Observer-Dependent Entropy; Cognitive Linguistics; Information Retrieval; Quantum Information; Benchmarking

1. Introduction

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. Yet 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 per se.

• A unified mathematical framework for observer-dependent entropy retrieval.
• Explicit retrieval (Eq. 2) and transition (Eq. 5) 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 model ambiguity and interference without claiming 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.

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

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.

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

2. Mathematical Framework

This section formalizes ODER’s observer-centric retrieval law, defining the entropy kernel, contextual gradient, density-matrix state, and inverse decoder that together generate the trace-level metrics used throughout the paper. For reference, the core observer parameters are attention, working-memory capacity, and prior knowledge.

A Note on Reading This Section

Readers unfamiliar with quantum formalism may focus on the high-level interpretations of Eqs. 1–10. Density matrices capture multiple possible interpretations at once, and the operators below model how those interpretations evolve under linguistic input and observer constraints. Full derivations appear in Appendix A.

2.1. Observer-Dependent Entropy

$$S_{\mathrm{obs},j}\left(\tau \right)=-\sum _i{P}_{\mathrm{obs},ij}\left(\tau \right)log{P}_{\mathrm{obs},ij}\left(\tau \right)+{\int }_0^{\tau }f\left(L_{\mathrm{hier}}\left(t\right),I_{\mathrm{trans}}\left(t\right),\nabla C\left(t\right)\right)dt,$$

(1)

where j indexes the observer trace. The first term parallels Shannon entropy, whereas the integral introduces observer- and time-dependent factors:

• $L_{\mathrm{hier}}$: hierarchical syntactic complexity
• $I_{\mathrm{trans}}$: information-transfer efficiency
• $\nabla C$: contextual gradient (captures reanalysis effort; spikes correspond to increased retrieval load)

2.2. Retrieval Kernel

$$f(L_{\mathrm{hier}},I_{\mathrm{trans}},\nabla C)=A{\left[\alpha L_{\mathrm{hier}}+\beta I_{\mathrm{trans}}\right]}^{\delta }exp\left(-\tau /{\tau }_0\right),$$

(2)

(notation is quantum-inspired only; no physical quantum computation is assumed). Here A is a scaling constant and $\delta$ modulates nonlinearity in the joint influence of $L_{\mathrm{hier}}$ and $I_{\mathrm{trans}}$.

2.3. Contextual-Gradient Operator

$$\nabla C\left(t\right)={\displaystyle \frac{dC\left(t\right)}{dt}},|\nabla C|\le M,C\in C^1[0,\tau ].$$

(3)

Spikes in $\nabla C$ correspond to rapid reanalysis events—for example, P600 or N400 peaks. Unlike signal-level surprisal gradients, $\nabla C$ is computed in observer space: it reflects coherence retrieval constrained by memory, attention, and entropy access; therefore it cannot be derived from the stimulus alone.

2.4. Quantum-Inspired Density Matrix

$${\rho }_{\mathrm{obs}}\left(\tau \right)=\left(\begin{array}{cc}\alpha \left(\tau \right)& \mu \left(\tau \right)\\ {\mu }^{*}\left(\tau \right)& \beta \left(\tau \right)\end{array}\right),$$

(4)

where $\alpha \left(\tau \right)$ tracks attention, $\beta \left(\tau \right)$ tracks memory load, and $\mu \left(\tau \right)$ encodes coherence.In particular, μ encodes cross-interpretation coherence and may modulate retrieval interference in future variants; see Sec. Section 6 for planned extensions. The density-matrix notation is quantum-inspired only; no physical quantum computation is assumed. It serves as a compact way to represent ${\rho }_{\mathrm{obs}}$, a weighted mixture of competing interpretations under observer constraints.

2.5. State Transition and Unitary Evolution

$$\begin{array}{cc}\hfill {\rho }_{\mathrm{obs}}(\tau +\Delta \tau )& =T\left({\rho }_{\mathrm{obs}}\left(\tau \right),L_{\mathrm{hier}},I_{\mathrm{trans}},\nabla C\right),\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill T({\rho }_{\mathrm{obs}},L_{\mathrm{hier}},I_{\mathrm{trans}},\nabla C)& =U\left(\tau \right){\rho }_{\mathrm{obs}}{U}^{†}\left(\tau \right),\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill U\left(\tau \right)& =exp\left[-i\left(H_0+H_{\mathrm{int}}(L_{\mathrm{hier}},I_{\mathrm{trans}},\nabla C)\right)\tau \right],\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill H_{\mathrm{int}}& ={\gamma }_1{L}_{\mathrm{hier}}{\sigma }_x+{\gamma }_2{I}_{\mathrm{trans}}{\sigma }_z+{\gamma }_3\nabla C{\sigma }_y.\hfill \end{array}$$

(8)

2.6. Forward Retrieval Law and Inverse Decoder

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

(9)

(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 $30\text{–}40\%$ under the constant-$\gamma$ assumption; allowing $\gamma \left(\tau \right)$ to vary piece-wise raises this ceiling (see Discussion Section 6), but we retain this parsimonious form to preserve sharp falsifiability of the baseline hypothesis.

Inverse decoder:

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

(10)

2.7. Implementation Algorithm

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).

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 Eq. 4   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:    $S_{\mathrm{obs}}\leftarrow S_{\mathrm{obs}}+f(L_{\mathrm{hier}},I_{\mathrm{trans}},\nabla C)$▹ Eq. 2   7:    ${\rho }_{\mathrm{obs}}\leftarrow T({\rho }_{\mathrm{obs}},L_{\mathrm{hier}},I_{\mathrm{trans}},\nabla C)$▹ Eq. 5   8:end for   9:return$S_{\mathrm{obs}}$

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

3. 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.

Benchmark Metrics

Metric	Interpretation
ERR	Entropy-reduction rate (slope of $S_{\mathrm{obs}}$)
${\tau }_{\mathrm{res}}$	Retrieval-collapse point (resolution time)
$R^2$	Overall model–trace fit quality
$\Delta$AIC	Parsimony advantage over baselines
$\nabla C$	Contextual gradient (reanalysis effort)
$\gamma$	Entropy-retrieval rate coefficient
CV Error	Mean absolute error across k folds
$\Delta \gamma$	Observer-class divergence in $\gamma$

As summarized in Table Section 3, the benchmarking protocol evaluates both fit quality and neurocognitive interpretability.

3.1. 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–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-grain correlate of local ERR fluctuations.

3.2. Protocol

• Compute baseline entropy with Eq. 1 for all Aurian stimuli.
• Run ODER retrieval dynamics (Eqs. 2–5) and log stress flags whenever $R^2<0.60$ or ${\tau }_{\mathrm{res}}>2$ seconds.

3.3. 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 5):

• 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 boxed metrics to observable brain dynamics (boundary-case discussion in Section 6.2).

3.4. 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, 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.

4. 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. Appendix D provides the full calibration roadmap and code pointers for these noninvasive procedures.

4.1. 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.

4.1.1. Aurian Grammar Specification

Core syntactic rules

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

(11)

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

(12)

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

(13)

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

(14)

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: Kem vora fel (“He/She sees the object”)
• Medium entropy: Kem vora fel ren (“He/She sees the object quickly”)
• High entropy: Kem daz sul tir fel vora (“He/She thinks that the object falls”)
• Very high entropy: Kem daz sul tir fel sul ren vora poli zul (“He/She thinks that the object that quickly falls does not move”)

Table 2 summarizes how token count maps to cumulative $L_{\mathrm{hier}}$ across these four entropy tiers.

4.1.2. 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 serves as a minimal-pair generator that isolates structural factors while holding lexical semantics constant. This controlled start 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.

4.2. 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 Eq. 10. These checks quantify how robust ODER predictions remain under realistic measurement noise.

5. 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 (Eq. 9) 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.

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

5.1. 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 3. 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	$\mathit{\Delta }$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 3. 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	$\mathit{\Delta }$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 6.

5.2. 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.

5.3. Interpreting the 31% Convergence Rate

A 31% success rate may appear low, but 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 4 groups these cases by stimulus class, observer profile, and failure symptom; the clusters motivate concrete modifications discussed in Section 6.

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.

5.4. 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 6).

Table 5. 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 5. 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

5.5. 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 one–two positions, shifting predicted ERP windows by ≈400 ms.

5.6. Representative Trace Comparison

Figure 1. 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 1. 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. Same sentence, O3 observer. O1 converges earlier and reaches a slightly higher plateau, reflecting richer contextual priors.

Figure 2. Same sentence, O3 observer. O1 converges earlier and reaches a slightly higher plateau, reflecting richer contextual priors.

5.7. 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 (piece-wise $\gamma$).

5.8. 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 in the Zenodo bundle cited in the Data-Availability statement.

6. 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.

6.1. Theoretical Contributions

ODER reframes comprehension as observer-specific entropy convergence rather than prediction or static syntactic complexity. The framework introduces a quantitative retrieval law that makes observer divergence both measurable and falsifiable, while the collapse point ${\tau }_{\mathrm{res}}$ supplies a time-resolved marker that can be aligned with ERP windows. In uniting modular entropy retrieval, observer class, and temporal processing signatures, ODER explains not only what is difficult but also when and for whom. Finally, by treating Aurian as a controlled precursor to natural-language corpora (Section 4), the model 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]. Subsequent work showed that permitting variable stack depth removes this blind spot [30]. ODER’s present 31% ceiling therefore marks a typical, not anomalous, stage in model evolution.

Figure 3. 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 3. 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.

6.2. ERP Anchoring and Observer Diversity

Bootstrap analysis (10,000 resamples) yields the following retrieval-rate estimates for the five convergent traces:

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

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, not a stimulus-determined timestamp.

6.3. Parameter Diversity and Observer-Class Variation

Although the present study does not 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 mirror phonological-loop constraints characteristic of developmental dyslexia [34]. These hypotheses remain to be empirically tested, but ODER provides a formal architecture in which such lawful divergence can be expressed without ad-hoc tweaks, turning the 31% convergence ceiling into a diagnostic asset.

Appendix F formalizes these conjectures by specifying provisional parameter bands and their expected ERP correlates; traces that resist fit under baseline bounds thus become prime candidates for mapping onto these neuro-parameter profiles, converting model non-fit into structured empirical signal.

6.4. Failure Taxonomy

A systematic Failure Taxonomy (Table 4) reveals two principled breakdown modes:

(a)

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

(b)

Flat-ambiguity plateaux: 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 expose where ODER is currently falsified and motivate two remedies: piece-wise $\gamma \left(\tau \right)$ and attention-gated transitions. Flat $\nabla C$ traces and delayed ${\tau }_{\mathrm{res}}$, for example, may reflect lawful retrieval limits tied to phonological-memory or attentional bottlenecks, potential diagnostic markers of observer-class divergence.

6.5. Known Limitations and Boundary Conditions

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

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

Rather than conceal failures, we surface 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. That record is the framework’s power.

6.6. Open Questions and Future Experiments

Key empirical questions remain:

• 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?
• 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?
• 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. A key implication is that ODER does not enforce universal fit: the 31% convergence rate reflects diagnostic fidelity; the framework preserves 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.

7. 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.

7.1. Tier 1 — Adaptive Interfaces and Reading Diagnostics

7.1.1. 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.

7.1.2. 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).

7.2. Tier 2 — Pilot-Ready Extensions

7.2.1. 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.

7.2.2. Translation and Cross-Linguistic Semantics

Parallel-corpus resources such as OpenSubtitles [25] enable immediate testing of ODER’s semantic superposition term $\mu$ (µ, 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 low-$\delta$ bilinguals.

7.3. Summary Table

Table 6. Near-term ODER constructs mapped to publicly testable outcomes. All metrics derive from publicly available corpora or open 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 6. Near-term ODER constructs mapped to publicly testable outcomes. All metrics derive from publicly available corpora or open 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

8. Conclusions and Future Directions

By formalizing comprehension as a process of observer-dependent entropy retrieval, ODER reframes linguistic understanding not as passive syntactic decoding, but as an 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 ceiling3 and the mean $R^2$ of 0.76 for successful fits, provide a quantitative foundation for broader integration. The application tiers below follow directly from the trace-based methods and observer-specific parameterization introduced in Section 7:

• Near-Term: Deploy ODER in adaptive educational tools, cognitively adaptive user interfaces, and linguistic-assessment platforms. 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 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 its limits.

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 hyperbolic tangent yields early linear growth and late saturation. Parameter-estimation note: throughout 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, $0.01\le \gamma \le 1.5$, $0.05\le {\tau }_{\mathrm{char}}\le 0.15\mathrm{s}$).

Appendix A.2. Variable Definitions

• $\gamma$ — constant retrieval-rate coefficient for the current sentence;
• ${\tau }_{\mathrm{char}}$ — characteristic time (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 with $S_{\mathrm{ret}}\left({\tau }_{\mathrm{res}}\right)\ge 0.95S_{\max }$.

Appendix A.3. Derivation Outline

(a)

Begin with logistic growth, $d{S}_{\mathrm{ret}}/d\tau \propto S_{\max }-S_{\mathrm{ret}}$.

(b)

Replace the constant factor with $\gamma tanh(\tau /{\tau }_{\mathrm{char}})$ to capture early-late regime change.

(c)

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

Appendix A.4. ERP Alignment via Collapse Point τ res

Let k be the smallest token index satisfying $S_{\mathrm{ret}}\left(k\Delta t\right)\ge 0.95S_{\max }$, with $\Delta t=400\mathrm{ms}$. Define ${\tau }_{\mathrm{res}}=k\Delta t$, then

• 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 entropy collapse co-occurs with semantic integration (N400) and syntactic reanalysis (P600).

Appendix A.5. 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 Eq. (4) 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:     $S_{\mathrm{ret}}\leftarrow S_{\mathrm{ret}}+f(L_{\mathrm{hier}},I_{\mathrm{trans}},\nabla C)$▹ Eq. (2) 7:     ${\rho }_{\mathrm{obs}}\leftarrow T({\rho }_{\mathrm{obs}},L_{\mathrm{hier}},I_{\mathrm{trans}},\nabla C)$▹ Eq. (5) 8: end for 9: return$S_{\mathrm{ret}}$

Note: Entropy here represents retrievable semantic uncertainty; comprehension proceeds by increasing retrieval of meaning, not by discarding information.

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

• aurian: decay modulated by hierarchical complexity $L_{\mathrm{hier}}$, delaying convergence for deeper embeddings.
• flat: initial plateau followed by delayed decay, modelling syntactically correct but semantically anomalous items.
• gpath: non-monotonic trace with a mid-sentence spike that simulates garden-path reanalysis.
• ambig: plateau with shallow decline, representing lexical ambiguity where competing parses persist.
• delayed: flat plateau until token four, then exponential decay; serves as a control for late retrieval onset.
• normal: monotonic exponential decay with slope set by $\gamma$ and mild Gaussian noise.

Appendix B.3. Observer Class Bias

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

Higher decay constants and lower noise give O1 a faster trajectory toward collapse with smaller residual variance.

Appendix B.4. Trace Generator: Logic Summary

Appendix B.9.9.1. Purpose

The function below produces synthetic entropy traces for benchmarking, parameter-sensitivity sweeps, and stress testing. Empirical analyses in Section 3–4 use retrieval traces derived directly from corpus structure and observer parameters.

Appendix B.9.9.2. Key Points

• observer_class (“O1” or “O3”) is mapped to $k_1$, $k_2$, and $\sigma$ via the bias table.
• The optional lhier_score modulates delay only in aurian mode.
• Output values are clipped to $[0,1]$ to respect entropy bounds.

Appendix C.1. Failure Matrix

Table A2. Rows list every trace that triggered at least one stress flag. “R” denotes $R^2<0.60$, “A” denotes $\Delta$AIC $>+2$ relative to the best baseline, and “P” denotes parameter inversion or pegging. “Method” indicates the collapse-token rule that located ${\tau }_{\mathrm{res}}$ (90% threshold unless stated otherwise). Blank entries (—) indicate 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. Rows list every trace that triggered at least one stress flag. “R” denotes $R^2<0.60$, “A” denotes $\Delta$AIC $>+2$ relative to the best baseline, and “P” denotes parameter inversion or pegging. “Method” indicates the collapse-token rule that located ${\tau }_{\mathrm{res}}$ (90% threshold unless stated otherwise). Blank entries (—) indicate 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 (left, eng_1/O1) and a non-convergent garden-path trace (right, gpath_2/O1). Convex valleys signal identifiable minima, whereas flat ridges and secondary bumps reveal the non-identifiability patterns documented in Table A2.

Figure A1. $\gamma \times {\tau }_{\mathrm{char}}$ error contours for a convergent trace (left, eng_1/O1) and a non-convergent garden-path trace (right, gpath_2/O1). Convex valleys signal identifiable minima, whereas flat ridges and secondary bumps reveal the non-identifiability patterns documented in Table A2.

Appendix C.3. Threshold Criteria

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

These criteria surface retrieval failures without suppressing them, providing concrete checkpoints for model refinement and future falsification.

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).
  Cause: early retrieval growth is interrupted by a spike, violating the single-phase tanh assumption.
  Remedy: replace the constant $\gamma$ kernel with a piecewise $\gamma \left(\tau \right)$ or spline basis (see Appendix A, Fig. S4).
• ${\tau }_{\mathrm{char}}$ pegging at lower bound
  Symptom: parameter hits $0.05\mathrm{s}$ ceiling, especially on short sentences (flat_1).
  Cause: trace length under-constrains the saturation regime; optimizer collapses.
  Remedy: enforce a minimum eight-token input or add a weak hierarchical prior on ${\tau }_{\mathrm{char}}$ centered at $0.08\mathrm{s}$.
• AIC under-performance vs. linear baseline
  Symptom: $\Delta$AIC $>+2$ despite visually plausible fit (aur_complex_2, O3).
  Cause: parameter-count penalty outweighs small error gains for very flat traces.
  Remedy: introduce an attention-gated transition term that defaults to a linear model when $\nabla C\to 0$.
• Parameter inversion (${\gamma }_{\mathrm{O}3}>{\gamma }_{\mathrm{O}1}$)
  Symptom: inversion on ambig_1.
  Cause: lexical ambiguity drives superposition ($\mu$) more than memory limits, reversing rate ordering.
  Remedy: couple $\gamma$ to $\mu$ via an interference term, or model lexical-versus-syntactic $\gamma$ separately.

These annotations convert raw failure codes into actionable hypotheses, ensuring that non-convergent cases serve as checkpoints, not exclusions.

Appendix D.1. Core Functions

• Real-time entropy trace fitting with nonlinear least squares or bootstrap resampling.
• Side-by-side observer comparison of retrieval curves, parameter estimates, and residuals.
• Automated collapse-token detection using threshold, inflection, and derivative criteria.
• Mapping from the detected collapse point to predicted N400 and P600 latency windows.
• Bootstrap validation that yields confidence intervals for $\gamma$, ${\tau }_{\mathrm{char}}$, and $R^2$.

Appendix D.2. Usage Notes

• Default parameter bounds and solver settings match those used in the simulations.
• The notebook reads and writes only to a sandbox directory and leaves the publication data untouched.

Appendix D.3. Access

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