IC–IIa: Formal Consolidation of the Informational Dynamic Calculus in the Theory of Informational Emergence (TIE)

IC–IIa: Formal Consolidation of the Informational Dynamic Calculus in the Theory of Informational Emergence (TIE) Adolfo J. Céspedes Jiménez ORCID: 0009-0003-3026-7611 University of Groningen – Faculty of Arts Abstract This document formalizes the stable version of the Informational Calculus II (IC–II), the dynamic computational layer of the Theory of Informational Emergence (TIE). IC–IIa provides the finalized notation, operators, dynamical rules, coherence functions, informational derivatives, and transition laws linking internal configurations (Iₛ), matrix-field configurations (Iₘ), and emergent perspectives. It also establishes the formal bridge to TIE–Dialog, the empirical implementation of IC–II for the measurement of conversational coherence. This consolidated version serves as the foundation for IC–III and the topological phase of the Informational Calculus. This version (v1) provides the mathematically consolidated formulation of IC–II as used in TIE–Dialog. Future versions will expand the topological treatment introduced in IC–III. 1. Introduction IC–II defines how informational systems evolve, synchronize, diverge, and self-repair. It provides: • a dynamic coherence function 𝒞ₜ • operators for resonance, contrast, and fusion • rules for informational stabilization and repair • a minimal differential ∂ᵢ capturing perspectival change • and a general transition cycle Iₛ → Iₘ → Iₛ′ IC–IIa consolidates these components into a single, internally consistent and empirically testable formal structure. 2. Core Entities and Notation Internal configuration (Iₛ) State vector of the system at time t. Matrix-field configuration (Iₘ) Information projected from the broader field/matrix at time t; represents constraints, context, and environmental structure. Coherence (𝒞ₜ) Degree of dynamic alignment between Iₛ and Iₘ, modulated by volatility and asynchronous inter-systemic difference: 𝒞ₜ = f(sim(Iₛ, Iₘ), ∂ᵢIₛ, Δₐₛyₙc) High coherence implies that Iₛ and Iₘ evolve in near-synchrony with minimal temporal offset. Informational differential (∂ᵢ) Rate of change of Iₛ across the perspectival timeline: ∂ᵢIₛ(t) = Iₛ(t) − Iₛ(t−1) Repair operator (ℛ) Minimal intervention that restores coherence after breakdown. Resonance (⊗) Normalized alignment between two configurations. Fusion (⊕) Stable combination of two configurations. Contrast (⊖) Difference between configurations. 3. Informational Operators 3.1 Fusion (⊕) Iₐ ⊕ Iᵦ = η·Iₐ + (1−η)·Iᵦ 3.2 Contrast (⊖) Iₐ ⊖ Iᵦ = Iₐ − Iᵦ 3.3 Resonance (⊗) Iₐ ⊗ Iᵦ = cos(Iₐ, Iᵦ) 4. Temporal Dynamics 4.1 Informational Differential (∂ᵢ) ∂ᵢIₛ(t) = Iₛ(t) − Iₛ(t−1) Measures internal volatility. 4.2 Extended Dynamic Coherence Function (𝒞ₜ) 𝒞ₜ = σ( α·cos(Iₛ(t), Iₘ(t+δ)) − β·|∂ᵢIₛ(t)| − γ·Δₐₛyₙc(t) ) Where: • cos(Iₛ, Iₘ) → semantic alignment • ∂ᵢIₛ → volatility penalty • Δₐₛyₙc → temporal-perspectival misalignment • δ → intrinsic perspectival offset • σ(x) = 1 / (1 + e^(−x)) • α, β, γ ∈ ℝ⁺ Coherence becomes the joint minimization of semantic misalignment, internal volatility, and asynchrony. 4.3 Asynchronous Inter-System Difference (Δₐₛyₙc) Δₐₛyₙc(t) = ∥ Iₛ(t) − Iₘ(t+δ) ∥ • δ ∈ ℝ⁺ models perspectival offset • δ = 0 recovers synchronous coherence • δ > 0 captures staggered informational flows Δₐₛyₙc reflects mismatches between: • lexical vs compositional latencies • prediction vs integration cycles • turn-taking and repair • perspectival emergence IC–IIa adopts Δₐₛyₙc as a primitive dimension of coherence. 4.4 Dimensionality-Driven Synchronization The rate at which a system reduces asynchronous difference depends on its informational dimensionality (Dₛ): dΔₐₛyₙc(t) / dt = −k(Dₛ)·Δₐₛyₙc(t) Where k(Dₛ) is monotonically increasing. Interpretation: • Low-dimensional systems → fragile coherence • High-dimensional systems → robust synchronization capacity Dimensionality determines the system’s ability to integrate dispersed and temporally displaced flows. 4.5 The Matrix as a Negative System The matrix-field configuration Iₘ acts as a negative system: • Iₛ(t) → positive projection • Iₘ(t+δ) → negative stabilization Approximation: Iₘ ≈ − ∂Iₛ/∂t + contextual structure Interpretation: Coherence emerges from a positive–negative interplay: • the system generates states • the matrix constrains, counterbalances, and stabilizes them This prevents runaway divergence and anchors perspectival coherence. 5. Repair and Stability 5.1 Breakdown condition 𝒞ₜ < Φ_low 5.2 Law of Minimal Repair ΔIₛ = argmin Δ | 𝒞ₜ₊₁ − 𝒞ₜ | subject to 𝒞ₜ₊₁ > 𝒞ₜ 5.3 Stabilization zone Φ_low ≤ 𝒞ₜ ≤ Φ_high 6. The Triadic Transition Cycle Iₛ(t) → Iₘ(t+δ) → coherence check → repair (if needed) → Iₛ′(t+1) Formal update: Iₛ′(t+1) = Iₛ(t) ⊕ g(Iₘ(t), ℛ(t)) 7. Minimal Pseudocode for t in range(1, T - delta): # asummed delta >= 0 # informational differential (∂ᵢ I_s) dI = I_s[t] - I_s[t-1] dI_norm = norm(dI) # semantic similarity + asynchronous difference sim = cosine(I_s[t], I_m[t + delta]) Delta_async = norm(I_s[t] - I_m[t + delta]) # extended coherence function C_t = sigmoid(alpha * sim - beta * dI_norm - gamma * Delta_async) # Law of Minimal Repair if C_t < Phi_low: I_s[t] = repair(I_s[t], I_m[t + delta]) store(C_t) 8. Empirical Implementation: TIE–Dialog TIE–Dialog instantiates IC–IIa by: • using semantic embeddings as Iₛ and Iₘ • computing 𝒞ₜ turn-by-turn • detecting breakdown (𝒞ₜ < Φ_low) • identifying repairs (minimal increases) • computing Δₐₛyₙc(t) • tracking 𝒞ᵢ per participant • generating full analytical reports IC–IIa becomes fully empirical and testable. 9. Numerical Example Given: Iₛ = (0.6, 0.8) Iₘ = (0.5, 0.7) Iₛ(prev) = (0.7, 0.6) sim = cos(Iₛ, Iₘ) = 0.992 ∂ᵢIₛ = (−0.1, 0.2) → |∂ᵢIₛ| = 0.224 𝒞ₜ = σ( 1.2·0.992 − 0.8·0.224 ) If 𝒞ₜ < Φ_low → repair triggered. 10. Figure (Conceptual) Iₛ(t) → Iₘ(t+δ) → coherence → repair (if needed) → Iₛ′(t+1) 11. Conclusion IC–IIa consolidates the dynamic layer of the TIE, providing a complete, stable, and reproducible formal basis for modeling coherence, rupture, repair, and perspectival evolution. It additionally incorporates the asynchronous inter-systemic difference Δₐₛyₙc, grounding coherence in the minimization of temporal desynchronization between informational flows. This aligns IC–IIa with neurocognitive evidence and provides a unified mechanism for synchronization across natural and artificial systems. References Céspedes Jiménez, A. J. (2025). The Informational Calculus Vol. II: A Formal Operational Framework for Coherence and Repair Dynamics in the Theory of Informational Emergence (TIE). 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