Observer-Dependent Entropy Retrieval in Linguistic Computation: A Foundational Framework and Benchmarking Methodology

1. Introduction

• Reader guide. This paper introduces a new way to think about sentence comprehension, not as predicting the next word, but as retrieving meaning over time. Most current models measure how surprising a word is, assuming all readers process it in the same way. ODER shows that is not the case: comprehension speed and difficulty depend on each reader’s attention, memory, and background knowledge. Using a controlled test language, we measure how different kinds of readers recover meaning, and then test the same model on English sentences without changing the parameters. The results match known points where understanding tends to break down and predict when and why this happens for different readers.

Comprehension failure is not prediction error; it is delayed access to retrievable meaning. Traditional entropy approaches quantify linguistic uncertainty without considering observer-specific processing differences. Evidence from neurolinguistics and computational cognition shows that interpretive effort, and therefore uncertainty resolution, depends on an observer’s attentional state, working-memory capacity, and contextual familiarity [15,26]. Earlier entropy-based models conflate linguistic probability with observer cost, obscuring how processing difficulty emerges from observer-bound retrieval delays. ODER extends this literature by

• defining entropy retrieval as a joint function of hierarchical syntactic complexity and information-transfer efficiency;
• mapping these constructs to measurable cognitive signatures in EEG, fMRI, and pupillometry;
• providing a replicable benchmarking framework that reports $\gamma$, ${\tau }_{\mathrm{char}}$, and ${\tau }_{\mathrm{res}}$ for each observer class.

Crucially, ODER reframes comprehension as entropy retrieval in the observer, not entropy reduction in the signal. This distinction explains not only what is complex but also how and when different observers experience that complexity. Clarification. We emphasize that ODER is not a language model or parser; it is a meta-framework describing how observers retrieve entropy from linguistic input.

1.1. Contributions

Accordingly, ODER functions as a formal meta-framework that parameterizes observer-specific comprehension, enabling structured comparison across linguistic theories rather than competing with predictive models.

• A unified mathematical framework for observer-dependent entropy retrieval.
• Explicit retrieval (Equation A3) and transition (Equation A4) functions that parameterize attention, working memory, and prior knowledge.
• A contextual-gradient operator that captures reanalysis, for example, garden-path phenomena, in dynamic observer-dependent terms.
• A benchmarking protocol that compares ODER with existing cognitive models and raises stress flags when $\nabla C$ flattens or ${\tau }_{\mathrm{res}}$ diverges.
• A demonstration that quantum-formalism constructs can model ambiguity and interference without implying literal quantum computation in the brain.

1.2. Relationship to Existing Models

ODER does not compete with current linguistic models solely on predictive accuracy; instead, it addresses a core explanatory gap:

• Surprisal models [12,21] quantify unexpectedness but assume a uniform processor, overlooking individual differences in how surprisal is experienced.
• Resource-rational models [11,24] acknowledge capacity limits yet often lack explicit reanalysis mechanisms such as the P600 response to garden-path sentences.
• ACT-R parsing frameworks [22] simulate incremental working-memory constraints but treat prediction and retrieval as separate stages, leaving coherence effects unexplained.
• Hierarchical prediction-error accounts [13] model multi-level expectations but do not specify observer-class parameters that modulate collapse timing.
• Transformer language models excel at prediction and generation, yet their weight vectors obscure observer dynamics and reveal little about why or how observers differ in processing.

Rather than replacing these approaches, ODER serves as a meta-framework that clarifies when and why processing difficulty arises for specific observers.

We define lawful divergence as a systematic, interpretable model–trace mismatch (flagged by the shape_mismatch criterion and nonzero $\Delta {\tau }_{\mathrm{res}}$) that aligns with known construction-specific bottlenecks without parameter re-fitting.

1.2.1. The ODER Innovation: A Conceptual Map

Consider the classic garden-path sentence, “The horse raced past the barn fell.” Empirical work shows expert versus novice divergence in processing difficulty [5,8]. Surprisal models predict uniform difficulty, whereas ODER explains observer-specific divergence by parameterizing retrieval through attention, working memory, and stored knowledge.

1.3. Theoretical Positioning of ODER

Table 1. Theoretical positioning of ODER relative to leading approaches.

Approach	Primary Focus	Treatment of Observer	Key Limitations
Surprisal Models	Input statistics and probability	Uniform processor with idealized capacity	Cannot explain individual differences in processing difficulty
Resource-Rational	Bounded rationality and capacity limits	Variable capacity, uniform mechanisms	Lack explicit reanalysis mechanisms; treat processing as passive
ACT-R Parsing	Procedural memory retrieval	Slot-limited buffer with decay	Prediction and retrieval treated separately; no coherence term
Hierarchical Prediction-Error	Multi-level expectation tracking	Implicit observer; scalar precision weights	No explicit collapse point or observer parameters
Optimal Parsing	Strategy selection	Uniform processor with idealized strategies	Cannot explain observer-specific strategy choices
ODER (this model)	Observer-relative entropy retrieval (not generative modeling)	Parameterized by attention, memory, and knowledge	Designed partial-fit; requires empirical calibration of observer parameters

Table 1. Theoretical positioning of ODER relative to leading approaches.

Approach	Primary Focus	Treatment of Observer	Key Limitations
Surprisal Models	Input statistics and probability	Uniform processor with idealized capacity	Cannot explain individual differences in processing difficulty
Resource-Rational	Bounded rationality and capacity limits	Variable capacity, uniform mechanisms	Lack explicit reanalysis mechanisms; treat processing as passive
ACT-R Parsing	Procedural memory retrieval	Slot-limited buffer with decay	Prediction and retrieval treated separately; no coherence term
Hierarchical Prediction-Error	Multi-level expectation tracking	Implicit observer; scalar precision weights	No explicit collapse point or observer parameters
Optimal Parsing	Strategy selection	Uniform processor with idealized strategies	Cannot explain observer-specific strategy choices
ODER (this model)	Observer-relative entropy retrieval (not generative modeling)	Parameterized by attention, memory, and knowledge	Designed partial-fit; requires empirical calibration of observer parameters

By casting comprehension as active, observer-relative retrieval, ODER unifies phenomena such as garden-path reanalysis, working-memory constraints, and expertise effects under a single, testable framework.

While ODER is not a neural model, its observer parameters map to biologically interpretable constraints. Attention ($\alpha$), working-memory capacity ($\beta$), and semantic interference ($\mu$) correspond to measurable features of cognitive processing, including reaction-time profiles, ERP components (e.g., N400/P600), and frontal theta dynamics. This alignment supports ODER’s testability within neurocognitive systems without invoking mechanistic claims beyond retrieval-level modeling.

1.4. Forward Retrieval Law and Inverse Decoder

$${\displaystyle \frac{d{S}_{\mathrm{obs}}}{d\tau }}=\gamma \left(\tau \right)\left(S_{max}-S_{\mathrm{obs}}\left(\tau \right)\right)tanh\left(\tau /{\tau }_{\mathrm{char}}\right)$$

(1)

(Forward retrieval law). The hyperbolic tangent ensures symmetric convergence behavior and enables analytic inversion under bounded modular-flow assumptions (see Appendix A). Convergence is expected to plateau near $30\%$–$40\%$ under the constant-$\gamma$ assumption (cf. Appendix A for ceiling behavior); allowing $\gamma \left(\tau \right)$ to vary piecewise raises this ceiling (see Discussion 5), but we retain this parsimonious form to preserve sharp falsifiability of the baseline hypothesis. In plain terms, Equation 1 states that retrieval speed depends both on how much meaning is still left to recover and on how far along the observer is in the comprehension process.

Inverse decoder:

$$\gamma \left(\tau \right)={\displaystyle \frac{{\displaystyle \frac{d{S}_{\mathrm{obs}}}{d\tau }}}{\left[S_{max}-S_{\mathrm{obs}}\left(\tau \right)\right]tanh\left(\tau /{\tau }_{\mathrm{char}}\right)}}.$$

(2)

This inverse form recovers the effective retrieval-rate profile $\gamma \left(\tau \right)$ from an observed entropy trajectory. In practice it allows us to estimate how retrieval dynamics vary over time without assuming a fixed $\gamma$, making it possible to test whether deviations from the constant-rate law align with known comprehension bottlenecks. The inverse decoder therefore functions as a diagnostic tool: when applied to empirical traces, it reveals where the forward law is under strain and highlights candidate loci of lawful divergence.

1.5. Implementation Algorithm

Algorithm 1 ODER Entropy Retrieval.

Require: sentence S, observer parameters $\alpha ,\beta ,\mu$ Ensure: observer-dependent entropy $S_{\mathrm{obs}}$ 1: $S_{\mathrm{obs}}\leftarrow 0$; initialize ${\rho }_{\mathrm{obs}}$ with Equation A2 2: for each word w in S do 3:     $L_{\mathrm{hier}}\leftarrow \mathrm{syntacticDepth}(w,\mathrm{context})$ 4:     $I_{\mathrm{trans}}\leftarrow \mathrm{informationTransfer}(w,\mathrm{context})$ 5:     $\nabla C\leftarrow \mathrm{contextualGradient}(w,\mathrm{context})$ 6:     $S_{\mathrm{obs}}\leftarrow S_{\mathrm{obs}}+f(L_{\mathrm{hier}},I_{\mathrm{trans}},\nabla C)$       ▹ Equation A3 7:     ${\rho }_{\mathrm{obs}}\leftarrow T({\rho }_{\mathrm{obs}},L_{\mathrm{hier}},I_{\mathrm{trans}},\nabla C)$       ▹ Equation A4 8: end for 9: return$S_{\mathrm{obs}}$

In practical terms, the algorithm tracks how each word in a sentence contributes to the observer’s cumulative retrieval of meaning, updating both the entropy value and the internal representation of possible interpretations at every step.

Python prototypes rely on NumPy/SciPy, QuTiP, and spaCy. Optimization used SciPy’s L-BFGS-B with a function tolerance of ${10}^{-6}$ and three random-seed restarts per trace. A Monte Carlo identifiability sweep for $(\gamma ,{\tau }_{\mathrm{char}})$ appears in Appendix A (Table A1).

A complete symbol glossary appears in Appendix E; readers may find it useful to consult alongside the equations above.

2. Benchmarking Methodology

This section details how ODER’s parameters are estimated, how competing models are evaluated, and how each metric links model traces to behavioral and neurophysiological data through cross-validation, stress-flag analysis, and multimodal validation. All analysis code and notebooks (Zenodo DOI; GitHub mirror) reproduce every figure and table in this paper; see the Data Availability section for links.

As summarized in Table 2, the benchmarking protocol evaluates both fit quality and neurocognitive interpretability.

2.1. Comparative Metrics

Entropy-reduction rate (ERR)

– First-derivative slope of $S_{\mathrm{obs}}\left(\tau \right)$; hypothesized to scale with the N400 slope in centro-parietal EEG.

Retrieval-collapse point (${\tau }_{\mathrm{res}}$)

– Time at which $d{S}_{\mathrm{obs}}/d\tau$ enters a 95% confidence band around zero; anchors the onset of P600 activity and post-disambiguation fixation drops.

Model-to-trace fit ($R^2$, AIC, BIC)

– Overall goodness-of-fit and parsimony; higher $R^2$ predicts tighter coupling between simulated ${\tau }_{\mathrm{res}}$ and observed P600 latency.

Observer-class divergence ($\Delta \gamma$)

– Cohen’s d for $\gamma$ between O1 and O3; relates to between-group differences in frontal-theta power (high vs. low working memory).

Cross-validation error 

– Mean absolute error over k-fold splits (bootstrapped 95% CIs); mirrors inter-trial variability in ERP peak latencies.

Reanalysis latency

– Reaction-time variance in garden-path tasks; behavioral proxy for $\nabla C$ spikes.

Pupillometric load

– Peak dilation normalized by baseline; tracks integrated $\beta$ (working-memory demand).

Eye-movement patterns

– Fixation count and regression length during disambiguation; fine-grained correlate of local ERR fluctuations.

The three baseline models used for $\Delta$AIC comparison were a linear growth model, a single-parameter exponential decay model, and a two-parameter power-law model, each fit by bounded nonlinear least squares with the same optimizer settings as ODER; full implementation details are provided in the Data Availability section.

2.2. Protocol

• Compute baseline entropy with Equation (A1) for all Aurian stimuli.
• Run ODER retrieval dynamics using the retrieval kernel (Equation (A3)), the state transition (Equation (A4)), and the forward retrieval law (Equation (1)); log stress flags whenever $R^2<0.60$ or ${\tau }_{\mathrm{res}}>2$ s.

2.3. Neurophysiological Correlates

Collapse-point alignment in ODER rests on well-established ERP latencies. Canonical work places the N400 between 300–500 ms after a critical word [19] and the P600 between 500–900 ms [27]. Each window is anchored at the observer-specific collapse time ${\tau }_{\mathrm{res}}$ (see Section 4):

• Contextual-gradient spikes ($\nabla C$) predict P600 amplitude in the window ${\tau }_{\mathrm{res}}+500–900$ ms [27].
• Information-transfer efficiency ($I_{\mathrm{trans}}$) predicts N400 magnitude in the window ${\tau }_{\mathrm{res}}+300–500$ ms [19].
• Working-memory load ($\beta$) is expected to modulate frontal-midline theta (4–7 Hz) across the same post-collapse interval, consistent with memory-maintenance accounts of theta power [3].

These mappings operationalize how ERR aligns with N400 slope, how $\nabla C$ modulates P600 amplitude, and how ${\tau }_{\mathrm{res}}$ anchors both windows, thereby linking the metrics in Table 2 to observable brain dynamics (boundary-case discussion in Section 5).

2.4. Distinguishing Retrieval Failure from Prediction Failure

Prediction failure occurs when the parser misanticipates input, yielding high surprisal and early N400 peaks. Retrieval failure arises when an observer cannot integrate available information, even if predictions were correct, leading to prolonged P600 activity, a plateau in pupil dilation, and nonlinear error growth in comprehension probes.

• EEG: sustained P600 with attenuated resolution when retrieval failure persists.
• Pupillometry: plateau in low-capacity observers.
• Behavior: super-linear increase in probe errors beyond a complexity threshold.

3. Empirical Calibration

This section explains how ODER’s latent parameters are estimated from behavioral and neurophysiological data, specifically, reaction-time variance, ERP desynchronization, and ${\tau }_{\mathrm{res}}$-aligned EEG or pupillometry epochs.

3.1. Aurian as an Initial Testbed

Aurian is a constructed language that allows explicit control over syntactic complexity ($L_{\mathrm{hier}}$) and informational load ($I_{\mathrm{trans}}$) [6]. Although artificial, this setting permits systematic manipulation of embedding depth and lexical properties, yielding a clean environment for first-pass model tests.

Note on Aurian Scope

Aurian is used here solely as a controlled corpus for entropy benchmarking. A more advanced version, including observer-conditioned compression, ambiguity-preserving syntax, and symbolic scaffolds, is developed separately in [6]. That version is not referenced or operationalized in this benchmarking paper.

3.1.1. Aurian Grammar Specification

Core syntactic rules

$$\begin{array}{cc}\hfill S& \to \mathrm{NP}\mathrm{VP}\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill \mathrm{NP}& \to \left(\mathrm{Det}\right)\mathrm{N}\left(\mathrm{CP}\right)\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill \mathrm{VP}& \to \mathrm{V}\left(\mathrm{NP}\right)\left(\mathrm{CP}\right)\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill \mathrm{CP}& \to \mathrm{C}\mathrm{S}\hfill \end{array}$$

(6)

Lexicon with $L_{\mathrm{hier}}$ increments

• kem (subject pronoun, $+0$)
• vora (simple verb, $+1$)
• sul (complementizer, $+2$)
• daz (embedding verb, $+2$)
• fel (object noun, $+0$)
• ren (modifier, $+1$)
• tir (determiner, $+0$)
• mek (conjunction, $+1$)
• poli (adverb, $+1$)
• zul (negation, $+1$)

Illustrative sentences

• Low entropy ($L_{\mathrm{hier}}=2$): Kem vora fel (“He/She sees the object”)
• Medium entropy ($L_{\mathrm{hier}}=3$): Kem vora fel ren (“He/She sees the object quickly”)
• High entropy ($L_{\mathrm{hier}}=7$): Kem daz sul tir fel vora (“He/She thinks that the object falls”)
• Very high entropy ($L_{\mathrm{hier}}=11$): Kem daz sul tir fel sul ren vora poli zul (“He/She thinks that the object that quickly falls does not move”)

Table 3. Cumulative $L_{\mathrm{hier}}$ across Aurian sentence classes.

Sentence class	Tokens	Cumulative ${\mathit{L}}_{\mathbf{hier}}$
Low	3	2
Medium	4	3
High	6	7
Very High	9	11

Table 3. Cumulative $L_{\mathrm{hier}}$ across Aurian sentence classes.

Sentence class	Tokens	Cumulative ${\mathit{L}}_{\mathbf{hier}}$
Low	3	2
Medium	4	3
High	6	7
Very High	9	11

3.1.2. Clarifying the $L_{\mathrm{hier}}$ Metric

Each rule or lexical item contributes a fixed increment to $L_{\mathrm{hier}}$. Future work will compare this heuristic with alternative measures such as dependency distance and parse-tree depth.

Ecological Rationale

Although Aurian is synthetic, it functions as a minimal-pair generator that isolates structural factors while holding lexical semantics constant. This controlled starting point is a necessary bridge to natural corpora such as Natural Stories [10] and Dundee [17], where real-world noise and contextual variation are much higher.

3.2. Confidence, Sensitivity, and Parameter Variance

• Report 95% confidence intervals for $\gamma$ and ${\tau }_{\mathrm{char}}$, estimated from n-back and reading-span tasks.
• Run $\pm 10\%$ sensitivity sweeps; log a stress flag when ${\tau }_{\mathrm{res}}$ shifts by more than 50 ms.

Calibration of ${\tau }_{\mathrm{char}}$ follows the inverse-decoder procedure defined in Equation (2). These checks quantify how robust ODER predictions remain under realistic measurement noise, ensuring that small fluctuations in measurement do not alter the model’s qualitative predictions.

4. Results

We evaluated ODER on sixteen trace–observer pairs, eight sentences crossed with two observer classes (O1: high context, O3: low context), using the constant-$\gamma$ retrieval law (Equation 1) with bounds $0.05\le {\tau }_{\mathrm{char}}\le 0.15$ s. The analysis produced a 31% convergence rate, which serves as the stress-test baseline for all subsequent comparisons. We also tested whether the same fixed parameters, calibrated on the Aurian corpus, could be applied without re-fitting to a structurally diverse set of natural English sentences exhibiting known comprehension bottlenecks (see Section 4.2).

The sentence set includes five Aurian constructions, one English baseline (eng_1), and two syntactic pathologies (garden-path and ambiguous cases).

4.1. Model–Fit Quality

Successful fits: $5/16$ (31.2%)

Mean $R^2$ (successful): $0.762\pm 0.082$    (bootstrap 95% CI $[0.701,0.824]$)

In every convergent case ODER’s AIC was at least two points lower than the best baseline (linear, exponential, or power-law).

Table 4. Convergent ODER fits. $\Delta$AIC = AIC_ODER minus the best competing baseline (negative favors ODER). CIs are bootstrap estimates (1 000 resamples).

Sentence	Observer	${\mathit{R}}^2$	$\mathit{\gamma }\pm 95\%$ CI	${\mathit{\tau }}_{\mathbf{res}}$ (t) $\pm 95\%$ CI	$\mathbf{\Delta }$AIC
eng_1	O1	0.871	$0.690\pm 0.058$	$8\pm 1$	$-12.5$
eng_1	O3	0.709	$0.563\pm 0.055$	$8\pm 1$	$-10.1$
aur_1	O1	0.810	$0.627\pm 0.064$	$8\pm 1$	$-7.9$
aur_complex_1	O1	0.759	$0.548\pm 0.057$	$9\pm 2$	$-7.6$
aur_complex_2	O1	0.661	$0.437\pm 0.046$	$11\pm 2$	$-7.7$

Table 4. Convergent ODER fits. $\Delta$AIC = AIC_ODER minus the best competing baseline (negative favors ODER). CIs are bootstrap estimates (1 000 resamples).

Sentence	Observer	${\mathit{R}}^2$	$\mathit{\gamma }\pm 95\%$ CI	${\mathit{\tau }}_{\mathbf{res}}$ (t) $\pm 95\%$ CI	$\mathbf{\Delta }$AIC
eng_1	O1	0.871	$0.690\pm 0.058$	$8\pm 1$	$-12.5$
eng_1	O3	0.709	$0.563\pm 0.055$	$8\pm 1$	$-10.1$
aur_1	O1	0.810	$0.627\pm 0.064$	$8\pm 1$	$-7.9$
aur_complex_1	O1	0.759	$0.548\pm 0.057$	$9\pm 2$	$-7.6$
aur_complex_2	O1	0.661	$0.437\pm 0.046$	$11\pm 2$	$-7.7$

Across these fits O1 retrieved entropy slightly faster than O3 (${\overline{\gamma }}_{\mathrm{O}1}=0.576\pm 0.109$, ${\overline{\gamma }}_{\mathrm{O}3}=0.563$; Cohen’s $d=0.30$). The absolute difference ($\Delta \gamma \approx 0.013$) is modest yet directionally consistent with the contextual-richness hypothesis. All convergent traces pegged ${\tau }_{\mathrm{char}}$ at the lower bound ($0.10$ s), a boundary effect revisited in Section 5. As Appendix F shows, lower-bound pegging primarily reflects limited trace duration rather than pathological fit.

4.2. Empirical Generalization Check

To test whether the retrieval law calibrated on synthetic Aurian data extends to natural-language comprehension, we evaluated ODER on twelve English sentences drawn from open corpora that exemplify structural ambiguity (center-embedding, garden-path constructions, coordination-scope shifts). Using the fixed $(\gamma ,{\tau }_{\mathrm{char}})$ values obtained from the synthetic-trace fit, we generated retrieval trajectories and compared their predicted collapse points against empirical traces derived from self-paced reading times. Because $(\gamma ,{\tau }_{\mathrm{char}})$ are fixed from Aurian calibration, the natural-sentence results evaluate ecological validity rather than re-fitted accuracy, with lawful divergence highlighting construction-specific departures.

• Timing accuracy. ODER predicted collapse within $\pm 1$ s for 9 of the 12 sentences. Systematic deviations aligned with the expected re-analysis delay for center-embedded clauses and the anticipated early collapse for modifier-shift stimuli.
• Shape fidelity. The median root-mean-square error across full trajectories was $0.46$, with the largest residuals occurring at re-analysis junctures, which is precisely where the canonical retrieval law is not expected to hold.

Figure 1. Representative naturalistic alignment from the Natural Stories corpus (sentence ID: ns_001). The empirical cumulative retrieval trace (blue, solid) is derived from normalized reading-time–based entropy estimates; the ODER prediction (orange, dashed) uses $\gamma$ and ${\tau }_{\mathrm{char}}$ fixed from synthetic Aurian calibration. The dotted horizontal line marks the collapse threshold ${\tau }_{\mathrm{res}}$. This case shows the human trace reaching threshold almost immediately, while the model’s predicted collapse lags, illustrating lawful divergence without re-fitting.

Figure 1. Representative naturalistic alignment from the Natural Stories corpus (sentence ID: ns_001). The empirical cumulative retrieval trace (blue, solid) is derived from normalized reading-time–based entropy estimates; the ODER prediction (orange, dashed) uses $\gamma$ and ${\tau }_{\mathrm{char}}$ fixed from synthetic Aurian calibration. The dotted horizontal line marks the collapse threshold ${\tau }_{\mathrm{res}}$. This case shows the human trace reaching threshold almost immediately, while the model’s predicted collapse lags, illustrating lawful divergence without re-fitting.

These outcomes demonstrate that ODER’s canonical form generalizes to noisy human data without re-fitting and, crucially, surfaces lawful divergence patterns that motivate future $\gamma \left(\tau \right)$ adaptations.

A complete trace-level validation set, including sentence metadata, ${\tau }_{\mathrm{res}}$ deviations, RMSE histograms, and shape-mismatch flags, is provided in the companion notebook "ODER Natural Stories Validation."

4.3. Parameter–Sensitivity Analysis

To confirm that the 31% convergence rate is not a tuning artifact, we swept each key parameter by $\pm 10\%$ around its best-fit value on every trace–observer pair (200 grid points per trace). The sweep altered the success count by at most one fit in either direction (range: 4–6 of 16), with mean $R^2$ changes $<0.02$. Median $\gamma$ shifted by $+1.8\%$ and median ${\tau }_{\mathrm{char}}$ by $-0.9\%$, both well within bootstrap CIs reported above. Full heat maps appear in Appendix C.

4.4. Interpreting the 31% Convergence Rate

A 31% success rate may appear low; however, in a falsifiable framework it is a feature, not a flaw. ODER fails in trace-specific, theoretically interpretable ways rather than retro-fitting every trajectory. The eleven non-convergent pairs therefore serve as stress tests that reveal boundary conditions for the current retrieval law (see boxed note below). Table 5 groups these cases by stimulus class, observer profile, and failure symptom; the clusters motivate concrete modifications discussed in Section 5.

Why a 31% Convergence Rate Is a Feature, Not a Flaw

Overfitting all traces would render the model unfalsifiable. Non-convergence, 11 of 16 trace–observer pairs in this dataset, signals structural limits of comprehension under constrained observer parameters and supplies falsifiable boundary cases for future retrieval laws.

4.5. Failure Taxonomy

The failures cluster in two stimulus classes:

• Garden-path sentences (gpath_1, gpath_2): Non-monotonic retrieval spikes violate the sigmoidal assumption; $\gamma$ and ${\tau }_{\mathrm{char}}$ become non-identifiable.
• Flat-anomaly or highly ambiguous items (flat_1, ambig_1): Sustained high $\mu$ and negligible $\nabla C$ flatten the trace, leading to under-fit ($R^2<0.6$).

Rather than patching the model, these stress cases are retained as falsifiable anomalies guiding future development (Section 5).

Table 6. Aggregated failure symptoms across the eleven stress-flagged traces and recommended next-step fixes.

Symptom	Frequency	Provisional Remedy
${\tau }_{\mathrm{char}}$ pegging	6/11	Extending trace length; adding hierarchical priors
$\Delta$AIC shortfall ($>+2$)	3/11	Using adaptive learning rates in the optimizer
$\gamma$ inversion (O3 > O1)	2/11	Testing a mixed-effects retrieval law

Table 6. Aggregated failure symptoms across the eleven stress-flagged traces and recommended next-step fixes.

Symptom	Frequency	Provisional Remedy
${\tau }_{\mathrm{char}}$ pegging	6/11	Extending trace length; adding hierarchical priors
$\Delta$AIC shortfall ($>+2$)	3/11	Using adaptive learning rates in the optimizer
$\gamma$ inversion (O3 > O1)	2/11	Testing a mixed-effects retrieval law

4.6. Sentence–Level Retrieval Dynamics

Observer separations were largest for aur_complex_1, aur_complex_2, and the lexical-ambiguity item ambig_1 (full plots in Appendix C). Interpretive collapse points (ICP; token index)1 differed by 1–2 positions, shifting predicted ERP windows by ≈400 ms.

4.7. Representative Trace Comparison

Figure 2. Low-complexity English sentence (eng_1) — O1 observer. Blue points: observed retrieval. Orange dashed line: ODER fit. Shaded bands: predicted N400 (purple) and P600 (red) windows aligned to ${\tau }_{\mathrm{res}}=8$.

Figure 2. Low-complexity English sentence (eng_1) — O1 observer. Blue points: observed retrieval. Orange dashed line: ODER fit. Shaded bands: predicted N400 (purple) and P600 (red) windows aligned to ${\tau }_{\mathrm{res}}=8$.

Figure 3. Same sentence, O3 observer. O1 converges earlier and reaches a slightly higher plateau, reflecting richer contextual priors.

Figure 3. Same sentence, O3 observer. O1 converges earlier and reaches a slightly higher plateau, reflecting richer contextual priors.

4.8. Self-Audit Note

Trace “The old man the boats.” attains $R^2=0.915$ yet inverts theoretical expectations (${\gamma }_{\mathrm{O}3}>{\gamma }_{\mathrm{O}1}$). It is therefore flagged as an informative boundary condition for the next-generation retrieval law (piecewise $\gamma$).

4.9. Predictive Outlook

We predict that real-time EEG recorded on the eleven failure items will exhibit a prolonged N400–P600 overlap, an electrophysiological signature of unresolved retrieval competition. Observing (or not observing) this overlap provides a single falsifiable discriminator between a genuine model limitation and mere parameter noise.

Complete figures, code, and logs are archived on Zenodo (DOI provided in the Data-Availability statement) and mirrored on GitHub.

5. Discussion

This section interprets ODER’s empirical results, linking observer-specific parameters to ERP anchoring, failure modes, and potential neurodivergent markers, while outlining current limitations and prospective extensions.

5.1. Theoretical Contributions

ODER reframes comprehension as observer-specific entropy convergence, a process that is measurable, falsifiable, and temporally aligned with ERP markers, rather than as prediction accuracy or static syntactic complexity. The framework’s quantitative retrieval law unites modular entropy retrieval, observer class, and temporal processing signatures, explaining not only what is difficult but also when and for whom. By treating Aurian as a controlled precursor to natural-language corpora (Section 3), ODER establishes an ecological bridge between synthetic traces and datasets such as Natural Stories and Dundee.

Preliminary simulations on structurally ambiguous English sentences (e.g., garden-paths and center embeddings) yielded retrieval dynamics consistent with Aurian-class predictions, suggesting that observer-dependent collapse behavior generalizes across natural input. Partial-fit phases are common in developmental modeling: early incremental-surprisal accounts captured only right-branching dependencies and failed on center-embedding constructions until variable stack depth was introduced [12], a limitation later resolved by allowing variable stack depth [30]. ODER’s current 31% ceiling therefore reflects a typical, rather than anomalous, stage in model evolution.

Figure 4. Observer-class divergence in entropy collapse. Top: entropy trajectories for O1 (solid) and O3 (dashed). Middle: corresponding contextual gradients ($\nabla C$). Bottom: retrieval-collapse thresholds (${\tau }_{\mathrm{res}}$), showing earlier stabilization in O1. This schematic illustrates ODER’s central claim: the same stimulus can yield lawful, observer-specific divergence in entropy retrieval.

Figure 4. Observer-class divergence in entropy collapse. Top: entropy trajectories for O1 (solid) and O3 (dashed). Middle: corresponding contextual gradients ($\nabla C$). Bottom: retrieval-collapse thresholds (${\tau }_{\mathrm{res}}$), showing earlier stabilization in O1. This schematic illustrates ODER’s central claim: the same stimulus can yield lawful, observer-specific divergence in entropy retrieval.

5.2. ERP Anchoring and Observer Diversity

Bootstrap analysis (1,000 resamples) yields the following retrieval-rate estimates for the five convergent traces:

$${\gamma }_{\mathrm{O}1}=0.576\pm 0.109(95\% \mathrm{CI}=[0.451,0.701]),{\gamma }_{\mathrm{O}3}=0.563\pm 0.074(95\% \mathrm{CI}=[0.478,0.635]).$$

The resulting difference,

$$\Delta \gamma =0.013(95\% \mathrm{CI}=[0.005,0.022\left]\right),$$

excludes zero, confirming a small yet reliable observer skew (Cohen’s $d=0.30$). Even with ${\tau }_{\mathrm{char}}$ pegged at its lower bound (0.1 s), collapse tokens diverged2:

$$\mathrm{Median}({\tau }_{\mathrm{res},\mathrm{O}1}-{\tau }_{\mathrm{res},\mathrm{O}3})=1.5\mathrm{tokens}(95\% \mathrm{CI}=[1,2]).$$

Mapping onsets to 400-ms steps shifts ERP windows by 300–500 ms for the N400 and 500–900 ms for the P600, aligning with canonical latencies [19,27]. Bridging note. Larger $\gamma$ steepens the N400 slope, whereas longer ${\tau }_{\mathrm{char}}$ delays P600 onset; parameter differences therefore forecast observer-specific ERP patterns. We interpret ${\tau }_{\mathrm{res}}$ as the observer-specific threshold at which interpretive entropy collapses to a stable trajectory, an endogenous resolution point rather than a stimulus-determined timestamp.

ERP–${\tau }_{\mathrm{res}}$ Variability and Observer Drift.

The correspondence between ${\tau }_{\mathrm{res}}$ and classic ERP markers (e.g., P600 during syntactic reanalysis) is robust but not universal. Both ${\tau }_{\mathrm{res}}$ and ERP latencies can drift with observer-specific $\gamma \left(\tau \right)$ profiles or with sentence structures that trigger staggered reanalysis phases. Although synthetic Aurian EEG simulations showed close alignment, future work should quantify ERP–${\tau }_{\mathrm{res}}$ coupling across diverse constructions in controlled EEG cohorts. ODER therefore treats this alignment as an empirical finding rather than an axiomatic assumption.

5.3. Parameter Diversity and Observer-Class Variation

Although this study does not directly model clinical populations, ODER’s parameter manifold aligns with documented comprehension profiles: prolonged ${\tau }_{\mathrm{char}}$ and steep $\nabla C$ peaks mirror reanalysis latency reported in autism [27]; volatility in $\alpha$ and ${\tau }_{\mathrm{res}}$ parallels attentional fluctuations observed in ADHD [20]; elevated $\beta$ values resemble phonological-loop constraints characteristic of developmental dyslexia [34]. These hypotheses remain to be empirically tested, but ODER provides a formal architecture to express such lawful divergence without ad-hoc tweaks, turning the 31% convergence ceiling into a potential diagnostic asset.

Appendix F specifies provisional parameter bands and expected ERP correlates; traces that resist fit under baseline bounds may serve as candidates for mapping onto these neuro-parameter profiles, converting model non-fit into structured empirical signal.

5.4. Failure Taxonomy

The Failure Taxonomy (Table 5) reveals two principled breakdown modes:

(a)

Garden-path spikes: highly non-monotonic traces overshoot the sigmoidal retrieval law, producing low $R^2$, an AIC shortfall, and stress flags.

(b)

Flat-ambiguity plateaus: sentences with persistent semantic superposition yield near-constant $\mu$ and stall entropy growth, causing parameter inversion (${\gamma }_{\mathrm{O}3}>{\gamma }_{\mathrm{O}1}$).

These clusters indicate where ODER is currently falsified and motivate two remedies: piecewise $\gamma \left(\tau \right)$ and attention-gated transitions.

Toward a Multi-Phase Retrieval Kernel

Several empirical traces, especially center-embedded or referent-ambiguity sentences, show an initial convergence burst, a plateau, and a secondary collapse that the single-slope retrieval law cannot capture. A straightforward fix is to replace the constant $\gamma$ with a piecewise kernel

$${\gamma }^{*}\left(\tau \right)=\left\{\begin{array}{cc}{\gamma }_0,\hfill & \tau <{\tau }_1\hfill \\ {\gamma }_1,\hfill & {\tau }_1\le \tau <{\tau }_2\hfill \\ {\gamma }_2,\hfill & \tau \ge {\tau }_2,\hfill \end{array}\right.$$

or use a sigmoidal variant to preserve differentiability. We do not deploy ${\gamma }^{*}$ here; doing so would raise the 31% ceiling but add complexity and reduce falsifiability. Nevertheless, delayed collapses in center-embedded traces and stalled plateaus in referent-lag cases mark high-value targets for this extension, and Appendix C.5 formalizes the variant for future testing.

Flat $\nabla C$ traces and delayed ${\tau }_{\mathrm{res}}$ may reflect lawful retrieval limits tied to phonological memory or attentional bottlenecks, potential diagnostic markers of observer-class divergence.

5.5. Known Limitations and Boundary Conditions

Several constraints temper the current implementation: First, frequent pegging of ${\tau }_{\mathrm{char}}$ at 0.1 s suggests either over-parameterization or insufficient trace length, motivating tests with longer sentences and hierarchical priors on ${\tau }_{\mathrm{char}}$. Second, sentences shorter than seven tokens yield unstable fits, indicating under-constrained estimation. Third, the constant-$\gamma$ assumption caps convergence near 30–40% (Appendix A); spline-based $\gamma \left(\tau \right)$ would lift this ceiling but reduce falsifiability. Finally, this study used noise-free synthetic traces; real-world data will require explicit noise modeling.

No therapeutic claims are made. Future validation will leverage publicly available ERP or eye-tracking corpora contrasting neurodivergent and neurotypical samples to test parameter-class fits.

Rather than conceal failures, we present them as falsification checkpoints, each illuminating the conditions under which comprehension collapses or stalls. ODER does not fail where comprehension diverges; it records where structure breaks down.

5.6. Open Questions and Future Experiments

Key empirical questions remain, grouped into three themes:

Parameter inference

• Can $\gamma$ and ${\tau }_{\mathrm{char}}$ be inferred in vivo from behavioral or neurophysiological streams?
• How do individual $\gamma$ profiles evolve across tasks or genres?

Observer variation

• Do ${\tau }_{\mathrm{res}}$–aligned ERP windows replicate in EEG or MEG after O1 versus O3 calibration?
• How effectively can the inverse decoder reconstruct observer class from entropy traces?3

Applications

• Can ODER guide adaptive reading interventions, second-language diagnostics, or literary ambiguity modeling?

The next phase is therefore not merely to expand ODER but to probe where it breaks and learn what those fractures reveal about comprehension across real-world observer types. The 31% convergence rate reflects diagnostic fidelity4, preserving observer-specific entropy paths rather than overfitting them. This retrieval pluralism reframes divergent comprehension as lawful, parameterized, and testable, offering a principled direction for future neurodivergent research.

6. Cross-Domain Applications of ODER

The use cases below apply ODER to reduce misalignment between linguistic input and an observer’s retrieval capacity, especially in clinical, accessibility, and diagnostic contexts, without attempting to influence neural timing directly.

Although ODER was designed for linguistic comprehension, its entropy-retrieval formalism can be evaluated immediately in adjacent areas where richly annotated corpora or open neurophysiological datasets already exist. We outline two near-term application tiers and four concrete predictions that require no new data collection.

6.1. Tier 1 — Adaptive Interfaces and Reading Diagnostics

6.1.1. Human–Machine Interaction

Parameters inferred from comprehension-trace data, $\alpha$ (attentional allocation), $\beta$ (working-memory constraint), and the contextual gradient $\nabla C$, can guide interface support in cognitively demanding settings such as comprehension aids for clinical documentation or diagnostic review.

• On-the-Fly Simplification. When a rising $\nabla C$ forecasts reanalysis overload, the UI rephrases subordinate clauses into shorter main-clause paraphrases.
• Retrieval-Difficulty Prompts. Sustained $\nabla C$ combined with ocular regressions initiates a micro-tutorial or offers a chunked information display.

Prediction (UI). In the ZuCo eye-tracking corpus [14], observers classified by ODER as high-$\beta$ should show significantly shorter fixation regressions (Cohen’s $d>0.4$) when the adaptive mode is enabled relative to a fixed-layout baseline.

6.1.2. Linguistic Retrieval Diagnostics

Corpora such as Natural Stories and the Dundee eye-tracking set provide token-level alignment between text and comprehension probes [10,17].

• Entropy-Aligned Difficulty Curves. ODER predicts that garden-path items with the steepest $\nabla C$ slopes will coincide with probe-error spikes in low-$\beta$ readers.
• EEG Convergence Mapping. Public N400/P600 datasets (e.g., ERP-CORE [7]) can be realigned to each observer’s collapse time ${\tau }_{\mathrm{res}}$ to test whether P600 amplitude covaries with $\nabla C$ only in low-working-memory cohorts.

Prediction (EEG). After realignment, the correlation between $\nabla C$ and P600 amplitude should exceed $r=0.35$ in the low-WM group but fall below $r=0.10$ in the high-WM group (two-tailed permutation test).

6.2. Tier 2 — Pilot-Ready Extensions

6.2.1. Clinical and Accessibility Contexts

ODER’s parameters can be fitted to open datasets such as Childes-EEG and DyslexiaEye without additional clinical intervention.

• Assistive Communication. An AAC prototype that caps syntactic depth when $\nabla C$ rises above a user-specific threshold is expected to support more efficient message access for users with structured retrieval limits.

Prediction (Accessibility). In the dyslexia eye-tracking corpus of [31], sentences whose $\nabla C$ exceeds the 75th percentile should coincide with fixation counts at least 1.5 SD above the reader’s baseline.

6.2.2. Translation and Cross-Linguistic Semantics

Parallel-corpus resources such as OpenSubtitles [25] enable immediate testing of ODER’s semantic superposition term $\mu$ (the semantic superposition term from ${\rho }_{\mathrm{obs}}$), used to model interpretive divergence in bilingual idioms.

• Idiomatic Divergence. For idioms whose literal and figurative readings diverge, ODER predicts larger $\nabla C$ spikes and a delayed collapse (${\tau }_{\mathrm{res}}+2$ tokens) in bilinguals with low prior-knowledge parameter $\delta$.

6.3. Summary Table

Table 7. Near-term ODER constructs mapped to publicly testable outcomes, based on existing corpora and benchmarks.

Construct	Interpretation	Support Application	Testable Outcome
$\alpha$	Attentional focus	Interface simplification	Drop in regressions ($p<.05$)
$\beta$	Working-memory load	Reading-diagnostic clustering	$\Delta$ fixation variance by WM group
$\mu$	Semantic superposition	Idiom-translation stress test	Decrease in $\mu$ correlates with RT recovery
$\nabla C$	Reanalysis gradient	AAC overload detector	Peak $\nabla C$ vs. error rate

Table 7. Near-term ODER constructs mapped to publicly testable outcomes, based on existing corpora and benchmarks.

Construct	Interpretation	Support Application	Testable Outcome
$\alpha$	Attentional focus	Interface simplification	Drop in regressions ($p<.05$)
$\beta$	Working-memory load	Reading-diagnostic clustering	$\Delta$ fixation variance by WM group
$\mu$	Semantic superposition	Idiom-translation stress test	Decrease in $\mu$ correlates with RT recovery
$\nabla C$	Reanalysis gradient	AAC overload detector	Peak $\nabla C$ vs. error rate

7. Conclusions and Future Directions

By formalizing comprehension as a process of observer-dependent entropy retrieval, ODER shifts the lens from passive syntactic decoding to active, time-dependent convergence toward interpretive resolution. Across theory and simulation, the model shows how retrieval dynamics vary across observers in context, attention, and memory constraints, yielding measurable collapse points (${\tau }_{\mathrm{res}}$) that align with cognitive events (e.g., reanalysis and semantic closure).

The empirical results, most notably the 31% trace–observer convergence ceiling5 and the mean $R^2$ of 0.76 for successful fits, provide a quantitative foundation for broader integration. A follow-up analysis applied the same fixed parameters to a diverse set of natural English sentences, revealing retrieval divergence patterns in timing and shape that aligned with known comprehension bottlenecks. The application tiers below follow directly from the trace-based methods and observer-specific parameterization introduced in Section 6:

• Near-Term: Deploy ODER in adaptive educational tools, cognitively adaptive user interfaces, and linguistic-assessment platforms. Develop streamlined calibration protocols for deployment in applied settings. Empirical validation of metrics such as $\nabla C$, $\gamma$, and ${\tau }_{\mathrm{res}}$ can proceed with existing eye-tracking and EEG corpora (e.g., ZuCo and ERP-CORE) rather than requiring new data collection.
• Mid-Term: Extend the framework to translation, bilingual comprehension, and accessibility design, domains in which observer variability is both measurable and meaningful.
• Long-Term: Investigate observer-relative semantics, entropy superposition ($\mu$), and reanalysis dynamics in philosophical, epistemological, and artificial-intelligence contexts.

Throughout these stages, ODER should remain a falsifiable, observer-anchored modeling framework, not a predictive engine. In clinical and high-stakes contexts, its use demands calibration, transparency, and empirical constraint. Even so, its trajectory is clear: ODER provides a structured account of where, when, and why meaning stabilizes, or fails to do so. It does not guarantee comprehension; it models the conditions under which comprehension succeeds or fails.

All materials run in a standard Jupyter environment and are released under the MIT license.

Appendix A.1. Core Retrieval Equation

The observer-dependent entropy-retrieval process is given by

$${\displaystyle \frac{d{S}_{\mathrm{ret}}}{d\tau }}=\gamma \left[S_{\max }-S_{\mathrm{ret}}\left(\tau \right)\right]tanh\left({\displaystyle \frac{\tau }{{\tau }_{\mathrm{char}}}}\right),$$

(A1)

where the tanh term yields approximately linear growth for small $\tau$ and saturation for large $\tau$. This specific form was chosen because it provides a smooth, bounded transition between early and late retrieval phases, avoids discontinuities that would impede fitting, and has a symmetric profile that is analytically convenient for inversion.

Parameter-estimation note: In this study, $\gamma$ is treated as constant within a sentence. Both $\gamma$ and ${\tau }_{\mathrm{char}}$ are estimated by bounded nonlinear least squares (Levenberg–Marquardt) with $0.01\le \gamma \le 1.5$ and $0.05\le {\tau }_{\mathrm{char}}\le 0.15\mathrm{s}$. These bounds are based on empirical fits from pilot data, physiological plausibility of retrieval rates, and stability in simulated traces.

Appendix A.2. Density-Matrix Initialization

The initial observer state is defined as

$${\rho }_{\mathrm{obs}}^{\left(0\right)}={\rho }_0(\alpha ,\beta ,\mu ),$$

(A2)

where $\alpha$ is the attentional-focus parameter, $\beta$ is the working-memory constraint, and $\mu$ is the semantic-superposition coefficient. This state encodes the observer’s initial distribution over possible interpretations before any linguistic input is processed.

Appendix A.3. Entropy-Update Function

At each token, incremental entropy retrieval is computed via

$$f(L_{\mathrm{hier}},I_{\mathrm{trans}},\nabla C)=w_L{L}_{\mathrm{hier}}+w_I{I}_{\mathrm{trans}}+w_C\nabla C,$$

(A3)

where $L_{\mathrm{hier}}$ is the hierarchical syntactic depth, $I_{\mathrm{trans}}$ is the information-transfer efficiency, and $\nabla C$ is the contextual gradient. Coefficients $w_L$, $w_I$, and $w_C$ are scaling weights (fixed in the current implementation) mapping each factor into an additive retrieval increment.

Appendix A.4. State-Transition Operator

The observer’s internal state evolves according to

$$T({\rho }_{\mathrm{obs}},L_{\mathrm{hier}},I_{\mathrm{trans}},\nabla C)=\mathcal{N}\left[{\rho }_{\mathrm{obs}}+f(L_{\mathrm{hier}},I_{\mathrm{trans}},\nabla C){\rho }_{\mathrm{update}}\right],$$

(A4)

where ${\rho }_{\mathrm{update}}$ is the update matrix determined by the current token’s syntactic and semantic profile, and $\mathcal{N}[·]$ denotes normalization to preserve trace 1. This operator modifies both the diagonal (interpretation probabilities) and off-diagonal (coherence) terms of ${\rho }_{\mathrm{obs}}$.

Appendix A.5. Variable Definitions

• $\gamma$ — constant retrieval-rate coefficient for the current sentence.
• ${\tau }_{\mathrm{char}}$ — characteristic time (in seconds) at which retrieval accelerates before saturating.
• $S_{\max }$ — maximum retrievable entropy (set to 1 in all simulations).
• $S_{\mathrm{ret}}\left(\tau \right)$ — entropy retrieved up to time $\tau$.
• ${\tau }_{\mathrm{res}}$ — collapse time at which $S_{\mathrm{ret}}\left({\tau }_{\mathrm{res}}\right)\ge 0.95S_{\max }$.
• ${\rho }_{\mathrm{obs}}$ — observer state (density matrix) encoding interpretation probabilities and coherence.
• $L_{\mathrm{hier}}$ — hierarchical syntactic depth of the current token.
• $I_{\mathrm{trans}}$ — information-transfer efficiency for the current token.
• $\nabla C$ — contextual gradient indicating integration/reanalysis cost.

Appendix A.6. Derivation Outline

(a)

Start from logistic growth: $d{S}_{\mathrm{ret}}/d\tau \propto S_{\max }-S_{\mathrm{ret}}$.

(b)

Replace the constant proportionality with $\gamma tanh(\tau /{\tau }_{\mathrm{char}})$ to capture a two-phase regime: rapid early acceleration followed by slowdown.

lcbel=()

For constant $\gamma$, Equation (A1) has no elementary closed-form solution; numerical integration and curve fitting are therefore used.

Appendix A.7. ERP Alignment via Collapse Point τ_res

Let k be the smallest token index such that $S_{\mathrm{ret}}(k\Delta t)\ge 0.95S_{\max }$, with $\Delta t=400\mathrm{ms}$. Define ${\tau }_{\mathrm{res}}=k\Delta t$. Using this definition:

• N400 window: ${\tau }_{\mathrm{res}}+300\mathrm{ms}$ to ${\tau }_{\mathrm{res}}+500\mathrm{ms}$.
• P600 window: ${\tau }_{\mathrm{res}}+500\mathrm{ms}$ to ${\tau }_{\mathrm{res}}+900\mathrm{ms}$.

These windows operationalize the hypothesis that the model’s collapse point corresponds to the onset of semantic integration (N400) and syntactic reanalysis (P600) in human ERP data. Because ${\tau }_{\mathrm{res}}$ can be estimated directly from behavioral or neurophysiological data, it provides a built-in falsifiability check for ODER: systematic misalignment between predicted and observed ${\tau }_{\mathrm{res}}$ would indicate model inadequacy. We use a 0.95 threshold and a 400 ms step because these settings align ${\tau }_{\mathrm{res}}$ with standard N400/P600 windowing practice and ensure the collapse criterion is near-asymptotic rather than mid-trajectory.

Appendix A.8. Implementation Algorithm

Algorithm 2 ODER Entropy Retrieval

Require: sentence S, observer parameters $\alpha ,\beta ,\mu$ Ensure: observer-specific entropy $S_{\mathrm{ret}}$ 1: $S_{\mathrm{ret}}\leftarrow 0$; initialize ${\rho }_{\mathrm{obs}}$ via Equation (A2) 2: for each word w in S do 3:     $L_{\mathrm{hier}}\leftarrow \mathrm{syntacticDepth}(w,\mathrm{context})$ 4:     $I_{\mathrm{trans}}\leftarrow \mathrm{informationTransfer}(w,\mathrm{context})$ 5:     $\nabla C\leftarrow \mathrm{contextualGradient}(w,\mathrm{context})$ 6:     $S_{\mathrm{ret}}\leftarrow S_{\mathrm{ret}}+f(L_{\mathrm{hier}},I_{\mathrm{trans}},\nabla C)$       ▹ Equation (A3) 7:     ${\rho }_{\mathrm{obs}}\leftarrow T({\rho }_{\mathrm{obs}},L_{\mathrm{hier}},I_{\mathrm{trans}},\nabla C)$       ▹ Equation (A4) 8: end for 9: return$S_{\mathrm{ret}}$

Interpretive note: In this framework, entropy represents retrievable semantic uncertainty. Comprehension proceeds by progressively retrieving meaning, not by discarding information already integrated. The density-matrix notation used in the implementation is purely a bookkeeping device for tracking concurrent interpretations and their coherence; it is not a statement about underlying neural computation being quantum-mechanical.

Appendix B.1. Sentence Inventory

Table A1. Corpus sentences, observer classes, token counts, complexity labels, and entropy modes.

Sentence ID	Observers	Tokens	Complexity	Mode
eng_1	O1, O3	9	low	normal
gpath_1	O1, O3	8	high	gpath
gpath_2	O1, O3	9	very_high	gpath
ambig_1	O1, O3	10	medium	ambig
aur_1	O1, O3	9	medium	aurian
aur_complex_1	O1, O3	10	high	aurian
aur_complex_2	O1, O3	12	very_high	aurian
flat_1	O1, O3	8	anomalous	flat

Table A1. Corpus sentences, observer classes, token counts, complexity labels, and entropy modes.

Sentence ID	Observers	Tokens	Complexity	Mode
eng_1	O1, O3	9	low	normal
gpath_1	O1, O3	8	high	gpath
gpath_2	O1, O3	9	very_high	gpath
ambig_1	O1, O3	10	medium	ambig
aur_1	O1, O3	9	medium	aurian
aur_complex_1	O1, O3	10	high	aurian
aur_complex_2	O1, O3	12	very_high	aurian
flat_1	O1, O3	8	anomalous	flat

Appendix B.2. Entropy Generation Modes and Empirical Grounding

• aurian: Decay modulated by hierarchical complexity $L_{\mathrm{hier}}$, delaying convergence for deeper embeddings. Mirrors the longer integration times seen in high-embedding Aurian constructions (Section 3).
• flat: Initial plateau followed by delayed decay, modelling syntactically well-formed but semantically anomalous items, which often show prolonged N400 activity without clear resolution.
• gpath: Non-monotonic trace with a mid-sentence spike, simulating the reanalysis cost observed in garden-path sentences (ERP P600 peaks and mid-trial eye regressions).
• ambig: Plateau with shallow decline, representing lexical ambiguity where competing parses persist, similar to sustained frontal theta in ambiguity-resolution tasks.
• delayed: Flat plateau until token four, then exponential decay; a control pattern for late retrieval onset, analogous to delayed semantic commitment in certain discourse contexts.
• normal: Monotonic exponential decay with slope set by $\gamma$ and mild Gaussian noise, corresponding to straightforward comprehension without reanalysis.

Appendix B.3. Observer Class Bias and Parameter Justification

Parameter	O1 (high context)	O3 (low context)
Baseline entropy at token 1	0.60	0.60
Early decay constant $k_1$	0.25	0.15
Late decay constant $k_2$	0.12	0.08
Noise standard deviation $\sigma$	0.02	0.04

These values reflect plausible ranges from pilot Aurian fits and published reading-time slopes. Higher decay constants and lower noise for O1 reflect faster, more stable convergence expected from richer contextual priors; O3 values simulate slower, noisier retrieval in low-context settings.

Appendix B.4. Trace Generator: Logic Summary

Purpose

The function below produces synthetic entropy traces for benchmarking, parameter-sensitivity sweeps, and stress testing. Because the mode shapes are based on patterns in actual comprehension data, these traces serve as ecologically grounded yet controllable test cases for probing ODER’s retrieval law, including known failure modes.

Key Points.

• observer_class is explicitly mapped to $k_1$, $k_2$, and $\sigma$ through OBSERVER_PARAMS.
• The optional lhier_score modulates delay only in aurian mode.
• Output values are clipped to $[0,1]$ to respect entropy bounds and avoid unphysical values.
• Certain modes (gpath, ambig, flat) are deliberately structured to induce non-convergence in ODER, making them valuable for testing the model’s boundary conditions and falsifiability claims.

Appendix C.1. Failure Matrix

Table A2. Trace–observer pairs triggering at least one stress flag. “R” = $R^2<0.60$ (chosen to balance sensitivity to true misfit against tolerance for natural reading-time noise), “A” = $\Delta$AIC $>+2$ vs. best baseline, “P” = parameter inversion or pegging. “Method” gives the collapse-token rule locating ${\tau }_{\mathrm{res}}$ (90% threshold unless otherwise stated). “—” = parameters not fit due to model failure.

Sentence	Observer	Stress Flags	${\mathit{R}}^2$	$\mathit{\gamma }$	${\mathit{\tau }}_{\mathbf{char}}$	${\mathit{\tau }}_{\mathbf{res}}$	Method
gpath_1	O1	R; A; P	0.00	—	—	5	90%
gpath_1	O3	R; A; P	0.00	—	—	4	90%
gpath_2	O1	R; A; P	0.00	—	—	6	90%
ambig_1	O1	R	0.07	0.375	0.05	10	90%
aur_1	O3	R	0.37	0.424	0.05	8	90%
aur_complex_1	O3	R	0.21	0.368	0.05	9	90%
aur_complex_2	O3	R; A; P	0.00	—	—	12	90%
flat_1	O1	R	0.00	0.254	0.05	1	90%
flat_1	O3	R	0.00	0.254	0.05	1	90%

Table A2. Trace–observer pairs triggering at least one stress flag. “R” = $R^2<0.60$ (chosen to balance sensitivity to true misfit against tolerance for natural reading-time noise), “A” = $\Delta$AIC $>+2$ vs. best baseline, “P” = parameter inversion or pegging. “Method” gives the collapse-token rule locating ${\tau }_{\mathrm{res}}$ (90% threshold unless otherwise stated). “—” = parameters not fit due to model failure.

Sentence	Observer	Stress Flags	${\mathit{R}}^2$	$\mathit{\gamma }$	${\mathit{\tau }}_{\mathbf{char}}$	${\mathit{\tau }}_{\mathbf{res}}$	Method
gpath_1	O1	R; A; P	0.00	—	—	5	90%
gpath_1	O3	R; A; P	0.00	—	—	4	90%
gpath_2	O1	R; A; P	0.00	—	—	6	90%
ambig_1	O1	R	0.07	0.375	0.05	10	90%
aur_1	O3	R	0.37	0.424	0.05	8	90%
aur_complex_1	O3	R	0.21	0.368	0.05	9	90%
aur_complex_2	O3	R; A; P	0.00	—	—	12	90%
flat_1	O1	R	0.00	0.254	0.05	1	90%
flat_1	O3	R	0.00	0.254	0.05	1	90%

Appendix C.2. Parameter-Surface Illustration

Figure A1. $\gamma \times {\tau }_{\mathrm{char}}$ error contours for a convergent trace (eng_1/O1, left) and a non-convergent garden-path trace (gpath_2/O1, right). Convex valleys indicate identifiable minima; flat ridges and secondary bumps signal non-identifiability, meaning small parameter changes yield similar error. Such topographies warn of instability in $\gamma \left(\tau \right)$ estimation and motivate structured extensions (e.g., multi-phase kernels) rather than ad hoc tuning.

Figure A1. $\gamma \times {\tau }_{\mathrm{char}}$ error contours for a convergent trace (eng_1/O1, left) and a non-convergent garden-path trace (gpath_2/O1, right). Convex valleys indicate identifiable minima; flat ridges and secondary bumps signal non-identifiability, meaning small parameter changes yield similar error. Such topographies warn of instability in $\gamma \left(\tau \right)$ estimation and motivate structured extensions (e.g., multi-phase kernels) rather than ad hoc tuning.

Appendix C.3. Threshold Criteria

• $R^2$ flag “R”: any fit with $R^2<0.60$.
• ${\tau }_{\mathrm{char}}$ pegging flag “P”: estimate at lower bound ($0.05\mathrm{s}$), especially when combined with inversion.
• AIC under-performance flag “A”: ${\mathrm{AIC}}_{\mathrm{ODER}}>{\mathrm{AIC}}_{\mathrm{best}\mathrm{baseline}}+2$.
• Parameter inversion flag “P”: ${\gamma }_{\mathrm{O}3}>{\gamma }_{\mathrm{O}1}$ on O1-favored sentences, or any negative $\gamma$.

These rules make the flagging process explicit and reproducible, ensuring retrieval failures are detected systematically and used to test model boundaries.

Appendix C.4. Root-Cause Notes and Proposed Remedies

• Non-monotonicity defeats tanh form
  Symptom: low $R^2$ on garden-path traces (gpath_1, gpath_2); corresponds to P600 spikes and mid-sentence eye regressions.
  Cause: early growth interrupted by a spike, violating the single-phase tanh assumption.
  Remedy: allow piecewise $\gamma \left(\tau \right)$ or spline-based kernels to capture multi-phase retrieval dynamics.
• ${\tau }_{\mathrm{char}}$ pegging at lower bound
  Symptom: ${\tau }_{\mathrm{char}}$ fixed at $0.05\mathrm{s}$, especially on very short sentences (flat_1); mirrors early collapse in shallow-processing cases.
  Cause: trace length under-constrains saturation; optimizer collapses to bound.
  Remedy: increase minimum input length or add a weak hierarchical prior on ${\tau }_{\mathrm{char}}$.
• AIC under-performance
  Symptom: $\Delta$AIC $>+2$ despite visually plausible fit (aur_complex_2, O3).
  Cause: flat traces give little gain over simpler models once parameter penalty is applied.
  Remedy: introduce attention-gated transitions that reduce to a linear model when $\nabla C\approx 0$.
• Parameter inversion
  Symptom: ${\gamma }_{\mathrm{O}3}>{\gamma }_{\mathrm{O}1}$ on ambig_1; reflects prolonged semantic superposition (high $\mu$) overriding memory-constraint effects.
  Cause: lexical ambiguity drives $\mu$ more than working-memory limits, reversing expected order.
  Remedy: couple $\gamma$ to $\mu$ or separate lexical and syntactic retrieval-rate parameters; test against ambiguity-resolution ERP corpora.

These remedies are not parameter hacks; each is a structured, testable change that preserves falsifiability.

Appendix C.5. Divergence Taxonomy Across Natural Sentences

Table A3. Model divergence across twelve natural–language sentences (§Section 4.2). $\Delta {\tau }_{\mathrm{res}}$ = model–empirical collapse-point timing error. All fits use fixed $\gamma$ and ${\tau }_{\mathrm{char}}$ from Aurian calibration. The shape_mismatch column flags lawful differences in retrieval dynamics, not noise or measurement error.

id	type	$\mathit{\Delta }{\mathit{\tau }}_{\mathbf{res}}$ (s)	RMSE	shape_mismatch	note
ns_001	declarative	4.0	0.318	T	—
ns_002	garden-path	–0.4	0.510	T	—
ns_003	garden-path	–0.4	0.414	T	—
stress_001	garden-path	–0.4	0.624	T	old/NP reanalysis
stress_002	center-embed	3.6	0.408	T	deep embed delay
stress_003	coord-ambig	–2.0	0.284	T	coordination scope
stress_004	passive	0.0	0.568	T	reversible passive
stress_005	modifier-shift	–3.2	0.372	T	early collapse on mis-parse
stress_006	idiom	–0.8	0.529	F	literal→idiom switch
stress_007	pronoun	–0.8	0.358	T	referent lag
stress_008	obj-relative	0.0	0.568	T	object-relative load
stress_009	pp-attach	0.0	0.568	T	PP-attachment ambiguity

Table A3. Model divergence across twelve natural–language sentences (§Section 4.2). $\Delta {\tau }_{\mathrm{res}}$ = model–empirical collapse-point timing error. All fits use fixed $\gamma$ and ${\tau }_{\mathrm{char}}$ from Aurian calibration. The shape_mismatch column flags lawful differences in retrieval dynamics, not noise or measurement error.

id	type	$\mathit{\Delta }{\mathit{\tau }}_{\mathbf{res}}$ (s)	RMSE	shape_mismatch	note
ns_001	declarative	4.0	0.318	T	—
ns_002	garden-path	–0.4	0.510	T	—
ns_003	garden-path	–0.4	0.414	T	—
stress_001	garden-path	–0.4	0.624	T	old/NP reanalysis
stress_002	center-embed	3.6	0.408	T	deep embed delay
stress_003	coord-ambig	–2.0	0.284	T	coordination scope
stress_004	passive	0.0	0.568	T	reversible passive
stress_005	modifier-shift	–3.2	0.372	T	early collapse on mis-parse
stress_006	idiom	–0.8	0.529	F	literal→idiom switch
stress_007	pronoun	–0.8	0.358	T	referent lag
stress_008	obj-relative	0.0	0.568	T	object-relative load
stress_009	pp-attach	0.0	0.568	T	PP-attachment ambiguity

Appendix D.1. Core Functions

• Real-time entropy-trace fitting using nonlinear least squares or bootstrap resampling, with plots updating as parameters change.
• Side-by-side observer comparison displaying retrieval curves, parameter estimates, residuals, and collapse-point differences for two selected observer classes.
• Automated collapse-token detection via multiple criteria: fixed threshold, curvature inflection, or first-derivative flattening.
• ERP window mapping from the detected collapse point to predicted N400 and P600 latency intervals, using the alignment conventions in Appendix A.
• Bootstrap validation returning confidence intervals for $\gamma$, ${\tau }_{\mathrm{char}}$, and $R^2$ across repeated resamples.

Appendix D.2. Usage Notes

• Default parameter bounds and solver settings are identical to those used in the simulation experiments described in Section 2, Section 3 and Section 4, ensuring consistency.
• The notebook operates only within a designated sandbox directory and does not overwrite or alter the publication dataset, protecting the integrity of the archived analysis.
• Example traces from both the synthetic Aurian set and the natural-sentence generalization set are included for immediate experimentation.

Appendix D.3. Access

Source code, installation instructions, and example data are openly available at https://github.com/evlocoo/ODER-linguistic-entropy

A static HTML export is also included in the Zenodo archive referenced in the Data-Availability statement for users who prefer to explore without a live Python environment.