u003cp\u003eWelch\u0026rsquo;s ANOVA confirmed statistically significant between-group differences for all four indicators (Table\u0026nbsp;3). The effects are exceptionally large, especially for KL₍gram₎ and Hₗₑₓ, showing that proficiency level is strongly associated with the distributional organization of learner writing. The omnibus pattern therefore supports the central claim that interlanguage development involves coordinated reorganization across lexical choice, grammatical sequencing, structural regularity, and discourse-level information packaging.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab4\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 3\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003e\u003cem\u003eOmnibus tests of group differences (Welch ANOVA)\u003c/em\u003e\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"5\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eMetric\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eOmnibus test\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eF\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003edf\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003ep\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eHₗₑₓ\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eWelch ANOVA\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e4770.09\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e3, 104.98\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e\u0026lt;\u0026thinsp;.001\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eKL₍gram₎\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eWelch ANOVA\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e54286.40\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e3, 91.97\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e\u0026lt;\u0026thinsp;.001\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eCR\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eWelch ANOVA\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e1584.40\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e3, 107.02\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e\u0026lt;\u0026thinsp;.001\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003ePCI\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eWelch ANOVA\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e2270.41\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e3, 106.77\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e\u0026lt;\u0026thinsp;.001\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eConsidered together, the omnibus results justify detailed pairwise interpretation. Rather than isolating a single dimension of development, the findings reveal a coherent multi-metric profile in which some indices rise with proficiency and others fall. This mixed directional pattern is theoretically important because it indicates reorganization across subsystems rather than a simple one-directional increase in general complexity.\u003c/p\u003e \u003c/div\u003e \u003c/div\u003e \u003cdiv id=\"Sec24\" class=\"Section2\"\u003e \u003ch2\u003ePost-hoc comparisons and developmental reorganization\u003c/h2\u003e \u003cdiv id=\"Sec25\" class=\"Section3\"\u003e \u003ch2\u003eLexical entropy (Hₗₑₓ): progressive expansion of lexical dispersion\u003c/h2\u003e \u003cp\u003eGames\u0026ndash;Howell comparisons showed that all pairwise differences in Hₗₑₓ were statistically significant (Table\u0026nbsp;4A). The pattern is strictly monotonic (B1\u0026thinsp;\u0026lt;\u0026thinsp;B2\u0026thinsp;\u0026lt;\u0026thinsp;C1\u0026thinsp;\u0026lt;\u0026thinsp;L1), indicating progressive expansion in lexical dispersion as proficiency increases. The largest cumulative gain appears between B1 and C1, but the C1\u0026ndash;L1 contrast also remains significant, showing that advanced learner writing still differs from native writing in the breadth and balance of lexical choice.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab5\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 4\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003e\u003cb\u003eA.\u003c/b\u003e \u003cem\u003eGames\u0026ndash;Howell post-hoc tests for Hₗₑₓ\u003c/em\u003e\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"4\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eComparison\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eMean diff (G1\u0026ndash;G2)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eSE\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003ep\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eB1\u0026ndash;B2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e-0.313\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.013\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e\u0026lt;\u0026thinsp;.001\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eB1\u0026ndash;C1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e-0.710\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.013\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e\u0026lt;\u0026thinsp;.001\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eB1\u0026ndash;L1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e-1.130\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.014\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e\u0026lt;\u0026thinsp;.001\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eB2\u0026ndash;C1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e-0.397\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.006\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e\u0026lt;\u0026thinsp;.001\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eB2\u0026ndash;L1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e-0.817\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.008\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e\u0026lt;\u0026thinsp;.001\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eC1\u0026ndash;L1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e-0.419\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.008\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e\u0026lt;\u0026thinsp;.001\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eThe lexical results therefore point to continuing repertoire expansion rather than early plateauing. Higher proficiency is associated not simply with more words, but with a wider and more evenly distributed lexical choice space.\u003c/p\u003e \u003c/div\u003e \u003c/div\u003e \u003cdiv id=\"Sec26\" class=\"Section2\"\u003e \u003ch2\u003eGrammatical divergence (KL₍gram₎): progressive convergence toward target norms\u003c/h2\u003e \u003cp\u003eGames\u0026ndash;Howell comparisons for KL₍gram₎ again showed that every pairwise contrast was significant (Table\u0026nbsp;4B). Grammatical divergence declines at each step from B1 to L1, indicating progressively stronger convergence toward the native POS-trigram distribution. The large B1\u0026ndash;L1 and B2\u0026ndash;L1 gaps show that grammatical sequencing remains one of the most sensitive markers of interlanguage distance, while the significant C1\u0026ndash;L1 difference confirms that even advanced learner production does not fully collapse into the native distributional baseline.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab6\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 4\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003e\u003cb\u003eB.\u003c/b\u003e \u003cem\u003eGames\u0026ndash;Howell post-hoc tests for KL₍gram₎\u003c/em\u003e\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"5\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eComparison\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eMean diff (G1\u0026ndash;G2)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eSE\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003ep\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c5\"\u003e\u0026nbsp;\u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eB1\u0026ndash;B2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.377\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.013\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e\u0026lt;\u0026thinsp;.001\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e\u0026nbsp;\u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eB1\u0026ndash;C1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.892\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.013\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e\u0026lt;\u0026thinsp;.001\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e\u0026nbsp;\u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eB1\u0026ndash;L1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e1.737\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.013\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e\u0026lt;\u0026thinsp;.001\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e\u0026nbsp;\u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eB2\u0026ndash;C1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.516\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.005\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e\u0026lt;\u0026thinsp;.001\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e\u0026nbsp;\u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eB2\u0026ndash;L1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e1.361\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.004\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e\u0026lt;\u0026thinsp;.001\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e\u0026nbsp;\u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eC1\u0026ndash;L1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.845\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.004\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e\u0026lt;\u0026thinsp;.001\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e\u0026nbsp;\u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eThis is one of the clearest developmental signatures in the dataset. As proficiency rises, learner texts become less distributionally distant from the native reference, which directly supports the claim that interlanguage development can be modeled as probabilistic convergence rather than only as reduction of overt error.\u003c/p\u003e \u003cdiv id=\"Sec27\" class=\"Section3\"\u003e \u003ch2\u003eCompression ratio (CR): increasing structural regularity with proficiency\u003c/h2\u003e \u003cp\u003eGames\u0026ndash;Howell comparisons also showed significant differences for every group contrast in CR (Table\u0026nbsp;4C). Compression ratio declines monotonically from B1 to L1, meaning that texts become increasingly compressible and structurally regular as proficiency develops. The B1\u0026ndash;B2 drop is already substantial, which suggests that important gains in global regularity emerge relatively early; subsequent reductions remain significant but appear more incremental.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab7\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 4\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003e\u003cb\u003eC.\u003c/b\u003e \u003cem\u003eGames\u0026ndash;Howell post-hoc tests for CR\u003c/em\u003e\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"4\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eComparison\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eMean diff (G1\u0026ndash;G2)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eSE\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003ep\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eB1\u0026ndash;B2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.049\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.003\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e\u0026lt;\u0026thinsp;.001\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eB1\u0026ndash;C1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.084\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.003\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e\u0026lt;\u0026thinsp;.001\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eB1\u0026ndash;L1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.145\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.003\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e\u0026lt;\u0026thinsp;.001\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eB2\u0026ndash;C1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.035\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.002\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e\u0026lt;\u0026thinsp;.001\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eB2\u0026ndash;L1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.096\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.002\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e\u0026lt;\u0026thinsp;.001\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eC1\u0026ndash;L1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.061\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.002\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e\u0026lt;\u0026thinsp;.001\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eThese results indicate that development involves not only richer lexical choice but also tighter global organization. Learner texts become easier to compress because they display more stable patterning, more predictable sequencing, and less redundant structural noise.\u003c/p\u003e \u003c/div\u003e \u003c/div\u003e\n\u003ch3\u003ePositional concentration (PCI): stronger early information packaging at higher proficiency\u003c/h3\u003e\n\u003cp\u003eFor PCI, Games\u0026ndash;Howell comparisons again showed significant differences for all pairs (Table\u0026nbsp;4D). The index rises steadily from B1 to L1, indicating that more proficient writers concentrate a larger share of informational load in the opening portion of the text. The strongest early shift occurs between B1 and B2, suggesting that discourse planning begins to change noticeably at the intermediate stage, after which the same front-loading tendency continues in more refined form.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab8\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 4\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003e\u003cb\u003eD.\u003c/b\u003e \u003cem\u003eGames\u0026ndash;Howell post-hoc tests for PCI\u003c/em\u003e\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"4\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eComparison\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eMean diff (G1\u0026ndash;G2)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eSE\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003ep\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eB1\u0026ndash;B2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e-0.170\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.005\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e\u0026lt;\u0026thinsp;.001\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eB1\u0026ndash;C1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e-0.300\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.005\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e\u0026lt;\u0026thinsp;.001\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eB1\u0026ndash;L1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e-0.387\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.005\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e\u0026lt;\u0026thinsp;.001\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eB2\u0026ndash;C1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e-0.130\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.003\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e\u0026lt;\u0026thinsp;.001\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eB2\u0026ndash;L1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e-0.218\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.004\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e\u0026lt;\u0026thinsp;.001\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eC1\u0026ndash;L1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e-0.088\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.004\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e\u0026lt;\u0026thinsp;.001\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eThe PCI results therefore complement the lexical and grammatical findings by showing that development also involves improved discourse packaging. More advanced writers do not merely choose different forms; they distribute information across the text in increasingly native-like ways.\u003c/p\u003e \u003cdiv id=\"Sec29\" class=\"Section2\"\u003e \u003ch2\u003eSummary of developmental reorganization\u003c/h2\u003e \u003cp\u003eAcross the four focal indicators, the results converge on a single developmental profile. Lexical entropy rises, grammatical divergence falls, compression ratio falls, and positional concentration rises. This combination is crucial: learner language becomes more lexically expansive while simultaneously becoming more grammatically aligned, more globally regular, and more strategically organized across discourse position. The overall pattern is therefore best described as coordinated probabilistic reorganization rather than as a single monotonic change in complexity.\u003c/p\u003e \u003c/div\u003e"},{"header":"Discussion","content":"\u003cp\u003eThe present study examined whether interlanguage can be meaningfully modeled as a probabilistic system and whether its development can be traced through distribution-sensitive information metrics. The findings strongly support this view. Across lexical entropy, grammatical divergence, compression ratio, and positional concentration, the results revealed systematic and statistically robust differences across proficiency levels. More importantly, the developmental pattern did not reflect a single linear increase in \u0026ldquo;complexity.\u0026rdquo; Instead, it showed coordinated reorganization across multiple representational layers, which is precisely what the proposed model predicts. In the model, interlanguage development is not the simple accumulation of rules or forms; rather, it is a gradual reallocation of probability mass across lexical, grammatical, phraseological, and discourse-positional options. The empirical results are consistent with this account.\u003c/p\u003e \u003cp\u003eAt the lexical level, entropy increased significantly across proficiency levels, indicating that more advanced learners distribute probability mass across a broader range of lexical choices. This lexical expansion is consistent not only with information-theoretic accounts of dispersion, but also with recent corpus-based evidence that richer lexical patterning continues to distinguish more advanced EFL writing (Yang \u0026amp; He, \u003cspan citationid=\"CR22\" class=\"CitationRef\"\u003e2025\u003c/span\u003e) and with NLP-oriented approaches that model lexical knowledge through distribution-sensitive representations rather than simple counts (Crossley \u0026amp; Holmes, \u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e2023\u003c/span\u003e). This pattern corroborates information-theoretic accounts in which complexity is defined as dispersion or unpredictability in outcome distributions rather than mere structural accumulation (Ehret \u0026amp; Szmrecsanyi, \u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e2016\u003c/span\u003e).\u003c/p\u003e \u003cp\u003eIt also aligns with usage-based perspectives on SLA, according to which increased exposure and entrenchment gradually expand the learner\u0026rsquo;s choice space and weaken reliance on a narrow set of high-frequency default forms. The present results therefore answer the first research question affirmatively: interlanguage development can indeed be captured probabilistically, and lexical entropy is one meaningful indicator of such development. At the same time, the remaining gap between C1 and L1 suggests that advanced learners still differ from native benchmarks in the balance of lexical distributions, which confirms that lexical development remains probabilistically incomplete even at higher proficiency levels.\u003c/p\u003e \u003cp\u003eIn contrast to lexical entropy, grammatical divergence decreased sharply as proficiency increased. This is one of the clearest findings of the study and provides a strong response to the second research question concerning whether development involves convergence toward target norms. The answer is clearly yes, but this convergence is subsystem-specific. Using KL divergence, the study measured the directed distance between learner distributions and the target grammatical reference model. The significant downward trend across proficiency groups indicates that learners\u0026rsquo; grammatical selections become progressively less divergent from target-like distributions.\u003c/p\u003e \u003cp\u003eThis result directly corroborates Sun and Wang\u0026rsquo;s (\u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e2021\u003c/span\u003e) argument that relative entropy can serve as a sensitive indicator of developmental change in L2 proficiency. It also supports the model\u0026rsquo;s claim that development involves redistribution and pruning, whereby competing non-target probabilities are gradually reduced and more contextually appropriate sequential patterns become more strongly preferred. The fact that lexical entropy rises while grammatical divergence falls is especially important theoretically, because it demonstrates that interlanguage does not become uniformly more variable or uniformly more constrained. Rather, it becomes more diversified in lexical choice and more stabilized in grammatical patterning. This asymmetry is fully consistent with recent cautions in SLA complexity research that developmental change must be interpreted at the level of the subsystem being examined rather than through a single global metric (Ehret et al., \u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e2023\u003c/span\u003e; Han et al., \u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e2023\u003c/span\u003e).\u003c/p\u003e \u003cp\u003eThe findings for compression ratio provide further support for the model. Compression ratio declined significantly across proficiency levels, indicating that learner language becomes more compressible and therefore more structurally regular and predictable. This result corroborates earlier work showing that compression-based measures can capture holistic structure and redundancy in learner language in ways that traditional feature counts cannot (Ehret \u0026amp; Szmrecsanyi, \u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e2019\u003c/span\u003e; Wang et al., \u003cspan citationid=\"CR21\" class=\"CitationRef\"\u003e2022\u003c/span\u003e; Alzahrani, \u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e2024\u003c/span\u003e). Within the present theoretical framework, this means that development is not only a matter of increasing lexical range or decreasing grammatical divergence; it also involves the emergence of broader regularity in how texts are assembled. Importantly, the decrease in compression ratio does not contradict the increase in lexical entropy.\u003c/p\u003e \u003cp\u003eOn the contrary, the coexistence of these trends is one of the strongest pieces of evidence for the probabilistic model. It indicates that the system becomes globally more organized even as it becomes locally more diversified. In other words, learner language appears to achieve greater communicative efficiency by coordinating lexical expansion with structural regularization. This pattern is compatible with the distinction drawn by Ehret and Szmrecsanyi (\u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e2016\u003c/span\u003e) between entropy as uncertainty over distributions and compression as a proxy for description-length complexity.\u003c/p\u003e \u003cp\u003eThe discourse-positional findings are equally revealing. The positional concentration index increased with proficiency, showing that more advanced learners increasingly concentrate informational load earlier in the text. This pattern extends the model beyond lexical and grammatical distributions to discourse-level packaging and provides a clear answer to the third research question regarding whether information-density development is also visible at higher discourse levels. The answer is again affirmative. These results corroborate prior research showing that information density is position-sensitive and related to how speakers and writers manage processing demands over unfolding discourse (Jaeger, \u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e2010\u003c/span\u003e; Levy, \u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e2008\u003c/span\u003e; Trujillo \u0026amp; Holler, \u003cspan citationid=\"CR19\" class=\"CitationRef\"\u003e2025\u003c/span\u003e).\u003c/p\u003e \u003cp\u003eWithin the present model, PCI indexes discourse-position ID and therefore reflects how learners package information across textual segments. The sharpest shift occurring from B1 to B2 suggests that intermediate learners begin to reorganize discourse planning so that greater informational load is placed earlier in the text, while more advanced learners continue this trajectory toward native-like positional concentration patterns. This is an important contribution because it suggests that discourse organization may reorganize earlier than full grammatical convergence, thereby reinforcing the model\u0026rsquo;s claim that interlanguage development is multi-layered and non-synchronous across subsystems.\u003c/p\u003e \u003cp\u003eIn combination, these findings provide strong empirical support for the proposed probabilistic model of interlanguage information density. The model predicts that development will involve subsystem-specific redistribution of probability mass rather than uniform growth, and that prediction is borne out by the data. In this respect, the present findings also resonate with broader arguments that linguistic systems exhibit trade-offs across complexity domains rather than straightforward one-directional growth (Bentz et al., \u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2023\u003c/span\u003e).\u003c/p\u003e \u003cp\u003eIn addition, lexical entropy increases, grammatical divergence decreases, compression ratio declines, and positional concentration rises. These are not isolated trends. Rather, they describe a coherent developmental ecology in which learners gradually widen the lexical choice space, reduce divergence from target grammatical norms, increase the structural regularity of output, and reorganize discourse packaging in more target-compatible ways. This interpretation also helps explain why traditional complexity measures are often insufficient. If complexity were treated as a single scalar property, the present results would appear contradictory: some indices increase while others decrease. Yet from the perspective of the present model, such divergence is precisely what should be expected. Development is not a unidirectional rise in complexity but a coordinated recalibration of probabilities across layers of the system.\u003c/p\u003e \u003cp\u003eThe findings also contribute to broader debates about the nature of interlanguage. Rather than supporting a deficit view in which learner language is judged only in terms of missing target features, the results suggest that interlanguage is better understood as a dynamic distributional system shaped by competing pressures of expressivity, predictability, efficiency, and cognitive constraint. This interpretation is consistent with Bradlow\u0026rsquo;s (\u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e2022\u003c/span\u003e) view that L1 and L2 production differ not only in grammatical selection but also in how information is encoded and transmitted. It also aligns with pedagogically oriented evidence showing that information density affects comprehension and discourse processing in instructional settings (Mekheimer \u0026amp; Fageeh, \u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e2025\u003c/span\u003e). From this perspective, learner language is not simply \u0026ldquo;less advanced\u0026rdquo; than native language; it is differently organized, and its development consists in increasingly adaptive redistribution of linguistic probabilities. The use of a stochastic L1 baseline strengthens this interpretation. By treating native-speaker production as a range of probabilistic variation rather than a zero-variance ideal, the study offers a more realistic model of convergence and avoids reducing target-like performance to a fixed categorical endpoint.\u003c/p\u003e \u003cdiv id=\"Sec31\" class=\"Section2\"\u003e \u003ch2\u003eLimitations\u003c/h2\u003e \u003cp\u003eSeveral limitations should be acknowledged. First, although the L1 baseline was modeled stochastically through leave-one-out estimation, the benchmark remains tied to a specific genre and task ecology, namely argumentative writing. Because entropy and divergence measures are sensitive to register and discourse conditions, the baseline established here should not be assumed to generalize directly to other genres or communicative settings. Second, some measures depend on automated annotation procedures such as lemmatization and POS tagging.\u003c/p\u003e \u003cp\u003eAlthough these procedures were suitable for large-scale analysis, non-canonical learner language, especially at lower proficiency levels, may introduce annotation noise that affects entropy and divergence estimates. Third, the fixed-window approach improved comparability across texts, but it may have reduced the visibility of discourse-level phenomena that emerge more fully in longer texts. Fourth, learners\u0026rsquo; L1 backgrounds were not explicitly modeled, which means that some distributional tendencies may reflect cross-linguistic transfer in addition to proficiency-related development, a possibility that has been highlighted in recent work using Kolmogorov-complexity measures across L2 groups and L1 backgrounds (Alzahrani, \u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e2024\u003c/span\u003e). Finally, the data were derived from a balanced, controlled corpus, which strengthens internal comparability but may underrepresent the variability and non-linearity of naturalistic interlanguage development over time.\u003c/p\u003e \u003cdiv id=\"Sec32\" class=\"Section3\"\u003e \u003ch2\u003eImplications\u003c/h2\u003e \u003cp\u003eThe findings have important theoretical, methodological, and pedagogical implications. Theoretically, they support a reconceptualization of interlanguage as a probabilistic, multi-layered system rather than a collection of isolated forms or errors. This moves the discussion of learner language beyond static structural counts and aligns SLA more closely with information-theoretic and usage-based approaches to development.\u003c/p\u003e \u003cp\u003eMethodologically, the study demonstrates that entropy-family measures can function as bridge constructs linking uncertainty, divergence, compressibility, and discourse packaging within a single analytic framework. This complements calls in SLA complexity research for multiple, explicitly theorized measures rather than single omnibus indices (Housen et al., \u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e2019\u003c/span\u003e; Kuiken, \u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e2023\u003c/span\u003e). This provides a richer diagnostic account of development than conventional complexity indices alone.\u003c/p\u003e \u003cp\u003ePedagogically, the results suggest that learner progress should not be evaluated through a single proficiency score or a narrow notion of complexity. Learners may advance differently across lexical dispersion, grammatical convergence, structural regularity, and discourse-level information packaging. Entropy-based profiling may therefore offer a more informative way to diagnose developmental strengths and weaknesses. More broadly, the findings imply that instructional design should recognize that lexical expansion, grammatical stabilization, and discourse packaging do not necessarily develop at the same rate and may require different types of pedagogical support.\u003c/p\u003e \u003cp\u003eOverall, the study shows that information density is a theoretically meaningful and empirically tractable lens for understanding interlanguage development. By demonstrating that learner language changes through coordinated redistribution of probability mass across several subsystems, the study provides evidence that interlanguage is best understood not as a deficient approximation of native competence but as a dynamically reorganizing probabilistic system.\u003c/p\u003e \u003c/div\u003e \u003c/div\u003e"},{"header":"Conclusion","content":"\u003cp\u003eThe present study offers a stronger basis for understanding interlanguage development as a process of probabilistic reorganization rather than as the mere accumulation of linguistic forms or the gradual reduction of error. Across proficiency levels, learner language was shown to change through systematic redistribution of probability mass across multiple subsystems: lexical entropy increased as the range of lexical choices broadened; grammatical divergence decreased as sequential patterns moved closer to target-like distributions; compression ratio declined as texts became more structurally regular, predictable, and compressible; and positional concentration increased as informational load was packaged more efficiently toward earlier parts of the text.\u003c/p\u003e \u003cp\u003eRead alongside prior work on relative entropy, compression-based complexity, and multidimensional complexity modeling, this pattern reinforces the view that development is best understood as coordinated distributional change rather than isolated gain in a single domain (Sun \u0026amp; Wang, \u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e2021\u003c/span\u003e; Wang et al., \u003cspan citationid=\"CR21\" class=\"CitationRef\"\u003e2022\u003c/span\u003e; Housen et al., \u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e2019\u003c/span\u003e). These converging patterns demonstrate that development is not adequately captured by a single scalar notion of complexity. Instead, it unfolds as a coordinated reshaping of dispersion, constraint, regularity, and discourse organization within an evolving linguistic system.\u003c/p\u003e \u003cp\u003eIn combination, these findings substantiate the view that interlanguage is best conceived as a dynamic information-density architecture governed by interacting pressures of learning, exposure, communicative need, and cognitive resource management. This perspective helps refine theoretical accounts of second language development by showing that expansion and constraint are not opposing outcomes but complementary dimensions of maturation within different representational layers.\u003c/p\u003e \u003cp\u003eAt the same time, the study illustrates the value of entropy-family metrics as analytically powerful tools for revealing developmental change that traditional count-based indices may obscure. By capturing how learners reorganize probabilities across lexical, grammatical, structural, and discourse-positional domains, the present framework opens a more precise way of describing proficiency growth and a more nuanced basis for interpreting learner performance. In this sense, the study not only strengthens the conceptualization of interlanguage as a probabilistic system, but also provides a practical foundation for more sensitive evaluation of development, one that recognizes learner progress as multidimensional, adaptive, and increasingly efficient rather than simply more or less complex.\u003c/p\u003e"},{"header":"Declarations","content":"\u003cp\u003e \u003cstrong\u003eEthics approval and consent to participate\u003c/strong\u003e \u003cp\u003eThe study protocol was reviewed and approved by the Institutional Review Board of the Faculty of Education, Beni-Suef University, Egypt (Approval No. BSU-FoE-004-01-01-2016). All methods were carried out in accordance with relevant guidelines and regulations. Informed consent was obtained from all participants and/or their legal guardians.\u003c/p\u003e \u003c/p\u003e \u003cp\u003e \u003cstrong\u003eConsent for publication\u003c/strong\u003e \u003cp\u003eNot applicable.\u003c/p\u003e \u003c/p\u003e \u003cp\u003e \u003cstrong\u003eCompeting interests\u003c/strong\u003e \u003cp\u003eThe author declares no competing interests.\u003c/p\u003e \u003c/p\u003e\u003ch2\u003eAuthor Contribution\u003c/h2\u003e\u003cp\u003eM.M. conceived the study, conducted the analysis, interpreted the results, and wrote the manuscript.\u003c/p\u003e\u003ch2\u003eAcknowledgement\u003c/h2\u003e\u003cp\u003eThanks to STDF/EKB for potential payment of APCs upon acceptance.\u003c/p\u003e\u003ch2\u003eData Availability\u003c/h2\u003e\u003cp\u003eThe data are not publicly available because they contain learner-generated materials collected under consent conditions that do not permit open public release. 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Lexical richness in Chinese university students\u0026rsquo; EFL writing: A corpus-based comparison. \u003cem\u003eHumanit. Social Sci. Commun.\u003c/em\u003e \u003cb\u003e12\u003c/b\u003e, 1199. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttps://doi.org/10.1057/s41599-025-05560-x\u003c/span\u003e\u003cspan address=\"10.1057/s41599-025-05560-x\" targettype=\"DOI\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e (2025).\u003c/span\u003e\u003c/li\u003e\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"scientific-reports","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"scirep","sideBox":"Learn more about [Scientific Reports](http://www.nature.com/srep/)","snPcode":"","submissionUrl":"","title":"Scientific Reports","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Scientific Reports","inReviewEnabled":true,"inReviewRevisionsEnabled":true},"keywords":"information density, entropy, interlanguage, second language acquisition, KL divergence, probabilistic systems","lastPublishedDoi":"10.21203/rs.3.rs-9295874/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-9295874/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eInterlanguage development is often assessed through structural counts that only partially capture how learner language is organized probabilistically. This study proposes a multi-level framework for measuring interlanguage information density using entropy-based metrics. A corpus of 150 L2 English argumentative essays from B1, B2, and C1 learners was compared with a genre-matched native-speaker corpus of 50 essays. Four indicators were examined: lexical entropy (Hₗₑₓ), grammatical divergence from a native reference distribution via POS trigrams (KL₍gram₎), compression ratio (CR), and positional concentration index (PCI). To model native variability more defensibly, KL₍gram₎ for each L1 text was calculated against a leave-one-out L1 reference distribution. Results showed a clear developmental gradient: lexical entropy and positional concentration increased with proficiency, whereas grammatical divergence and compression ratio decreased. Mixed-effects models confirmed that these shifts were robust effects of proficiency. The findings support a probabilistic view of interlanguage development and offer a principled diagnostic framework for evaluating communicative efficiency in L2 writing.\u003c/p\u003e","manuscriptTitle":"Measuring the Information Density of Interlanguage: An Entropy Analysis","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2026-04-16 20:47:23","doi":"10.21203/rs.3.rs-9295874/v1","editorialEvents":[{"type":"communityComments","content":0},{"type":"decision","content":"Revision requested","date":"2026-04-23T04:52:03+00:00","index":"","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2026-04-20T09:27:57+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"240271722583821208437519468568551713952","date":"2026-04-14T08:33:48+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"74065207702063348253669425847738387644","date":"2026-04-13T02:33:29+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"329022194994514493168191960534739903750","date":"2026-04-10T13:04:39+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"324378574221458016161506321643935683916","date":"2026-04-09T12:56:53+00:00","index":"hide","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2026-04-09T06:56:49+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"138199077996601286198092861962486287310","date":"2026-04-09T06:35:16+00:00","index":"hide","fulltext":""},{"type":"reviewersInvited","content":"","date":"2026-04-08T12:53:39+00:00","index":"","fulltext":""},{"type":"editorAssigned","content":"","date":"2026-04-08T12:46:52+00:00","index":"","fulltext":""},{"type":"editorInvited","content":"","date":"2026-04-07T11:59:35+00:00","index":"","fulltext":""},{"type":"checksComplete","content":"","date":"2026-04-06T13:20:12+00:00","index":"","fulltext":""},{"type":"submitted","content":"Scientific Reports","date":"2026-04-06T12:14:18+00:00","index":"","fulltext":""}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"scientific-reports","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"scirep","sideBox":"Learn more about [Scientific Reports](http://www.nature.com/srep/)","snPcode":"","submissionUrl":"","title":"Scientific Reports","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Scientific Reports","inReviewEnabled":true,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"89657cad-5469-41b5-b625-2466645375b9","owner":[],"postedDate":"April 16th, 2026","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"under-review","subjectAreas":[{"id":66276147,"name":"Humanities/Language and linguistics"},{"id":66276148,"name":"Social science/Language and linguistics"},{"id":66276149,"name":"Physical sciences/Mathematics and computing"}],"tags":[],"updatedAt":"2026-05-19T14:10:34+00:00","versionOfRecord":[],"versionCreatedAt":"2026-04-16 20:47:23","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-9295874","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-9295874","identity":"rs-9295874","version":["v1"]},"buildId":"XKTyCvWXoU3ODBz1xrDgd","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}