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Appendices The following appendices provide supplementary material supporting the analyses, visualizations, and theoretical interpretations presented in the main text. They are organized to ensure full transparency, replicability, and conceptual integration within the framework of the Theory of Informational Emergence (TIE). Appendix Title Purpose A Computational Parameters and Pipeline Details the full computational setup used to generate and analyze the coherence trajectories ( ), including model configuration, metric computation, and Dynamic Time Warping (DTW) parameters. B Supplementary Visualizations Provides illustrative figures of coherence curves, rupture–repair cycles, DTW alignments, and frequency spectra that complement the quantitative analyses. C Data Tables Presents the complete numerical results (F1, κ, lag, DTW values) for all dialogues, along with descriptive statistics across analytical levels (micro, meso, macro). D Theoretical Bridge: Proto-Coherence and Structural Coupling Interprets the empirical findings within the TIE framework, linking measurable coherence dynamics to informational coupling, perspectival flow, and emergent synchronization. Together, these appendices expand the empirical foundation of the TIE–Dialog pilot and demonstrate how coherence can be both quantitatively measured and conceptually grounded as an emergent informational phenomenon.
Appendix A — Computational Parameters and Pipeline This appendix details the full computational setup used to generate, process, and analyze the coherence trajectories ( ) in the TIE–Dialog pilot. All procedures were implemented in Python and executed in Google Colab to ensure transparency and reproducibility. A.1 Model and Embedding Configuration ● Framework: TIE–Dialog v1.4 ● Sentence embedding model: sentence-transformers/all-MiniLM-L6-v2 ● Embedding dimensionality: 384 ● Similarity computation: cosine similarity between consecutive turns ( ) ● Temporal smoothing: Exponential Moving Average (EMA) with α = 0.3 ● Normalization: z-score normalization per dialogue before thresholding The coherence function was defined as: where and represent the internal and external informational configurations respectively, and ∂t captures their temporal derivative.
A.2 Extraction of Coherence Curves ( ) For each dialogue, the model produced a continuous coherence trajectory reflecting moment-to-moment informational alignment. ● Φ-thresholds: Dynamic Φ_low / Φ_high boundaries estimated from the 25th and 75th percentiles of the distribution per dialogue. ● Event detection: Ruptures (valleys) and repairs (peaks) identified with scipy.signal.find_peaks, using: ○ prominence = 0.03 ○ distance = 2 turns between peaks ● Smoothing window: rolling mean (window = 3 turns) applied for visualization only, not for metric computation. This ensured that both human and model events were derived from the same underlying informational signal. A.3 Human Annotations and Aggregation ● Annotators: 5 trained raters. ● Aggregation rule: majority consensus (≥ 3/5 agreement) per event and per dialogue. ● Tolerance windows: comparisons tested under ±1, ±2, and ±3 turns. ● Output vectors: binary event vectors (0 = no event; 1 = event) for ruptures and repairs separately. A.4 Metric Computation For each dialogue and event type (rupture/repair): ● Precision, Recall, and F1 were computed relative to majority human annotations.
● Cohen’s κ was calculated for both inter-annotator agreement and human–machine agreement. ● Cross-correlation (ρ) was used to estimate the mean temporal lag (lead/lag asymmetry). ● Dynamic Time Warping (DTW): used to assess global alignment between human and model coherence curves (see Section A.5). All metrics were computed using scikit-learn 1.5 and SciPy 1.13. A.5 Dynamic Time Warping Parameters To evaluate macro-scale structural similarity between trajectories, each pair of curves (human vs. model ) was aligned using DTW. ● Implementation: fastdtw (radius = 1) ● Distance metric: Euclidean ● Normalization: total distance divided by (n + m) ● Extracted parameters: ○ dist_norm – normalized distance ○ lag_α – average alignment lag (in turns) ○ who_leads – dominant direction of synchronization (“human” / “model”) ○ β – local elasticity of the warping path ○ r_warped – Pearson correlation between warped signals A typical configuration aligned 20–26 turns per dialogue, producing one DTW summary entry per conversation.
A.6 Software Environment All computations were executed in a controlled environment: Library / Framework Version Python 3.11 NumPy 1.26 SciPy 1.13 scikit-learn 1.5 sentence-transforme rs 2.3 matplotlib 3.9 Analyses were run in Google Colab (2025-04 build) using seed = 42 for reproducibility. A.7 Reproducibility and Data Access The complete analysis pipeline—including preprocessing scripts, metric computation, and DTW alignment—has been archived at: Zenodo DOI: https://doi.org/10.5281/zenodo.17516211 All scripts are provided under an open MIT license. This ensures full reproducibility of the results and enables future replications under the TIE framework.
Appendix B — Supplementary Visualizations This appendix presents additional visual materials that complement the quantitative analyses reported in the main text. The figures illustrate the temporal dynamics of informational coherence ( ) across representative dialogues, showing how rupture–repair cycles and structural alignment patterns emerge between human and model trajectories. B.1 Example of Coherence Trajectory and Event Detection Figure B1. Illustrative coherence curve ( ) with annotated rupture and repair events. ● Each point corresponds to a conversational turn. ● Colored markers indicate detected ruptures (valleys) and repairs (peaks). ● Horizontal dashed lines represent the adaptive Φ thresholds (Φ_low / Φ_high). ● The resulting pattern reveals alternating phases of stability (S), breakdown (B), and recovery (R)—the fundamental S–B–R triad defining the Quantum of
Coherence (𝒬ₐ). This visualization confirms that both human and model signals exhibit rhythmic coherence cycles rather than random fluctuations. B.2 Human–Model Coherence Curves (Representative Cases) Figure B2. Dialogue 2 — High structural alignment (r₍warped₎ = 0.98, distₒᵣ = 0.98). Reconstructed visualization based on the original TIE–Dialog output for Dialogue 2. The human (blue) and model (orange) coherence curves overlap closely, showing minimal temporal distortion. Peaks and valleys coincide within ±1 turn, illustrating near-isomorphic informational trajectories and human-led synchronization (lag ≈ +0.4 turns).
B.3 Proto-coherence and transient inversion of perspectival alignment. Figure B3. Dialogue 4 — Phase inversion and proto-coherence (r_warped = 0.98, lag = –1.23). Here, the model’s curve anticipates human coherence fluctuations by roughly one turn. This anticipatory behavior exemplifies a proto-coherent signature, in which the model’s informational configuration reorganizes before explicit human recognition. Together, these figures visualize the bidirectional coherence dynamics described in the main DTW analysis: most dialogues show human-led alignment, while rare inversions reveal spontaneous model-driven resonance.
B.4 Dynamic Time Warping Alignment Path Figure B4. DTW warping path for Dialogue 2 (human ↔ model). The matrix plot displays the optimal alignment path (white line) connecting human and model time steps. The nearly diagonal path indicates a stable one-to-one correspondence between the two trajectories, confirming minimal temporal distortion and strong coupling. B.5 Conceptual Schema: The Informational Heartbeat
