Axiodynamics v5.9 — Angular Semantics and Coordinate Conventions
Abstract
This technical note addresses a practical requirement for making Axiodynamics computable on real-world corpora: assigning an angular orientation (θ ∈ [0, 2π)) to observed mémotions in a telotopic space, relative to a situational telos. In the axiodynamic framework, affective-cognitive dynamics are represented by vectorial forces—conative (Fc) and inhibitory (Fi)—and key metrics such as telotopic negentropy depend on the directional coherence of these vectors. This methodological addendum extends the axiodynamic framework (v5.7: core axioms, v5.8: telos formalization) by operationalizing the protective-belt layer for empirical falsifiability. The document formalizes four complementary estimation routes designed to be used alone or in combination: (A) structured expert annotation based on behavioral inference, (B) semantic embeddings coupled with circular multidimensional scaling, (C) anchoring on stable axiological reference axes, (D) an axiological alignment coefficient (β) that combines semantic proximity (derived from embeddings) with affective valence to enable semi-automated (with valence annotation) or fully automated (with sentiment models) inference at scale. To prevent "black-box" angle assignment, the note specifies validation requirements for reliability and convergence across methods (inter-annotator agreement, cross-method consistency), and it defines the associated circular-statistics toolkit (Rayleigh tests, angular correlations, bootstrap confidence intervals). The note is accompanied by a minimal Python implementation (modules for circular statistics, β computation, and method comparison), intended to integrate directly into the PoC Telotopic Signatures pipeline. Overall, this deposit functions as a methodological bridge between the formal axiodynamic axioms and corpus-based testing, enabling large-scale experiments (N > 5000 utterances) on historical social-media data to evaluate whether telotopic negentropy anticipates shifts in collective stability and coordination.
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Page 1/7 Axiodynamics v5.9 — Angular Semantics and Coordinate Conventions Alan Kleden Independent researcher Email: [email protected] Paris, France Reference core: Axiodynamics v5.7 (Axioms 1–9 unchanged) Status: computational specification (empirical calibration pending) Abstract: Axiodynamics formalizes affect-cognition dynamics through vectorial forces (conative Fc, inhibitory Fi) projected in telotopic space. A critical operationalization gap remains: how to assign angular orientations 𝜃k∈[0,2𝜋) to observed mémotions relative to a situational telos. This technical note formalizes four complementary estimation methods: (A) expert annotation via behavioral inference, (B) semantic embedding with circular multidimensional scaling, (C) axiotypic anchoring on universal axiological axes, and (D) axiological alignment coefficient 𝛽. Method D decomposes angular position into semantic proximity ( 𝜃_axio via embeddings) and affective valence, enabling automated large-scale inference. Validation protocols require: inter-annotator reliability k_circ ⩾0.70, crossmethod convergence 𝜌_circ ⩾ 0.60, and predictive correlation 𝜌( N_tel, behavioral outcome) > 0.40. Essential circular statistics (Rayleigh test, bootstrap confidence intervals, angular correlation) are formalized with Python implementation (circular_stats.py, beta_coefficient.py, method_comparison.py). These methods provide empirical infrastructure for the PoC Telotopic Signatures validation program, testing whether telotopic negentropy predicts collective stability in historical social media corpora. Keywords: Axiodynamics, angular estimation, circular statistics, beta coefficient, telotopic negentropy, validation protocols DOI: License statement : Ajouter "CC-BY 4.0 Preprint published on Zenodo (December 16, 2025)
Page 2/7 1. Introduction Axiodynamics v5.7 formalizes affect-cognition dynamics through nine axioms (Kleden, 2025), defining conative forces (Fc) and inhibitory forces (Fi) as vectorial projections in telotopic space. A critical operationalization gap remains: How do researchers assign angular orientations θₖ ∈ [0, 2π) to observed memotions? This technical note formalizes four complementary estimation pathways: (A) expert annotation, (B) semantic embedding with MDS, (C) axiotypic anchoring, and (D) axiological alignment coefficient β. Focus is placed on Method D (β coefficient) and circular statistical validation protocols. Full methodological treatment appears in the companion manuscript (Kleden, Affective Forces and the Structure of Interaction: A Telotopic Model of Negentropic Organization, 2025). 2. Circular Statistics for Angular Data Angular data require specialized metrics respecting periodicity (θ = 0° ≡ 360°). Essential formulas: Circular mean: 𝜃‾=atan2(∑𝑘=1 𝑁 sin(𝜃𝑘),∑𝑘=1 𝑁 cos(𝜃𝑘)) Mean resultant length (concentration metric R ∈ [0, 1]): 𝑅‾=1𝑁√(∑𝑘=1 𝑁 cos(𝜃𝑘))2+(∑𝑘=1 𝑁 sin(𝜃𝑘))2 Rayleigh test for uniformity (proto-telos gate, H₀: angles uniformly distributed): 𝑍=𝑁⋅𝑅‾2(𝑍>5.99,𝑝<0.05⟹ reject 𝐻0) Circular Cohen's kappa (inter-annotator agreement, tolerance ε = 20°): 𝜅circ=𝐴𝑜−𝐴𝑒 1−𝐴𝑒, 𝐴𝑜=1𝑁∑𝟙[|𝜃𝑘(1)−𝜃𝑘(2)|≤𝜖] 𝑁 𝑘=1 Circular correlation coefficient ρ_circ (Jammalamadaka & SenGupta, 2001):
Page 3/7 𝜌circ =∑𝑘 sin(𝜃𝑘(1)−𝜃‾(1))sin(𝜃𝑘(2)−𝜃‾(2)) √∑𝑘 sin2(𝜃𝑘(1)−𝜃‾(1))∑𝑘 sin2(𝜃𝑘(2)−𝜃‾(2)) Implementation: Python module circular_stats.py (GitHub: Discursive-telotopicsignatures/scripts/v5.9_methods/). Requires: numpy, scipy. 3. Method D — Beta Coefficient The axiological alignment coefficient β decomposes angular position into two orthogonal components: (1) axiological distance θ_axio (semantic proximity to telos), and (2) affective valence (positive/negative polarity). Definition: 𝛽(𝑎,𝜏)=cos(𝜃axio)⋅sign(valence) where θ_axio = arccos(similarity(mémotion, telos)) is computed via semantic embeddings (e.g., sentencetransformers all-MiniLM-L6-v2), and valence ∈ {-1, +1} classifies affective polarity. Angular assignment formula: 𝜃(𝑎,𝜏)=sign(𝛽)⋅𝜃base⋅|𝛽|+Δ𝜃(𝑐)⋅(1−|𝛽|) Defaults: θ_base = 0° (perfect conative alignment), Δθ(c) = 90° (orthogonal ambivalence). Example — Heinz Dilemma (Kohlberg, 1984): Should Heinz steal drug to save dying wife? Telos: θ*_T = 0° = "steal drug". • Mémotion: "I took an oath to help anyone in medical distress" → Similarity = +0.88, Valence = +1 → β = +0.88 → θ ≈ 11° (Strong Fc) • Mémotion: "I'm terrified of going to jail" → Similarity = -0.35, Valence = -1 → β = +0.34 → Manual override required (dual-obligation) → θ = 175° (Strong Fi) Implementation: Python class BetaCoefficientCalculator in beta_coefficient.py. Full treatment: "Affective Forces" Section 6.5.
Page 4/7 4. Validation Protocols Inter-annotator reliability (Method A): Require κ_circ ≥ 0.70 across N ≥ 3 independent coders. Training protocol: 2-hour session (Fc/Fi distinction, angular compass, bin coding), followed by calibration on gold standard scenarios. Bootstrap stability for telos θ*_T: Resample N mémotions with replacement (B = 1000 iterations), compute circular mean θ_T for each sample. Acceptance criterion: 95% CI width < 45° (quarter-circle). Cross-method convergence: Methods A/B/C/D must satisfy pairwise ρ_circ ≥ 0.60 on common corpus (N ≥ 30 mémotions). Root Mean Squared Angular Error (RMSAE): RMSAE(𝜃(1),𝜃(2))=√1𝑁∑𝑑circ(𝜃𝑘(1),𝜃𝑘(2))2 𝑁 𝑘=1 where d_circ(θ₁, θ₂) = min(|θ₁ - θ₂|, 360° - |θ₁ - θ₂|). Criterion: RMSAE < 40° (acceptable), RMSAE < 20° (excellent). Predictive validation: Compute telotopic negentropy N_tel from inferred angles, test correlation with behavioral outcomes (action taken, response time, physiological markers). Acceptance: ρ(N_tel, Outcome) > 0.40 (p < 0.01). Implementation: Module method_comparison.py provides MethodComparator class for automated convergence assessment, Bland-Altman plots, and validation reports.
Page 5/7 5. Computational Implementation Repository: GitHub https://github.com/Alan-Kleden/Discursive-telotopic-signatures/tree/main/scripts/v5.9_methods/ Modules: • circular_stats.py — Circular statistics (κ_circ, Rayleigh, bootstrap CI) • beta_coefficient.py — Method D (β coefficient, semantic similarity) • method_comparison.py — Cross-method convergence metrics • README.md — Complete usage documentation with examples Dependencies: Python 3.11+, numpy ≥ 1.24, scipy ≥ 1.10, sentence-transformers ≥ 2.2, matplotlib (visualization). Installation: pip install numpy scipy sentence-transformers matplotlib Integration with PoC Telotopic Signatures: Methods formalized here are deployed in the empirical validation project "Discursive Telotopic Signatures without Recruitment" (OSF: https://osf.io/kjpmz , https://osf.io/rm42h ). Method D (β coefficient) enables large-scale automated inference of θₖ from historical Twitter/Reddit corpora for predictive validation of telotopic negentropy N_tel as collective stability marker.
Page 6/7 6. References Jammalamadaka, S. R., & SenGupta, A. (2001). Topics in Circular Statistics. Singapore: World Scientific. Kleden, A. (2025). Affective Forces and the Structure of Interaction: A Telotopic Model of Negentropic Organization. doi:10.5281/zenodo.17950264 Kleden, A. (2025, 12 8). Axiodynamics: A Nine-Axiom Computational Framework for Affect-Cognition Dynamics (v5.7). doi:10.5281/zenodo.17853441 Kohlberg, L. (1984). The Psychology of Moral Development (Vol. 2). San Francisco: Harper & Row. Mardia, K. V., & Jupp, P. E. (2000). Directional Statistics. Chichester: John Wiley & Sons.
Page 7/7 7. Summary 1. Introduction .......................................................................................................................................... 2 2. Circular Statistics for Angular Data ..................................................................................................... 2 3. Method D — Beta Coefficient .............................................................................................................. 3 4. Validation Protocols ............................................................................................................................. 4 5. Computational Implementation ............................................................................................................ 5 6. References ............................................................................................................................................ 6